MAY 1955 321

ENTROPY IN 'SCIENCE AND TECHNOLOGY

Ill. EXAMPLES AND APPLICATIONS (continued)

by J. D. FAST.

As in the previous article (II), the author demonstrates with examples the important partplayed by entropy in widely differing fields of science and technology. This article deals withthe theory of the specific heat of solids, 'th« thermotlynamic equilibrium of certain latticedefects in solids and their relationship with diffusion phenomena, and the equilibria ofmixtures, both in a solid form (e.g. alloys). and in the form of liquid solutions, including'the extreme case of solutions of polymers. The statistical aspects of rubber elasticity arealso discussed, a phenomenon that can be completely described as an entropy-effect.

The first article of this series 1), referred to belowas I, gave some general observations on the conceptof entropy; the second 2), referred to as I1, presentedsome applications of this concept in chemistry,physics and technology. In the present, third article,some further examples are discussed. A fourtharticle, to be published later, will be of a somewhatdifferent nature, and will discuss the application ofthe concept of entropy to information theory.

The specific heat of an Einstein solid

In I (page 262) we discussed a simplified model ofa solid, in which the identical.atoms behave aslinear harmonic oscillators, executing their vibra-tions around fixed centres practically independentlyof one another, and capable of ahsorhing equalquanta of· a value hv: Suppose that this solid,

Iconsisting of N atoms, absorbs a number q ofenergy quanta hv (starting from absolute zero).This energy can be distributed among the oscillatorsin a great number 'of ways (it is assumed thatN'}> 1 and q'}> 1). Each distrihution represents onemicrostate and the total number of micro-states mis given, according to (I, 9) by:

(q+N-1)!m=

. q! (N-1)!

For the sake of convenience it was not taken intoaccount when deriving this formula that an atom

. in an Einstein solid has to he assigned three degreesof vibrational freedom. An actual crystal of Natoms in this model can be regarded as a systemof 3N linear oscillators. Since real crystals as a rulecontain more than 1019 or 1020 atoms, unity is

1) J. D. Fast, Entropy in science and technology, I, Theconcept of entropy, Philips tech. Rev. 15, 258-269, 1954/55.

2) J. D. Fast, Entropy in scienceand technology, I1, Examp-les and applications, Philips techno Rev. 16, 298-308,1954/55.

536.75

negligible compared to N, and the formula becomes:

(q+ 3N)!m=

q! (3N)!. . . . (III,l)

The energy of the crystal, after q vibrational quantahave been absorbed, has risen to an amount

U=qhv . . . . . . (Ill,?)

above the zero-point energy, whilst the entropy,according to (1,12) and (Ill, 1) and using Stirling'sformula in the approximation (I, 3) is 'given by

S = k ~(q+ 3N) In (q+ 3N) - q In q - 3Nln 3N~.(III,3)

The Helmholtz free energy,. F = U - TS, maytherefore he written as

F(q) = qhv-kT~(q + 3N) In (q+ 3N)-. - qlnq-3Nln3N~. (III,4)

Until now the number' of absorbed quanta hasbeen assumed to be known. In fact it is usually notprimarily this number of quanta which is known,but the rempcrature to which the crystal is heatedby bringing it into thermal contact with surround-ings at constant temperature. What we wish toknow, therefore, is the number of quanta q presentat a given temperature T in the Einstein solid. Theknowledge of q as a function of T will enable us tocalculate the specific heat at any temperature.If the solid could entirely submit to its "striving"

towards a minimum value of of the energy, it wouldnot absorb any quanta at all. Conversely, if it couldentirely submit to its "striving" towards a maximumvalue of the entropy ~the number of quanta absorbedwould continue to .increase, The compromise (thestate of equilibrium) lies,. according to I, at thepoint where the Helmholtz free energy is a

322 PHILlPS TECHNICAL REVIEW VOL. 16, No. 11

minimum, r.e, where

dF(q) = O.dq

Differentiating (Ill, 4) and equating to zero,

q+3Nlw- kTln = 0

q

or, after re-arranging,

3N. (III,5)q,=, ehv/hT -1.

The specific heat at constant volume is given by:

dQ' dU d(qhv)Cv = dT = d T = ,dT

Substituting for q from (Ill, 5) gives the relation-ship sought:

. ( h» )2 ehv/kT

'Cv = 3 kN kT (ehv/kT -:- 1)2' (111,6)

This formula was deduced by Einstem in 1907.For the part of the internal energy that varies withtemperature we can write, according to (Ill,' 2)and (Ill, 5):

3NlwU = h 'kT ••••• (111,7)e v,' -1

If kT is very much greater than lw, then

ehvlhT ~ 1+ lw/kT

and therefore,U~3NkT,

Cv ~ 3Nk.

At relatively high temperatures, therefore, thespecific heat of an Einstein solid reaches a constantvalue which is not only independent of the temper-ature but also of v. If this idealized solid representedthe behaviour of actual (elementary) solids, thenthe latter would all have the' same specific heatper gram-atox;n at not too low temperatures. Thiswould amount to Cv = 3 Nok = 3R, .in which' Norepresents Avogradro's number and R the gasconstant. According to the experimentally estab-lished law of Dulong and Petit, many solid elementsdo in fact have an approximately equal specific heatof about ·6 cal per degree per gram-atom (R =approx. 2 cal per degree) at not toolow temperaturesIt is also in agreement with observation that (Ill, 6)requires tb-at Cv, at decreasing temperature, finallyapproaches zero; cf. fig -. 1. which represents therelationship between cv/R and kT/hv according to(Ill, 6).

To a first approximation, Einstein's formulathus provides a good expression ,for the thermalbehaviour of the solids considered here. The formof the experimental cv(T) curves does not agree indetail with the formula, however. The largest

4cv/RI3

2V fo-

1 I)

2 2,5 3----kT/hl)

0,5 1,582411

Fig. 1. c,/R as a function of kT/bv for an Einstein solid (formula(III, 6».

deviations occur at very low temperatures. Thisis due to the fact that a real crystal cannot, infact, be considered as an asssembly of nearlyindependent oscillators of equal frequency asassumed in the Einstein derivation. A crystal hasa great number of vibration modes at widely diver-gent frequencies; they may. not be regarded as thevibrations of individual atoms, for they are intrinsicto the lattice as a.whole: they can be pictured as apattern of standing waves in the crystal. A theorybased on. this .model and in better agreement withexperiment has been formulated by Debye. Weshall not enter into' this theory in view of the factthat, considered purely thermodynamically, itcovers no fresh ground. One of its consequences,which is in good agreement with experiment, maybe mentioned: at low temp'eratures the Stefan-Boltzmann law (cf. I1) is applicable to the vibrationsin a solid. According to this law, the energy is propor-tional to T4, so that the specific hèat is proportionalto T3. This consequence is due to the fact that boththe radiation inside a hollow body and the vibrationsin a solid can be considered as 'a broad spectrum of. modes of vibration. It is true that the hollow spacehas an infinite number of modes, whereas the num-ber of modes in the solid is restricted by the numberof atoms N (it amounts to 3N). At low temperatures,however, this difference is irrelevant since then,according to quantum mechanics, the high-freq-uency modes play no part,

We have already seen that to consider a solid as an'assembly of independent oscillators of equal fre-quency is rather an inexact model. A 'diatomic orpolyatomic gas, on the other hand, conforms very

MAY 1955 ENTROPY, III 323

satisfactorily to this model. In a diatomic gas eachmolecule can vibrate iri such a manner that theatoms oscillate along their line of centres. (Trans-lational and rotational degrees of freedom alsocontribute to the specific heat of a gas, but will nothe discussed here.) Exchanges of energy are broughtabout by collisions b~tween the gas molecules.Apart from this the vibrations are mutually inde-pendent, Formula (Ill, 6), with the omission ofthe.factor 3, is therefore' applicable with considerablygreater accuracy to the vibrations of a diatomic gasthan to tho~e of a solid. The specific-heat valuescalculated' on this basis provide the necessarycorrections to give validity to the calculation ofchemical equilibria (see the heginning of Il) evenat very high temperatures.

Vacancies and diffusion in solids

Atomic diffusion plays an important part inseveral processes occurring in solids. Diffusion in

. metals and alloys in particular has been the subjectof extensive research during the last few decades.This has revealed that the presence of defects in theperiodic crystal lattice, particularly vacancies orinterstitial atoms, is essential for the occurrence ofdiffusion.As we have seen in the foregoing discussion of an

Einstein solid, the' vibrations of the atoms aroundtheir positions of equilibrium become more 'and moreviolent as the temperature rises. Even before themelting point is reached, a fraction of the atoms willpossess sufficient energy to leave their lattice posi-tions completely. These atoms form new latticeplanes on the outside of the crystal, since the spacesbetween the other atoms, the interstices, are toosmall to accommodate them. Starting, then, witha perfect crystal (no defects), any vacancies whichoccur in the crystal as the temperature rises mustoriginate in the boundary layer. One can imaginethat a few atoms in the boundary layer leave theiroriginal positions and occupy new positions on thesurface (fig. 2). Atoms from deeper layers can sub-sequently jump into the newly created vacancies,and so on. We may equally well say that the vacan-cies. arise at the surface and subsequently diffuseto the interior. The fact that at high temperatureslattice defects are bound to arise even in the stateof equilibrium, is due to the fact that theÏ! occur-rence represents an increase in the entropy. Disreg-arding the extremely small macroscopie volumechanges occurring with the formation of the vacan-cies, it can be said that at a certain temperature Tvacancies will continue to be formed until ultimatelythe Helmholtz free energy F= U - TS has reached

its minimum value. Although the internal energy Uwill rise by an amount LIU due to the introduetionof vacancies, it is nevertheless possible that undercertain conditions the value of the free energy will 'drop, viz. if TLIS> LIU.

Q230~00000000'0000000000000I!. 00000000000000000000000000000000000 0000

II 000000000000000000Fig; 2. The vacancies in' a solid are assumed to form at theouter surface (a) and subsequently diffuse to the interior (b),this being equivalent to the outward diffusion of atoms (1, .2, 3).

If the energy e required for forming a vacancy isknown, the percentage of vacancies can be evaluatedfor any temperature ..This is done as follows.If a perfect crystal of N atoms' changes into an-

imperfect state with n vacancies, then the corres-ponding increase in entropy, LIS = k In rn, is givenby the' number of different ways, m in which Nidentical atoms can he accommodated in a latticewith (N + n) available positions. This number is

m=(N+n)!N!n!

(IlI,8)

Using Stirling's formula in the approximation (I, 3),we find: '

LIS= k ~(N+ n) In (N + n) - N In N - n In n~.

As long as the concentration of the vacancies is so.small that their interaction is negligible, the increasein the energy is determined by LIU = ne and thusthe change of thè free e.nergy by ,

LlF . LlU - TLIS == ne-kT~(N +.n) In (N + n) -NlnN-nln n~.

(IIl,9)

In the equilibrium state, the free energy is a mini-mum; thus àF/àn = 0, ox, from (Ill, 9):

e-kT ~In(N + n) -.In n~= 0,

i.e.

324 PHILlPS TECHNICAL REVIEW 'VOL. 16, No. 11

Since n is negligible with respect to N, this becomes:

n -BikT -EIRT-=e =e ,N

(IlI,10)

where E is the energy required to make 6 X 1023

(Avogadr?'s number) vacancies. According to thecalculations of Hunrington and Seitz 3) e has thevalue of approximately 1 eV for copper, i. e. E ~23000 cal per "gram-atom of vacancies". For1000 "K, therefore, we arive at:

!:__ = e-23000/2000 = 10-5:N

In this case one' in each 100000 lattice positionswould be unoccupied. This corresponds to an averagedistance between adjacent vacancies of the order ofthirty interatomie distances.

The additional contribution to ~he internal energy owing tothe formation' of vacancies is accompanied by an additionalcontribution to the specific heat, ~givenby:

dAU dus d(NEe-BlkT) E 2 -.kT-- = - = ----- = R(-) e I per gram-atomdT <iT dT kT·' .

The above is based on the assumption that the increase inthe entropy during the transition from the ideal to the disturb-ed arrangement is exclusively based on the number ofpossible"distrthutions m, given by formula (Ill, 8). In reality thevibration frequencies of the atoms near the vacancies alsochange; this contributes to thc entropy.This may be explained as follows. An atom adjacent to a

vacancy is bound more weakly than an atom completelysurrounded by similar atoms. Consequently it will have alowervibration frequency, at least in the direction of the vac-ancy. Since, from (Ill, 3) and (Ill, 5).

kTSF:::!3Nklnh; for kT}>hv, (IIJ,l1)

is applicable to an Einstein solid, thc introduetion of vacanciescauses not only the entropy increase as evaluated in (Ill, 8),bul' also an increase in the vibrational entropy. The coneen-trution of the vacancies 'can, therefore, be considerably greaterthan the value calculated above,

At high temperatures the vacancies move atrandom through the lattice. A displacement of avacancy over one interatomie distance is broughtabout if an adjacent atom jumps into the vacantsite. For such a jump an amount of energy, at leastequivalent to the activation energy q is required.This energy will, as a rule, be large compared to themean thermal energy of the 'atoms: The fraction ofthe total number of atoms which possesses at leastthe activation energy q is given by

f = e-qlkT = e-QIRT,

3) H. B. Huntington and F. Seitz, Phys. Rev. ,61, '315-325,1942.

where Q = Noq is the "activation energy per gram-atom". The diffusion constant D of the vacancieswill be approximately proportional to this factor:'

D -:- Do e-QIRT, •••• (III,12)

where Do is a constant which IS independent or onlyslightly dependent on the temperature.

Primarily, however, we are often not so muchinterested in the diffusion of the vacancies as in thediffusion of the atoms of the pure metal, i.e. the self-diffusion, or the diffusion of foreign atoms occupyingthe lattice sites ( substitutional atoms]. The self-.diffusion can he studied with the aid of certainradioactive isotopes of the atoms of the pure metal.The diffusion. constant for this type of diffusionshould be smaller than that for the vacancy diffusion,because the atoms are capable of jumping only atthat moment when there happens to be a neigh-bouring vacancy .. The number of times per secondthat this condition occurs is proportional to thediffusion constant of the vacancies and to theirrelativenumber, and hence to (Ill, 12) and (Ill, 10).

According to this reasoning, the diffusion coeffi-cient .will be given by an expression of the form:

D - D' -Q/RT -EIRT- 0 e e

or(III,13)

where. . . (III,14)

and Do' is a constant which is independent or onlyslightly dependent upon the temperature.

The foregoing also applies to substitutionalatoms, provided that the properties and size of theseatoms differ so little from those of the solvent thatthe vacancies have no preference for either type ofatom, and that the necessary activation energy isthe same for both.

All~ys

Entropy of mixing and energy of mixing

An enormous number of different alloys with agreat diversity of properties can be made, owing to. the almost limitless combinations which are possible:many metallic elements can he combined in severalways in!o binary, ternary, .etc. alloys, the relativequantities in each combination can be varied, andeach alloy can he subj ected to a great variety ofheat and mechanical treatments.

The central problem in the metallurgy of alloysis that of the solubility of the different metals, moreparticularly the solubility in the solid state. If we

MAY 1955 ENTROPY, III 325

confine ourselves to the alloys of two metals (binaryalloys), we find that the majority of comhinations,no matter in what proportion they are mixed, formit homogeneous mixture in the liquid state, whereasin the solid state they have only a limited miscibility.Miscibility in all proportions in only possible in thesolid state' if both metals have the same crystalstructure and if their atoms show only slight differ-ences in size and electron configuration.. Homogeneous mixing is in nearly all casesaccompanied by an increase of the entropy, i.e. itis nearly always promoted by the entropy effect. Theseparation of the mixture into two phases can onlyoccur if also the energy increases during the homo-geneous mixing. If the energy does not change oreven decreases in the course of mixing, then only onehomogeneous phase can exist in the state of equili-. brium. The entropy increase occurring when twometals A and B are homogeneously mixed, can againbe directly established by a statistical reasoning,provided that all . configurations are equallyprobable. If we consider a total number of N metalatoms, of which Nx are of type B and henceN(I-x)are of type A, then the number of possible ways inwhich these can be distributed among the availablelattice points is given by:

N!m = (Nx) QN(I:'_xH!" . (I1I,I5)

The increase in entropy that accompanies the mixingi.e. the entropy of mixing LIS = k In m, thus be-comes:

LIS= Nk ~-x Inx - (I-'x)-In (1- x)~. (I11,16)

Strictly speaking, this expression only gives theentropy of mixing at 0 "K. Ifit is.assumed that thevibrational entropy (cf, Ill, 11) changes little ornot all in the course of mixing, (Ill, 16) is alsoapplicable at other temperatures. '

To find a mathematical expression for the energyof mixing (also called the heat of solution) we startfrom the over-simplified assumption that the internalenergy at zero temperature can be written as thesum of the binding energies of nearest neighboursonly. As a consequence ofthis only three interactionenergies 8AA' 8BB and 8AB between neighbouring

_ pairs A-A, B-B and A-B will occur in the termsof this sum 4). In order to do the summation we

4) We would ultimately find the same formula (Ill, 19) - stilIto he derived - ifwe also took into account, the interactionbetween more remote pairs. The energies EAA,EBBand EAB'would then represent the sums of the interaction energiesbetween pairs A-A, B-B and A-B. Fundamentally theformula would only change ifwe considered the interactionbetween e.g. sets of three or four atoms.

have to evaluate the numbers of the three types ofneighbour configurations.

The number of nearest neighbours to anyoneatom in alloy is denoted by z (z = 8 for the body-centred cubic strl!cture and z = 12 for the close-packed cubic and hexagonal structures). If theatoms ofthe type A and·B are distributed at randomamong the lattice points, then the chance of anatom. A occupying any given lattice point is (1- x)whilst that of an atom B is x. The probahility offinding the .combination A-A in two adjacentpositions is further given by (1- X)2, that offinding B-B by x2 and that of finding either A-B. or B-A by 2x(I - :x) .. In total there are Nz/2 bonds ,between the N' atoms of the alloy; in this particular .case they are distributed as follows:

Nz .T(I-x)2 A-Abonds,

Nz 2-x2

B-B bonds,

Nzx(I-x) A-Bbonds.

The energy of the alloy at 0 "K is therefore given by:

NZ8AA . NZ8BB .UAB= -2- (I-x)2 +-2- x2+~lz8ABX (I-x) ..

(III,17)For pure A (x = 0) the formula gives:

U_ NZ8BB·.

AA---2-'

for pure B (x = 1):

U _ NZ8BBBB---

2-'

For a heterogeneous mixture of the pure metals, theinternal energy is given by:

NZ8AA NZ8BB--. (I-x) +--x ... (I11,18)2 2

The heat of solution at 0 "K is given by the differen-ce between (Ill, 17) and (Ill, 18):

~ 8AA+28~B~.LIU -' x(I-x)'Nz rAB ~ (III,I9)

As a rule the vibrational energy will not have much. influence during the mixing process, so that; (IIr~ 19).is also valid at higher temperatures.

Since x(I - x)Nz. is always positive, the' sign ofLIU is determined by that of èAB-t(8AA + 8BB)'

Solution, ordering, segregation and precipitation. .

If in the foregoing 8ABwere to equal the mean of8AA and 8im (which will probably never be exactly

326 PHILIPS TECHNICAL REVIEW VOL. 16, No. 11

true), then LtU = O. As regards energy, there is nopreference whatsoever for a specific hond, and the'entropy ~ffect causes a homogeneous mixing in thestate of equilibrium.If 8AB is more negative thaD; the mean of 8AA

and 8BB' then zlU is negative. In that case there isa stronger attraction hetween the atoms A and Bthan' hetween atoms of the same type. Duringthe mixing, therefore, heat is liberated. The entropyeffect is here supported hy the energy effect and thehomogeneous solution, just as in the previous case,is the thermodynamically stahle condition. Contraryto the previous case, however, there here occurs (ata suitahle atomic ratio) a tendency towards ordering

. in the sense that each A atom is preferentially sur-rounded hy B atoms and each B atom hy A atomsAt temperatures such that the absolute value of8AB~ (8AA + 8BB) is large compared to kT, theatoms can submit to this ordering provided that aperceptible d~usion can still occur at this tempera-ture.

À typical example 'of an alloy in which orderingoccurs, is p-hrass. This phase occurs in the systemcopper-zinc. The stahility range extends from ahout40 to 50% zinc. The system .has a body-centredcuhic structure. At high temperatures the entropy-effect prevails, so that the copper and zinc atoms aredistributed at random among the lattice points. Atlower temperatures (helow approx. 450°C) theenergy effect predominates, so that ordering occurs.Roughly speaking, each Zn-atom surrounds itselfby eight Cu-atoms and each Cu-atom hy eightZn-atoms (fig. 3). This ordered structure is desig-nated P' -hrass.If 8AB is less negative than the mean of 8AA and

8BB' then LtU is positive. The attraction hetweenidentical atoms is then stronger than the attraction

~=cu or Zn

b8230:;

Fig. 3. The unit cell of brass with the composition 50% Cu+ 50% Zn. .a) At high temperature (p-brass), the atoms are distributed atrandom among ,the lattice points.b) At lower temperatures (p'-brass), the atoms are ordered, asthis is more favourable from the point of view of the internalenergy.

hetween different atoms. During isothermal mixing,the internal energy increases, i.e. heat is ahsorhed.The entropy effect promotes the mixing, hut" theenergy effect promotes separation, At low tempera-tures the latter effect prevails and the system con-sists of two phases: virtually pure A and virtually

/lF

1

° 0,2 0,4 0,8

----..x1,00,6

82306

Fig. 4. Frce cnergy of mixing LlF, according to (Ill, 20) and(Ill, 21), as a function of the atomic fraction x for variousvalues of kT/C. At low temperatures, segregation occurs asthis is more favourable from the point of view of internalenergy. For a given temperature, the compositions of thetwo co-existing phases are defined by the points of contact Pand Q of the (dashed) double tangent to the LlF-curve.

pure B.At rising temperature, however, the entropyeffect causes increasing mutual soluhility. For asystem satisfying the conditions of the previoussection the mutual solubility as a function of thetemperature can he evaluated as follows.

The increase in the free energy due to homogene-ous mixing is given, according to (Ill, 16) and (Ill,19) hyLtF= NCx (I-x) + NkT)xlnx+ (I-x) In (l-x)~,

(111,20)where

~ 8AA + 8BB~ •••C=Z?8AB. 2 ). (111,21)

Fig. 4 shows LtF as a function of x for various _values of the ratio kT/C. The curve for T.= 0(kT/C = 0) is ohviously identical to the LtU-curve 5).

5) It will be clear that the expressions (Ill, 20), (Ill, 21) areonly justified for small values of CfkT. In fact, if Cdiffersfrom zero, someordering will immediately occur, causing theentropy of mixing to become smaller than that relatingto a random distribution; the energy of mixing will alsochange.

MAY 1955 ENTROPY, III 327

from which it follows:x T1001-------;H----I------1-\---+--~._l

--- . e-C(1-2x)/kT. (Ill 23)I-x .,..,

Throughout a wide temperature range the LlF-curves show two minima, which approach each otheras the temperature rises. The LlF-curves are symme-trical with respect to the vertical axis x = 0.5, whichis evidently due to the symmetrical nature of xand (1- x) in (Ill, 20). The double tangent PQto a LlF-curve is thus parallel to the abscissa. Thefree energy has a minimum in the state of equili-brium, i.e. the homogeneous mixtures (solutions)

. represented by the branches of the curve to theleft of P and to the right of Q, are stablè. Theportion ofthe z1F-curves situated between Pand Q,on the other hand, repreaent non-stable solutions.By a separation into two phases with compositionsPand Q, the free energy of a homogeneous alloy canbe lowered to a point on the straight line PQ. Thepoints Pand Q, therefore, indicate the solubilityvalues sought. The position of 'these points can befound by putting the differential coefficient of (Ill,20) equal to zero:

d(LlF) .~ = Ne (I-2x) + NkT ~lnx-In (l-x)~ = 0,

(I1I,22)

For the construction of the curve representing the.sol~bility as a function of the temperature it is moreconvenient to put the relationship between x and Tin the form

e (I-2x)T= ---;-----,

k In~(I- x)/x~ .. (I1I,24)

82JOI

ex.

1,0o__,..x

Fig. 5. Solubility curve (segregation curve) according to (Ill,23) or (Ill, 24), giving the relationship between kT/C and thecomposition (P and Q in fig. 4) of the co-existing phases ai' a2'

Fig. 5 gives the solubility curve according to (Ill;23) or (Ill, 24). The diagram shows that above acertain temperature complete mis~ibility. occurs.

1800°C:r--~...---..----r- r--_---' N

;;N0)

L

9001---~--+-----4-----~--~-+----~

80%~L--~20~--4~0~--6~O~-~~--~WO%Pt Au

-__,..xFig. 6. Phase diagram of the system platinum-gold, which,'even in' the solid state, shows a complete curve of the typeof fig. 5. a, al and a2 indicate solid phases; the liquid phaseis denoted by L. •

This "critical" temperature can be evaluated.' asfollows. A LlF-curve with two minima will also havetwo points of inflection, at which we have

d2(LlF) . (1 1) ,=~2Ne+NkT -+-- =0,dx2 x I-x -

whence ~e= kT(!.. +-'_1_). . . (111,25)_ x I-x

At the temperature rises, the points of inflectionapproach each other to coincide ultimately. atx = 1/2, when the critical temperature· Tk is reached.From (Ill, 25) it follows:

eTk =-. ..... (Ill, 26)

2k

.328 PHILIPS TECHNICAL REVIEW VoL. 16, No. 11

. Actmil solubility curves never show the symmetri-cal form of fig. 5. This is mainly dué to the fact thatthe assumption made above, viz. that the energymay be taken' as the sum of interactions betweenpairs only, involves a certain approximation 4). Acomplete solubility curve in the sense describedabove occurs, e.g., in the system platmum-gold(fig. 6). In most cases of binary systems with apositive LIU~value, however, melting phenomenaoccur long before the critical mixing temperatureis reached. An example of this is the system silver-copper (fig. 7).

7700 .'

L /IV I

~

\~, V L + (/2 Q'!

/'a,\ "../

:j .\I \

a,+az

I

1000

900

800

700

600

500

4000Ag

Fig. 7. Phase diagram of the system silver-copper. In thiscase melting phenomena occur long before the critical mixingtemperature is reached. . .

20 40 60_--.!.--I ......X

80 .

The decreasing solubility at decreasing temperatureoccurring in systems of the type discussed here, is'often utilized industrially, where it is known asprecipitation hardening. This is done by rapidlycooling an alloy from the homogeneous region (e.g.from point A in figs 6 and 7) down to a tempera-ture at which there are two phases in the state ofequilibrium and at which virtually no , diffusionoccurs. Owing to the rapid cooling the thermodyna-mically required separation cannot occur and asuper-saturated homogeneous solution is obtained.By subsequently heating the alloy at a suitable, not

excessively high temperature, incipie;nt precipitationoccurs, which is accompanied by a substantialincrease in the hardness.

, Rubber elasticity

Rubber ànd rubber-like materials belong to theclass .of substances known as high polymers. Theirmolecules take the form of long chains, the links 'ofwhich consists of groups of relatively few atoms(monomers). There may be several hundreds ofidentical monomers in the molecule. These mono-meric groups are bound together by' real chemicalbonds. The atoms in these molecules neverthelesshave a certain mobility with respect to each otherin the sense that one bond can rotate around theother at a constant angle. In fig. 8 this is shownschematically for a simple case, the black spotsrepreaenting atoms' of the chain. Owing, to this

........

t-,i '\.' ,:\ \., .. \ ,: \ \., ," ., ,

\\ \,. \

\ \

82JOIJ

Fig. 8. Macro-molecules often possess a great flexibility 'owingto the fact that the chemical bonds, though of constant valenceangle, can occupy different positions with respect to eachother.

100%Cu

rotational freedom, the chain molecules can assumean enormous number of different forms. Against theone chance of the molecule occurring in the form of 'a stretched chain there exist countless other possibi-lities of each molecule occurring in a coiled or tangledform (seefig; 9). .

The entropy of a high-polymer' substance will begreater when the chain molecules have an irregular-ly coiled form than when they are stretched out and

82309

a

Fig. 9. Part of a molecular chain: à) coiled at random, b)stretched. For simplicity case (a) is depicted ,in a flat plane(the same applies to fig.l0).

MAY 1955 ENTROPY, III ,329

lie in parallel arrangement: this follows from therelationship S = k In m, discussed in I (m = numberof micro-states). In some polymers, e.g. cellulose,the latter condition is so much more favourableenergetically, than the former, that the tendencytowards a maximum value of the entropy is over-come by the tendency towards a minimum value of,the energy,so that the ordered (stretched)' conditionprevails. In rubber the difference in energy betweenthe two conditions is only very slight, so that theentropy effect prevails and the molecules have acoiled form. By exerting a tensile stress in a rubberrod, the chains can be stretched, during whichconsiderable elongation of the whole rod occurs.After the stress is removed, the thermal motion ofthe links of the chain restore the chains into thecoiled and thus shortened form. The fact that alsothe rod as a whole returns to its initial form isexplained by the presence of cross-links between thepolymer chains. These links are established duringthe vulcanizing process of the rubber, which givesrise to a wide-meshed network (fig. la). In normaleases about 1% of the monomeric groups are linkedby cross-links in this process. .

112310 •

Fig. 10. Vulcanizationof rubber has the effectthat the longrubber molecules(-) are bound together by short cross-links(===). Thesecross-linksmayconsiste.g.of two sulphuratoms: - S - S -. '

Some of the remarkable thermal properties ofrubber can be explained by the foregoing, e.g. thenegative coefficient' of expansion of stronglystretched rubber. The contraction at rising tempera-ture and constant load is caused by the increasingthermal agitation of the links of the chain molecules,i.e. the increased tendency of the molecules toassume their shortened, coiled form. The parallelalignment of the molecules of natural rubber underconsiderable stress is a kind ?f crystallization pheno-mena. The crystalline nature of the rubber in thiscondition can be established by X-ray diffraction.After removal of the load the crystalline parts"melt" again.

Foor a thermodynamical treatment of the subjectwe start from the first law

dU=dQ+dW, .. (IlI,27)

where dQ represents the heat supplied to the systemand dW the work exerted upon it. Suppose that thesystem is a rubber rod of unit diameter and lengthI" and it is stretched to a length I+ dl., If we dis_'regard the very small amount of energy spent tobring about the small volume change, dWis given by: '

dW= a dl,

where 0' is the applied stress. If the stretching takesplace reversibly, dQ = TdS (if S is the entropy ofthe rubber rod); 0' is the equilibrium value of thestress in that case. Equation (Ill, 27) can now bewritten as:

dU = TdS + 0' dl .... (IlI,28)

At constant temperature, therefore, we' have

d(U-TS)='dF , a dl,and hence

aF au asa = al,=al - Tal' " (IIl,29)

For many types of rubber it has been found ex-perimentally that in the case of a large, constantelongation LJl, a is approximately proportional to theabsolute temperature T. This confirms the state-merrtonade above that aU/al (the increa~e of theinternal energy with the elongation and therefore.with the uncoiling of the polymer chain) is verysmall for this class of substances. This means thethe elastic properties of rubherlike substances area result merely of the decreasing entropy, duringstretching and not on the accompanying smallchange in the energy. For the majority of sölids, e.g.metals, the reverse is true. These materials becomecool slightly when adiabatically and elastically.stretched, whereas rubber heats up slightly underthe same conditions. This rise in temperature canhe partly explained by the fact that the totalentropy does not change with a reversible adiabaticprocess, because dS = dQrev/1.'.The decrease in theentropy, corresponding to the increasing order

, during strain, is therefore compensated by an in-crease in the vibrational entropy of the molecules;this is accompanied by an increase in temperature.

The behaviourof a "perfect" rubber, for which oU/ol = 0and whoseelasticitymay be regardedas a pure entropyeffect,is completelyanalogousto that of a perfect gas.For a rever-siblevolumechangeof a perfectgas,

dU= TdS-pdV,

330 PHILlPS TECHNICAL REVIEW VOL. 16. No. 11

so that at. constant temperature

d(U-TS) = dF= -pdV,oF oU es

P =-èJV = -oV+ToV'

At constant volume, the pressure of a gas is found to beapproximately proportional to the absolute temperature, i.e.the internal energy U of gases is nearly independent of the.volume. If the gas is "perfect" 0Ulo V = O. The pressureof such a gus can thus also be described as an entropy effect.

Solutions of polymers

We have, seen that the entropy of mixing of ahomogeneous solid alloy, so far as it depends ,uponthe configuration of the atoms concerned, is givenby formula (Ill, 16). Strictly speaking this formulais only applicable when the energy of mixing zl U' O.If zl U differs from zero, the various configurationsno longer possess the same probability. In this casethe' formula provides a sufficient approximationonly for t.emperaturea at which kT is considerablygreater than eAB-t(eAA + eBB)'

No reference was made above to any differencein size between the atoms or molecules of themixture. This was not necessary for the solid alloysunder considération, in' view of the fact that twometals with greatly different atomic radii will notshow any appreciable miscibility in the solid state.

Liquid homogeneous mixtures, however, oftenconsist of molecules (or atoms) of widely differingsizes. Solutions of polymers are extreme cases ofsuch mixtures. In the following we shall derive. anapproximate expression for the entropy of mixingof solutions of this kind.

Let us first consider a liquid mixture of moleculesA and B, which have the same shape and size andfurthermore show no preference for similar or dis-similar neighbouring molecules (Lt U = 0). In sucha mixture, a molecule B may be suhstituted for amolecule A without any effect on the immediate. surroundings. As is known from research on liquids,the immediate surroundings differ little from thoéein the crystalline state. The arrangement is merelyslightly less ordered, so that the order does notextend over such a long range as in the crystallineform. Since we are mainly interested in the bondsbetween neighbours, it is permissible to use aquasi-crystalline model (fig. 11) to represent liq-uids of the type under consideration. Formula(Ill, 16) is then applicable witho~t modifloation.This formula assumes the interchangeability of themolecules A and B. If however the molecules differ·.oonsiderably in size or shape, then' the entropyof mixing, even when the energy of mixing LtU iszero, is no longer given by (Ill, 16). For example,

a direct Interchangeability of molecules of A and Bis out of the question if the molecules A are sphericaland occupy one site in the diagram of fig. 11 andthe inolecules B are dumb-bell shaped and occupytwo sites (fig. 12).

The extreme case ofpolymers (macro-molecules)dissolved in a micro-molecular 'solvent can be ideal-ized as follows 6). It is assumed that' a polymerbehaves as if it were divided into a large number of

00.0.00.00000000000000.0.0.000000000000..0000000.00.0000.0000000000.00••000000.0.0.0.00000000.000.0.82311

Fig. 11. Schematic representation of a solution of moleculesB(black) in a liquid of molecules A (white), The molecules Aand B are of about the same size and are statistically distrib-uted in the quasi-lattice.

mobile segments, each having the same size as onemolecule of the solvent. It is further assumed thateach segment occupies one site in the quasi-latticeand that the adjacent segments of a chain occupyadjacent sites (-fig. 13). If the solution contains NImolecules of solvent and N2 polymer molecules, eachconsisting of p segments, then there are, (NI +pN2) sites in the sense of figs 11-13. The fractions

'000000000000000 ..00000.00 ..0.000.00000.0000000000000000000.00000..00.000..0000000000.000000000...000(12312

Fig. 12. Schematic representation of a statistical distributionof dumb-bell shaped molecules B (black) in a liquid of sphericalmolecules .4 (white). The dumb-hell shaped molecules occupytwo adjacent sites of the quasi-lattice.

6) Cf. e.g, P. J. Flory, Principles of polymer chemistry,Cornell University Press, New York 1953.

MAY 1955 ENTROPY, III 331

of the t~tal volume occupied by both componentsare given by .

NI pN2'

epI= NI + pN2' ep2'= NI + pN

2' (I1I,30)

An ~xpression for the entropy of mixing, in whichepI and ep2 are to play an important part, is nowderived as follows. First consider a state in which all(NI + pN2) sites are unoccupied. The first segment "of the first polymer molecule can thus be accommo-dated in (NI + pN2) different ways. Once a certain

. site hasb een decided upon, the second segment ofthe same polymer molecule can be accommodatedin the quasi-lattice in z different ways, z being thenumber of adjacent sites of any given site in the'lattice (the so-called co-ordination number).

Each of the subsequent segments (the 3rd, 4th,••• ,pth of the same polymer-molecule has (z -1)sites are available, because one of the z adjacentsites is already occupied by the first' segment.The first polymer molecule can, therefore, be ac-commodated in . ,

(NI +pN2)z(z-I)P-2 ... (I1I,31)

different ways 7) ..

82313

Fig, 13. Schematic representation of an arbitrarily coiledpolymermolecule (black) in a solvent of the same typc as thatin figs. Ll and 12.

Let us now assume that the quasi-lattice alreadyaccommodates (k-l) polymer molecules. In howmany different ways can the kth polymer moleculenow be accommodated? The first segment of thismolecule can be placed in anyone of

NI +pN2- (k-l)p = NI +P (N2-k + 1)

different sites. The second segment of the kth poly-mer molecule, unlike. that of the first, cannot beaccommodated in z different ways, since there is apossibility that one of the adjacent sites is occupied

7) Here certain factors have been overlooked. We shallreturn to these points at the end of this article,

by a segment of one of the (k - 1) .polymer mole-cules already accommodated. This possibility can beaccounted for with a reasonable accuracy by multi-plying z by the ratio of the number of sites stillunoccupied to the total number of sites '

NI + p(N2-k+ 1)NI +pN2

A similar reduction factor can be applied to eachfactor (z - 1) for the number of ways in which the.3rd, 4t

\ .... pth segment can be accommodated.The kth polymer molecule can, therefore, be placedin the lattice in

different ways. For k = 1 this expression reduces tothe form (Ill, 31). The total number of ways inwhich N2 identical polymer molecules 'can be accom-modated in the lattice is given by the productofthe N2 terms (Ill, 32) which are obtained by sub-'sequently substituting for k the values 1, 2.... , N2Since in our case the polymer molecules are actuallyidentical, so that no new situation is created whentwo of them are interchanged, the resultingnumberhas to be divided by N2!. The total number ofpossible different arrangements of N2 'Identicalpolymer molecules among the (NI +pN2) sites ofour model is therefore

~z(z_I)P-2~N. .m= N' X

2 •

(NI +pN2)P[(Nl +p(N2-1)]P ... (NI+p )P

(Nl' +pN2) (p-I)N. • (Ill, 33)

In each of these arrangements a number of sites(NI) remain unoccupied. Since in each case thesesites can be occupied by molecules of the solvent in 'one way only, the addition of these molecules doesnot add to the number m. In other words, theexpression (Ill, 33) also gives the number of possibleconfigurations m12 of the mixture of NI solventmolecules and N2 polymer molecules. After re-arr~nging, we obtain the expression:

~z(z-1)p-2rpN. {(~+N2 )!rm12=N

2'( :1 t N

2YP_I)N. ~( :1) !f (I1I,34)

The entropy of ~ixing is given by

LIS= klnm12-kln~ -klnm2, (Ill, 35~

where mI is the number .of possible arrangementsof NI solvent molecules in the solvent itself (Nt

. sites), and m2 for the number of possible arrange-

332 PHILIPS TECHNICAL REVIEW VOL. 16, No. 11

ments of the N2 polymer molecules in the polymeritself (pN2 sites).It is clear that m1 = 1 and hence k In m1 = O.For m2 this does not apply because of the manypossible coiled eonfigurations of the polymerchains. If it is assumed that the polymer molecu-les in the undiluted state behave in an analogousway as in the solution, then m2 is found by puttingN1= 0 in (Ill, 34):

~z (z-1)P-2 tNI pN'(N2!)Pm2 = N: I N, (p l)N, (111,36)

2' 2

From the last three formulae it follows that

Employing Stirling's formula in the approxima-tion (I, 3), we find:

kN2ln pN2

NI +pN2

(IIl,37)

If in the previous evaluation we had erroneouslyconsidered the polymer molecules as being equiv-alent to the micromolecules, then the earlier for-mula (Ill, 16) would agafu have been obtained,which in the same notation as (Ill, 37) has theform:

N2kN2ln N N' (.111,38)1+ 2

This is obtained from (Ill, 37) by putting p = 1.If, on the other hand, the pN2 segments 'of

the polymer molecules had, heen considered asseparate molecules, then we would have. obtainedthe relationship

N1 ' pN2LIS== -kN1In------kpN2In .N1+pN2 N1+pN2 '

(IlI,39)

The conclusions from (Ill, 37) will be intermediatebetween those of (Ill, 38) and (Ill, 39).'Using equation (Ill, 30), we mayalso write, (Ill,37) as:

LIS=- kN1ln CP1 - kN2ln CP2' • (IlI,40)

where CP2 = 1 - CPl'This formula has a similar form to that of the

earlier formula (Ill, 16) for the entropy of mixing;instead of the mole fractions Xl and X2 = 1- Xl'

however, here the volume fractions CP1 .and rp2 =1 - CP1 o~cur.

From the derivation offormula (Ill, 37) it appears'that any symmetry factor may be disregarded in(Ill, 31), since this factor would occur both in theexpression for' m12 and in that for m2, so that itwould be cancelled out in the expression for LJS (cf.formula (Ill, 35)). For the same reason we mayoverlook the fact that for the 3rd, 4th ••• , pth seg-ments of a polymer molecule less than (z -1) sitesare available, because a polymer molecule is not asa rule perfectly flexible and, moreover, one or moreof the (z - 1) sites may be occupied by previoussegments.

Several thermodynamical properties of solutionsof polymers, such as osmotic pressure and miscibilitycan be understood by means of formulae( IlI,37),which also gives us a picture of the differencebetween this type of solution and that of commonmicro-molecular solutions.