Contents of the section: Nonisothermal kinetics by TA and utilization of fractals

RATIONALE AND FALLACY OF THERMOANALYTICAL KINETIC PATTERNS: How we model subject matter Jaroslav Šesták J Therm Anal Calorim 2012 - DOI 10.1007/s10973-011-2089-1 THE STUDY OF THE KINETICS OF MECHANISM OF SOLID-STATE REACTIONS AT INCRESING TEMPERAURE. J. Šesták, G. Berggren Thermochimica Acta, 3 (1971) l-12, PHILOSOPHY OF NON-ISOTHERMAL KINETICS Jaroslav Šesták, 3. Thermal AnaL 16, 1979 JUSTIFICATION AND FICTIONS OF THERMOANALYTICAL KINETICS: Evaluation footing Jaroslav Šesták, NATAS plenary lecture, proceedings COMPARISON OF CRYSTALLIZATION KINETICS DETERMINED BY ISOTHERMAL AND NON-ISOTHERMAL METHODS T. Kemeny, J. Šesták Thermochimica Acta, 110 (1987) 113-129 APPLICATION OF FRACTALS IN CONDENSED‐STATE PROCESSES Peter Šimon, Oldřich Zmeškal, Jaroslav Šesták Invited lecture at Thermophysics 2012

Rationale and fallacy of thermoanalytical kinetic patterns

How we model subject matter

J. Sestak

CEEC-TAC1 Conference Special Issue

� Akademiai Kiado, Budapest, Hungary 2011

Abstract Modeling tradition is reviewed within its his-

torical maturity from Plato do Penrose. Metaphors in

nonisothermal kinetics achieved a wide application mostly

employing models derived by means of undemanding

isothermal descriptions. Geometrical basis of such model-

ing is revised and discussed in terms of symmetrical and

asymmetrical (pentagonal) schemes. The properties of

interface (reaction separating line) are found decisive in all

cases of heterogeneous kinetics. Application of fractal

geometry is accredited, and associated formal kinetic

models based on nonintegral power exponents are

acknowledged. Typical erroneous beliefs are dealt with

showing common kinetic misinterpretation of measured

data and associated mathematical manipulability of kinetic

equations. The correction of a measured DTA peak is

mentioned assuming the effects of heat inertia and tem-

perature gradients.

Keywords Nonisothermal kinetics � Reaction mechanism

� Geometrical modeling � Fractals � Evaluation �Misinterpretation � DTA � Heat inertia

Introduction

More than a quarter century ago, I published in this journal

a discussion article on kinetic evaluations [1] emphasizing

various questions, which is not worth to repeat even if

many of inquires have remained unanswered [2–5]. Since

then, many thousands of articles were published showing

that the kinetic subject has persisted to be the best cited

area in thermal analysis [6]. Albeit being a doyen of such a

novel [7–13] and now cross-boundary research area [14], I

was proud that I was given yet another space to revaluate

and reassume my standpoints toward the thermal analysis

kinetics, the subject thoroughly explored in my books

[14–19]. Nonetheless, I have decided to do it in a more

generalized way pointing out the pitfalls and yet unac-

cepted site issues to show different ways of possible and

untraditional assessments even falling back upon the

beliefs of Greek philosophers [18, 19].

Some philosophical thoughts on a general exercise

toward mathematical models

The four basic elements fire, air, earth, and water (intro-

duced by Empedocles 492–432) were the first known

models and metaphors to signify the substantiality of which

all subsistence is composed (i.e., quantities as well as

interconnecting qualities like warmness, dryness, coldness,

and humidity). They were thought to possess the integra-

tive and structural essence ether or better form in the sense

of an imperishable firmament (which in the modern world

is interpreted as in-form-ation) [18, 19]. Today however,

mathematical modeling ensues differently not only from a

further approved physical existence but also from an

existence that is assigned by our more cultured mental

Devoted to the 90th birth anniversary (July 2012) of Vladimir Satava

(Prague) and also written on the occasion of passing away (October

2011 at the age 85) of Ivo Proks (Bratislava) both the foremost

pioneers in the fields of thermodynamics and thermal analysis, to

whom this paper is dedicated.

J. Sestak (&)

New Technology—Research Centre in the Westbohemian

Region, West Bohemian University, Universitnı 8, 30114 Pilsen,

Czech Republic

e-mail: sestak@fzu.cz

123

J Therm Anal Calorim

DOI 10.1007/s10973-011-2089-1

perceptions. It is not just the precision but also the subtle

sophistication and mathematical beauty of successful

models that is profoundly mysterious. Mathematics is

crucially concerned about truth and philosophers would

agree that there are some other fundamental (almost

absolute) concerns, namely that of beauty and of good,

which exist since the Platonic geometrical world of math-

ematical forms [20].

The early modeling elements were first depicted by the

metaphors of simple, equilateral triangles either pointing

up, in order to escape like air or fire, or down, to rest like

water or earth. Later Plato (427–347) used more explicit

geometrical models spatially arranging multiple triangles,

i.e., 3 triangles formed tetrahedron (*fire), 8 triangles—

octahedron (*air), 6 squares or 12 triangles—cube

(*earth). Water, however, was represented by a more

complex geometrical body called icosahedra (20 triangles,

cf. Fig. 1). In Greek culture the term symmetry was

interpreted as the harmony of different parts of an object.

Symmetria (*common measure) is composed of the prefix

sym (*common) and metres (*measure). The Greek

Gaius Plinius Secundus (23–79) provided the early funda-

ment for crystallography (derived from Greek cristallos =

piece of ice) as he gave primary rules for the plan-metric

faces of crystals and their visually imagery shaping.

However, the most impressive historical treatise on

crystallography was written by Johann Kepler (1571–1612),

while residing in Prague (during 1600–1612), and was

devoted to the description of snowflakes. In analyzing their

numerous forms bearing a steady hexagonal symmetry,

Kepler suggested a certain generalization model for the

densest arrangement of rigid balls. Factually, he introduced

the coordination-like number for a ball environment and

declared the consistency of angles between analogical

crystal planes and edges. It can be assumed that some

implication of Platonian geometry was also inherent in

Kepler’s applications [20]. Though Auguste Bravais

(1811–1863) was not sure that crystals are internally

arranged in a repeatable manner, he mathematically mod-

eled the 14 geometrical figures which can be spatially

arranged in a periodic mode [20–22]. They can be charac-

terized by a combination of one or more rotations and

inversions in a lattice that is understood as a regular array of

discrete points representing individual structural units

(atoms, molecules, species, etc.), which thus appear exactly

the same when viewed from any point of the array [20]. This

discovery allows us to classify crystal shapes nowadays in

the seven geometrically basic schemes: area and/or space

can be filled completely and symmetrically with tiles of

three, four, and six sides. This is close to the Platonian

conceptions of geometrical bodies but excludes, however,

any pentagonal arrangement (also involved in the early

Platonic bodies) because it is not possible to fill any area

completely with its fivefold symmetry. In the early 1970s,

however, Roger Penrose (1931–) discovered that a surface

can be wholly tiled in an asymmetrical but non-repeating

manner [21, 22] providing some constructions similar to the

cluster structure of liquid water [23] or glassy state of non-

crystalline materials (like metallic glasses), which has been

for a long run in the core of attention [19], see Fig. 1.

These relations can be followed far back to history when

the geometry of pentagon (and the pentagram inscribed

within it) bore its high metaphysical association as explored

by the Pythagoreans (after Pythagoras 586–506): they

considered it as an emblem of perfection. It was a doctrine

that all things compose and proceed from numbers and the

middle number five, as being formed by the union of the odd

and the first event was deemed of a peculiar value. In China,

the central number five, similarly, represented the fifth

additional element—the earth as the allied symbol of the

great China [18, 19]. In astrology, geometrical figures kept

engendering mystical and occult connotations such as with

the supposed magical powers of pentagons and pentagrams.

This effect lasted until recently as various occult guilds are

often symbolized by five leave rose.

One consequence is the way how we fragment real-world

entities into several categories [20]: things, events, and

processes. By things, we typically mean those entities which

are separable, with identifiable shapes and size, and which

persist in time. Events, on the other hand, have a relatively

short duration and are composed of the interactions of sev-

eral things of various sizes. Processes are, in this last

property, similar to events but, like things, have a relatively

long duration. However, many other entities may have a

transient character, such as vortices, flames, clouds, sounds,

ceremonies, etc. There is an obvious difference between

generic categories and particular entities because a category

may be scale-thin in two different ways: generically (atoms,

birds, etc.) or individually (geometrical concepts, etc.).

In the case of complex objects, there is a close rela-

tionship between their distribution over scales and a hier-

archy of their structural, functional, and describable levels.

We tend to assign objects of our concern into structural

levels and events as well as processes into functional lev-

els. Obvious differences of individual levels yield different

descriptions, different terminologies (or languages), and

eventually different disciplines. Two types of difficulty,

however, emerge, one caused by our limited understanding

of whether and how distinct levels of a system can directly

interact and, the other, related to the communication

(words) barriers developed over decades of specialization

of scientific disciplines [24] (providing the urgent need for

a topic interdisciplinarity).

One of the first mathematical theories in science that

dealt with inter-level interactions was Boltzmann’s statis-

tical physics, which is related to thermodynamics and the

J. Sestak

123

study of collective phenomena. It succeeded in eliminating

the lower (microscopic) level from the macroscopic laws

by decomposing the phase space to what is considered

macroscopically relevant subsets and by introducing new

concepts, such as the mannered entropy principle. It

requested to widely adopt the function of logarithm that

was already and perpetually accustomed by nature alone

(physiology, and psychology [18]). In comparison, another

scaled sphere of a natural process can be mentioned here

where the gradual evolution of living parts has been

matured and completed in the log/log relations, called the

allometric dependence, often penetrating into the kinetic

evaluation methods [18, 19, 25].

Another relevant area is the study of order/disorder phe-

nomena [18, 19], acknowledging that microscopically tiny

fluctuations can be somewhat ‘‘immediately’’ amplified to

a macroscopic scale. What seems to be a purely random

event on one level can appear to be deterministically lawful

behavior on some other level. Quantum mechanics may

serve as an example where the question of measurement is

actually the eminent question of interpreting macroscopic

images of the quantum-scale events [19]. Factually, we

construct ‘‘things’’ on the basis of information; in resem-

blance we may call engines as information transducers

because they transform energy without changing itself (not

accounting on wearing).

Modeling roots applied in reaction kinetics

In solid-state reaction kinetics [25–40], it is convenient to

postulate a thought (gedenken’’) model visualizing thus the

Vertex

Edge

Face

4-rayed vertex

3-rayed vertex

Fig. 1 Building faces (blocks) available for modeling. Pentagon and

hexagon in the upper row shows the incommensurability to compose

a continuous web because of uncovered/overlapping areas (left dark).

This can only be harmonized by curving their edges or adjusting

angels (five-sided ? convex and seven-sided ? concave) or

employing asymmetrical tiles (far right). Classical symmetrical

network can only be satisfied with a collection of triangular,

tetragonal and hexagonal faces (middle) and their combinations.

For an array of equal balls and/or cubes (even when crimped), the

restrictions are faced because of a strict Euclidean dimensionality, not

found, however, in any actual images, which is often characterized (or

observed) by typical 2D cross-sections (bottom right). Irregular grain

structures (middle bottom) possess distinctive faces, which can best be

characterized by the degree of vertices (four-rayed vertex decompos-

ing to three as a spontaneous growth occurs). Some constructions

(associable, e.g., with the clustered structure of liquid water [23]) are

tiling in fivefold symmetry, which were thought for long run as

impossible to fill areas completely and regularly (middle right). Upperright is shown dodecahedron (which Plato associated with firmament)

interconnecting with the larger clusters of icosahedron (which Plato

associated with water). Right and below are visualized the diamond

(Penrose) basic tiles [21, 22] with a specific shape called ‘‘rhombus,’’

where upper thick rhomb has longer diagonal equal to the ‘‘Golden

ratio’’ phi (u = 1.618034, which is related to the number 5 by

formulae (1 - H5)/2), fascinatingly, playing a crucial role in various

aspects of natural livelihood and also man-made art constructions

[20]. The thinner rhomb has his shorter diagonal equal to 1/u. Both

rhombs can be derived from a pentagon, which five diagonals match

/ and which five-side structure leaves gaps when used to be

continuously repeated in space. On the other hand, the rhombs can fill

the surface in an asymmetrical and non-repeating manner, which is

known as continuous but non-repeating structure (sometimes called

quasicrystals). On expanded tiling when covering greater areas, the

ratio of the quantity of thick rhombs to thin ones approaches u again,

and if the rhombs are marked by shadow strips, then they form the

unbroken structure (middle) where we can localize both the chains

(like polymers) and pentagons (like water clusters), where the below

connectivity map shows the molecules’ orderliness within an

icosahedron). Such a structuring can also be applied to a spatial

distribution if the two kinds of rhombohedra are assembled to form

icosahedrons matching thus the larger clusters (water again), but were

never employed in the modeling of reaction solid-state species

Rationale and fallacy of thermoanalytical kinetic patterns

123

feedback, which is usually separated into a sequence of

possible steps then trying to identify the slowest event,

which is considered to be the rate-determining process

[25]. Such models (among others, e.g., [14, 19, 26–42])

usually incorporate (often rather hypothetical) description

of consequent and/or concurrent processes of interfacial

chemical reactions and diffusion transport of reactants,

which governs the formation of new phase (nucleation) and

its consequent (crystal) growth. Such a modeling is often

structured within the perception of simplified geometrical

bodies, which are responsible to depict the incorporated

particles, and such visualization exemplifies the reaction

interfaces by disjointing lines. Such a derived kinetics then

depends on all such physical, chemical, and geometrical

events focused to the behavior of interface acting between

the product and the initial reactant. Accordingly the space

co-ordinates become rate-controlling elements, which cre-

ate heterogeneity consequence inevitably to be incorpo-

rated. At the moment when interfaces are created, they

should be identified with the underlying principle of defects

conveniently symbolized by a pictographic contour (bor-

derline curves) at our graphical representation (cf Fig. 1).

Hence, the mathematical description turns out to be much

more complicated because no mean measure (such as bulk

concentration and temperature) but the spot/site/defect

assessment (extent of phase interface or gradients) carries

out the most considerable information undertaking thus the

posture of true rate-controlling process/execute associable

with the reaction progress (cf. Fig. 1), not omitting the

delivery task of reaction species moving to and/or from

reaction boundary.

Early in 1950s, Smith [43, 44] already proposed a

classical approach emphasizing ‘‘normal grain growth

result for the interaction between the topological require-

ments of space-filing and the geometrical needs of surface

tension equilibrium.’’ We can distinguish that in both 2D

and 3D (dimensional) arrangements, the structure consists

of vertices joined by edges (sides), which surround faces

and in the 3D case, the faces surround cells, see Fig. 1. The

cells, faces, edges, and vertices of any cellular structure obey

the conservation law (Euler’s equation), i.e., F - E ? V =

1 (for 2D plane) and F - E - C ? V = 1 (for 3D space).

Here C, E, F, and V are, respectively, the number of cells,

edges, faces, and vertices. Moreover, the number of edges

joined to a given vertex settles its coordination number, z.

For a topologically stable structure, i.e., for those in which

the topological properties are unchanged by any small

deformation, z = 3 (for 2D) and z = 4 (for 3D), is legiti-

mate everywhere. This can be best illustrated for 2D struc-

ture by a four-rayed vertex, which will tend to be unstable

decomposing into two vertices, each of three-rays, which

process is often termed as neighbor-switching. For a 2D

structure, in which all boundaries have the same surface

tension, the equilibrium angels at a vertex are 120�. The

tetrahedral angle at 109o280 is the equilibrium angle at a

four-edged vertex in 3D having six 2D faces.

The grain growth in 2D is inevitable unless a structure

consists of an absolutely regular array of hexagons. If even

1 five-sided polygon is introduced and balanced by a

seven-sided one then the sides of the grains must become

curved to maintain 120� angles at the vertices. Grain

boundary migration then tends to occur because of the

curvature maneuver reducing boundary surface tension so

that any grain with the number of edges above six will tend

to grow because of concave sides and below six will incline

to shrink because of convex sides.

It is clear that any reaction rate, particularly at the

beginning of its ‘‘acting-ion-exchange,’’ must depend upon

the size of the solid grains which undergo transformation

(growth or dissolution). Reaction rate, r?, should thus be

inversely proportional to the particle size, r, in the form of

a certain power law: r! ¼ rDr�3, where Dr is the charac-

teristic reaction dimension, which can be allied with a

nonintegral fractal [45–48]. It is obvious that a mere use of

strict integral dimensions, typically r1 and r2, would be an

apparent oversimplification. Moreover, we have to imagine

that the initial rate is directly proportional to the extent

(true availability) of ‘‘ready-to-react’’ surface and/or

interface as well as to its coarseness (i.e., roughness as a

kind of another characteristic with a non-integral dimen-

sion, again). It seems that such a concept can be discrim-

inated as rather useful to describe the responding behavior

of a reacting object toward the reaction impact character-

ized by fractal dimension. It recounts in -self summing all

events occurring during the overall heterogeneous process.

There, however, is not a regular polyhedron with plane

sides subsisting exactly tetrahedral (angle 109o280 between

the edges). The nearest approach to space filling by a

regular plane-sided polyhedron in 3D is obtained by the

Kelvin ideal tetracaidecahedra spaced on a body-centered

cubic lattice. Even then, the boundaries must become

curved to assure equilibrium at the vertices so that a grain

growth is likely to occur. It can be even illustrated by beer

frost, which can be of two kinds: at-once draft beer with

more interfacial fluid possessions (enabling mutual bubble

slipping) and the already aged beer with a more rigid

interfacial structure (*‘‘dry’’ hexagonal-like make up).

Apparently, both are unlike in experts’ taste, being capable,

however, to self-adjust by boundary migration and gas

permeation through the cell membranes to equalize pres-

sure of adjacent bubbles.

The most common kinetic models are associated with the

shrinking core of a globular particle, which maintains a sharp

reaction boundary [26–36]. Using a simple geometrical

representation, the reacting system can be classified as a set

of spheres [18, 19] where each reaction interface is

J. Sestak

123

represented by characteristic curve. We assume that the

initial reactants’ aggregation must be reached by (assum-

ingly) well-distributed (homogenization of reacting) com-

ponents (often through various transporting means)

otherwise possibly initiating certain self-organization [18,

19]. Any of such created interfacial (separating) layers, y,

endures thus the role of a kinetic impedance, and the slower

of the two associated elementary processes, i.e., diffusion to/

from acting along with the chemical interface, then becomes

the rate-controlling process [18–42, 49–51] responsible for

the over-all reaction progression. We may indicate that that

the above discussed kind of ‘‘as-belief’’ models depict both

the ideal situation of only single-reaction controlling mode

as well as a rigid spherical representation for all reacting

particles. Though this simplification has no any investiga-

tional authorization, such a theoretical fashion sometimes

(and from time to time even routinely) provides a surpris-

ingly good fitting for thermoanalytical data kinetics ignoring

a common inspection misfit often adjusted by means of

simultaneously accomplished direct observations (such as

microscopy). Rationalized approach can be accomplished

when assuming a certain model coincidence for improved

geometrical fit incorporating thus some additional symmetry

features such as a regularity adjustment of pattern-similar-

bodies (globe $ prism $ cube $ block $ hexahedral

$ dodecahedral $ etc.). It somehow helps us to authorize

the relation truthfulness and applicability of such (oversim-

plified) models when put into operation on more cogent

(irregular) structures, which we often decline, or at least, are

anxious to observe. Even symmetry generalization does not

facilitate above modeling to the full-scale matching of real

morphologies (customarily witnessed in practice).

In our kinetic practice, we can either survive with a

simplified model-free description using a ‘‘blank’’ model-

ing pattern (as below shown SB equation) or we ought to

adapt another philosophy of modeling whichever reaction

mechanisms, learning how to employ more complex

mathematics and/or providing a range of functions instead

of single numerical values (typically activation energies

often pointlessly precised to decimal places). This tactic,

however, may auxiliary interfere within the limiting cases

of experimental setup: either by diminishing the sample

size to a certain threshold (thus being incapable of distin-

guishing the measured response of bulk behavior from that

of sample surface) or by accelerating the imposed tem-

perature changes (�/) probably getting in touch with an

effect analogous to the uncertainty principle (unable to

correspondingly determine each one of the independently

measured parameters with an adequate precision, i.e.,

temperature and/or and its change—heat flux). Therefore,

the future development of thermal analysis may become

different than that we presuppose today.

Use of yet atypical fractal geometry

Always existing perturbations on the reaction interface can

be imagined to encounter a driving force to accelerate

growth that is usually expressed by the negative value of

the first derivative of the Gibbs energy change, DG, with

respect to the distance, r. For small super-heating/cooling,

we can still adopt the concept of constancy of the first

derivatives, so that dDG equals to the product of the

entropy change, DS, and the temperature gradient, DT,

which is the difference between the thermodynamic tem-

perature gradient (associated with transformation) and the

heat-imposed gradient at the reaction interface as a con-

sequence of external and internal heat fluxes. Because

DS is often negative, a positive driving force will exist to

allow perturbations to grow, only if DT is positive. This

pseudo-thermodynamic approach gives the same result as

that deduced from the concept of zone constitutional und-

ercooling [52, 53], and its analysis is important for the

manufacturing advanced nano-materials [54–57] such as

fine-metals, nano-composed assets, formation of quantum

low-dimensional possessions (dots), composite whiskers,

tailored textured configurations, and growth of oriented

biological structures.

The physical–geometrical models also neglect other

important factors such as interfacial energy (immediate

curvature, capillarity, tensions, nano-grains radius [54–57]),

and particularly undistinguish internal and external trans-

ports of heat and mass (to and from the localized reaction

boundary) resulting in a breakdown of smooth (planar)

reacting interface [19], which, at the process terminations,

are anyhow responsible for complex product topology [52,

53]. Variously activated disturbances are often amplified

until a marked difference in the progress of the tips and

depressions of the perturbed reacting interface occurs,

making the image of resultant structures irregular and inde-

finable [19, 52–54, 58]. It creates difficulties in the correla-

tion of traditional morphology observations with anticipated

structures, becoming rather different from the originally

assumed (simple, planar, 3D, etc.) geometry. Depending on

the directional growth conditions, the so-called dendrites

(from the Greek dendros = tree) develop their arms of

various orders and trunks of different spacings because of the

locally uneven conditions of heat supply. This process is well

known in not only metallurgy (quenching and casting of

alloys [52, 53, 58], water, and weather precipitates (such as

snow-flakes formation or crystallization of water in plants)

but also for less frequent types of other precipitation, crys-

tallization, and decomposition processes associated with

dissipation of heat, fluids, etc.

First, we should notice that there are sometimes fussy

effects of particle radius, r, encompassing a wide range of

Rationale and fallacy of thermoanalytical kinetic patterns

123

reacting compacts. Besides affecting measurable point of

phase changes [55–57], the most of the derived model

relations stay either for a simply reciprocity (*1/r, if the

whole reacting surface is exposed to ongoing chemical

events) or for the inversely proportional square (*1/r2, if

the diffusion across the changing width of reactant/product

layer became decisive). It is clear that for a real instance,

we can imagine such a situation when neither of these two

limiting cases is unfailing so that the relation 1/rn becomes

effective, and a new non-integral power exponent, n, comes

into view falling to the fractal region 1 B n B 2. It is

somehow similar to the case of heat transfer across the

layer d, which can similarly fall in between two optimal

cases limited by 1/d and 1/d2. The associated cooling rate /is essentially influenced by the heat transfer coefficient, K,

and the thickness of cooled sample, d, and relatively less by

its actual temperature, T. At the condition of ideal cooling,

where we assume infinitely high coefficient of heat trans-

fer, the cooling rate is proportional to 1/d2, while for the

Newtonian cooling controlled by the character of phase

boundary, / correlates to 1/d, only. In practice, we may

adopt the power relation / = 1/dn (where n is a noninte-

gral experimental constant 1 B n B 2).

We can presuppose that transport properties, because of

fractal nature of percolation changes, incorporate into the

physical laws. For an enough randomly diluted system, we

can even admit that the localized modes occur for larger

reacting frequencies, which can be introduced on basis of

bizarrely called fractons [18, 59]. Inherent state density

then shows an anomalous frequency behavior and, again,

the power laws can characterize their dynamic properties.

On fractal conductors, for example, the density is propor-

tional to Ld and approaches zero for L ? ?. If we increase

L, we increase the size of the non-conducting holes, at the

same time decreasing the conductivity, r, which, because

of self-similarity, decreases on all length scales, leading to

the power law dependence defining the critical exponent, l,

as r * L-l . Owing to the presence of holes, bottlenecks,

and dandling ends, the diffusion is also slowed down on all

length scales. Assuming the common example of ‘‘random

walker’’ [60–63] and its probability to stay in place (using

the standard relation, x2(t) = 2dDt, where D is the diffu-

sion constant and d is the dimension of lattice), the clas-

sical Fick Law loses its orthodox validity. Instead, the

mean square displacement is described by more general

power law: x2 tð Þ ffi t2=dw , where the new exponent, dw, is

always greater than two. Both exponents can be related

through the Einstein relation, r = e n D/(kBT), where e and

n denote, respectively, the charge and density of mobile

particles, and kB is a rate constant. As a result, dw =

d0 - d ? 2 ? l, where d0 can be substituted by the ratio

relation, log3/log2, so that dw becomes proportional to

log5/log2, which, however, is not so easy to ascertain in the

standard integer-like cases.

It is widely shown [60–63] that many dynamic systems

can regularly produce a chaotic behavior. One set of

associated problems for us is the investigative concern in

the direction of a difference equation called logistic map-

ping obviously being the quadratic transformation, which

comes in different forms, typical succession as x ? a

x(1 - x). This name sounds a little peculiar in modern

science as its origin subsisted in economics from which it

gives us the term logistic to describe any type of a planning

process. It derives from the consideration of a whole class

of problems in which these two factors control the size of a

changing population, x varying between 0 and 1. This

population passes through a succession of generations,

labeled by the suffix n, and so we denote the population in

the nth generation by xn. There is a birth process in which

the number of populated species (nuclei, insects, and even

people) would deplete resources, and prevent survival of

them all. There is a negative depletion term proportional to

the square of the population. Putting these together, we

have the nonlinear difference equation. By defining the

iteration as xn?1 = a xn(1 - xn), we can illustrate the

process graphically upon the superimposed parabola (x2)

and straight line (x) in the interval 0 B x B 1 [18]. We can

arrive to the two types of iterations by adjusting both the

initial point, xo, and the multiplying coefficient, a. It can

either exhibit a sensitive-irregular pattern or non-sensi-

tively stable behavior. Very important phenomenon is thus

sensitivity, which either can magnify even the smallest

error or dump the larger errors, if the system is finally

localized in the stable state. This behavior is called the

sensitive dependence on initial conditions and is central to

the problematic of chaos [18, 49, 60–63], though it does not

automatically lead to disorder.

The crucial difference between the discrete logistic

system and its continuous derivative-like counterpart sub-

sists in the fact that it is plainly impossible for the

dynamics of the differential equation to behave chaotic.

The reason is that in the 1D system, no two trajectories, for

the limit Dt ? 0, can cross each other, thus typically

converging to a point or escaping to infinity, which, how-

ever, is not a general consensus in 3D system often dis-

playing chaos. These types of differential equations are the

most important tools for modeling straightforward (‘‘vec-

tor’’) processes in physics and chemistry, though, no single

particular analytic solution is regularly available. The

associated relationship can be transposable to write the

form of a common kinetic equation: da/dt % a(1 - a),

where a can be the normalized extent of chemical

conversion. Factually, this is a well-known form of Prout-

Tompkins’ self-catalyzed kinetic model [64, 65] naturally

J. Sestak

123

related to various aspects of contradictory chemical reac-

tivity. However, life in the real science is not simple and

almost in any physical–chemical system, the state cannot

be described by a single variable or equation characterized

by an integral power exponent (=1), as shown in detail in

the previous paragraph. Therefore, it was obvious that for

generalized purposes of chemical kinetics, this logistic-like

equation has got to be completed by the non-integral

exponents, m and n, as completed in the advanced form of

Sestak–Berggrenn (SB) equation [66–69] da/dt % am(1 - a)n.

The involvement of reactants (a) and products (1 - a) is

attuned to their actual chemical transience (m) and fertility

(n), which is timely adjusted by this type of power-law

[18, 19].

Yerofeevev [70] has already shown that if a solid is

involved with acting interfaces, the formation and growth of

nuclei are describable by the above SB equation with m = 2/

3 and n = 2/3. It may include certain extension and corre-

lation, for example, for: m = 1/2, 2/3, 3/4, 4/5, …, and 1:

n = 0.774, 0.7, 0.664, 0.642, …., and 0.556, respectively. It

was shown [32, 33, 66–68] that such a two-parameter model

retains its physical meaning only for m B 1 which can be

even correlated with the classical nucleation-growth equation

(often abbreviated as JMAYK) staying factually the special

case of SB equation (when approximating {ln(1 - a)}p

through expansion by am [18, 19]). The increasing value of

the exponent m indicates ‘‘mortality’’—a more important role

of the precipitated phase on the overall kinetics. It also

appears that a higher value of the second exponent (n [ 1)

indicates increasing reaction complexity; however, the

temptation to relate the values of m and n to a specific

reaction mechanism can be doubtful and should be avoided

without complementary measurements [18, 19]. Besides

Malek shown [71, 72] that there exists a relationship between

the JMAYK robust exponents r = 0 ? 1.5 ? 2 ? 3 and

SB exponents keeping on the respective values m–n =

0–1 ? 0.35–0.88 ? 0.54–0.83 ? 0.72–0.76. Nevertheless

the best interpretation depicts n and m as mere fractals

without any correlation to a reaction mechanism.

Manipulability of the kinetic equation

Chemical kinetics is based on the experimentally verified

assumption that the reaction rate r?, based on degree of

conversion a, is signified by time (t) derivative da/dt, which

is a function, f, of the state alone, i.e., da/dt % f0(a, T). The

state dynamics of a studied system is characterized by the

mean and dimensionless measure of reaction progress (a)

[18–42, 49–51]. The long-lasting practice has accredited a

routine in which the function f0(a, T) is divided into the

two mutually independent functions, k(T) and f(a). Using

this traditional postulation, the appropriate constitutional

approach to inaugurate the desired constitutive equation [73,

75] depicts a principal form of the product of two separate

functions, i.e., the rate constant k(T), dependent solely on

the temperature, T, and the mathematical portrayal-model of

the reaction mechanism, f(a), reliant on the variation of the

degree of conversion, only [18, 19, 74]. It may be compli-

cated by interference of a changing equilibrium background

in which a became a product of two kinetic k and equilib-

rium keq degrees [75] (a = k/keq). The degree of conversion

may eventually involve multiple kinetic degrees (such as

simultaneous phase separation [76] etc.) and may also

become pressure dependent, etc.

For a mathematical treatment, we have any number of

possibilities of mathematical manipulations: we can adjust

modification of either term: da/dt (*a0), f(a) or k(T). We are

all familiar that the derivative can possess only an integer

order (1,2,3…), which is a common language for formulating

and analyzing many laws of physics. In accordance with

above discussion, even the calculus of fractional derivatives

[77, 78] may become adequate to be employed for kinetic

applications (factually being an old technique selected by

Gottfried von Leibnitz already in 1695), i.e., dax/dty (explicitly

for a case of x = 1/2 it makes equal to aH(da/a) [77, 78]). It

has already affected the classically derived Fick law of dif-

fusion, showing certain exceptions, which is termed as a

strange kinetics [79]. It is based on the portrayal of a random

walker concept (see above) and on an unconventional distri-

bution of functions. In standard kinetics, however, fractional

derivatives did not come into sight as yet which does not

exclude a chance that in future a mathematically based

application would surface conquering its apparent contro-

versy embedded nowadays. Worth noting is another curiosity

associated with derivatives, which appeared when the con-

stitutive rate da/dt was incorrectly treated as a multiple of two

partial terms, i.e., temperature (da/dt)T and time (da/dt)t

dependent [2, 3, 73–75]. This implication created a long-

lasting anarchy in the scientific literature fortunately clarified

in time [18, 19, 73] though some strange commentary issues

keeps persistent until today.

A long-lasting kinetic practice employs both functions

k(T) and f(a) after their analytic implementation. Traditional

approach habitually employs the Arrhenius exponential

constant because kineticists believes [19, 80, 81] in the

exponential law of energy distribution, k(T) = A exp (-E/

RT), derived by Arrhenius for the stochastic process of

evaporation (and approved by Boltzmann statistics). How-

ever, experimentalists often think it can be proved mathe-

matically while the mathematicians believe it has been

established by observations. Parameter E is the so-called

activation energy identified as the energy barrier (or

threshold) that must be surmounted to enable the occurrence

of the bond redistribution steps required to convert reactants

to products. The pre-exponential term, or frequency factor,

Rationale and fallacy of thermoanalytical kinetic patterns

123

A, provides a measure of the frequency of occurrence of the

reaction situation. Although the Arrhenius equation has been

widely (and often successfully) applied to innumerable

solid-state reactions, its use factually lacks a theoretical

justification (merely acknowledged in homogeneous sys-

tems) because the energy distribution, particularly among

the immobilized constituents of crystalline reactants, may

not be adequately represented by the Maxwell–Boltzmann

equation. The interracially reacting species, however,

encompass a certain degree of freedom to adjust their energy

distribution along the surface reaction zone affecting the

energy sharing in reactive sites (reaction interfacial contact),

moreover exaggerated by strain (between the differently

juxtaposed networks), nonstoichiometry, defects, catalytic

activity of newborn sites, irreversible re-crystallization, or

perhaps even by local volatilization (if not accounting for

the thickness of such a reacting zone contoured by local heat

and mass fluxes across and along the interfaces). An alter-

native framework for this theory of solid-state decomposi-

tions [82] is available through the so-called L’vov’s [83]

congruent dissociative vaporization (CDV) mechanism.

Here the super-saturation (S) of the vapor of the low-vola-

tility component at the instant of decomposition is related

through a transfer parameter s by proportionality

(s % 0.351 log log S ? 0.017). However, this unusual

(doubly logarithmic) relationship between s and S may

become a key point in understanding the mechanism of

energy transfer in condensation (for low volatility compo-

nent distribution between reactant and product solids)

depending on the ratio characterized by the s-coefficient.

When f(a) function is routinely modeled on basis of

simplified physical–geometrical assumptions [14–19,

25–42], the direct application of the basic differential

equation is straightforward. However, the differential mode

of evaluation can even provide a better matching up with the

true experimental conditions when the actual non-uniform

heating and factual temperatures are considered. In such a

case, we ought to introduce the second derivatives, T00 and a00

leading to a more complex equation correlating a00T2/a0T0

with df(a)/da/f(a) {T2a0/T0} ? E/R. It occurs that it is almost

impossible to evaluate this relation with a satisfactory pre-

cision because of its extreme sensitivity to noise, particularly

affecting the second derivatives (as well as heat flow chan-

ges, q00, see below) A more convenient and rather popular

method of kinetic data analysis is based on expressing the

maximum value (index max) on the dependence of a0 versus

T, for which it holds, a00 = 0 = a0max {E/RTmax/Z// exp(-

E/RTmax) df(a)/da}. The result provides simple but rather

useful dependence, often called Kissinger plot (known since

1959 [84]), which in various modifications and reproves

shows the basic relation between the ratio of heating rate /and the peak maximum temperature Tm along with the

activation energy E.

The so-called integral methods of evaluation [14, 18, 19]

have become recently more widespread. They are based on a

modified, integrated form of the function f(a), which is

determined by the following relation, gðaÞ ¼R a

0da=f ððaÞ

and rather intricate integration of the rate constantR T

T0k Tð ÞdT=/ ¼ AEð Þ=ð/RÞexp �E=RTð Þp xð Þ=x where /

is the constant heating rate applied, p(x) is an approximation

term of the temperature integral (a common target of

numerous publications [42, 85]). In many cases its value is

simplified and even neglected, which can be visualized by

simple withdrawal in front of the integral, namely: constantRk Tð Þdt � k Tð Þ

Rdt. This mathematical handling is habit-

ually obscured within complicated mathematics but got

involved in the nonisothermal derivation mode of traditional

nucleation-growth (JMAYK) equations [76, 86, 87].

It is evident that the derivation increases discrimination

introducing, however, too high sensitivity to experimental

noise and, on contrary, the integration decreases noise

sensitivity but launches a lower indifference toward dif-

ferent f(a)-resolution (discriminability) [18, 19]. In the

history of kinetic appraisal, there appeared curiosity when

three similar but also disparate ways of evaluation seeks for

a linear correlation of the logarithmic form of kinetic

model function In g(a) either lnT, T and 1/T; all making

available a equivalent outlook for the corresponding values

of activation energies, E, evaluated from the slope of

respective plots. Such an inbuilt disparity is caused by the

different approximation of p(x) function applied differing

in the multiplication parameter, which by itself well indi-

cates a certain extent of inherent inexactness of such a kind

of kinetic evaluation. A better insight of this inquisitiveness

was provided by the use of an asymptotic expansion [88] of

a series with a dimensionless parameter replacing Eapp. It

revealed that the ln g(a) versus 1/T plot is twice as good as

that of the former two dependencies. In this way, we can

also substantiate the numerical results of several authors

who found that the plain dependence of ln g(a) versus

T yields E with an error of at least 15%, whereas the 1/

T plot can decrease this error by half .

Important practical correlations were adjusted for vari-

ously interpreted E [15, 19]. For example, a rather

sophisticated correlation provides interrelation between the

experimental activation energy, EDTA, and those for shear

viscosity, Eg, on the basis of the relative constant width of

glass formation interval, Tg (i.e., difference between the

onset and outset temperatures). It reveals a rough temper-

ature dependence of the logarithm of shear viscosity g on

the measured temperature, T, using the simple relation log

g = 11.3 ? {4.8/2.3 D(1/Tg)}(1/T - 1/Tg) [89] detailed

discussion of which is beyond the scope of this text.

The basic JMAYK equation reveals that the apparent

(overall) values of activation energies, Eapp (particularly

J. Sestak

123

being the center of interest when determined on the basis of

DTA/DSC measurements, EDTA) can be conveniently

correlated to the partial activation energies [14, 18, 19, 42,

90, 91] of nucleation, EN, growth, EG, and/or diffusion, ED

employing a simple relation Eapp = (a EN, ? ba EG)/

(a ? bd) where a and b are characteristic multiplying

constants providing that the denominator (a ? b d) equals

to the robust power exponent of the integral form of

JMAYK equation [18, 19, 71, 72], and the value b corre-

sponds to 1 or 1/2 related to the movement of growth front

controlled by either chemical reaction (1) or diffusion (1/

2). Moreover, the coefficients d and b are associable with

the nucleation velocity and the growth dimension, respec-

tively. I am proud to mention that I was the first person

pointing out such a likelihood [90], though it may become

a persuasive but often misleading tool in an effortless

interpretation of reaction mechanism.

The above evaluation quandary can be eased upon the

application of the sample controlled thermal analysis

(SCTA) method or constant rate thermal analysis (CRTA),

where it is not monitoring the constant increase of tem-

perature but the constant rate of reaction (a0 = da/

dt = constant) [92–94]. It, however, does not help in

avoiding undesired mathematical interrelation between the

pre-exponential factor A and the activation energy E in the

standard exponential law (the so-called kinetic compensa-

tion effect—KCE) [95–98]. A complementary help can be

achieved when employing simultaneous experimental

techniques such as optical microscopy [99–101] or dila-

tometry [100, 101].

Only plausible aid is introducing a novel kind of rate

constant [102] or treating the kinetics in the novel so-called

model-free or nonparametric mod of evaluation [103–106]

where the traditional functions k(T) and f(a) are replaced by

a joint function f(a, T), thus going back to the roots of

kinetic evaluations. It may avoid customary exploitation in

terms of ill-reputed and almost religious constants, mostly

linked with the activation energies that never express the

ease of reaction (to be desirably related to the reactivity as

a kind of ‘‘tolerance’’ and to the reaction mechanism as a

kind of ‘‘annexation’’).

In conclusion, I would like to draw attention to yet other

outstanding issues that we outlined 40 years ago when

interpreting a compositional case of a measured DTA peak

assuming the inherent effect of heat inertia [107–110] (i.e.,

the term dDTDTA/dt which is a standard part of the DTA

and/or heat-flux-DSC equation [14, 19, 107, 108, 110]).

Such an expediency rather of great importance has

remained overlooked in most books [30, 31, 112–114]

(with few exceptions [18, 19, 115]). Our recent studies

[111] indicate that this feature is not a fiction but bears a

true consequential impact arising from the real process of

heat transfer (q) which has the greatest impact for moments

where the heat flux is immediately changed (�q00). Our

numerical stimulations [111] show that the heat inertia

effect is even amplified because of such temperature gra-

dients affecting thus the shape of corrected DTA peak.

Associated mistakenness is not reduced by shrinking the

sample size to micro-level [111]. It is questionable whether

such difference between the measured (as-determined) and

authentic (as-rectified) DTA peak impinges on the sup-

plementary derived data especially when calculating the

reaction progress (a) from the enveloped peak areas and/or

on the kinetic data evaluated from the shift of peak apexes

[84] or fractionally determined at successive a’s. It is an

open sphere for further examination and DTA evaluation

scrutiny together with hidden and yet unidentified prob-

lems [111] brought about by lessening the sample micro-

size [19, 55–57] and escalating the employed heating/

cooling rates [19, 52–54].

Acknowledgements The results were developed within the CEN-

TEM project, reg. no. CZ.1.05/2.1.00/03.0088 that is co-funded from

the ERDF within the OP RDI program of the Ministry of Education,

Youth and Sports. I feel also indebted to my scientific friends,

coworkers, and uppermost kineticists: late Joseph H. Flynn

(Bethesda), Jerry Czarnecki (Fullerton), Klaus Heide (Jena), Pavel

Holba (Prague), Nobuyoshi Koga (Hiroshima), Jirı Malek (Pardub-

ice), Eugene Segal (Bucharest), Peter Simon (Bratislava), Donald

R. Uhlmann (Tucson) and Zivan Zivkovic (Serbian Bor).

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123

likmhhka Aa7 Ekvkr PnMhhing company. Xmstcr&m PrinvdinBcIginm

1

STUDY OF THE KINETICS OF THE MECH.4NISM OF SOLID-STATE REACTIONS AT INCREASING TEMPERATURES

Institute of Solid State Physics of the Czech Academy of Sciences, Prague (Czecizoslocakia)

GUNSARBERGGRES

AB Aronrenergi, Sfudrrik (Stceden)

(Received February 23rd. 1971)

Possible discussed. The cif the reaction

reasons for the misinterpretation of non-isothermal kinetics are importance of the correct selection for the assessment of the progress and the acquisition of representative experimental data, as well as the

effect of non-isothermal conditions and possible change of the equitibrium on the kinetic equation are stressed. Detailed attention is given to the probable mechanisms of individual cases of solid-state reactions as expressed in integral and/or differential forms of kinetic equations. Reactions controlled by the movement of phase boundaries, by simp!e nucleation. by nucleation followed by nucIei growth and by diffusion are discussed; a combined form of differential equation suggested for the preliminary appraisal of possible mechanisms is

nz - = kC(l -z)“(-br(1 -a))p dr -

INTRODUCTiO;*;

The study of reaction kinetics in the solid state (and/or Iiqilids) is mainly

designed to gair. information about the kinetic parameters and associated mecha-

nisms of the process. In dynamic experiments with increasing temperaturelW4

dT= @ and t==-=a

dt @ (1)

where @ is the linear heating rate and T, the temperature of the equilibrium of conversion6, the caIcuIation of kinetic parameters is often simplified and may be based on the assumption that the course of the reaction can be described by the differential equation

dr - = F(T) f(a) dz

(2)

where 3~ is the degree of conversion, d.z/dt the rate of reaction, k’( 7) the temperature

T;hernwchinz. Acra, 3 (1971) l-12

2 J. &ST.iK, G. BERGGREN

dependent rate constant and f(r) a function* which represents the hypothetical

mode1 of the reaction mechanism. Apart from direct investigational methods9-32*33,

mat.hematicaI modeIs based on hypothetical reaction mechanisms may be ,~3.9.33-53

The correlation between the reacted fraction (2) and time of reaction (t) is usually derived for isothermal studies and the variabIes are given in a comp!ex form

5.r =.f(K N, 0. r, p. t- ---)r=&Jnst_

where

(3)

V = reacting volume of material

X = number of nucIei (or spots avaiIabIe for nucleation)

U = specific surface r = geometric factor of particIes (radius erc.)

p = partia1 pressure of the gaseous products

PC6!3BLE SOURCES OF ERRORS AX3 hlI!%WXRPRETATIOX5 IX SOS-ISOTHERMAL KIKFXICS

a. Selection of a correct expression for the degree of reaction Contrary to gases (or liquids), concentration is not a vaIid representative

parameter for sohd-state processes as it may vary through the sampIe3. Therefore,

a nondimensional ten-n, degree of conversion a, should be introduced.

The probIem is, in what sense is the value of z a linear function of the physical

property measured6 and how does it represent the state of the system investigated. In basic thermoanalytical methods, such as thermogravimetry (and dilatometry) this

function is defined as

where Y and W are D-eight (volume) and weight loss (volume contraction) of the

SampIe, and indexes 0, t and 03 indicate initial, instantaneous and final stages,

respectively. Differential thermal analysis is still of questionable vaIidity3’ because a representative value which wouId unambigously define the change in the system

from the initia1 or from the final state is not yet available from a DTA peak.

b. Selection of rhe experiznental condition for a representatitre experinrental process model

All variable factors describing experimenta conditions, such as grain size

distribution, sample packing and geometry as well as the effect of dynamic environ-

-_- *f(z) is frequently oversimplified as (1 -zy or .zF assuming analogy with homogeneous kinetics of w’“. and setting R and m as orders of reaction. This simplification has, however, no general validity for solid-state reaction? but as an approximation it may be used to investigate the %xiation of apparent activatien energy under different experimental conditionszg.

SOLID-STATE REACTIONS AT IKCREASISG TEMPERATURE 3

ment and vacuum, shouId be defined in terms of the equilibrium of the process6 and be included in the function f(z) if they alter the reaction mechanism_ Temperature deviation from a chosen temperature increase (e.g. due to heat of reaction in the sample’*30) should be minimized. Uniform and programmed temperature throughout the sampIe body is required for the successful kinetic investigation of solid state reactions’_ Heat and mass transport should be facilitated as much as possible by a suitable experimental arrangement, unless they are true and valid rate controlling processes.

Two possible approaches when invest&tin, n solid-state processes can be distinguished. The physicochemical approach’ is to separate and stress that process usually associated with the re-arrangement of molecules and ions when forming a new phase. Specially developed experimental conditions are applied (e.g. thin layers of solid samples in good contact with large heat reservoirs’) in order to obtain information about the nature of physical and/or chemical changes. On the other hand, the engineering approach” is used to analyse the esperiments (employing large solid sampIes with well defined outer geometry) in terms of therma and concentration gradients in order to obtain information about the practical application of the process.

c. Adnritfing a continuous change of lhe equilibrium of the process during temperature

increases

Contrary to the invariant processes, where the equilibrium state of the system changes by a jump. in monovariant processes equilibrium can be a monotonous function of temperature within the temperature interval of the reaction_ Thereiore a non-isothermal degree of conversion i. should be introduced6

where i.,, is the propagation of the equilibrium. The ordinary rate of reaction dr/dr must then be replaced by the corrected rate of the process d</dt6

d5

z=

’ (6)

Thus it is necessary to establish the dependence’ of i,, on T.

d. Construction of a kinetic equation of solid-state reactions under non-isothermal

conditions There are some doubts6, *O-r 2*19*z0 as to the validity of the currently used

kinetic equation (Eqn. (2)) either when describing heterogeneous and/or solid-state reactions or when considering the non-isothermal character of Frocesses.

It is the total concept of the specified rate constant ((Eqn. (Z)), frequently e.xpressed in the form of the Arrhenius equation,

k(T) = A exp(-E/RT)

ThamaCti~. Acru. 3 (1971) I-12

(7)

4 J. &!Xk, G. BEFIGGRJS

which has been successfully employed and derived for homogeneous kinetics. A is the frequency factor proportional to the number of successful collisions of the reacting

moIecules and E is the activation energy_ Polanyi and Wigner’ ’ attempted a quanti-

tative treatment of the rate of solid state reactions using X-(T; (Eqn. (7)). They

assumed that the molecules which achieved the critical energy E could czcape from the rezctant surface (here cf is the atomic frequency of vibration)_

GeneraUy the exponential character of the rate constant’3-26 for solid state

reactions can be accepted, as ail true solid state processes are activated processes’.

The rate constant represents the impedance which the process must overcome to

achieve the new thermodynamically favourable arrangement in the solid state. The

exponent represents the statistical probability for the energy barrier E to be surpassed

when the system undergoes an effort with sufficient activation energy (E). The pre-

exponential factor A can then be correlated with the number of successful attempts.

The missing term T” for multiplying A (where 0 < m < 1 is derived in different ways

depending on the theoretical application of k(T)) can be ne&cted when working in namow temperature intemaIs, and included in A (together with the initial weight and the molecuku weight of the sampIe)_

For a particuIar analysis of soiid state reactions as applied to the therma

decomposition of carbonates, the activated complex theory has been employed”. A in Eqn. (7) is then proportional to the Maxwell distribution coefficient kT/h and

the ratio Q*/Q, where Q* and Q are complete partition functions for the activated

complex and reactant, respectiveIy_ The partition function can be expressed using the

partial partition function for translation, rotation and vibration for each unit of the

crystal lattice undergoing the change, with regard to their possible activities_ A good

agreement between experimentally calculated and numerically predicted rate constants

for various types of decompositions has been found’-‘.

A recent pubiication I9 claims that the current state of knowledge of solid state kinetics is inadequate7.32-‘3. New assumptions are made, nameIy that heterogeneous

rtzzctions are not activated processes (with no activation energy) and that the only rate

controlling process is heat exchange between the reacting mass and its surroundings.

Furthermore, it is suggested that the reaction can proceed only at a fixed

(reaction) temperature T R, given by the equiiibrium of the process (where the free enthaIpy d G = 0). The corrected rate constant is then found to be proportional to the

hezing rate, transfer of heat (heat conductivity) and to the reciprocal value of the

heat of reaction_ Good agreement is found for several types of endothermic decom- positionX’*zO_

The objection is, however, that in this case the physical property chosen

(e-s_ weight) for measuring reaction advancement is controlled by a heat supply

corresponding to the instantaneous thermodynamicaI need for the interface reaction

equilibrium (TR at dG = 0). This becomes true for specSc conditions of vohuninous samples, where the progress of reaction is governed by the interface advancement

(or crystal growth in certain cases), as will be considered in the next section. In other cases activated types of process which can proceed at temperatures higher than TR

SOLID-STATE REACTIONS AT INCREASING TE?HPERATURE 5

can also be accepted (where AG -c 0), but no proof is yet available for either of these

assumptions (by measuring and distinguishing whether rhe reaction proceeds at T,

or above TR).

Another source of doubt is the time-temperature dependence of r

isothermal conditions.

z = f(t, ?-(2))

under non-

(8)

In accordance with established mathematical rules for partial derivation, Eqn. (8)

can be rewritten as

(9)

Some authors * 2. I9 claim that the formal isothermal kinetic equation is valid in the

form

T = W-3 f(r) (10)

contrary to the current formulation of Eqn. (2). Using pure mathematical operations,

the combination of Eqns. (9) and (IO) results in the corrected non-isothermal rate

constant k’(T) in the form’

This correction is calculated to give 2 50% in E for (T- To) -C 10 and E c 10”.

Although it is difficult to imagine the physical meaning of the change of the degree of

conversion with temperature when time is kept constant’ *, particularly when

temperature can change within a definite time interval, isochrones (t = const.) can be

drawnrg, This, in its wav a mathematically correct approach needs more examination _ as there is little experimental evidence as well as IogicaI justification5.

There is still a confusing deviation in the activation energy values calculated

from data derived from the first part of experimental resuits of the process. If probable

experimental errors are neglected, the best mathematical approach seems to be the appreciation of a correction for the proximity to equilibrium’3B’S

k’(T) = k(T)(l -exp(-AGIRT)) (12)

where the change of free energy for the reaction (AG) approaches zero at the equili-

brium temperature T,-,(exp (-A GIRT,,) -B 1) and is unchanged for stages distant from equilibrium (A G 4 0).

This correction has a Iogical justification in the obvious “activation energy” dia-

gram, A G ZIS reaction travel, and can also be derived by means of irreversible thermo-

dynamics. The expression was originally derived for transformations passing through

the vapour phase’ 3, and satisfactorily applied to describe heterogeneous kinetics.

2?kmmsm Acm, 3 (1971) 1-12

6 J. ikST,k, G. BERGGREN

It must be admitted that the theoretical concept of k’(T) for solid state reactions studied under non-isothermal conditions is still not satisfactorily resolved, but it is

difficult to believe that a new concept will be introduced. The proportionality constant

(KG integral form. k= dex-ivative form) in the next section will, however, be used

without a closely defined meaning.

2. Physical meaning of er-alrrared kinetic parameters

Many theories have been advanced7*8*z’~*2~38 in order to give a true meaning

to the pre-exponential constant /i (see Eqn. (2)). Generally it can be said that A can

include many constants describing the initial state of the sample, such as three-

dimensional shape factors of initial particles, molecular mass, density, stoichiometric

factors of chemical reaction, active surface and number of lattice imperfections, etc.,

as well as factors arising from the surface adsorption of gas, and pressure.

The activation energy E can be understood only ix relation to the rate control- ling processes_ For exampIe, in the case of diffusion. E can be associated with the

activation energy of intrinsic diffusion as well as for substitutional reactions. Pro-

portionality between k(T) and the specific electronic conductivity can be established

by the activation ener,T of electrical conductivity. Also, for nucieation and crysta! growth the value of E is composed of activation energies of &ass supply (diffusion)

and factors arising from the thermodynamical potential of nucleation and/or crystal

growth. erc.

fi Corzsrruction of a hyporherical i?rarhemaricaJ description of rlre reaction mecJzanism

A formal equation in general use (simiiar to Eqn_ (2)) and applicable to solid-

state reactions shouId represent the relation between dr/dt and r. It sometimes in-

cludes a term f(r) as wel1. This type of differential equation impks pure mathematical

fitting of experimental data and may be treated in two ways:

1. As a pure mathematical fittin, 0 of a poIynom to the experimental curve, the

expression

f(2) = &i Al riA,r’i __. _._ + A,$ (I31

is usually employed in computer calculations of mathematical derivations (process

lXites)=-3O, without considering the specific reaction mechanism. Also, the use of the

term “reaction order-’ is included, having a certain physical and engineering

application.

2_ By using a certain logical model to form the function f(z) in accordance

with the suggested reaction mechanism.

This equation includes the basic geometrical and physicochemical aspects of

processes_ A detaiied analysis of this problem for classifying tial equations in a

simplified way is the aim of this article and will be dealt with in the following section (ali thermoclynamicaIly directed aspects discussed in this section were given in detail

ifi a previous article “Kinetics with regard to the thermodynamics of processes

studied by use of non-isothermal techniques”“).

SOLID-STATE REACI-IONS AT INCRE,\SIXG TEMPERATCXE 7

MATHEXATICAL DESCRIPTION OF POSSIBLE MECHAXISWS IN SOLID-STATE PROCESSES

The reaction mechanism can be expressed in two ways_ The ordinary forinuia-

tion Y m. f as empIoyed for basic isotherma concepts (see Eqn. (3))7-p,32--50 is here

mathematically transferred into the relation dr/dt LX z in accordance with the

requirement given by the differential Eqn. (2)3-“W’o. The final and simplified forms

of forma1 kinetic equations are listed below for individual types of mechanism_ The

left-hand side presents the equations frequentIy called the integrated form*5-*8 and

provided with function g(g). These functions 1 s-’ * were satisfactorily employed in

the only known method for the estimation of reaction mechanisms from dynamic

TG traces by using monograms “- 18_ The right-hand side gives the equivalent forms

of the function f(r).

a_ Phase-bomdary controllen processes Surface nucleation takes place extremely rapidly and the total process is gover-

ned by movement of the resulting interface* towards the center7*3g.

A. One-dimensional movement (and some cases of evaporation)“’

z=Kt; dr - = kz” = constant dt

B. Two-dimensional movement (for cylinder or prism)

1-(1-r)“’ = Kr; $ = k(l _?$lt

C. A sphere reactin g from ail surfaces inwards (three-dimensional movement)

1 -(I -Y)I’~ = Kt; dr _ = k(l _r)2/3

dt

b. Reactions controlled by nucleation

A. Nucleation according to the power law7sB

(14)

where nz c 1 and p = I , 2, 3, 4 according to the spherical symmetry36 of nuclei.

*T’iie rate controlling process might be the chemical reaction on this interface as given by Eqns. (14), (15). (16) or the mass and.kr heat transport to or from this surface, such as diffusion or heat exchange I9 as a special case of a non-activated process following the equationlg*fo:

1-(1-~)*‘3 = K. 2Lt~ AH

This represents volatile products during the endothermal decomposition of a spherical sample where K0 is the linear constant and dH is the heat of the reaction.

8

B. Nucleation according to exponential Iaw7m4’

J. %S-LiK, G. BERGGREX

J = I -exp(--KP); dr - = kz”(I -2)” dt

08)

where n lies between l/2 and I and m between 0.774 and 0.556 corresponding to integer numbers of p = Z 3.4. For both m and n = l/3, the rate of reaction was found

to be proportional to the difference ‘between interfaces of reacting solids36_ For

further reztction progress see Eqns. (30) and (21).

c. Process is goremed by nucleation fol/oured b)l the bulk growth of nuclei

(Nucleation proceeds accordin g to the exponential law, see Eqn. (18)).

A_ induction stage’-“*43

.z = hrtp; dz - = h” dt

where p z 4 and n g 3j4. This equation is also suitable for the description linear rate of nuclei growth when the rate of nucleation is negIigible36.

B. Tu-o-dimensionaI growth of nuclei7-‘3 (Avrami equation)

-In(l-2) = A’?; dr - = k(1 -z)(-ln(1 -z))‘~’ dt

C. Three-dimensional growth of nuclei7.s-f3 (Avrami equation)

-ln(l -cx) = Kz’; * = k(1 -z)(-ln(1 -z))2’3 dt

D. Later stages (unimolccuku decay law)“’

-ln(l -x) = K!: dz - = k(l-z) dt

(1%

of the

(20)

VI)

(2)

or random nucleation with one nucleus on the individual particle’ 7*32. This form of

the equation may also describe some cases of the decomposition and/or volatilization

of pure condensed phase systems in which the reaction occurs homogeneously36 and

without dilution of the reactant by reaction products.

[E_ fnteraction of nucleation frequency (for p> 4 increasing and for 3 _(p < 4 decreasing) and time”

-ln(l -r) = KtP; dz - = k(l-~)(-ln(l-z))-“P dr

(23)

u-here (p- l)!p is between 213 and 3/4_ For p = 0.63 this equation may also be used to describe bimolecular reactions3 6_ Mathematically, this is the appropriate form of the differential expressing Eqns_ (18) and (31).]

SOLID-ST’ATE REACi-iONS AT INCEEASING TEMPERATURE

d. Process is controlleri by nucleation follorced by the linear grozcth of nuclei

A. Chain growth of nucIei including the possibility of their branching7*“s

z=Kexp(-K’t); -$=li, (24)

also explosive reactions where time has a large power, e = kxn for 11 3 14”*“*‘” dr >

B. Branching nuclei interacting during their g~owt.h’~~~

In L = Kt; dz

l--z - = ka(l-z) dt

(25)

C. Later stages of A and B’.“’

In cc -=KKogt; dr - = k$(l--z)”

l-a dt f 26)

wherent 1 andm> 1.

e. D&msion-controlled reactions

A. One-dimensional transport process with constant diffusion coefficient (parabolic law) 32-35

a2 = Kt; da _ kz-’

dt (27)

A similar equation, dr/dt = kc-“, may also be valid for some cases of sintering where 0 < n -K 1, or possibly in the form of Eqns. (30) and (31)9.

B. Two-dimensional transport process (e.g. for a cylinder with no volume change during the reactionJ6)

(l-z)In(l --z)+a = Ki; z = k(-In(l -z))-* (28)

C. Three-dimensional transport process in a sphere* (Jander’s equation32-3’aJ7)

(1 -(l -x)1’3)2 = Kt; k’(l -a)‘13((l _a)-‘13 -1)-l

SG k(1 --r)1’3(-In(l -a))-’ (29)

*These integral equations are difficult to simplify into a suitable differential iorm and thus an appro.ximation is necessary:

(l-a)-“3 g 1t(l~3)a+(4j18)a2+(21/162)rx3 i- _._...

ln(l-a) g -(z+(1/2)a’+(1/3)a3+(lj4)x4 t . . . ...)

7Ihermocfzim. Acre 3 (1971) 1-12

IO J. ikXiK, G. BERGGREN

The GinstIing and Brounshtein equation3 3.48*59

1 -~21;33c-(~-z)2~3 = Kt; dz - = /./((I _~j-x/3- I)-’ dt

g k(--In(l-z))-’ (30)

[D. Growth of spherical particIes during their precipitation determined by diffusion5’ and expressed by an approximate form of the differen&I equation36

-In(l -;r) = KtP; dz - = kzfyl-2)” dz

(31)

where p = 312, 2, 513, 3 dependin g on interface reaction and particIe size and geo- metr$O. Some other authors give p a value below 0.5 in the diffusion region3’.]

A consideration of aII cases of the mechanisms mentioned above with regard to the function f(x) in the differential Eqn (2) stems to indicate that it is possible to express this function in a more anaIytica1 fomr (there has been no attempt to give a compfete enumeration)_ The most convenient approximation can be considered to be as folloNs:

Jx - = k(l-~)“~~(-ln(~-~I))P dt

thus rendering a proper combination of exponent-factors n, 111 and p and a mathe- mati- description of the majority of possible mechanisms_

If ;thc relationship

((I -r)- I.‘3 -1) = B,(-In(l-2))

is assumed, it is possible to establish the tendency of B_- The value oi B,, was computed as a wnstant using 5 terms of the series m-r&en above by mans of IBM/360 for TX from 0.1 (step 0.1) to 0.9

2 0.1 0.2 0.3 0.4 OS 0.6 0.7 0.8 0.9

B, 0.339 0.346 0.354 0.363 0.373 0.384 0.396 0.409 0.423

The maximum deviation from the average vaIlit for B is about f IO?&. This is quite suitable having relprd to the accuracy required for the experimental data and the activation energy_ Thus the average value of B (eO.376) may be inchxded in the pre-exponential factor of k. see Eqns. (29) and (30). The above series of B, may aIso be adopted in the actual extluation.

SOLID-STATE REACTIONS AT INCREASIXG TEMPERATURE II

Combination of

nmp is suitable for:

0 0 0 Phase boundary reaction (linear Eqn. (14))

x 0 0 Phase boundary reaction (Eqns. (I5), (16))

Unimoiecular decay law (Eqn. (22))

0 x 0 Nucleation (Eqns. (I7j, (19)), linear growth of nuclei (Eqn. (24)),

linear diffusion (Eqn. (27))

00x Diffusion (Eqns. (28j, (30))

X X0 Nucleation (Eqns. (I 8), (Bj), latter stages of linear growth of

nuclei (Eqns. (25), (26)), diffusion (Eqn. (31))

x0x Growth of nuclei (Eqns. (20), (21), (23)), diffusion (Eqn. (30))

0xX Unjustified as yet

x x x Any complicated case, unjustified as yet

This equation appears to be suizable for the algorithmization of the preliminary estimation of the probable reaction mechanisms either when fitting experimental data

&/dt and z into different combinations of two exponent factors of Eqn. (32) in order

to get a constant value for k, or by direct computation of the combination of two

factors (e.g. by matrix evaluation). The suitability of such a treatment can be justified

only by the evaluation of known experimental data. This discussion touches only

upon theoretical points of view.

This procedure, however, cannot give the direct answer in the search for a true

reaction mechanism. The overlapping of the values of the exponent-factors, as well as

the expected difFicuIties wi*& the numerical solution of Eqn. (32) will provide only a rough idea of a spectrum of possible reaction mechanisms. Therefore a comple-

mentary direct investigation of the process by means of X-ray diffraction and/or microscopy becomes necessary. This should be included in alI kinetic investigations.

REFERENCES

1 J_ SEST~K, TaZanra, 13 (1966) 567; and Silikdry (Prague), 7 (1963) 125.

2 J. H. FLYNN AND L. A. WALL, J_ Res. Nat. BUT. Smnd., 70A (1966) 487.

3 J. &STAK, Silikcfty (Prague), 11 (1967) 153.

4 J. H. FLYSX, in ?-hermrrl Analysis, Academic Press, New York, 1969, p. I I1 1.

5 G. W. BRINKLEY, J. H. SHARP AND B. N. N. ACHAR, Thermal Anaiysis 1965, MacMillan, London,

1965. p_ 180; and Proceedings of the In:ernatioaaI CZay Conference, Jerusalem, 1966, p_ 67.

6 P. HOLEA ohm J. &s-r& J. Phys. Chem., to be published. 7 P. W. IU. JACOBS AND F. C. TO.WPKINS, in W. E. GARSER (Ed.), fhemistry of the Solid Sture,

Academic Press, New York, 1955, p. 184.

S D. A. YOIJXG. Decomposition ofSoIids, Vol. 1. Pergamon Press, Oxford, 1966. 9 W. D. KISGERY, Kinetics of High Temperature Processe s, Wiley, New York, 1959; and Infroduclion

ro Ceramics, Wiley, New York, 1960.

IO E. S. FREESIAN ASP B. C_ERROL, 1. Phys. Chem., 73 (1969) 751; and J. R. MACXALLUSI ASD J. TASSER, ibid., 73 (1969) 751.

Thermochim. Acra, 3 (1971) l-12

5. kSIiK, G. BEFXXREN

Ii R. A. W. HILL, ;Varure, 227 (1970) 703.

I2 J. R. %f~CCIJ*tixs AND J. TASG;ER, hirrurc, 225 (1970) i 127. 13 R. S. &UDLEY, J. P&s. Chcnr., 60 (1956) 1347.

14 J. H_ Sm. G. W. BRISDLEY AXD B. Ei. N. AC-, 1. Amer. Ceram. SOL, 49 (1966) 379. 15 J_ ihito, .I _ Phys_ Chem., 72 (I 968) 2406.

16 J. ZZAKO, I_ Thermal AnaL, 2 (1970) 145.

17 V. SATAVA AND F.~wARA,J.A~T. Ce~urn Sot., 52(1969) 591.

IS F. %X.%X4 AXD V_ ~TAVA. 1. Thumak Anal_, 2 (1970) 325. 19 A. L DRAPER, 3rd Proc_ Toronro Symp. Thermnl Anal., 1970. p. 63.

20 A. L. DRAPER ASD L. K. SVEKM, TIxermachim_ Acra, 1 (1970) 345.

21 I_ G. MLFXGCXESCU ASD E. I. SEGAL. Rec. Roum. Chim., 10 (1965) 307. 22 H. F. CORD- J_ Phys_ Chenr, 72 (1965) 21S5. 23 R. I_ LEROY, J_ Phj-s. Chem., 73 (1969) 4338. 24 R. D. S~UXKON. Tranr. Fma&y Sot.. 60 (1964) 1902. 25 H. S. i&'%lJRTy, D. L. BEDEFMA ri ASD E. A. HEUXZ, 1. Phys_ C/rem.. 72 (1968) 746.

26 H. D. PAM-, Irrr. J. Chem. Kinetics, 2 (I9iO) 7’7.

27 C. Y. WEN AXD S. C. WAXG, Ind. Eng_ Chem., 62 (8) (1970) 30.

28 J. &~%a, V. SATAVA ASD V. &HAIL, Silikdfy (Prague), 11 (1967) 315. 29 J. %sr&, A. BROW%, V. ~IHAL ASD G. BERGGRES, in Thernzzl Analysis, Academic Press, New

York, 1969. p. 1035. 30: J_ VACHU~KA AND M. Vomoitri, Thermochim. Acfa, 2 (197X) 379. 31 J. ?&sr.& AXD G_ BERGGRES, Chem_ LisIy (Pruguej, 64 (1970) 695.

32 V. SATAVA, Introduction to Physical Chemistry of Silicates, SNTL, Prague, 1966 (in Czech).

33 P_ P. B~~DSIKOV ASD A. M. GISSTLISG, Principles of Solid State Chemistry- Gordon and Breach

New York, 196s (transl. from Russhn).

34 P. G. SHEWON, DigiLFion in Soliak, McGraw-Hill. New York, 1963. 35 D_ A_ FRAXK-.KAMEXTSKI~. Dijksian und Hear Tramfzr in Chemical Kinetics, Plenum Press,

N’e= York, 1969 (transl. from Russian).

36 hi. hf. PAVLYUCHESRO, Heterogeneous Chemical Reucrions, Nrtuk. Tcchn., Minsk, 1965 (in

Russian).

3T B. DELUOS ASD Y. C. JUS;CERS. lnfrudarfion U ia Cinerique Heferogene, Tczhnip, Prlris, 1969.

35 D. NICHOLSOS, Irazz Fua&- Sot_. 61 (1965) 990. 39 J_ HLME ASD J. Corvrs, Proc. Roy_ Sot., 125 (1929) 635; I32 (1931) 248.

4Cl B. V. EROFEEV, Compr. Reqd. Acad. Sci. URSS, 52 (1946) 51 I ; and Reacticiry of Solids, Elsevier,

Amsterdam, 1961, p. 273. II K. L. ;CIAMPEL, Z. P;~_rsik. Chem., AISi (1940) 235.

42 W. A. JVHSS~S ASD R. F. MEHL, Trans. Amer. Inst. Mining Mer. Eng., 135 (1939) 416.

43 hf. A~RXW, J_ Chem. Phys_, 7 (1939) 1103; S (1949) 212; 9 (19$1) 177.

M W. E. GARSER ASD H. R. HAIUS, Proc. Roy. Sot., Al39 (1933) 576.

45 E. G_ PROL? ASD F. C. TOMPKINS, Truns. Faradq Sot.. 40 (1944) 455; 42 (1936) 468. 36 G. VALESSI, Compr. Rend., tOi (1935) 602; ibid., 202 (1936) 309.

47 W. JAXDER, Z. Amrg. Allgem. Chem.. 163 (1923) 1.

4C A. %I_ GISXAUSG ASD EL I. BROUSSHTEIX, Zh. Prikf. Khim. (Leningrad), 23 (1950) 1249.

49 R. E. CARTER. J_ Chem. Phys., 34 (1961) 2010; 35 (1961) I137_

50 H. L. FRISH ASD F. C. COLLISS, J. Chem. Phys., 21 (1953) 2158. 51 M. Pouxn ASD E WIG~XR, Z. P&-s_ Chcm., 139 (1928) 439. -

52 G. F_ Hiirn~, Morz.t_sh. (TH Graz), 85 (19%) 365. 53 E. G. FROLT, 1. Inorg_ Xuci. Chem., 7 (i958) 368. 54 Fcf. hf. PAVLYUCHESKO, Zhur. Fiz. Khim.. 29 (1955) 996.

Special Review

P H I L O S O P H Y OF N O N - I S O T H E R M A L KINETICS

J. gEST.~K

Institute of Physics of the Chechoslovak Academy of Sciences, 18040 Pra#ue, Czechoslovakia

(Received August 8, 1978)

The most debatable and discrepant viewpoints of non-isothermal kinetics are dis- cussed in the form of twelve questions and answers. The reputation of non-isothermal kinetics when carried out by thermoanalysts; the consequences of simplified concepts transferred from the kinetics of homogeneous reactions; the physical meaning of basic kinetic parameters in solid-state processes; the kinetic compensative effect and inter- dependence of kinetic parameters using the Arrhenius rate constant; the mutual use- fulness of differential and integral methods of kinetic data evaluation; their accuracy and correctness; the reliability of DTA measurements; non-isothermal versus isother- mal investigations; equilibrium and kinetic data and their mutual effect; the extended discussion initiated by MacCallum and Tanner; non-isothermal data publication policy; and finally the use of computers.

Instead of an introduction

An article which would contain merely questions with their answers on some notoriously discussed but not yet clear topics of the fashionable kinetics of non- isothermal processes first came into my thoughts on the basis of the remarks presented at the panel discussion at the TA seminar on kinetics held in Budapest, in July 1972 (Berggren, Jesenak, Pungor, Zsak6). This idea, however, survived untouched until the meeting of Hungarian thermoanalysts in Balatonfiired in October 1976, where it matured during the round-table discussion (Dollimore, Meisel, Oswald, Rouquerol). The written form was finally encouraged by J. Simon, editor of JTA.

The problematics of description of non-isothermal processes fall into two cate- gories. The part dealing with the more rigorous thermodynamic aspects and definitions of experimental conditions and thermophysical experiments in general was published in TCA 28/2 in 1978 in the form of a classical type of review. The present part intends to pay attention to some controversial viewpoints regarding non-isothermal kinetics, which sometimes intentionally and sometimes by mistake veil the consequences of dynamic experiments. This untraditional form for a review seeks, last but not least, for possibly better answers to those questions which are felt not to have been explained in a satisfactory manner, or which have

3. Thermal AnaL 16, 1979

504 SESTAK: NON-ISOTHERMAL KINETICS

not been taken into account as yet. This also remains a task for the reader's own consideration when studying the article.

It should further be mentioned that this approach was affected by the discus- sions held during the German autumn school on kinetics, in Eisenach in 1977 (Anderson, Heide, Ludwig, Schultze). My thanks are also due to my friends and co-workers, particularly V. Satava and P. Holba, for a long, cheerful, as well as most creative cooperation, and for their frank discussions on all aspects (reasonable and even crazy) of philosophy and science.

Questions and answers

(1) Q: Why has non-isothermal kinetics gained so bad a reputation? Is it due to experiments carried out mainly by thermoanalysts?

A: Any advent of commercial apparatus production sophisticated stage of automatization and even pressing development of evaluating techniques makes it easier to produce nice curves and other digital or punched output ready for further mathematical treatment. As already long ago, this has tempted many investigators to determine easily attainable parameters of processes called kinetic data [1 ]. The result is well known: a number of scientific reports containing infor- mation with varying degrees of reliability. It is evident that those making classical investigations of kinetics have attacked non-isothermal studies. This has con- sequently resulted in a rather great disrespect for non-isothermal studies, and in some restrictions with respect to the publicability of such data in some of the more recognized journals.

Let us take an example. In the literature we can find recipes on how calculate basic kinetic parameters, regardless of their true physical meaning. What is more, this is often applied to a most complicated case, i.e. DTA measurements of solid- state processes. For instance we can draw by hand a curve in the form of a DTA peak and then calculate certain numerical data: the activation energy by Piloyan's method [2] and the reaction order by Kissinger's method [3]. There is certainly something out of order, as nobody can extract the similarly important thermo- dynamic data so easily without some additional stzecifications as to the curve origin (material and experiment). The puzzle of such a non-isothermal kinetic treatment is a fairly good coincidence with the analogous data established with complementary measurements. In spite of this we can easily see some lack of logistics or, at least, the insufficiency of the physico-chemical grothwork in some branches of non-isothermal kinetics. However, I by no means want to increase any consciousness of thermoanalytical society that all non-isothermal data,ought to be treated with great caution, as was felt during the 5th ICTA in Kyoto. I am aware that any critique of the present kinetic procedures because of they unsatis- factory description of the reality of e.g. heterogeneous reactions is much easier than making a new proposal how to substitute it by more suitable theory. Therefore

3. Thermal Anal 16, 1979

SEST.~K: NON-ISOTHERMAL KINETICS 505

the rejection of the present mode of the calculation of certain parameters which we want (although not always may) compare is not a solution at all.

(2) Q: Non-isothermal kinetics, in fact, is liable to pay for the mode of its histor- ical introductions. Where are the roots of possible misfits and where can we best learn the basis of non-isothermal treatments?

A: Probably the first description of a non-isothermal treatment of TG data was reported by Van Krevelen, Van Heerden and Hutjens in 1951 [4]. The real progress was started by the work of Murray and White, Tsuzuki and Nagasava (for detail see [1]) Kissinger [3] and Freeman and Carroll [5] in the late fifties. The first concise reviews came in the middle sixties [6-8]. However, we should not forget the pioneering work of Akahira [9] who introduced the integration of the tem- perature-dependent exponential in the late twenties.

Unfortunately, all the above articles carried along the basic terminology by analogy with the adopted kinetics appropriate for the description of homogeneous reactions in gases and/or liquids. However, this was an understandable result of an easy treatment of the concept of reaction-order to surmount the initial mathe- matical difficulties. Later on, the continuous schematic and sometimes blind transmission of such premises led to problems in the application of the non- isothermal treatments to heterogeneous processes. This has been the subject of severe criticism [1, 10], but many of these notorious oversimplifications survive. Nevertheless, I do hope never to read again introductory sentences such as "solid- state process may be represented by a simplified kinetic relation defined by reac- tion o r d e r . . . "

The present state still needs a monograph truly devoted to non-isothermal kinetics. The existing literature deals either with homogeneous reactions or with a formal description of heterogeneous processes* [1, 11 ]. Some reviews have been published in JTA [12, 13] as well as in the Proceedings of ICTA conferences [14].

Otherwise, much useful information is scattered throughout all these scientific journals which still dare to print articles on non-isothermal kinetics. The ratio of the contributions on the different kinds of kinetic analysis was searched for in two most representative TA journals (see Table 1). Although no steady trend towards a definite sort of kinetics has developed as yet, it is worth noting that the first signs of a new, actually non-isothermal approach are to be seen [15, 16].

(3) Q: What do kinetic parameters such as activation energy and reaction-order really mean ? Are these terms 9enerally applicable, particularly assuming solid-state processes?

A: From the textbooks we know well what these terms express, i.e. we can ascribe a definite meaning to the activation energy E, e.g. for the reaction of

* There also was published a book by E. Koch: "Non-isothermal Reaction analysis" which unfortunately I have not seen as Plenum Press refused to provide it for reviewing. A book by K. Heide about non-isothermal kinetics of heterogeneous processes is announced to be pub- lished in Berlin. Another book including kinetics of solid state reactions is in the course of preparation by author to be published jointly by Academia and Elsevier.

16 J. Thermal Anal. 16, 1979

506 ~ESTAK: NON-ISOTHERMAL KINETICS

gaseous iodine with hydrogen either by coordinate energy calculation or by the scholastic approach. It is, in fact, a potential barrier separating the initial and final states. At the moment when we admit one reactant to be a solid phase, the above reaction scheme presumably loses its validity; nevertheless it can still yield certain numerical values. Consequently the rate constant is affected by the choice of a kinetic model and may be found in the form of a power. Such an exponent is a function of nuclei formation, interface and mass transport, and thus the analyt- ical formulation of the kinetic equation becomes a decisive step. The parameter E fitted to the given kinetic equation is understood as a derived constant and is of limited validity. The obligatory expressing of E in the dimension of energy per mole is .retained as a practical result of multiplication by the gas constant (compare for example, phase-boundary processes). The overall (apparent) value of E is not always a primitive value for nucleation-controlled processes [17]. It is composed of partial energies associated with nuclei formation, crystal growth and diffusion, and altogether may reach a rather large value. Experimentally, such a process exhibits a very sharp, explosive-like course; its surprisingly high barrier thus does not permit the process to occur thermally until the barrier is finally surmounted the reaction then rocketing at full speed. On the other hand, the reactions which are limited by mass transport show a rather low value of E, although the diffusion seems to be a smooth process taking place within a compar- atively large time-temperature interval. For metals it may be well interpreted by a defect-walk mechanism [1, etc].

In contrast to the physico-chemical approach, we must admit that the reaction- order concept may be an adequate tool for finding the most favourable conditions to control and carry out an engineering process satisfactorily. It may be assumed that not much care is directed to the reliability of the mathematical description nor to the correctness of the kinetic parameters. The aim is merely a formal fit of the experimental data using the simplest form of polynomials, to be able to compare reaction rates or energy barriers for various experimental conditions in a sensitive enough manner.

No less important is an appropriate choice of the experimental set-up and measuring conditions to match the conditions of the process in question. It is evident that the thermogravimetric investigation of micro-samples in vacuum is of little applicability to the industrial process of the burning of limestone, while the study of the decomposition of a ball-like sample at least 10 mm in diameter may provide useful data as regards not only the chemical character of the process, but more likely to its geometric aspects, mass and heat transfers, etc.

In addition one must keep in mind that a bad kinetic result either due to pure and undefined sample and/or vague experimental conditions cannot be veiled by any resonant nor ingenious mathematics. That is, "experiment does not have to agree with our theory and process never obeys our equations" [10, 14b].

(4) Q: Some authors advocate either the separability or non-constancy of kinetic data. Is this true, and how is it related to the so-called kinetic compensation effect?

A: Although this is connected with the preceding question, it is worth discussing

J. Thermal Anal. 16~ 1979

SEST~.K: NON-ISOTHERMAL KINETICS 507

separately. Let us first talk about the "Arrhenius" rate constant, k(x), itself, which h as~a~general validity for thermally-activated processes, where either diffusion and/or phase-boundary chemical reaction are effective as basic rate-controlling processes, k(r ) is then composed of the pre-exponential factor, A, describing the frequency of attempts to surpass an energy barrier E of the process, and the expo- nential, exp ( -E/RT), expressing the probability of successful attempts. It was argued that the lack of an energy statistical distribution in real solids rules out the use of this concept [14b]. This certainly would deny the fifty years' work of famous scientists in the field of heterogeneous kinetics although a certain criticism of mechanically derived models upon oversimplified geometric systems is in its place. At least from a viewpoint of general relaxation processes we ought to admit that there always exist an (energetic) hinderance separating initial and final states which must be surmount if the process is to advance even it is located to certain sites.

For a fixed value of the exponential, the constant ratio (E/T) must be maintained by the equal compensation of changes in both the numerator (E + AE) and denom- inator (T + AT). Similarly, for a fixed value of the reaction rate, the change of the whole exponential must be compensated by the change of the we-exponential factor. For reasonable values of reaction rates, we thus cannot obtain the mutually independent values of the kinetic parameters (E and A). As consequence for high- temperature processes (T > 1500 K) we find greater values of E (E > 100 kcal) and A (A > 1016 s-l), which certainly lend to difficulties in their theoretical inter- pretation [18]. The interdependence of kinetic parameters can be best demon- strated through the theoretical construction of artificial TA curves with respect to their position within the T-coordinates and the magnitude of the kinetic par- ameters [19], which is a frequent and often recurring subject of kinetic reports of (cf. [1]). Nevertheless, it reflects sensitively any case where the rules of our kinetic game are not fulfilled. It may happen when the temperature-dependence of A (=.,IT b) is not accounted for and/or the inappropriate model relation F(c 0 (instead off(e)) is employed. In such a case the kinetic compensation effect can be understood as a misfit to be described analytically by equations: E = E - - b RTln Tot E = E - RT[ln F(e)/f(e)]. So far, it is still an open question how much the experimentally-found thermal dependence of E is an entirely physical property of it or merely a result of non-uniform experimental conditions and homo- geneity of material or inadequate mathematical treatment.Adonyi [20], for example, emphasizes that the examination of the function: log A versus E, together with the motion of coordinate axes, when taking into account the measuring conditions, can only give deeper information about the kinetics.

(5) Q: Which method of kinetic data evaluation is better, differential or integral? (quite a common question !).

A: Every method has its positive merits, as well as disadvantages, associated with the mathematical treatment itself and further affected by the demand for experimental data of a certain quality and extent. The principal classification can be seen from the following scheme:

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508 SESTA.K: NON-ISOTHERMAL KINETICS

variable model relation rate constant name

s, ~, ~, T, T d[f(s)]/[dc~f(s)l kT, dkT/dT double-differential o~, ~:, T Af(~) k~ differential As, A~, AT Af(z.) ~, n, m ~ E difference-differential s, T g(~) = J" d~f(s) p(x) ~ J" k(~r~ integral

The differential method of kinetic data evaluation can be understood as the basic method, as it utilizes the primitive form of kinetic equation, i.e., ~ = kTf(s). It requires threefold input data sets, the reaction rate usually being the derived one. Generally, any derivation or differentiation exposes the resulting equation to its becoming more distinguishable with respect to the experimental input data, but also more time-consuming and sensitive as regards their fineness (namely ~). A simple comparison of the results obtained by various formal forms o f f ( s ) function based on an increasing number of reaction-order-like exponents may not provide any deeper information about the mechanisms involved, because the closeness of the relation between the calculated and measured data is increased to a limited extent [20].

By integrating this primitive kinetic equation, one obtains such a form of kinetic treatment where only twofold experimental data sets are needed. Such a method is then presumably less laborious as well as less sensitive to experimental errors (scatter in ~ is avoided). This, of course, is on account of the model relation distin- guishability, i.e. the analytical forms of model relations g(s) overlap each other for the different cases of rate-controlling processes, f (s) [21]. The foundation of these methods is complicated by establishing the so-called exponential-integral function p(x) [9], which is obtained through the integration of the rate constant k T within the temperature limits.* The practical evaluation by integral methods is comparatively easy, but requires certain assumptions about the behaviour of the p(x) function [1]. It often results in the double-logarithm plot, because the g(~) function usually exhibits the logarithmic form itself, particularly for nucleation- growth processes. It is worth quoting an article in the Journal of Irreproducible Results saying that multilogarithmization is the most recommended way of all to diminish sensitivity. Certainly, this is not directed at the condemnation of the utilization of integral methods, but should rather be understood as suggesting caution in searching for the true limits of the applicability of integral methods.

In this place the uncorrect treatment based upon the derivation of Johnson- Mehl-Avrami-Kolgomorov-Yerofeev (nucleation-growth) equation is worth noting. It results from the origin of this equation as derived already by the integration at constant temperature. Introducing p(x)-function to this primary

* It is worth noting that Reich [22] once accomplished such an integration by simply sep- parating the exponential in front of the integral (,~, f d T = AT) , which was evidently subjected to severe criticism [6]. Nevertheless, recent articles showed that after long and complicated mathematical manipulations one reaches a similar result [1, 4, 6] indicating that nature always tends to simple solutions.

J. Thermal Anal. 16, 1979

SESTAK: NON-ISOTHERMAL KINETICS 509

integration makes it correct and leads, fortunately, to the same result differing only by a constant [29].

In contradiction with the preceding methods, where a linear temperature increase is assumed, a priori, the double-derivative method (which has not been officially named as yet) permits introduction of the actual temperature increase, through T and truly instantaneous values of T. This, of course, is manually too difficult and experimentally too sensitive for elaboration, and hence is not yet popular.

For successful kinetic evaluation we should always bear in mind the purpose of our kinetic analysis and the means we have at our disposal for its realization. Methods of kinetic data calculation should not be self-satisfying and nor should we be blind to their possible shortcomings.

(6) Q: What about the accuracy of kinetic data calculation? Is this only a con- sequence of the accuracy of the experimental input data, or can it be further affected by its mathematical treatment?

A: First of all we must distinguish not only the accuracy of the numerical values calculated, but also their correctness, i.e. the coincidence between our mathematical model (namely the f(c 0 function) and the actual process under investigation (only the data fulfilling both conditions are reliable). The diligent improving of numerical accuracy is typical evidence of how much importance is laid on mathematics and how little on logistics, as can be found in numerous articles trying to settle the accuracy after the decimal point [1]. In the light of some poorly-defined experiments such refinements are ludicrous if one imagines the possible misfit introduced by a false application of unreliable kinetic models. On the other hand, we cannot deny the definite mathematical skill and wittiness of the theoretically-oriented articles, which, however, sometimes falls beyond the range of normal kinetics, e.g. the reports refining the calculation of the p(x) function. Everyone can remember that once it was fashion to search for a new rearrangement in mathematical analysis and to be frank, many of us non-isother- mal kineticists must admit to the dream of discovering a new method of kinetic treatment to be named after the author. However, in most cases this ended in the knowledge that the arithmetics did not leave much space for success. As side- effect, the accuracy of different kinetic evaluations was listed mainly for different kinds of integral methods established through different ways of exponent-integral expansion. Finally there appeared the still underestimated article by Broido and Williamson [21], who diagnosed exactly what integral method yields the most accurate result by distinguishing the accuracy of individual polynomials hidden in individual kinetic treatments. For instance there are the common plots of log g(c~) vs. log T, ATvs. 1/T, the last more accurate than the first one. Nevertheless, further new such reports have very recently been published showing the notorious desire to increase the ways of deriving methods of kinetic data evaluation and to improve the numbers of resulting kinetic constants, not considering that any deviation in E under comparable conditions of the same g(e) always gets within the limit of _+ 10 rel ~o of E, well accepted in all kinetic studies. This is less impor-

Y. Thermal Anal: 16, 1979

510 ~ESTAK: NON-ISOTHERMAL KINETICS

tant than the reliability of the g(a) function, the correct establishment of which should be the real goal of chemical kinetics.

(7) Q: There are controversial view regarding the reliability o f the most popular TA method: DTA. Is it useful for kinetic studies or is it too dependent on its experi- mental set-up and thus more suitable for analytical applications only?

A: As already noted in the first question, the oversimplifications of kinetic analysis of DTA data led in many reports to the popular plot of log AT vs. 1/T [2] to extract E from the slope. It certainly provoked much criticism [10, 24, 25] directed to all unclear and questionable aspects of DTA, going sometimes so far as to condemn completely the use of DTA in kinetic studies. This is quite under- standable if we imagine the complicated relation between the measured (AT) and searched (d) properties. In this DTA equation [26] particularly the term containing A i" is often neglected, although it expresses the important fact that the heat cannot be delivered to the sample infinitely quickly. When searching in the classical articles about DTA (Borchard and Daniels [27], we can already observe that the Piloyan plot [2] ought to be written as log (Cp AT + kDT A AJ') vs. 1/T, where Cp and kDrA are heat capacity and DTA apparatus constant, respectively; this has been employed so far only twice [28, 29]. What is worth noting again, is the embar- rassing coincidence between DTA kinetic data and those obtained under compa- rable conditions. Is there a real picture of a chemical process in the study, or does it report the constants of mass and/or heat transfer. All the same time, however, we must emphasize that such experimental data as those of DTA contain actual information about the non-stationarity of the system, and it is only up to the inter- preter how to decipher the curves. This is not easy, as has been demonstrated in numerous articles utilizing gradient theory (Eriksson [30]), Laplace transforms (Takeo [31]), Green's functions (Akita [32]) and other high mathematics, that have not received the deserved attention. This is probably because the TA experi- menter has not much time to get through high theory, and even with the aid of computers he tends to deal with easily understandable approaches like the model based on the phenomenological description of heat-exchange between two bodies of uniform temperatures [26].

DTA measurements are, in fact closely dependent on our experimental inability to maintain a strictly linear temperature increase. If we diminish the temperature deviation, the DTA peak also disappears. An increasing amount of generated heat yields a relatively higher temperature gradient and heat-exchange which may contribute to the DTA peak formation. In the first approach this may be neglected when the different heating rates affect the value of E by less than + 15 %. The different sizes of DTA peaks also have no effect in the logarithmic representation if the peaks are geometrically similar. A small deviation of the true sample temper- ature (T) from the programmed linear increase (T=0 = const) may be negligible if the temperature interval of the peak is comperatively large. A good coincidence between DTA and optical microscopy data is shown in [29] on crystallization of glass.

(8) Q: I f there are so many troubles with the gradients, would it be better to

J. Thermal Anal. 16, 1979

SESTA.K: N O N - I S O T H E R M A L KINETICS 511

employ only isothermal methods? What, in fact, is the difference between iso- and non-iso-data?

A: One of the most important problems is the extraction of appropriate infor- mation from a given set of experimental data. Let us define an isothermal process as a progress of the state of the system at constant temperature as a function of time, where the measured property Z is defined by the equations: Z = Z(t), T = const. For the system exposed to externally-applied constant heating, the non-isothermal process is defined by time-temperature sequences of its state, i.e., Z = Z(t), T = constant = 0. Such resulting data are, of course, more com- pact and may provide a wider but also a more complicated basis for their analysis.

Numerous authors argue that the system being heated is distorted by the creation of temperature gradients, i.e. 2r r const. Fortunately, such a system reaches a quasi-stationary state (2P + A = eonst)under steady conditions of heating. The constant distribution of gradients makes it possible to define the state of the system by an average value of the measured property, which is in accordance with the theory of phenomenological thermodynamics and which, moreover, agrees with the single-valued data provided mostly by all TA measurements. If a thermal process occurs, the steady state becomes dependent on the time- temperature progress of the process, and thus T ~ const and T = T(t). In such a case, however, the isothermal course also becomes distorted, as the sample temperature is not constant either (T ~ const) due to self-cooling and/or self- heating of the sample. This fact is often neglected or forgotten, although it may have the same importance as in non-isothermal measurements. The advantage of non-isothermal experiments, however, is a continuous measurement of the instantaneous sample temperature, and its deviation from a predetermined course helps us to determine not only the progress of the temperature gradients but also the entire progress of the process itself, such as in the case of DTA (see the pre- ceding question). For ordinary measurements we may be satisfied with certain mathematical simplifications which are adequate for the simple kind of TA instru- mentation; but we always have to be aware of the limits and consequences of such simplifications. Any sort of treatment can thus be tolerated if the author recognizes its shortcomings and gives to it the necessary scientific aim. An intimate correlation of iso- and non-iso-methods may then lead to refining of the true meaning of the rate constant as well as the physico-chemical aspects of the process in question.

(9) Q: Can we ever speak about thermodynamics in terms of dynamic thermal experimentation, how compatible are kinetic and equilibrium data; and what then is the equilibrium background of the process?

A: Let us first put our system investigated by TA into the scale of classical thermodynamics (thermostatics) and rational thermodynamics as limiting cases. Assuming three basic quantities, T, P, Z, we may write for theoretically quasi- stationary TA process:

d. Thermal AnaL 16, 1979

5 1 2 S E S T A K : N O N - I S O T H E R M A L K I N E T I C S

class, thermodynamics

T(dT<) P(dP<) \~ Z(dZ<) \ \ \ ~

rat. thermodynamics

Td-, I; . . . . ) ,-~ p(p, lb, .) �9 . .

/ -

, , " z ( 2 , 2 , .) . J

thermal analysis T(AT, T =consO F(~e, P = o) Z(AZ, 2 ~ kinetics)

This, in fact, is in accordance with a well-known result following from the thermo- dynamics of an irreversible process, which says that for systems which are not too distant from their equilibrium state and in which the processes proceed quickly enough, the ordinary thermodynamic logistics and relations may be employed. Joint and concise mathematical descriptions of the dynamic (flux) and static (equilibrium) behaviours of such systems is not yet available and, anyhow, will be too complicated for normal use. Therefore, two separate disciplines have developed: thermodynamics, where we attempt to determine the equilibrium quantities by extrapolation to equilibrium conditions (T, P(Z) ~ 0), and kinetics, where we try to describe the time-temperature development of the system equi- libration (Z, AT, AP). It is evident that for the system in equilibrium (dT, dP, (dZ<) it is meaningless to speak about kinetics, as there is no driving force for any change (only time-temperature fluctuations). On the other hand, when the temperature of the system is continuously raised, the equilibrium conditions may also be gradually shifted, creating the so-called equilibrium background of the process. This is important for processes which proceed within a certain temper- ature interval (variant processes and/or diffuse first-order transformations), such as melting in a two-phase region [33]. This effect is often neglected during non- isothermal investigations, although it ought to be incorporated into the calculation in the form of the propagation of equilibrium under the idealized conditions of infinitely slow heating. High enough heating rates, however, enable us to evaluate the major part of a TA curve above the two-phase region, where ordinary kinetics is applicable again.

There is another influential factor: proximity to equilibrium, which should often be involved in our mathematical operations. Temperature integration of the rate constant yields, in fact, the difference of two p(x) functions [p(x) - p(x0)], in which the second part may be neglected only when the measurement is carried out at temperatures distant from the equilibrium temperature. This becomes important for reversible processes studied at low heating rates, so that a great portion of material reacts in the vicinity of equilibrium [34]. Inclusion of the driving force (AG) into the primitive form of the kinetic equation yields the rate constant in a more complex form: kT(1 --ka) . Accordingly, the integration becomes more complicated, giving as many as four p(x) functions [13]. Hence,

d. Thermal Anal.~16, 1979

SEST~,.K: NON-ISOTHERMAL KINETICS 513

it is evident that appropriate knowledge of the kind of process investigated, with an adequate choice of measuring conditions, particularly the heating rates [1], makes it possible to achieve reliable kinetic analysis.

(10) Q: The year 1970 was full of activity in seeking the fundamentals of the correct representation of the non-isothermal kinetic equation, which was initiated by MacCallum and Tanner's article. What in fact is the essence of this problem?

A: In principle this concerns the mathematical formulation of a basic kinetic equation valid under non-isothermal conditions. Two time-dependent variable (t and T) were put together to represent the state of the system (c 0. Upon a normal mathematical procedure of partial differentiation, this assumption leads to the dependence: ~ = (~)r + (0~/~3T)tT, in which theterm (~)r is assumed to describe the isothermal rate only. This puzzle was first raised by McCallum and Tanner [35 ]. The wide public reaction showed again the danger of mere mechanical application of the mathematical symbols involved [36]. Let us investigate the true physical meaning hidden in such a relation, imagining a heated container filled with water, about which we assert that the content of water (c 0 is dependent merely upon its instantaneous temperature (T) and the time (t) since its filling (t = 0), regardless of whether any water still remains in the container. It is evident that this logistic approach is wrong, as we do not describe effectively enough the actual dynamics of the boiling process in question, i.e. the volatility rate (~) as a function of ready- to-react material (c 0 at a given temperature (T) (compare question (8)).

Nevertheless, such discussion may be very instructive and such kinds of articles should from time to time, be accepted for printing. In this place the editorial policy should be noted. An example is the case of philosophically oriented reply notes [37-39] published in Nature and following the article by MacCallum and Tanner [35]. Although the problem has not been sufficiently clarified [37-39]. Nature stopped accepting additional remarks, leaving this subject to be further discussed in "local" journals [36, 40, 41], possibly regarding it as "thermoanalyt- ical business" only. In many cases it did not touche the point as yet [41, 42]. The wide kinetic community, however, may also feel the lack of the final word [42]. On the other hand, I touched on the above problem during my 1975 lecture at the Institute of Chemical Physics in Jerevan. To my surprise I learned that this puzzle was completely evident to those people dealing with the kinetics of adiabatic gaseous reactions. Here I probably approached one of the most painful sides of kinetics and nowadays possibly all scientific work in general: the lack of an appro- priate exchange of information.

(11) Q: Is it worth paying such attention to non-isothermal kinetics? Is it merely good for a narrow region of interested scientists, and how about the already-mentioned publication policy?

A: Well, it should be stressed once more that the problem associated with dynamic kinetic measurements has its significance in all branches of experimental kinetics, because truly constant experimental conditions are hard to achieve. Non-isothermal kinetics, as is obvious from recent articles printed in JTA and TCA, is primarily limited by the TA experimental technique, but is carried out

J. Thermal AnaL 16, 1979

514 ~EST~,K: NON-ISOTHERMAL KINETICS

Table 1

Survey of the number (and tel. ~) of kinetic articles* so far published in J. Thermal Anal. and Thermochim. Acta

Year and journals

1970 JTA TCA 1971 JTA TCA 1972 JTA TCA 1973 JTA TCA 1974 JTA TCA 1975 JTA TCA 1976 JTA TCA 1977 JTA TCA 1978 JTA TCA Total JTA TCA

Number of reviewed articles

The par t dealing

with kinetics

Theory of formal

approach

Applica- tion of formal

approach

Formal mechanism

(stoichio- metry)

Theory of mechanism

and kinetics

44 57

43 74

39 78

58 106

68 101

120 144

91 121

77 148

67 237

521 1066

12 (27 %) 12/21

3/7 17/23

10/26 13/17

23/40 20/19

15/22 8/8

24/20 35/24

34/37 21/17

26/34 26/18

19/28 49/21

166/32 201/19

4 (33 ~) 1/8

1/33 5/29

5/50 4/31

8/35 4/20

3/20 2/25

6/25 20/57

9/26 3/14

5/19 2/8

5/26 12/25

46/28 53/26

3 (25 5/41

0 2/12

2/20 2/15

5/22 6/30

6/40 2/25

10/42 2/6

15/44 5/24

4/15 7/27

3/16 15/31

48/30 46/23

%) 3(25 Z) 5/41

1/33 5/29

2/20 3/23

2/9

3/15

4/27 1/2o

2/8 2/6

7/21 7/33

8/31 5/19

2/11 8/16

31/19 39/19

1 (8 %) 1/8

0 3/18

o 2/15

2/9 5/20

1/7 3/30

4/17 5/14

1/3 1/5

3/12 5/19

2/11 5/lO

14/9 3o/15

Mechanism and

kinetic applica-

tions

1 (8 %) 2/17

1/33 2/12

1/10 2/15

6/26 2/10

1/7 0

2/8 6/17

2/6 5/24

6/23 7/27

7/37 14/29

27/16 40/20

* It presents a certain (subjective) view in ranking articles as those dealing with formal determination of kinetic data (reaction order) and its formal applications, analysis of reaction stoichiometry (mechanism), theory of reaction mechanism (kinetic models) and finally the applications of kinetic models in physical-chemical studies. (The number of articles compiled must not be identical with their actual number published in the year because of some missing issues.)

by numerous workers in different fields (cf. Table 1) (question (1)). The surpris- ingly low citation index is p robab ly rooted in our preoccupat ion which often leaves insufficient t ime for a patient understanding o f other authors work , logistics and part icularly symbolics. A good example is the series o f very na r row specialized

]. Thermal Anal 16, 1979

~ESTAK: NON-ISOTHERMAL KINETICS 515

articles on the p(x) function, following each other at an interval of about one year, but making little reference to the previous ones [42-49] although printed in similar journals (TCA or JTA). This, of course, is also subject to the quality of the referee's work, the importance of which is sometimes underestimated, partic- ularly when considering the still explosive trend of kinetic publications.

Our often very sectorized specialization leads, unfortunately, to a certain type of ignorance concerning the other works on a similar theme. This results in easy self-citing, because our own past work is most simple to be understood again "usually with the little effort of one re-reading". I once read that the usefulness of our citations may be evaluated by a quotient obtained by subtracting twice the self-citations form the remainder, which should at least be positive. (Ref. 26 is not the best example.)

Another side is the efficiency of the written text. The repetition of previous results and/or theories, so obvious in kinetics articles, is sometimes powerless and so subject to severe criticism, but, on the other hand, it may help the reader to catch the point very quickly. The basic fault of most introductory lines is their dispersity and/or length. It might be a good idea to introduce a key (or code) system (similar to those employed for abstracting), making it possible to write an introduction in a standard, concise but summarized way, even avoiding redun- dant derivations.

Finally, I should also present a diffident comment regarding the elegance and legibility of scientific language, as recently emphasized in a very nice manner by Keatch [50]. Such discussion belongs certainly to those who possess English as their native language. Besides, the quality and amount of our work the beauty of a well-written scientific report should be a (goal and entire pleasure) of our activity, similarly to an artistic work.*

(12) Q: The offensive of computing techniques in all branches of science and engineering has not certainly avoided the field of TA. Will computers be of real help in our kinetic work, or will they solve only the most painful numerical troubles?

A : This question must be answered in a broader sense, as we must surely admit that in the near future all TA laboratories will be fully or partly equipped with computing equipment (hardware, software, microcomputers, etc.). The question hence should read: are we ready to accommodate the new methods of treatment in our everyday practice? Sophisticated sets have already appeared on the market, but their furnishing with standard programs (good for the late sixties, compare kinetic articles in TCA 1977) has brought again the danger of employing good mathematics in a rather poor physical sense. On the other hand, some terms from the computer language (slang) have become so fashionable in accelerating sales, that their real information content may be discredited.

The most important fact in data treatment is how the output signal from a TA apparatus is maintained. A typical chart record will probably persist so as to give

* I would like to support the proposal as to create an annual prize for the best presented paper appearing in either of TA journals and or I C T A procceedings.

J~ Thermal Anal. 16, 1979

516 SESTAK: N O N - I S O T H E R M A L K I N E T I C S

an immediate, first-sight feeling as to whether the curve is of good or poor quality; however, the further elaboration of the curve then requires an enormous effort: an intermediate step of usually manual and time-consuming trace reading and numerical data preparation. Another problem arises for the preparation of short algorithms and for often changing mathematical procedures, where debudgeing, data punching and mechanical transfers are sometimes too slow and immobile in computing centers. The desk or pocket calculators provide a more direct and flexible service, and should become a new "logarithmic ruler" in everyday thermo- analytic work. Nevertheless, the most serious obstacle rooted in extensive numer- ical and statistical data treatment has been overcome. Of course, there may appear certain inequalities, hidden in the possibility of describing the system by sets of more sophisticated equations but where we may lack a sufficient amount of physical information relating to a single point measured (T - e versus the sets

i i

of ~ T i - ~ cq data). At the same time, we may not be ready to know how to 1 1

interpret properly a new sort of output data which may not be in the form of single-valued constants or numericals.

More work should be done in using methods of chemical and mathematical statistics, information content and capability including its theory, hypothesis testing and goodness of fit tests, etc [51 - 53]. It would be recommendable to all author to get acquainted with at least some books 'on statistics [54] prior writing an article on numerical results.

A big qualitative step is the direct digitalization of the TA apparatus signal, which will presumably become a basic requirement for all TA experimentation. Entire on-line computing systems and possible self-control of TA experiments seems to be not too effective, because it may keep the computer busy with inade- quate and rather mechanical work, such as temperature control, which is more suitable for cheaper microprocessors. We should not forget the analogous com- puters suitable for direct comparison of the experimental curve with the master curve or for solving differential equations, both of use in theoretical kinetics, whereas a digital computer must transfer this problem to the solution by matrices.

Looking back to the literature, probably the first computer polynomial fit was applied to a thermocouple signal [55]. It was followed by different tabulations, efforts either to make precise or list different functions, e.g. the p(x) function. The pioneering work in non-isothermal kinetics was probably that by Anderson [56], who solved three simultaneous equations for different heating rates. Main activity, however, was focused on the evaluation of reaction orders and activation energies by using different simplified models [57, 58]. Only the most recent years finally brought up the problem of selection of appropriate mechanisms by fitting different analytical forms off(a) and #(e) functions. This included correlation coefficients between the experimentally-determined and theoretical master curves as well as built in subroutines to evaluate the p(x) function. No heavy-capacity algorithm, however, could give a direct and unambiguous answer to the basic kinetic question what the true reaction kinetics and mechanism involved. They

J. Thermal Anal 16, 1979

SEST.&K: NON-ISOTHERMAL KINETICS 517

merely classified the output data to a list of more or less probable kinetic models for a given sort and quality of experimental data. Complementary information, additional analysis of their physical meaning or more sophisticated criteria are needed.

In conclusion, more space should be given to computing treatments in existing TA books such as [1, 60, 53]. Seminars should bring together specialists from computers and TA to teach other the ways and means of analogies and digital data elaboration and the process control in thermophysical experiments, the numerical treatment (derivation, integration, etc.), fitting and smoothing of exper- imental curves and the actual aim of analysis of TA data in order to face our new tasks. The first attempt was made by the Czech group on thermal analysis in Prague, at the meeting directed to the digital treatment and conversion of the TA output signal, numerical evaluation and information property of experimental data, and the theoretical basis of TA with regard to general problems of the extrac- tion of physico-chemical quantities when using computers [59].

In concluding I would like to re-employ the generally oriented words that V. Satava used in his introductory lecture on theory and experiment [59]. If we ask a laic what is the progress in natural science he would probably answer that it consists in the gradual growth of results of observations and experiments. But who is actually dealing with research knows that a mere data collection is of little value, and what more, would not be self-satisfactory because of the lack of excite- ment. The real aim of science is different; it seeks the knowledge about the rules which control the nature i.e. the understanding and consequent description of observed phenomena. The logistics and mathematics of facts with respect to their ordering is thus the most imFortant procedure [61- 64]. Complicated mathematics, however, is not alway the only way how to reach the answers. Philosophically oriented thoughts are sometime equally important if we can afford them in the present busy world. So that I personally much appreciate such aspects in our field of science: non-isothermal kinetics, as we can trace in the work of e.g. Boldyrev, Flynn, Garn, Ozawa Satava or Segal etc. We also may remember not often cited words of some famous scientists as "when I ever had some success in mathe- matical physics it was because I was able to avoid mathematical difficulties" (Gibbs) and/or "do not disturb such a nice thing like science by mathematics" (Fermi). But would it be ever said in our age of computers?

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518 SESTAK: NON-ISOTHERMAL KINETICS

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Anal. 13 (1978) 581. T. OZAWA: Proceedings of joint US -Japanese Symposium on TA, Kodanska and Halsted, Tokyo, 1974. p. 155.

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R~SUMI~ - - Les aspects les plus discutables de la cin6tique non-isotherme sont trait6s sous forme de douze questions et r6ponses, en particulier: r6putation de la cin6tique non-isotherme appliqu6e par les thermoanalystes, cons6quences des concepts simplifi6s d6daits de la cin6tique des r6actions en phase homog6ne, signification physique des param6tres cin6tiques fonda- mentaux dans les processus en phase solide, effet de compensation cin6tique et interd6pendance des param6tres cin6tiques lots de l'utilisation de la constante de vitesse d'Arrhenius, utilit6 mutuelle des m6thodes diff6rentielles et int6grales d'6valuation des donn6es cin6tiques, leur exactitude et leur degr6 de justesse, fiabilit6 des mesures par ATD, 6tudes non-isothermes vis4t-vis des m6thodes isothermes, donn6es h l'6quilibre et en r6gime dynamique et leurs effets mutuels, extension de la discussion engag6e par MacCallum et Tanner, politique de publica- tion des donn6es non-isothermes et finalement utilisation des ordinateurs et aide apport6e par ceux-ci.

Z U S A M M E N F A S S U N G - - In der Form yon zwOlf Fragen und Antworten werden die meist um- strittenen und auseinandergehenden Aspekte der nicht-isothermen Kinetik diskutiert. Ins- besondere das Ansehen der nicht-isothermen Kinetik bei Durchftihrung dutch Thermoanaly- tiker, die Konsequenzen der vereinfachten yon der Kinetik homogener Reaktionen tiber- tragenen Konzepte, die physikalische Bedeutung der kinetischen Grundparameter in Fest- phasenprozessen, der kinetische Kompensafionseffekt and die gegenseitige Abh/ingigkeit der kinetischen Parameter bei Anwendung der Geschwindigkeitskonstante nach Arrhenius, die gegenseitige Ntitzlichkeit der Differential- und Integralmethoden der Auswertung kinetischer Daten, ihre Genauigkeit und Richtigkeit, die Zuverlfissigkeit der DTA-Messungen, nichtiso- therme gegeni.iber yon isothermen Untersuchungen, Gleichgewichts- und kinetische Daten und ihre gegenseitige Wirkung, die Ausbreitung der von MacCallum und Tanner initierten Diskussion Politik der VerSffentlichung nicht-isothermer Angaben und, endlich, Einsatz und Hilfeleistung der Computer.

Pe310Me - - B qbopMe ~Bena~uaTn BonpocoB I,I OTBeTOB 06cym/~enb~ HarI60nee cnopn~,~e r~ npoT•- Bope~BbIe TOqX~I 3peh'qa~ ner~30TepMltqecxo~ KI~HeTnYd, L B ~faCTHOCT~I paCCMaTp~IBaK~TC~ r~3- BeCTHOCTTa HeH3oTepMR~ecKo~ IcmHeTI4~/e/, xor~a oua npOBO,/~HTC~I TepMoaHa~tTHKaMH, noc~te~o

J. Thermal Anal. 16, 1979

5 2 0 ESTAK: NON-ISOTHERMAL KINETICS

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J. Thermal AnaL 16, 1979

Rationale and Fallacy of Thermoanalytical Kinetics**

Jaroslav ŠestákInstitute of Physics of the Academy of Sciences,

Cukrovarnicka 10, CZ-16253 Prague, Czech Republice-mail: sestak@fzu.cz

INTRODUCTION TO CHEMICAL KINETICSChemical kinetics is based on the experimentally verified assumption that the rate of change,

dx/dt, in the state of a system (characterized by) x is a function, f, of the state alone, i.e., dx/dt = x’ = ƒx (x). Using this traditional postulation, the appropriate constitutional approach to inaugurate the desired constitutive equation can be written in the principal form of the dependence of the reaction rate, expressed as the time development of the normalized degree(s) of transformation, α = (x – xo)/(xo – x∞), on the quantities that characterize the instantaneous state of the reacting system.

The reaction change is traditionally depicted in the form of a generalized equations, α’ = ƒα (α, T) = k(T) f(α) f(α,T). Very important is to see the properties of the function, ƒα, which is liable for an appropriate match of temperature progression of reaction mechanism under study. The long-lasting practice accredited such a routine in which the function ƒα (α, T) is divided into the two mutually independent functions k(T) and f(α) when traditionally assuming that f(α, T) = 1 . This implies that the rate of change, α’, is assumed to be proportional to the product of two separate functions, i.e., the rate constant k(T), dependent solely on the temperature, and the mathematical portrayal of the reaction mechanism, f(α), reliant on the variation of the degree of conversion, only.

In chemical kinetics, the operation of k(T) function is customarily replaced with the known and well-defined analytical form of the Arrhenius exponential function, k(T) = A exp (− E/RT), thoughtfully derived from the statistics of evaporation. The inherent constants are the so-called activation energy, E, identified as the energy barrier (or threshold) that must be surmounted to enable the occurrence of the bond redistribution steps required to convert reactants to products. The pre-exponential term, or frequency factor, A, provides a measure of the frequency of occurrence of the reaction situation, usually envisaged as incorporating the vibrating frequency in the reaction coordinate. The time-honored problem, however, is the mutual interdependence of E and A, which arises from the mathematical consequences of the Arrhenius equation and which causes its use complicated.

There are also alternative depictions related to the activated complex theory and the partition functions of the activated complexes to give limited freedom along the reaction coordinate. The validity of the Arrhenius equation is so comprehensively accepted that its application in homogeneous kinetics normally requires no justification. Although the Arrhenius equation has been widely and successfully applied to innumerable solid-state reactions, this use factually lacks a theoretical justification because the energy distribution, particularly amongst the immobilized constituents of crystalline reactants, may not be adequately represented by the Maxwell−Boltzmann equation. Due allowance should be made for the number of precursor species within the reaction zone as well as for the changing area of the reaction interface. It can affect the enhancement of reactivity at the active sites (reaction contacts), exaggerated by strain (between the differently juxtaposed networks), defects, catalytic activity of newborn sites, irreversible re-crystallization, or perhaps even by local volatilization (if not accounting for the thickness of reacting zone contoured by local heat and mass fluxes across and along the interfaces). It brings multiple questions about its applicability and we can humorously cite: “Everybody believes in the exponential law of energy

* Dedicated to the 40th Anniversary of the Ozawa−Flynn iso−conversional method.

distribution, the experimenters because they think it can be proved mathematically, and the mathematicians because they believe it has been established by observations”.

Nevertheless, it was shown that energy distribution functions of a similar form arise amongst the most energetic quanta. For values significantly above the Fermi level, both electronic energy (Fermi−Dirac statistics) and phonon energy (Bose−Einstein statistics) distribution approximate to the same form as that in the Maxwell−Boltzmann distribution. Also the interface levels, capable of accommodating electrons, are present within the zone of imperfections where chemical changes occur. These are analogous to impurity levels in semiconductors, imperfections levels in crystals (F-centers) and, similarly, can be located within the forbidden range between the energy bands of the crystal. Such levels represent the precursor energy states to the bond redistribution step and account for the increased activity relative to similar components in more perfect crystalline regions. Occupancy of these levels arises from activation through an energy distribution function similar in form to that characteristic of the Maxwell-Boltzmann equation and thereby explains the observed fit of k(T) data to an Arrhenius-type equation.

However, for the resolution of the second function, f(α), we must act in a different way than above because the exact form of this function is not a priori known and we have to determine its analytical form (taking, in the same time, its diagnostic potential into our account). The specification of f(α) is thus required by means of the substitution of a definite analytical function derived on basis of modeling the reaction pathway usually by employing some physical-geometric suppositions. In contrast to the determination of predefined characteristic parameters of the k(T)−function, the true purpose of kinetic studies is to hit upon this pathway, i.e., to find our imaginative insight into the reaction mechanism. Thus it is convenient to postulate a thought model visualizing the reaction, usually splitting it into a sequence of possible steps and then trying to identify the slowest step, which is considered to be the rate-determining one. Such models usually incorporate (often rather hypothetical) description of consequent and/or concurrent processes of interfacial chemical reactions and diffusion transport of reactants, which governs the formation of new phase (nucleation) and its consequent (crystal) growth. Such a modeling is often structured within the perception of simplified geometrical bodies, which are responsible to depict the incorporated particles and such visualization exemplifies the reaction interfaces by disjointing lines. Such a derived function f(α) then depends on all such physical, chemical and geometrical relations, which are focused to the interface between the product and the initial reactant.

When not accounting on the interfaces or other in homogeneity phenomena we deal with homogeneous reactions determined by an averaged concentration in a whole reacting volume and thus the f(α) function is paying attention to a most simplified description only, which is associated with the so-called reaction order. However, for solid-state reactions the reactants are neither mixed on an atomic level not equally distributed in the whole reactant volume and they must, therefore, penetrate, flow or diffuse into each other if the reaction is to start and propagate within the given volume. Accordingly the space co-ordinates become a controlling element, which creates heterogeneity effects inevitably to be included and which are actually accounted for creating interfaces by means of ‘defects’ conveniently symbolizing a pictographic contour (borderline, curve).

The mathematical description turns out to be much more complicated due to the fact that no mean‘ bulk concentration’ but the spot ‘phase interfaces’ carry out the most significant information undertaking the true controlling role for the reaction progress. The most common models are derived isothermally and are associated with the shrinking core of a globular particle, which maintains a sharp reaction boundary. Using a simple geometrical representation, the reacting system can be classified as a set of spheres where each reaction interface is represented by curve. We assume that the initial reactants’ aggregation must be reached by (assumingly) well distributed (reacting) components (often through diffusion). Any such an interfacial (separating) layer, y, bears thus the role of kinetic impedance and the slower of the two principal processes, diffusion and interface chemical reaction become the rate controlling process within the limits of our simplified modeling..

MODELING APPLIED IN CHEMICAL KINETICS1) Traditional kinetics: the Arrhenius exponential parameter k(T) is introduced and the f(α)

function is modeled on basis of simplified physical-geometrical assumptions (growth/change of reaction interface)

2) Phenomenological kinetics: the Arrhenius exponential parameter k(T) is maintained but the f(α) function is replaced by a suitable fitting (polynomials) or by fractal-like equations.

3) Novel non−parametric kinetics: mutually independent functions k(T) and f(α) assume arbitrary function, i.e., both the Arrhenius parameter and model function are not connected with a specific reaction mechanism showing accidental-like relationship.

Let us first see the means of traditional kinetics schematically shown on the following figure:

Certainly, it brings a lot of simplifications regarding the true and idealized models, see bottom picture, but it provides a convenient list of models ready for application, which analytical forms are the traditional and repeated subject of most kinetic papers.

Model: Symbol: function f(α): integral form g(α)=o∫α dα/f(α):

Nucleation − growth Am (1−α)[− ln (1−α)]1−1/m [− ln (1−α)]1/m

(JMAYK eq.) m = 0.5, 1, 1.5, 2, 2.5, 3 and 4(Johnson-Meal-Avrami-Yerofeev-Kolmogorov)

Phase boundary Rn n(1−α)1−1/n [1 − (1−α)1/n ] n-Dim.: n = 1, 2 and 3(equivalent to the concept of reaction-order n)

Diffusion controlled Di

1-dim. diffusion D1 1/2α α2

2-dim. diffusion D2 − [1/ln(1−α)] α + (1−α) ln (1−α) 3-dim. diffusion D3 (3/2)(1−α)2/3 /[(1−α)−1/3– 1] [1– (1− α)1/3]2

(Jander) 3-dim. diffusion D4 (3/2)/[(1−α)−1/3 – 1] 1 − 2α/3 – (1−α)2/3

(Ginstling-Brounshtein) 3-dim.counter dif. (3/2)(1+α)2/3/[(1+α)−1/3 + 1] [(1+α)1/3 – 1]2

(Komatsu-Uemura, sometimes called ‘anti-Jander’ equation)

Normal grain − growth Gn (1 − α)n+1/(n ron) [ro/(1 − α)]n − ro

n

(Atkinson – long-range diffusion where ro is initial grain radius)

Unspecified − fractal approach ( extended autocatalytic ) fractal-dimension SB (1 − α)n αm no particular (Šesták-Berggren) analytical form

It is worth repetition that the above listed mathematical modeling can be broken into three major but reduced designs according to the downgraded reaction geometry. We can envisage either elementary process (a) diffusion of reactants to and/or from the reaction interface through a continuous product layer, (b) product nuclei formation and their consequent growth and (c) chemical reaction restricted to the phase boundary.

These events can roughly concur to what can specifically happen when we try to measure and study the overall reaction path. There are no assured means that suchlike aggregated form of reactants would conduct themselves within our simplified picture and that the selection of a particular model is acceptable; however, a nuclei growth analysis shows as a rather useful tool for a preliminary screening of reactions within powdered compacts.

However, the modern mathematical tools of thermal physics make available powerful theoretical models employing nonparametric evaluations or neural networks to be associated with the true reality of natural processes that are never at equilibrium neither without gradients by appreciating the decisive role of thermal fluxes. In the scientific intent (often directed to generate publications based on non-isothermal kinetics), this approach has not yet been applied, but in the more urgent technological processing, such as industrially significant cement firing, ingot casting, arc melting or welding, which adoption became a real necessity to overcome manufacturing

difficulties. In particular, we can generally assume that at some distance from the reaction zone where the phase transformation (solidification) is taking place, the viscous (often molten) material undergoes irregular (turbulent) motion. It creates a mushy zone consisting of cascade of branches and side branches of crystals and interspatial melts, that remains lying between the original reactant (fluid) and the product (fully solidified region). Some chemical admixtures (as the alloy solution) are concentrated in the interspatial regions and ultimately segregated in the resulting micro-texture pattern. Such a highly irregular microstructure of the final solid can become responsible for alternative properties, e.g., reduced mechanical strength that is a costly factor thus worthy of active search as to resolve the intricacy of the processes involved. It follows that small changes in the surface tension, microscopic temperature fluctuations or non-steady diffusion may determine whether the growing solid looks like a snowflake, seaweed or spongy sinters. The subtle ways in which tiny perturbations at the reacting interface are amplified then become important research topics bringing necessarily into play either higher mathematics or non-Euclidean geometry of fractals.

The classical diffusion laws derived by Fick dominated physicist’s view on transport for more than a century. The recent observations, however, demonstrated its clear exemptions, which anomalousity were termed as “strange or fractal kinetics”, usually solved on basis on random walk approaches and unconventional distribution functions. Such a need for so called “fractional diffusion-advection equations” helps to a better understanding and prediction of the long-termed impacts of pollution on sizeable ecosystems and, for contrarily micro-systems, concerning the problem of a particle escape over a rigorous potential barrier. For the continuous−time random walk (CTRW) the strict time interval between consecutive steps become governed by a waiting-time distribution, which stem from possible obstacles and traps that delay the particle’s permanent jumps and thus introduce memory effects into the transport motion. If the mean time diverges, as is the case for the common power-law waiting-time distribution, the mean squares displacement <x2(t)> grows proportional to tn with non-integral time power (n≠1). A consistent generalization of the standard diffusion continuity equation could still remain to be the second order in the spatial coordinates but is violated by exhibiting a fractional-order temporal derivative, such as (∂c(r,t)/ /∂t) = kn ∇2 c(r,t), where c(r,t) is the concentration of particles at some point r and time t (also corresponding the probability P(r,t) of finding a particle there). The diffusion coefficient kn holds the dimension of L2/T but may be replaced for the fractional diffusion by the coefficient L2/Tn and such a fractional modification generate the scaling behavior often seen in inter- and sub-diffusion systems, which follow a complex relation (∂n/∂tn){c(r,t)} = oD1

−n kn ∇2 c(r,t). It can be derived from CTEW scheme along the lines used in the Einstein’s work on Brownian motion linking normal diffusion with anomalous diffusion. A new mathematical description is thus introduced, which is called “fractional calculus” often applied to the derivative dny/dxn when n holds an unfamiliarly non-integral value. Fractional derivatives proceeds from the repeated differentiation of its power, i.e., dn/dxn(xm) ≅ Γ xm−n (where Γ is the Gamma function) or by a more elegant way based on the fact that the n−th derivative is an operation inverse to an n−fold repeated integration providing the symbolic formulae of a general fractional derivative of the function f(x), i.e. Dx

1−n f(x).It is clear that the reaction kinetics of solid-state processes involving time-dependent

diffusion and nucleation did not absorbed as yet all possible means of modern mathematics to tackle the complex properties necessary to properly describe the reaction mechanism involved. It is assumingly due to its very demanding calculus, which is not simply compatible with “easy-drawn” geometrical models based on regular special portrayal of aggregated particles of someway regular shapes. It is recently supplied by the introduction of so called model-free descriptions, which use arbitrary mathematical function regardless their associations to model processes or point out their detailed mechanism, which is often in the crucial interest of the analysis of the process in question.

It was found useful to introduce a kind of pervasive description by means of a simple empirical function, h(α), containing the smallest possible number of constant. It provides some flexibility, sufficient to match mathematically the real course of a process as closely as possible. In

such case, the kinetic model of a heterogeneous reaction is assumed as a distorted case of a simpler (ideal) instance of homogeneous kinetic prototype {f(α) ≈ (1−α)n}. It is mathematically treated by the introduction of a multiplying function a(α), i.e., h(α) = f(α) a(α), for which we coined the term ‘accommodation function’ and which is accountable for a certain ‘defect state’ (imperfection, non-ideality, error in the same sense as is treated the role of interface during the new phase formation).

It follows that the so-called empirical kinetic model function can be generally described by all-purpose, three-exponent relation: h(α) = αm (1 − α)n [−ln (1 − α)]p . It is practically applicable as either form, Sestak−Berggrenn (SB) equation, αm(1 − α)n, or modified Johnson− Mehl−Avrami −Yerofeev−Kolmogorov (JMAYK) equation, (1 − α)n [−ln (1 − α)]p.

Only the single exponents, such as, m, (classical reaction order) or, p, (traditional JMAYK equation with n=1) or the two appropriately paired exponents are suitable for meaningful evaluation such as the classical model of self-catalyzes (m=1 and n=1), which is thus a special case of the general SB equation. The justified and proper complementary couples are only (m+n) or (m+p), whilst the pair (n+p) exhibits an inconsistent role of two alike-functions, which are in an inadmissible (self-supporting) state.

The SB-equation αm(1−α)n subsists the generalized use of a basic logistic function, x(1 – x), which is customarily exploited to depict the case of population growth. It consists of the two essential but counteracting parts, the first responsible for mortality, x ≅ αm , (i.e., reactant disappearance and the product formation) and the other for fertility, (1−x) ≅ (1 − α)n (i.e., a kind of products’ hindrance generally accepted as an ‘autocatalytic’ effect). The non-integral exponents, m and n, play thus the role similar to all circumstances, which act for the true fractal dimensions.

Peculiarity of non-isothermal approachFor all generally assumed transformations, the most frequently applied isothermally derived

theories are directly applicable to non-isothermal kinetic evaluations. A more detailed analysis imposes, however, that such a non-isothermal generalization is only possible along the lines of the so called ‘isokinetic’ hypothesis, i.e., the invariance of the rate equation under any thermal conditions. It was shown that the non-isothermal derivation includes the operation of integrated Arrhenius exponential, i.e., the so called p(x) function. Application of suitable approximations lead to the change of the pre-exponential factor only maintaining thus the residual model function unchanged. Thanks to such coincidence, all functions tabulated above are applicable for a straightforward submission to kinetic evaluations under non-isothermal conditions.

So-called integral methods of evaluation became rather widespread due to its apparent simplicity. They are based on a modified, integrated form of the function f(α), which is determined by the following relation, g(α)= α∫αdα/f((α) = To∫Tk(T) dT/φ = (AE)/(φR) exp(−E/RT) π(x)/x = (AE)/(φR) p(x), where φ is the constant heating rate applied, π(x) is an approximation of the temperature integral (often used in the intact representation as the p(x) function, where x = E/RT). There are many approximations of this temperature integral arising from the simplest π(x) = 1/x and (1−2/x)/x (based on the expansion) to more accurate π(x) = 1/(x+2) up to the most sufficient π(x) = (x+4)/(x2

+ 6x + 6)(based on the rational approximations). However, the determination and sufficiently precise calculation of the p(x) function should be seen as a marginal problem though it has led to exceptionally wide publication activity. Despite the specific accounts on p(x) accuracy it can be shown that the mathematical nature tends to simple solutions, because in most kinetic calculations the effect of the actual value (of π(x) and/or p(x)) is often neglected or, at least, greatly diminished. This factually implies that k(T) is (often unintentionally) considered as a constant and, instead of complex integration, is simply withdrawn in front of the integral, namely: ∫k(T) dt ≈ k(T) ∫dt , which handling is habitually obscured within complicated mathematics (often associated within p(x) deciphering).

It is worth highlighting that already a plain application of a logarithmic function brings not only the required data smoothing but also introduces a higher degree of insensitiveness. For that reason, the popular double-logarithmic operation effectively paralyses the methods’ discriminability

towards individual kinetic models as well as our attached desire to attain as linear proportionality as possible. We may humorously make a note of curious but sometimes actual role of double-logarithmic plotting in sense of a drawing on ‘rubber-stretch’ paper.

To get a better insight, we can divide the so called integral methods into three groups according to the method of plotting ln g(α) against the three possible temperature functions (i.e., ln T , T or 1/T) giving thus the respective slopes of tan ϖ . Consequent estimation of E depends on the respective values, i.e., RTm (tan ϖ −1), RTm

2 tanϖ and/or (Rtan ϖ −2RT). The involved approximation can be tested using the asymptotic expansion of a series with a dimensionless parameter replacing E. It can be seen that the latter dependence is twice as good as that of the former two dependencies. In this way, we can confirm the numerical results of several authors who found that the plain dependence of ln g(α) vs. T yields E with an error of about 15% at best, whereas the ln g(α) vs. 1/T plot can decrease the error by half. This is relative to the entire series given by set of T’s and 1/T’s.

Another obstruction may be seen when employing a more general temperature program, which is proportional to the reciprocal temperature, i.e., dt = φ d (1/T). In this case the function p(x) simplifies in the form of a difference dT, which can be replaced by d(1/T) so that p(x) ~ (ZE/Rφ) [(−E/R)(1/T – 1/To)], which would be no complication but, on contrary, it would have allowed a more straightforward kinetic appraisal. Although such temperature regulation is rare, it would essentially facilitate the kinetic data calculation, requiring, however, another systematic. It would be feasible, depending just on the manufacturers’ will, as to consider somehow unproblematic extension of their commercially available digital programmers to become applicable for another, newly specified case of kinetic investigations.

Often neglected but one of the most important cases is the assumption of differences arising from the effect of changing the equilibrium background of the process . It can be illustrated on the example of various processes taking place in non-stoichiometric oxides (e.g. tetragonal-cubic phase transformation of spinels: MnxFe3-xO4 (t) → MnxFe3-xO4 (c)). The simple kinetic analysis of plotting the model relation against reciprocal temperature revealed that the change of the slope, often ascribed to the change of reaction mechanism, is just caused by switching between the part of the process proceeding under changing equilibrium background and the remainder invariant part controlled by the process itself.

Proximity to equilibrium plays an important but often underestimated role, which is effective in the beginning of a reaction where the driving force is minimal. In such a case some integration terms of p(x)−function cannot be neglected as conventionally accomplished in most kinetic studies. The resulting function [p(x) – p(xo)] is more complex and the associated general plot of g(α) vs.1/T develops into non-linearity. This consequence can be best illustrated on the decomposition of CaCO3

as an habitually misconducted case.

KINETIC EVALUATION PROCEDURESDistinctiveness graph below of the model functions of ln f(α) vs. α shows three characteristic

areas of self-similar models abbreviated as nucleation (A), phase-boundary (R) and diffusion (D) controlled processes. The middle plot displays the wider variety of possible diffusion-paths controlled processes exhibiting a non-integral (fractal {(1−D)/D}) geometry. On the left-upper shaded areas depict the models formed on basis of the strict Euclidean geometry. However, the models involve a modified course of characterization, which allows intermediate cases to exist (e.g. for 2<DD ≤3 it holds dα/dt = {(1− α) (1−D)/D−1}−1). Such approach seemingly enables to cover a larger area with the array of kinetic curves making possible their increased packing, e.g., from D=2, which coincides with the classical D2 curve (shown in the left graph), we can see the curves’ shift to the upper side of the plot with increasing D’s as far as to 1.9 and than their moving downwards from D=2.1 to 3.0 to crowd the normally unfilled area (defined between the conventional D2 and D4

curves). Thee same curve swelling applies to the other groups (A’s and R’s, see left) and it shows well the almost meaningless attempt to define precisely the solitary particularity of individual reaction as introduced by Riko Ozao (Kanagawa, Japan). Right is the characteristic graph of z(α) =

f(α) g(α), which is plotted against α showing again the individual types of such ‘master plot curves’ typical for the following three kind of processes: nucleation-growth (G1↔G3, light shadow range), phase-boundary (chemical, R1↔R3, shadow range) and diffusional (D1↔D4, dark shadow range). This three-stage separation and computer evaluation yield just three group of most probable process, which is an acceptable answer in solving reaction kinetics as introduced by Jří Málek (Pardubice, Czechia)

There is another yet unconventional treatment often called non−parametric kinetics as introduced by Serra, Nomen and Sempre. It is literally based on our previously accepted assumption that the reaction rate can be expressed as the product of two mutually independent functions k(T) and f(α), but the temperature dependence of k(T) need not be of the Arrhenius type and can thus assume an arbitrary function as well as the ‘model’ function f(α) may not be straightforwardly connected with a specific reaction mechanism and can again assume any accidental-like relationship. These both functions are just temperature and conversion components of the kinetic ‘hypersurface’. If the reaction rate is measured from several experiments at different heating rates and organized in a matrix system whose rows correspond to different (but constant) degrees of conversion and whose columns correspond (by interpolation) to different (but constant) temperatures than a suitable matrix decomposition algorithm can provide two characteristic vectors. These vectors can be further analyzed by examining the resulting plots of rates against conversions possibly capable to check the validity of Arrhenius –type behavior or the fittingness of another (more simple) functions. It, however, requires rather extensive number of experimentally determined points and wider working ranges of temperatures but such a model-free method allows the isolation of temperature dependences involved without making any assumption about the reaction model.

The parameters occurring in both functions are only certain perceived quantities (similarly to the above portrayal of ‘apparentness’), which, in general, possess no mechanistic interpretation. Such values are composite and their representation in terms of individual sub-processes and associative subordinated values can be found obscure. Since k(T) is not the standard rate constant, there is no reason to be restricted by an exponential relationship and thus other functions can be used leading to a closed form of the temperature integral.

This single-step kinetics approximation involves the imperative condition of separability of both temperature and conversion functions and any couple of such autonomous functions lead to an acceptable description of accommodating kinetics. However, it has been reasoned that if a couple of separable functions cannot be found, it indicates that the single-step kinetics approximation is too crude and the description of the kinetic hypersurface may become inappropriate. The temperature and conversion functions contain enough adjustable parameters so that their values are attuned in the procedure of fitting in order to reach the best fit between the experimental and calculated data. The

separability of temperature and conversion functions must thus imply that the values of adjustable parameters are supposed to be unvarying in the whole range of conversions and temperatures.

It is generally recognized that the isoconversional methods lead to the dependence of adjustable parameters in the temperature function on conversion. This fact has provided the concept of a variable activation energy. As shown in, dependence of activation energy on conversion violates the conditions separability and thus, in the case of variable activation energy, the basic assumption of the single-step kinetics approximation is incompetent for the associated description and calculated data are inadequate. This factually represents the “logical trap” of the concept of variable activation energy since it becomes mathematically incorrect and inherently self-inconsistent. Deductions drawn from the dependence of activation energy on conversion can hardly be considered trustworthy and should be judged very critically and carefully.

We should accentuate again that the main contribution of the above concept of single-step kinetics approximation is to elucidate the non-physical meaning of equations involved so that it is just a mathematical tool enabling a description of the kinetics of solid-state reactions without any deeper insight into their mechanism. The correct mathematical description should recover the values of conversion and the rate of the reaction under study for given values of time and temperature. Since the adjustable parameters represent just apparent quantities, no conclusions should be drawn from their values and so it is close to the use of fractal power exponents discussed beforehand but some useful conclusions can be drawn from the values of reaction rates or isoconversional temperatures and times such as it has been done in the study of induction periods.

BACKING LITERATURE:[*] T. Ozawa: “A New Method of Analyzing Thermogravimetric Data” Bull. Chem. Soc. Jpn. 38(1965)1881 [*] J.H. Flynn, L.A. Wall: “Quick Direct Method for the Determination of Activation Energy from thermogravimetric data” Polymer Letters B4 (1966)323 [*] P.D. Garn “Thermoanalytical Methods of Investigation” Academic Press, New York 1964[*] J. Šesták, V. Šatava, W.W.Wendladt „The Study of Heterogeneous Processes by Thermal Analysis“, a special issue of Thermochim. Acta, Vol. 13, Elsevier, Amsterdam 1973[*] J. Šesták “Thermodynamic Basis for the Theoretical Description and Correct Interpretation of Thermoanalytical Experiments“ Thermochim. Acta 28(1979)197[*] J. Šesták, NATAS/Mettler Award lecture „Rational Approach to the Study of Processes by Thermal Analysis“ in “Thermal Analysis”, Proc. of the 4th ICTA in Budapest, Vol.1, Academia Kiadó, Budapest 1975, p.3. and ICTAC/TA Award lecture “Thermal Treatment and Analysis: the art of near-equilibrium studies” in J. Thermal Anal. 40(1993) 1293 [*] S.F. Hulbert “Models for Solid-state Reactions in powdered Compacts” J. Brit. Ceram. Soc. 6(1969)11[*] H.V. Atkinson “Theories of Normal Grain Growth in Pure Single Phase Systems” Acta Metall. 36(1988)469[*] J. Šesták, G. Berggrenn „Study of the kinetics of the mechanism of solid-state reactions at increasing temperature“ Thermochim. Acta. 3(1971)1[*] P. Holba J. Šesták „Kinetics with regards to Equilibrium of Processes studied by nonisothermal techniques” Zeit. phys. Chemie NF 80(1972)1[*] J. Šesták, P. Holba “Kinetics of Variant Precipitation Processes in Non-stoichiometric Oxides” in TERMANAL ’73, SVST Bratislava 1973, p. S-67[*] J. Šesták, key lecture at the. 2nd ESTAC (Europ. Symp. on Thermal Analysis) “Integration of Nucleation-Growth Equation when Considering Nonisothermal Regime and Shared Phase Separation” in Proc. “Thermal Analysis”(D. Dolimore, ed) Heyden. London 1981, p. 115[*] J. Kemeny, J. Šesták “Comparison of Crystallization Kinetic Theories Derived by Isothermal and Nonisothermal methods” Thermochim. Acta 110(1987)113[*] J. Šesták, book: „Thermophysical Properties of Solids: their measurements and theoretical thermal analysis“, Elsevier, Amsterdam 1984 and Mir, Moscow 1987 (in Russian) [*] Šesták J.: “Diagnostic Limits of Kinetic Models when Introducing Non-integral Exponents through an Accommodation Function” J. Thermal Anal. 36(1990)1997[*] J. Blažejowski “Remarks on the Description of Reaction Kinetics under Nonisothermal Conditions” Thermochim. Acta 76(1984)359[*] J. Pysiak “Some Errors Committed in the studies of Kinetics of Thermal Dissociation of Solids” J. Thermal Anal. 43(1995)9[*] N.Koga, J.Málek, J.Šesták, H.Tanaka: “Limits of Phenomenological Models & Data Treatment in Non-Isothermal Kinetics” Netsu Sokutei (Jap. J. Thermal Anal.) 20(1993) 210[*] J. Šesták, J. Málek: “Analytical Limits of Phenomenological Models of Heterogeneous Reactions and Thermoanalytical Kinetics” Solid State Ionics 63/65(1993)254 and also N. Koga, J. Málek: Thermochim. Acta 282/283(1996)69

[*] E. Illeková “Kinetic Characterization of Nanocrystal Formation in Metallic Glasses” Thermochim. Acta 280/281(1996)289 and 387(2002)47[*] J. Málek, Thermochim. Acta “Kinetic Analysis of Crystallization Processes in Amorphous Materials” 355(2000)239 [*] J. Málek, T. Mitsuhashi, J.M. Criado “Kinetic Analysis of Solid-state Processes” J. Mater. Sci. 16(2001)1882 [*] J. Šesták, book: “Heat, Thermal Analysis and Society”, Nucleus, Hradec Kralove 2004 [*] N. Koga: “Kinetic Compensation Effect” Thermochim. Acta 244(1994)1 and 258 (1995)145[*] N. Koga, J. Šesták „Thermoananlytical Kinetics and Physico-geometry of the Nonisothermal Crystallization of Glasses“ Bol. Soc. Esp. Ceram. Vidr. 31(1992)185[*] J. Šesták „Use of Phenomenological Kinetics and the Enthalpy-temperature Diagrams (and its derivative DTA) for a better Understanding of Transition Processes in Glasses“ Thermochim. Acta 280/281(1996)175[*] V. Mamleev, S. Bourbigot, M. LeBras, S. Duquesne, J. Šesták „Modelling of Nonisothermal Kinetics in Thermogravimetry“ Phys.Chem.Chem.Phys. 2(2000)4708 and 4796[*] J. Šesták (ed), “Vitrification, Transformation and Crystallization of Glasses”, special issue of Thermochim. Acta, Vol. 280/281, Elsevier, Amsterdam 1996[*] A.K. Burnham „Application of Sestak-Berggrenn (SB) equation to organic and inorganic materials of practical interest“ J. Thermal Anal. and Calor. 60(2000)895[*] P. Šimon, “Isoconversional Methods: fundamentals, meaning and application” J. Thermal Anal. Calor.. 76(2004)123 [*] P. Šimon, “Single-step Kinetics Approximation Employing Non-Arrhenius Temperature Functions” J. Thermal Anal. Calor 79(2005)703 and 82(2005)651[*] R. Serra, R. Nomen, J. Sempere, “Non-Parametric Kinetics: a new method for kinetic study” J. Thermal Anal.Calorim. 52(1998)933 and Thermochim. Acta 316(1998)37[*] S. Vyazovkin “Model-free Kinetics: staying free of multiplying entities without necessity” J. Thermal Anal. Calor. 83(2006)45[*] M. Ochiai., R. Ozao.: „Fractal Reactions in Solids“ Bull.Ceram.Soc.Jpn. 26(1991)1181 and J. Thernal Anal. 38(1992)1901[*] R. Kopelman: “Fractal Reaction Kinetics” Science 241(1988)1620 [*] M.F. Shlesinger, G.M. Zaslavsky, J. Klafter “Strange Kinetics” Nature 366(1993)31[*] D. Avnir “Fractal Approach to Heterogeneous Chemistry” Wiley, New York 1989 and Nature 308 (1984) 261[*] N. Koga, J.M. Criado “Kinetic Anaalysis of Solid-state Reactions with a Particle-size Distribution” J. Amer. Cer. Soc. 81(1998)2901 [*] J. Šesták, book: “Science of Heat and Thermophysical Studies: a generalized approach to thermal analysis” Elsevier, Amsterdam 2005[*] E. Segal, D. Fatu, book “Introduction to Nonisothermal Kinetics” , Publ. House Romanian Acad. Sci., Bucharest 1983 (in Romanian) [*] A.K. Galwey, M.E. Brown, book “Thermal Decomposition of Solids” Elsevier, Amsterdam 1999[*] A.K. Galwey, M.E. Brown “Application of the Arrhenius Equation to Solid-state Kinetics: can it be justified?” Thermochim. Acta 68(2001)1[*] M.E. Brown “Steps in a minefield: some kinetics aspects of thermal analysis” J. Thermal Anal. 49(1997)17[*] M.E. Brown “Stocktaking in the Kinetics Cupboard” J. Thermal Anal. Calor. 82(2005)665[*] Z. Chvoj, J. Šesták, A. Tříska (edts), book: “Kinetic Phase Diagrams: non-equilibrium phase transitions“, Elsevier, Amsterdam 1991[*] J. Šesták. Z. Chvoj: “Irreversible Thermodynamics and True Thermal State Dynamics in View of Generalized Solid-state Kinetics” Thermochim. Acta 388(2002)427[*] J.J. Mareš, J. Stávek, J. Šesták, “Quantum aspects of self-organized periodic chemical reactions” J. Chem. Phys., 121(2004)1499[*] J. Stávek, M. Šípek, J. Šesták “Application of the Principle of Least Action to some Self-organized Chemical Reactions” Thermochim. Acta 388(2002)440[*] J. Málek, J. Šesták, F. Rouquerol, J. Rouquerol, J.M. Criado, A. Ortega „Possibilities of Two Nonisothermal Procedures for Kinetic Studies“ J. Thermal Anal. 38(1992)71

Thermochimica Acta, 110 (1987) 113-129 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

113

COMPARISON OF CRYSTALLIZATION KINETICS DETERMINED BY ISOTHERMAL AND NON-ISOTHERMAL METHODS

T. KEMBNY

Central Research Institute for Physics, P.O. Box 49, Budapest 1525 (Hungary)

J. SESTAK

Institute of Physics of the Czechoslouak Academy of Science, 18040 Prague (Czechoslovakia)

ABSTRACT

Theoretical results for nucleation and growth controlled transformations indicate that a non-isothermal generalization is only possible along the lines of the isokinetic hypothesis, i.e., the invariance of the rate equation under any thermal conditions. The alternative formalism, i.e., the general use of the isothermal law and the corresponding modification of the rate equation under non-isothermal conditions, is found to be incorrect. Different methods for the evaluation of kinetic parameters are compared, taking into account even thermal lag correc- tions.

It is illustrated by experimental results for oxide, chalcogenide and metallic glasses that all of the non-isothermal methods that yield kinetic parameters in good agreement with the appropriate isothermal results are easily derived from the usual rate equation.

INTRODUCTION

The crystallization of different glassy materials has been extensively investigated by various non-isothermal methods. It is evident that the kinetics of crystallization are important from the points of view of both fundamental and applied research. The theory and practice of this evalua- tion, however, are far from being universally accepted.

The general question of whether the usual rate equation is applicable to non-isothermal studies [l-5] is also relevant for crystallization. Apparently unrelated to this problem, many researchers [6-lo] freely applied the equa- tions derived for the isothermal case by Johnson and Mehl, Avrami, Kolmogorov and Yerofeev to non-isothermal conditions. When the time derivative of these JMAKY kinetics were calculated, new methods were suggested [ll-131 for the calculation of kinetic parameters that are signifi- cantly different from those generally applied. Despite a detailed criticism of

Thermal Analysis Highlights, 8th ICTA, Bratislava, Czechoslovakia.

0040-6031/87/$03.50 Q 1987 Elsevier Science Publishers B.V.

114

these attempts [14-321, the exact mathematical equivalence of the modified rate equation and the use of isothermal laws under non-isothermal condi- tions were not generally recognized 133,341.

In this paper, some results on the theory of nucleation and growth controlled transformation are reviewed to show that a non-isothermal gener- alization is only possible along the lines recommended by the isokinetic hypothesis, i.e., the invariance of the rate equation. If this treatment is to be believed, however, it must be shown that those evaluation methods which claim good experimental verification are consistent with this framework even if originally suggested on the basis of now obsolete theories.

THEORY OF NUCLEATION-GROWTH KINETICS UNDER GENERAL THERMAL CONDITIONS

Crystallization is a special case of the nucleation and grain growth controlled solid-state transformation processes, the theory of which is well known [35-381. If an embryo of the transformed phase nucleates at moment 7 and grows thereafter isotropically in three dimensions with a linear momentary growth rate Y( t’) in moment t’, then its volume u at moment t

(where 7 < t’ -c t) is

u(7, t)= y [ [Y( t’)dt’13

When the possible overlap of the grains is neglected, the so-called extended volume V,,, is calculated:

J&= V N,u(O, t) + J~~(T)u(T, t)dT] I where NO is the initial number of nuclei and I = I( T) is the nucleation rate per unit volume. As the change of the real transformed volume, dV,, and that of the extended volume, dV&,, is related by

dVt= I-; dl/,,, ( 1

the basic nucleation-growth equation for the transformed fraction (Y = V/V is

-ln(l - a) = F I3 + p)[lry(t9dtj3d~}

This equation is evidently valid under any thermal conditions. Up to this point no assumptions have been made regarding the origin of the time dependence of I and Y, so it could easily include also the implicit time dependences Y[T( t’)] and I[ T( r)]. An important limitation of this equa- tion, however, stems from the use of eqn. (3) which describes a completely

115

random overlap of growing crystallites. Surface crystallization is a clear case where this assumption must not be valid. Owing to the possible absence of overlap, this case is also easily treated even under general thermal condi- tions.

For any shape where the volume is expressed as the product of a geometrical factor and the cube of a characteristic distance, the calculation easily yields

1 - (1 - &3 = lfk[T(t’)]dt’ (5) 0

which is a clear example of nth order chemical reaction kinetics. Having established that the classical methods of kinetic parameter evaluation are to be used in this instance, we might return to our central problem: evaluation of kinetic parameters based on eqn. (4).

The two terms in eqn. (4) determine the two limiting processes of major interest under isothermal conditions when Y = constant:

(a) the growth of frozen-in nuclei (No + 0) without thermal nucleation (I=o):

[ - ln(l - a)] 1’3 = kt (6)

(b) homogeneous thermal nucleation (No = 0, I = constant # 0):

[-ln(l--)]“4=kt (7)

It is worth noting that in a number of instances, e.g., for the time-depen- dent transient nucleation where I( 7) = IoeW7’Q or for the diffusion-limited growth where the characteristic distance is proportional to t112, a form analogous to eqns. (6) and (7):

[-ln(l-(w)]““=kt (8)

holds as an acceptable approximation [39], where the exponent n gives an indication of the mode of transformation. This is the physical background of the wide applicability of the JMAKY equation under isothermal conditions.

The same treatment is also applicable to the non-isothermal case taking into account that now

y( t’) = yoepEJRT(“)

The generalization of case (a) is very easy, yielding

[ -ln(l - a)]1’3 = i’k[T(t’)]dt’

where

(9)

(10)

k(T) = Zge-Es/RT (11) zg = Y, (4nNo/3y3 (12)

116

For case (b) the calculation has already been carried out [40,41] by evaluat- ing the double integral in eqn. (4) by the usual methods, where T = To + /3t

I( T) = Zne-E”/RT(7) (13)

The result is

[ -ln(l - ‘~)]l’~ = z(Z,,Z,,E,,E,)s( ~)*e-~/“=JbR[T(r’)]dr’

where

E= (E, + 3E,)/4 (15) i = ge- @RT(t)

(16)

It should be clear from eqns. (10) and (14) that the non-isothermal equiv- alent of the widely used isothermal JMAKY relation [eqn. (S)] is

[ -ln(l - a)]l’” = J’k[T(t’)]dt’ 07) 0

as also shown by Henderson [42,43] and Meisel and Cote [44]. Some particular points following from the preceding analysis are worth noting:

(A) The time derivatives of eqns. (8) and (17) are the same, i.e., the rate equation

$ = nZ(l - (y)[ -ln(l - a)]Ye-FIRT (18) bears the same form for both isothermal and dynamic cases.

(B) The isothermal transformed fraction can naturally be written either as

a = I - e-(kr)” @a)

or as

(y = 1 - e-kt” (19b)

as the two equations use only a different definition of the activation energy. If

k = koe-E/Rr (2Oa) and

k, = ko.ePE*/RT (2Ob) then

E,=nE (21)

However, the frequently used [45,46] differential form

da - = k,n(l - a)t”-’ dt (22)

117

is not valid [51,52], as

-ln(l -a) = (/‘k[r(t’)]df’)l+ Jrk, [T(t’)]t’“-‘dt’ (23) 0 0

This fact was originally observed by De Bruijn et al. [40] and was recently re-emphasized by Louis and Garcia-Cordovilla [47-491 and by Kasap and Yuhasz [108]. The correct form of eqn. (22) where the rate equation is formulated as a function of T and a only is expressed as [51,52]

da n-l

- = nk’,/“(l - a)[ -ln(l - a)] n dt (24)

Later we shall return to establish that this freedom in defining the activation energy does not influence the comparison of isothermally and non-isother- mally determined kinetic parameters.

(C) It is evident from eqn. (17) that the isothermal form of the JMAKY relation [eqn. (S)] is not valid under non-isothermal conditions. The relation- ship derived [eqn. (17)] clearly shows that (Y depends on the whole T(t) path. In this way the theory of nucleation and growth controlled transforma- tion is completely incompatible with the existence of an (Y = a( t, T) func- tion that would predict (Y if only the end-point (t, T) is specified, i.e., with the assumption that we shall call the alternative formalism.

The fact that the transformed fraction is a path function does not give any problems when a is determined for a properly defined thermal route. For example, with linear heating [50-521

T= T,+pt

Equation (17) yields

d4 = [ -141 - 41 l/n_ 25 - pRPww (25)

where p( E/RT) denotes the exponential integral. Equation (25) is no doubt a functional relationship, i.e., for linear heating the cy = 8,,( P,T) function exists.

In this way the theory of nucleation and growth controlled transforma- tions makes an unambiguous choice between the two methods of kinetic parameter evaluation. The usual method is based on the isokinetic hypothe- sis, i.e., on the invariance of the rate equation

2 = k(T)f(a)

under any thermal conditions. The transformed fraction is

g(a) = lag = l’k[T(t’)ldt’

which yields g(a) = kt only under isothermal conditions. This hypothesis,

(26)

(27)

118

even if not exactly valid [43] (causing here only a deviation of z from its

isothermally calculated value), is a reasonable approximation also in our case.

The alternative formalism is based on the existence of an a = B(T, t) function. A significant modification of the rate equation is calculated under non-isothermal conditions [l--5]. The preceding analysis therefore excludes the applicability of this formalism to nucleation and growth controlled transformations. The exact mathematical equivalence of the modified rate equation and the use of the isothermal law under non-isothermal conditions [6-lo] will be shown in a separate paper [34].

EVALUATION OF KINETIC PARAMETERS FROM THERMOANALYTICAL EX- PERIMENTS

The evaluation of linear heating experiments is based on eqn. (25), where two different approximations of the exponential integral, namely the Doyle approximation

1,,,,(E/RT) = In p(E/RT) = -5.33 - 1.056 + 3%

and the asymptotic equation

will be used. The first term in eqn. (29) means typically a few per cent accuracy of p(E/RT). It should be understood, however, that the use of eqn. (28) for estimating p(E/RT) itself is irrelevant, as the 3% error in the exponent has very serious consequences, as shown in Table 1.

This is why the form of the rate equation proposed by Matusita et al. [53]:

* = K(1 - (Y)P” exp( -1.05gm) dt

is considered to be too rough an approximation to be recommended. The actual evaluation methods proposed [53], however, utilize only ln(da/dt)

TABLE 1

Effect of a 3% error in the Doyle equation for calculating p( E/RT)

I Doyle( E/RT) = - 5.33 - l.OS( E/RT); with an error of f 32, and E/RT = 40

In P( E/W

p(E/RT) P/Pequation

+35% Equation

- 48.75 - 47.33 6.74~10-~* 2.79x1O-2’ 0.241 1

-3%

- 45.91 1.15 x lo-*O 4.14

119

and In g(a) and in this way are also easily derived in the correct formalism: when the logarithm of eqn. (25) is connected with eqn. (28) then

ln[ -ln(l - a)] 5: n In $ -

results. If the logarithm of (29), then the result is [54]

Z” lng =ln -

/

P”-l -$T+o I-a,ln +

1

T\

0 !

1.05% + c (30)

eqn. (18) is combined with either eqn. (28) or

(31)

It is evident that these plots might reveal only nE. If n and E are to be determined separately from a single thermogram, then the three-term ver- sion of eqn. (29) is applied to yield

In [

- ln( 1 - a)l’n T2 ]=ln[g(l-%)]-A (32)

The separate determination of n and E in this way requires a very high precision if the variation of terms whose order of magnitude is In T (found to be almost negligible with the usual precision) is to be significantly established in addition to terms of the order of l/T.

As the typical variations are a(l/T)/l/T = aT/T- 10% and a In T/ In T s 2%, and the precision of (Y itself can hardly surpass 2%, the fulfill- ment of this requirement is far from trivial. This is why multi-thermogram methods, usually based on temperature shifts caused by a variation of the heating rate, & play a dominant role. It is a basic advantage of these methods that the determination of activation energy (as it is defined here) is independent of the transformation mechanism [50,55-591. The connection of eqn. (25) with eqn. (28) for any g( (u) and for any a0 fixed transformed fraction yields

P In 2 = C, - 1.05s a0

while the use of eqn. (29) results in

(33)

(34)

where C, and C, are independent of /I and T,,, which denotes the tempera- ture where the transformed fraction is a@. If Z = constant it is incorporated in the C coefficient and then these equations are the Ozawa and the Kissinger plots, respectively, which are seen here to be only different approximations of p(E/RT) and are thus equivalent [60]. It is also easily verified that the transformed fraction at the maximum transformation rate, (Y m, is independent of fl and is thus useful for the determination of

120

activation energy without integrating the thermoanalytical curve. The usual criticism that the thermal lag between da/dt and the measured AT invali- dates this method will be treated in the Appendix and will be shown to be inapplicable to modern thermoanalytical instruments. Using

da Z ---z-e dt P

-E’RTf( o) (35)

the maximum condition is

where the substitution of eqn. (35) and rearrangement yields

which is reduced with the help of the first term approximation in eqn. (29) to

g( ) dfb) a-

da a,

+1=0

This relationship, besides proving that (Y, is independent of jI for any process described by eqns. (26) and (27), is also applicable to the direct determination of (IL, with

n-1

f(a) = n(1 - CX)[ -ln(l - a)] 7 (see eqns. 18 and 26)

g(a) = [ - ln(l - a)] “’ (see eqns. 25 and 27)

functions, yielding

(36)

(37)

%I = 1 - i = 0.63 e

These results, which were also derived previously [42,61-631, show that statements sometimes found in the literature that “the linearity of the Kissinger plot shows the applicability of the first-order kinetics” [106] or “the Ozawa plot is valid for chemical reactions only” [107] are not substan- tiated and the shift of (Y, with /!I which is also sometimes discussed [23a] must be insignificantly small.

The maximum reaction rate is calculated as

da iIZ dt ,,,==ee

- E/RT, (39)

which is clearly different from the result derived for fixed n th-order chem- ical kinetics, cf. eqn. (26), where f( CX) = (1 - a)” and

da - -ql - a)/n

dt nlar= RT,2

121

These results, i.e., for transformations described by eqns. (18) and (25) under linear heating conditions: (a) (Y, = 0.63 (independent of p, n or any other parameters), (b) T, is independent of n [which follows directly from eqn. (25) by applying point (a)] and (c) the maximum reaction rate is directly proportional to the characteristic exponent, n, are also fully confirmed by a detailed numerical simulation [64].

A comparison of isothermal and non-isothermal temperature shift meth- ods is appropriate here. As eqn. (26) is established to be valid in both cases, the relation

da In dt (x0

= In Zf(cv,) - j& W)

is applicable. In addition, the condition for the maximum isothermal rate is

(41)

which is independent of T, i.e., CX, is used in eqn. (40) under any cir- cumstances. On the other hand the relationship evident from eqn. (8):

In taO = C + j& (424

is also valid with tm. All of these relationships predict the same activation energy that is obtained by eqns. (33) and (34) under non-isothermal condi- tions. It is evident that a different activation energy can also be defined by eqn. (19b). By this definition the isotherms

In t, = c”+ E,/nRT PW

result, but owing to eqn. (24) the relationships (33) and (34) also yield E,/n in this instance.

This section should be concluded by emphasizing that peak methods, in addition to being so useful and convenient, also give a significant loss of information. With the availability of cheap microcomputers it is advisable to evaluate the full thermoanalytical curve to establish that the same activation energy is really associated with any transformed fraction value, the assumed transformation mechanism is in fact valid and the same characteristic exponent is determined from different heating rate experiments. If these cross-checks are carried out succesfully then a series of different heating rate curves can give as reliable a characterization as a series of different tempera- ture isotherms, while (Y and da/dt might be determined with better accu- racy even faster and the problems of the initial transient and the difficult determination of the incubation time are better avoided. This means that non-isothermal measurements can be used even for the determination of the mechanism of crystallization, in contrast to the belief that it is restricted to isothermal methods. only.

122

EXPERIMENTAL RESULTS FOR THE CRYSTALLIZATION KINETICS OF DIFFER- ENT GLASSES

This paper does not attempt to cover the vast literature on crystallization. Some results on oxide, chalcogenide and metallic glasses are mentioned to reveal how and why the general methods outlined in the previous section are modified under special circumstances and to investigate whether the same kinetic parameters are indeed to be determined from isothermal and non- isothermal experiments.

Oxide glasses

A basic feature of oxide glasses is the separation of temperature ranges for nucleation and growth. Thermal experiments detecting heat evolution naturally observe the growth of pre-existing nuclei. If the glasses were heated uniformly in the nucleation temperature range and investigated later with different heating rates in the growth regime only, then the evaluation methods in the previous section can be used without any modifications. As the samples are usually heated at the same rate in both regimes they are not even equivalent before crystallization and therefore a modification arises

1631. The number of nuclei is described as [68]

N” = N1 + Nz + N3 (43)

where N, is the number of bulk nuclei initially present, N2 is that of nucleated during heating in the appropriate temperature range and N3 is the number of surface nuclei treated separately. If Ni z+ N2 (“nucleated glass”) then N, = N1 is independent of j3 and the usual equations can be used. If N, +z Nz (“as-quenched glass”), then

where T, and T2 are the lower and higher limits of the nucleation tempera- ture range, respectively, and crystallization is observed at T > T2. Under these circumstances

and through eqn. (12)

z z p-l/II

is obtained, which modifies eqns. (30)-(34) to

ln[-h-1(1 -ci)] = -(n+l) In B-l.O5~2+~ +C

(46)

(47)

In%=-nlnp-gT+o(l-a,lnT)

In /3 = ci - *1*05&

P ,I + 1

:I,-_= E 1

T 211 %

C2-RT,0

123

(48)

(50)

Surface nucleation was shown to follow fixed-order chemical kinetics [eqn. (5)] and therefore an apparent order n = 1 is expected here.

The applicability of these relationships was carefully tested for LiO, - 2Si0, glasses [53,69-731. It was verified that ln[ -ln(l - (Y)]vs. In p with T = constant yields ca. 3 and ca. 4 for the nucleated and as-quenched glass, respectively [eqn. (30) with Z = constant and eqn. (47)] [53]. It was also established that the activation energies determined from eqns. (33) and (49) are in good agreement with that of viscous flow. Similar agreement is also reported for the surface nucleated case [72,73], where the exponent n = 1 is observed. (Correspondence of the activation energies of surface and bulk crystallization was also verified for metallic glasses [74].) These results are also substantiated by independent observations [75,76]. Equations (30)-(34) are also found to be applicable to the crystallization of CaO-SiOz based glasses [77,78] and good agreement with isothermal values was reported for GeO, and 3BaO * 5Si0, also [79].

In the original publications [53,68,69-73,75-791, eqns. (30)-(34) and (47)-(50) were derived by an irregular procedure, either assuming the validity of the JMAKY kinetics [eqn. (S)] under non-isothermal conditions, or using the extremely rough e’kyle(E’RT) j p ( E/R T) approximation. These doubtful procedures do in fact lead to some ambiguous results, e.g., to the suggestion of an evaluation method for DDTA curves [8] that is not yet derived from correct principles [29] and to a modified exponent in eqn. (50) where T,‘, is calculated instead of the correct Ta:,“. This latter problem is irrelevant, however, as the variation of p is dominant and in fact both methods, eqns. (49) and (50) yield to a good approximation the same [n/( n + l)]( E/R) value [73].

It is emphasized here that eqns. (30)-(34) and (47)-(50) have now calculated from correct premises [63] and found to be experimentally appli- cable in some instances. The basic reason for the modification apparent in eqns. (47)-(50) should, however, be understood: it is only a correction due to the unequal heating rates applied in the nucleation range, resulting in samples with unequal kinetic parameters.

Metallic glasses

For metallic glasses the difference between the glass transition tempera- ture, Tg, and that of crystallization, T,,, is very small; in many instances Tp

124

is not even observed, being overlapped by T,,. Under these circumstances a separate temperature regime for nucleation cannot be defined. A general equation for the activation energy was suggested by Von Heimendahl and co-workers [80-821

where a = 0 for quenched-in nuclei and a = 1 for a constant isothermal nucleation rate, m is the number of growth dimensions and b = 1 for linear and l/2 for parabolic growth. The activation energies of nucleation, E,, and growth, E,, were separately determined by transmission electron mi- croscopy, and ,!? predicted by eqn. (51) was in good agreement with that determined by the usual methods. As a high value is obtained for E,, it is evident that changes in the concentration of the quenched-in nuclei signifi- cantly influence i? The effect of quenching rate on crystallization has been experimentally verified for a series of Fe-B glasses [89]. This fact makes the

TABLE 2

Comparison of activation energies determined by isothermal and non-isothermal methods

Isothermal Non-isothermal

Eqn. Ref. E(k.I mol-‘) Eqn. Ref. E(kJ mol-‘)

COX4XPIS 2 42

COXl &X4 42

%&‘~~ 19 a

Pd xoGezo 42

(Aus~Cu,~),&Ge,4 42

Zr75 Pt 25 40 Ni,,,Nb,,, h.’ 42

Al 23Te7, 40

(CJeSe,),,,(Sb,Te,)zo(GeTe),, 40

42 42 42 42 42 42 42 19a 42

83 435 34 84 426 33 86 286 33 85 376 33 84 385 27 250 33 = 89 239 34 93 240-245 34 96 231 34

360 94 195 34 94 190 34 44 221 34 90 212 34 91 250 34 95 339 34 92 629 34

433 100 270 34

99 176 34

83 448 87 429 87 270

87 367

27 250 89 240 93 243 96 244

355 94 213 94 221 44 227 90 218 91 240 95 325-345 92 626

446 100 270

99 168

.’ Refined Ckawa plot [59] for different transformed fractions and detailed analysis of the full dn/dt curve [65-671.

” Different stages of the transformation evaluated separately. Transformed fraction determined from electrical resistance.

125

comparison of results obtained on different ribbons very difficult and can even contribute to the significant dispersion obtained on different batches of nominally similar samples [SS]. Despite these difficulties, there are many examples in the literature where an acceptable agreement was observed between activation energies obtained by isothermal and non-isothermal methods (Table 2). It is only remarked here that agreement is found both for metallic and chalcogenide glasses and for a wide variety of a and b parameters. n = 3 is observed for Fe,5B,, and n < 3 is established for FesoBZO, which indicates the dominant role of the quenched-in nuclei. On the other hand, n z 4 is found for Fe,Ni,P,,B,, which indicates linear growth with concurrent nucleation, whereas parabolic growth is applicable to Fe,,Ni,,Cr,,P,,B, [80].

There are only a few instances where the characteristic exponent is determined by non-isothermal methods. Good agreement with the isother- mally determined values was reported for Pd-Be [97] and Pd-B [98], where a combination of eqns. (31) and (33) was used, and for Fe,,B,,, where a detailed evaluation of the full da/dt curve was utilized [27].

This section is concluded by mentioning an interesting piece of evidence on the fundamental problem of non-isothermal evaluation. The basic as- sumption of the usual formalism (which in our case is a direct consequence of the well established theory of nucleation and growth controlled trans- formations) is that the rate equation is of the same form for both isothermal and non-isothermal measurements. It was actually tested for (GeSe,),, (GeTe), glasses with eqn. (40) and the two sets of measurements were found to merge into a common plot [loll. This result is direct experimental proof of the validity of the isokinetic hypothesis and shows that the alternative formalism, where the rate equation is modified under non-iso- thermal conditions, is inapplicable.

CONCLUSION

A progress report has been given on the investigation of crystallization in the solid state. It was concluded that a non-isothermal generalization of the laws describing nucleation and growth controlled transformations is only possible along the lines of the isokinetic hypothesis. The alternative for- malism, where the transformed fraction is invariant and the rate equation is modified under non-isothermal conditions, is in contradiction with both the general theory and the experiments. ,

Different methods were reviewed for the isothermal and non-isothermal determination of the kinetic parameters. It was established that the usual precision makes the simultaneous determination of all kinetic parameters from a single non-isothermal experiment very problematic. In addition to methods that are capable of determining nE only, special attention was paid

126

to the determination of E using temperature shift methods. For a linear heating experiment the transformed fraction at the maximum rate is always

% = 1 - l/e = 0.63, and is independent of any experimental parameter. The Ozawa and Kissinger methods are thus equivalent and both are generally valid if otherwise identical samples are heated at different rates in their transformation region. In addition to the different quenching rates, the different heating rates in the nucleation region may also result in systemati- cally different samples and then an appropriate correction is necessary.

Experimental results for oxide, chalcogenide and metallic glasses clearly illustrate that identical kinetic parameters can be determined from both isothermal and dynamic experiments in many instances.

ACKNOWLEDGEMENTS

Fruitful discussions with Drs. B. Fogarassy and R. Schiller are gratefully acknowledged.

APPENDIX: EFFECT OF EXPERIMENTAL THERMAL LAG ON MAXIMUM RATE METHODS

As the temperature at which the transformation rate is maximal is usually determined by thermal methods, the reliability of this determination was seriously questioned [102] owing to the finite thermal inertia of any measur- ing system. While it was shown in some recent publications [103-1051 that this negative conclusion is exaggerated, the actual calculation is given here to show the real figures.

If we assume the basic DTA equation in the simplified form: &= -ATK/AH-~TC,/AH

(neglecting terms arising from experimentally unbalanced conditions for non-stationary calorimetry in the twin arrangement), then AT can be routinely expanded into a Taylor series around To where AT is maximal:

AT=ATo+~~~~r(T-To)+~(T-2q)2 +... 0 dT2

As AT is maximal at To, dAT dT r=rO=’

Introducing the notation

D 2

= d2AT dT2 rsr,

liT=(T-T,)D,j3

ii~=~,p~

127

the temperature of the maximum transformation rate is determined by

AH~~=o=(-KAT-c,BT)I,_~~= -K(T,-T,)D,P-C,D,P~

yielding T, = To - C’P/K. When the activation energy is determined by plotting In /3 vs. l/T,, then

l/T, = (VT,)/@ - C,B/T,K) = l/T,@ + C’P/T,K)

Parameters typical of, e.g., the Perkin-Elmer DSC-2 are 5 mg Fe,,B,, (C,=2.5 mJ K-l), T,= 700 K, K= 10 mJ K-’ s-l, p=O.l5 K s-l, thus CJ?/T,K = 5 x 10e5, which is negligible. Even in typical macro-DTA with a 300 mg sample and K = 20-50 mJ K- ’ s- ’ the order of magnitude of the correction never exceeds 10p3, which is still negligible.

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Thermophysics 2012 ‐ Conference Proceedings 

17th International Meeting of Thermophysical Society 

 

 

 

 

 

 

 

 

 

 

 

 

 

31st October to 2th November 2012 

Podkylava, Slovak Republic 

Slovak Academy of Science in Bratislava 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 Copyright © 2012 Brno University of Technology, Czech Republic 

 All  rights  reserved. No  part  of  this  publication  or  the  information  contained  herein may  be 

reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, 

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the publisher.  

Although  all  care  is  taken  to  ensure  the  integrity  and  quality  of  this  publication  and  the 

information herein, no responsibility is assumed by the publishers or the author for any damage 

to property or persons as a result of operation or use of this publication and/or the information 

contained herein.  

 

Book cover design:  Vladimír Dvonka, Michal Veselý 

Printed by:  Brno University of Technology, Faculty of Chemistry, 2012 

ISBN:  978‐80‐214‐4599‐4 

2 

Invited Lecture 

Application of fractals in condensed‐state processes 

Peter Šimon1, Oldřich Zmeškal2, Jaroslav Šesták3 

1Institute of Physical Chemistry and Chemical Physics, Faculty of Chemical and Food 

Tehnology, Slovak University of Technology, Radlinského 9, 812 37 Bratislava, Slovakia; e‐mail: 

peter.simon@stuba.sk 2Institute of Physical and Applied Chemistry, Faculty of Chemistry, Brno University of 

Technology, Purkyňova 118, 612 00 Brno, Czech Republic; e‐mail: zmeskal@fch.vutbr.cz 3New Technologies — Research Centre in the West Bohemian Region, West Bohemian 

University, Universitní 8, CZ‐30614 Pilsen, Czech Republic, e‐mail: sestak@fzu.cz  

Abstract: Heterogeneous condensed‐state processes take place at interfaces of different phases. In general, 

the surfaces are not perfectly flat and their properties are not homogeneous. They are mostly rough with 

many  irregularities and  inhomogeneities. Surface geometrical  inhomogeneity  is reflected  in  its chemical 

inhomogeneity. For  the description  of  the  structure  of physical  objects with  inhomogeneous properties 

(roughness, mass density, heat density, etc.), the methods of fractal geometry can be applied [1]. In this 

paper,  two phenomena will be analyzed,  i.e.  the heat  capacity of  fractal  structures and  the nucleation‐

induced crystallization. 

Keywords: fractal geometry, heat capacity, nucleation‐induced crystallizaton 

1. Introduction 

Heterogeneous condensed‐state processes take place at interfaces of different phases. In general, 

the surfaces are not perfectly  flat and  their properties are not homogeneous. They are mostly 

rough with many  irregularities  and  inhomogeneities.  Surface  geometrical  inhomogeneity  is 

reflected in its chemical inhomogeneity. For the description of the structure of physical objects 

with  inhomogeneous properties  (roughness, mass density, heat density, etc.),  the methods of 

fractal  geometry  can  be  applied  [1].  The  word  „fractal“  originates  from  the  Latin  word 

„fractus“, meaning  broken.  It  is  indicated  that  fractals  are widespread  and  that  the  fractal 

geometry  is  the geometry of Nature  [2]. Classical geometry provides a  first approximation  to 

the structure of physical objects;  it  is the  language that we use to communicate the designs of 

technological  products  and,  very  approximately,  the  forms  of  natural  creations.  Fractal 

geometry  is  an  extension  of  classical  geometry.  It  can  be  used  to make  precise models  of 

physical structures of rough surfaces, disordered layers on surfaces and porous objects (such as 

heterogeneous catalysts). Furthermore, gels, soot and smoke, and most macromolecules are also 

fractals [3]. 

Father of  the  fractal geometry  is Benoît Mandelbrot  [4]. He described geometric  fractals as  ʺa 

rough  or  fragmented  geometric  shape  that  can  be  split  into parts,  each  of which  is  (at  least 

approximately) a reduced‐size copy of  the wholeʺ. Fractals are self‐similar objects  that exhibit 

dilatational  symmetry.  Fractals  have  details  on  all  scales;  therefore  Euclidean  geometry  and 

classical calculus are insufficient for their description and the fractal geometry is required.  

Fractals are disordered systems; the disorder is described by generally non‐integer dimensions 

[3]. As long as surface irregularities show scale invariance ‐ that is, dilatational symmetry ‐ they 

can  be  characterized  by  a  single number,  the  fractal dimension. This means  that  the  surface 

exhibits self‐similarity over certain length scales. In other words, the structure exhibited at the 

207

scale of the basic building blocks is reproduced at the level of larger and larger conglomerates. 

Note that self‐similarity  implies that the features of a structure  look alike at different scales of 

length or  time. This self‐similarity of a process at different scales of  time can be characterized 

with  a  fractal dimension: A higher value of  the  fractal dimension  indicates  a higher  level of 

heterogeneity or state of disorder. 

Fractals  possess  non‐trivial  geometrical  properties;  in  other  words,  they  are  geometrical 

structures with  non‐integer dimensions. A  consequence  of  the  fractal  nature  is  a  power‐law 

dependence  of  a  correlation  function  (rate  constant, number  of  entities)  on  a  coordinate  (for 

example, time or scale). The repeating shape or form does not have to be identical [3]. 

In this chapter, two  illustrative examples of application of the fractal geometry on condensed‐

state problems are demonstrated. 

2. Heat capacity of fractal structures 

Within the fractal theory, the number of objects, N(r), in the circumscribed area with the radius r 

can be given as 

    (1) DrKrN )(

where K is the fractal measure and D stands for the fractal dimension.  

Applying  the  equipartition  theorem  [5]  heat  capacity  C  of  a  solid  mass  fractal  structure 

embedded in E‐dimensional Euclidean space can be expressed as 

  B Bd( ) ( )d

DUC r Ek N r Ek Kr

T   (2) 

Here, U is the internal energy and T is temperature. In Eq. (2) it is taken into account that each 

object in the E‐dimensional solid matrix possesses E vibrational degrees of freedom.  

Heat capacity of solids is mostly normalized as  

  V1 d

dU

Cm T

  (3) 

where m  is the sample mass. If the fractal structure  is composed of Nm(r) objects (atoms) with 

the average atomic mass u, the sample mass can be expressed as 

  mm m( ) ( ) Dm r uN r uK r   (4) 

where K means  the  fractal measure of  the mass distribution  in  the solid with  fractal structure 

and Dm is the fractal dimension characterizing the mass distribution. 

Combining Eqs. (2) ‐ (4), one can get 

  mBV

m

D DEk KC r

uK ,  (5) 

When multiplying both the numerator and denominator of Eq.(5) with the Avogadro constant, 

we obtain 

  mV

m

D DnERKC r

MK   (6) 

208 

where R is the gas constant, M is the molar mass of the solid under consideration and n is the 

number of atoms in a molecule. 

It has been derived that the distribution of temperature in the fractal structure embedded in E‐

dimensional space is given as [6‐8]  

 2

B

( )( 2

D EK c rT r

k D D E

)  (7) 

where  /(2 )h  

Combination of Eqs. (6) and (7) yields:  

 

m2

Vm A

( 2)

D D

D EnERK RTC D D E

MK KN c

  (8) 

where NA is the Avogadro constant. The continuous fractal structure has the fractal dimension 

between 1 and 3 and  is embedded  in  three‐dimensional space  (E=3). Taking  this  into account, 

Eq.(8) can be simplified as: 

 

m1

Vm A

3 ( 1)

D D

DnRK RTC D D

MK KN c

  (9) 

Under the assumption that the distribution of internal energy and the distribution of mass have 

the same fractal dimension (D=Dm), Eq.(9) takes the form 

  Vm

3nRKC

MK   (10) 

If the distribution of internal energy and the one of mass are identical (D=Dm and K=Km), Eq.(11) 

reduces to the form expected from the equipartition theorem for high temperatures: 

  V3nR

CM

  (11) 

However, the condition of  identical distributions of the  internal energy and the mass may not 

necessarily be allways fulfilled.  

The dependence of heat capacity of fractal objects on temperature, given by Eq.(11), differs from 

the  dependences  derived  by  Einstein  or Debye.  The  Einstein model was  derived  under  the 

assumption that each atom in a perfect crystal oscillates around its equilibrium position with a 

single frequency. Debye amended the model taking into account that the atoms in fact oscillate 

over a range of frequencies from zero up to a maximum value [5]. When deriving Eq. (9), fractal 

structure of the sample instead of perfectly regular crystalline matrix has been assumed. 

3. Nucleation‐induced crystallization 

Kinetics  of  condensed‐phase  processes  is  very  often  described  by  the  generalized  kinetic 

equation 

  d fd

k Tt

  (12) 

209

where  t  is  time and α  is  the degrese of conversion  (or simply conversion). The  functions k(T) 

and  f(α)  are  the  temperature  and  conversion  functions  and  should  be  separable  [9‐11].  This 

paragraph deals with  the  temperature and conversion  functions  in  the process of nucleation‐

induced crystallization and represents a modification of the approach used by Ozao and Ochiai 

[12]. 

  The  basic  concept  of  the  kinetics  of  nucleation‐induced  crystallization  processes was 

established by Avrami [13‐15]. His model takes into account the interconnection of the growing 

grains. The assumptions adopted are: (i) Nucleation occurs randomly and homogeneously over 

the entire untransformed portion of  the material;  (ii) The growth rate does not depend on  the 

extent  of  transformation.  (iii)  Growth  occurs  at  the  same  rate  in  all  directions. Hence,  the 

constant rate of nucleation can be expressed as 

  nddn

kt   (13) 

where n stands for the number of grains. Volume of one grain with the fractal structure, V1, is  

  1 1DDV Kr K v t   (14) 

Here, K  is  the volume  conversion  factor,  r  is  the  (effective)  radius of  the grains and v1  is  the 

growth  rate  of  grains.  First,  the  growth  of  grains  without  their  interconnection  will  be 

considered. Volume of the transformed matter, V, can be obtained as  

 +1n 1

1 n 10 0

d d dd 1

t t DD D Dk Kvn

V V t k Kv t t tt D

  (15) 

The degree of conversion is defined as the ratio of the volume transformed to the total volume, 

Vo. From Eq. (15) then follows: 

 

+1n 1w

o 1

DDk Kv

tV D

  (16) 

From Eq. (16) one can get 

 

1+1 1o +1

n 1

1 D

DwD

V Dt

k Kv

  (17) 

Differentiation of Eq.(16) gives 

  w n 1

o

dd

DDk Kv

tt V

  (18) 

Combination of Eqs. (17) and (18) leads to the result 

  1

1w n 1 1

wo

d 1d

DDDD D

k KvD

t V

  (19) 

If the grains grow sufficiently large as to interconnect one another, only a fraction of αw is real; 

some  portion  of  it  lies  on  previously  transformed material  and  is  virtual.  Since  nucleation 

210 

occurs  at  random,  the  fraction  of  overlapped  volume  should  be  proportional  to  the 

untransformed volume fraction 1‐ α. Hence, the actual degree of conversion, α, is related to αw 

by the relationship [12‐16] 

  wd 1 d   (20) 

After separation of variables and integration, from Eq. (20) it can be obtained: 

  w ln 1   (21) 

From Eq. (21) further follows: 

  wdd 1d dt t

  (22) 

Combining Eqs. (19), (21) and (22), one gets 

  1

11n 1

o

d 1 1 ln 1d

DD DD Dk KvD

t V

  (23) 

The fractal dimension of the grains can be 1 < D < 3.  

Comparing  Eqs.  (12)  and  (23),  it  is  obvious  that  the  temperature  function  is  a composite  of 

several quantities: 

  1

1n 1

o

1DD

Dk Kvk T D

V

  (24) 

The conversion function is 

  1f 1 ln 1D

D   (25) 

Integration of Eq.(23) for a constant temperature leads to an Avrami‐like equation: 

 

1

1 exp1

Dkt

D

  (26) 

Main diference with the Avrami derivation is that it is assumed here that the growth of grains 

may cease at any moment in any direction, not only by impinging on another growing grain. In 

this  way  the  fomation  of  grains  with  fractal  geometry  from  the  very  first  moment  of 

crystallization is admitted, not just the formation of spherical particles.  

When deriving Eq.(23),  it was  implicitely assumed  that  the process was  isothermal. For non‐

isothermal processes, kn and v1 obviously depend on temperature [17]. The dependence of k(T) 

on  temperature  is almost exclusively approximated by  the Arrhenius  function.  In  the  case of 

crystal growth, it would be perhaps advisable to employ the equation by Harecourt and Esson 

[18] 

    (27)  mk T AT

where A  and m  are  kinetic parameters. The  form  of  temperature power  exponent  in Eq.(27) 

indicates that the fractal nature of matter is taken into account.  

211

4. Conclusions 

The  two case  studies  shown above are  just  illustration of  the application of  fractal geometry. 

This chapter has no aspiration to present a complete picture on the subject. For example, Google 

provides 99900 responses to the keywords „fractal heat capacity“ and 167000 responses to the 

keywords  „fractal  crystallization“.  Brownian  motion  has  also  the  fractal  character  [19]. 

Applications of fractals for the problems of adsorption, permeability, aggregation and growth, 

diffusion and reactions, spectroscopy, chemical degradation and other topics are summarized in 

the monography by Rotschild [20]. Fractal Approach to Heterogeneous Chemistry is reviewed 

in [21]. 

However, a reaction medium does not have to be a geometrical fractal in order to exhibit fractal 

kinetics  [22]. Classical  reaction kinetics has been  found unsatisfactory when  the  reactants are 

spatially constrained on the microscopic level by either walls, phase boundaries or force fields. 

Among the examples of fractal‐like kinetics are chemical reactions in the pores of membranes, 

charge  recombination  in  the  colloids  and  clouds,  reactions  catalyzed  by  catalytic  islands  on 

noncatalytic supports, etc. When applying classical reaction kinetics to the processes mentioned, 

anomalously  large values of  the reaction order can be  found  (from 3  to 75). The rate constant 

may depend on time, where the dependence is power‐law one [22]. 

This  contribution  is  a  shortened  version  of  chapter  12  from  the  book  „Thermal  analysis  of 

micro‐, nano‐ and non‐crystalline materials“, edited by.  J. Šesták and P. Šimon  [23]. The book 

will appear this year.  

References 

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thermal analysis. Elsevier, Amsterdam, 2005, ISBN: 978‐0‐444‐51954‐2. 

[2]   BARNSLEY, M.F. Fractals everywhere. Academic Press, 1993. ISBN 978‐0‐12‐079069‐2. 

[3]   SADANA, A.  Engineering  Biosensors  ‐  Kinetics  and Design Applications. Academic  Press, 

2002. ISBN 0‐12‐613763‐3. 

[4]   MANDELBROT, B. The fractal geometry of nature. Macmillan, 1983, ISBN 978‐0‐7167‐1186‐5. 

[5]   ATKINS, P.W. Physical Chemistry. Oxford Univ. Press, Oxford 1998, ISBN 0 19 850102 1. 

[6]   ZMESKAL, O.‐ NEZADAL, M.  ‐ BUCHNICEK M. Chaos, Solitons  and Fractals  19  (2004) 

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954.: Thermal properties of bodies in fractal and cantorian physics 

[8]   ZMESKAL, O. – VALA, M.‐ WEITER, M.‐ STEFKOVA, P. Chaos, Solitons and Fractals 42 

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[13]   AVRAMI, M  Journal of Chemical Physics 7  (1939) 1103–1112: Kinetics of Phase Change.  I. 

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[14]   AVRAMI, M  Journal  of Chemical Physics  8  (1940)  212–224: Kinetics  of Phase Change.  II. 

Transformation‐Time Relations for Random Distribution of Nuclei.  

[15]   AVRAMI, M  Journal  of Chemical Physics 9  (1941) 177–184: Kinetics of Phase Change.  III. 

Granulation, Phase Change, and Microstructure.. 

[16]   JENA, A.K. – CHATURVEDI, M.C. Phase Transformations in Materials. Prentice Hall, 1992, 

ISBN 0136630553. 

[17]  KEMENY,  T.‐  ŠESTÁK,  J.  Thermochimica  Acta  110  (1987)  113‐129:  Comparison  of 

crystallization kinetics determined by isothermal and nonisothermal evaluation methods. 

[18]   FLYNN,  J.H. Thermochimica Acta 300  (1997) 83‐92: The „Temperature  integral“ –  its use 

and abuse. 

[19]   KAYE, B.H. A random walk through  fractal dimensions. VCH, Weinheim, 1994. ISBN 3‐527‐

29078‐8. 

[20]   ROTHSCHILD, W.G. Fractals in chemistry. John Wiley & Sons, 1998. ISBN 0‐471‐17968‐X. 

[21]   AVNIR, D. Fractal Approach to Heterogeneous Chemistry. Wiley, New York, 1989. 

[22]   KOPELMAN, R. Science 241 (1988) 1620‐1626: Fractal Reaction Kinetics. 

[23]   ŠIMON,  P.‐  ZMEŠKAL,  O.‐  ŠESTÁK,  J.  Fractals  in  solid  state  processes.  Chapter  12  in: 

Thermal  analysis  of  micro‐,  nano‐  and  non‐crystalline  materials,  Eds.  Šesták  J,  Šimon  P, 

Springer Verlag, 2012, in press. 

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