82 Nuclear Physics B (Proc Suppl ) 5A (1988) 82-85 North-Holland, Amsterdam

FRACTAL TIME SYMMETRY IN THE GLASS TRANSITION

John T. BENDLER

Polymer Physics and Engineering Branch, General Electric Corporate Research and Development, Schenectady, New York 12301

Michael F. SHLESINGER

Physics Division, Office of Naval Research, 800 North Quincy Street Arlington, Virginia 22217

A defect diffusion model is employed to describe relaxation in glassy materials, When the motion of the defects is governed by fractal time (no characteristic time scale existing) then the ubiquitous stretched exponential relaxation time law is derived as a probablhty limit distribution. In our theory, the stretched exponential law does have a characteristic time scale inversely proportional to the concentration of mobile defects. We derive a generalized Vogel law describing the divergence of the relaxation time scale with dimimshing temperature. We relate this divergence to a phase transition govermng the disappearance of mobile defects.

1. Three Hallmarks of the Glass Transition.

Since the 1920's much of physics has turned its attention

to quantum mechanical phenomena. However, many

classical problems remain unsolved mcluding the nature

of turbulent flow, flow through porous media, and the

freezing of a liquid into a glass. The latter is the topic of

this paper. We focus here more on mathematical aspects

of the problem, rather than on the wealth of experimen-

tal data

. (+)¢ ~(t) = e , f l < 1 (1)

Second, in many materials, r in eqn (1) follows a non-

Arrhenlus law named for various researchers including

Vogel, ~' Fulcher, ~ and Tammann. 2~ The empirical

expression for r is

( const. ]

Exceptions to this law exist, e g. several materials follow

the more familiar Arrhenius law. 3

Three major points must be addressed in any theory of

the approach to the glassy state. First, relaxation func-

tions (e.g., dielectric, mechanical, NMR, etc.) decay

according to the stretched exponential law, 1

The third observation is that T O is not equal to the glass

transmon temperature Tg. One consistently finds

T~ < Tg, so the relaxation time is not keyed to Tg, This

leads to the question' What is the physical significance of

To ?

0920 5632/88/$03 50 ~ Elsevmr Scmnce Pubhshers B V (North-Holland Physxcs Pubhshmg Division)

J T Bendler, M F. Shlesmger / Fractal time svmmeto' m the glass transmon 83

2. A Defect Diffusion Model of Relaxation. 4'5

Let us consider a glassy material of polar molecules with

dipole moments/z(t) in an electric field. The energy sup-

plied by the field can allow the system to attain a

configuration with all the dipole moments aligned along

the field hnes. Now remove the field. How and accord-

ing to what law will the dipole orientations relax to a ran-

dom configuration?

We assume that the dipoles are frozen-in and unable to

change their configuration due to constraints imposed by

the glassy structure. We also assume that the system sup-

ports a population of mobile defects. We need not

specify the precise nature of the defects at this point,

although candidates for defects include vacancies, dan-

ghng bonds, and high energy conformers on polymer

chains Our next assumption is that the constraints on a

frozen dipole are relaxed if it is reached by a mobile

defect, 1 e. the arrival of a defect induces instantaneous

relaxation of the dipole. Our model (in the dilute, non-

interacting limit) considers N mobile defects (initially

randomly distributed at t=0) randomly walking between

V possible lattice sates, one of which (the origin) holds a

frozen dipole. The relaxation function ~ t ) is the proba-

bility that none of the defects has reached the origin (and

hence the dipole) by time t. The relaxation function is

given by

t ~ IN ~ t ) = 1- v - l ~ f F ( l o , r ) d r to o

(3)

The term in brackets as one minus the probability that a

given defect did reach the origin by time t, i.e. this is the

probability that the defect an question did not reach the

origin by time t. If the defect started at ~0, F(~o,r) is the

probabdaty density that it reachedT = 0 for the first time

at t = r This is integrated over all times less than t and

averaged over all initial positions of the defect ( which

has a probability 1/V of initially being at a particular lat-

tice site). The bracket is raised to the N th power as this

is the probability that none of the N defects (non-

anteracting) has yet reached the origin. In the limit

N, V ~ ce with N/V = c, a constant concentration of

defects, eqn (3) becomes

f t ~ t ) = exp ~ f F ( f f o , r ) d r To o

(4)

which can be shown to be equal to

~ t ) = exp [- cS (t)] (5)

with S(t) the number of distinct lattice sites a defect visits

m a time t. Basically, S(t) accounts for all the distinct

sltes visited by a random walker, starting at the origin, in

a time t, while the term in brackets in eqn (4) finds all

sites from which a random walker will reach the origin

wathin a time t. The defect induced relaxation concept

was originally introduced by Glarum 6 who considered a

single defect and a single dipole in one dimension.

3. Fractal Time Motion.

To proceed in our calculation of ~ t ) we need to specify

how a mobile defect moves. Let us assume that the

84 J 7', Bend&r, M.F Shlesmger / Fractal ttrne symmet O" m the glass transmon

defects jump between lattice sites. Let ~ t ) be the proba-

bility density that a defect leaves a lattice site after being

there for a rime t. If the first moment <t > of ~ t ) is

fimte, then in three dimension 4

t S (t) - (6)

< t>

The relaxation law (from eqns (5) and (6)) follows a sim-

ple exponential decay with a time scale given by <t >.

However, if at long times

distribution in eqn (11) is quite reasonable for amorphous

materials. The ~b(t) in eqn (7) applies to each of the

mobile defects. The relaxation of the frozen dipole is

subordinated to the first passage-time of a defect reach-

ing the dipole. The algebraic behavior of ~(t) is renor-

mahzed through this subordination process into the

stretched exponential relaxation law of ~ t ) . Note that a

key strength of our model is that we only need find the

distribution of barrier heights f ( A ) leading to t/J(t). The

distribution of barrier heights which leads directly to the

stretched exponential is not sensible.

~b(t)= t q'# , /3< 1 (7)

then ~b(t) is infinite and it can be shown that 7

s (t) = t e (8)

using eqns (5) and (6). This result leads directly to the

stretched exponential law

~ t ) = exp ( - const, c t ~) (9)

The form of ~b(t) in eqn (7) can arise through thermally-

activated hopping over barriers of heights zx, with ~ being

a random variable. 8 Let the time t to go over a barrier of

height A follow the law

t = exp(~T ) (10)

and let the distribution of barrier heights f (zx) be

-A f(A) = exp(--~B ) (11)

The &stributlon of barrier heights induces a distribution

of wamng times ~(t). Employing ~ t ) d t = f ( a ) d A yields

eqn (7) with fl = T / T B for T < TB. The stochastic pro-

cess ~b(t) has been called fracta/time because this waiting

time distribution possesses no characteristic scale. The

4. A Generalized Vogel Law. 5,9

An important result of the defect diffusion model is that

the time scale r in the stretched exponential law,

~ t ) = exp(-ct ~) - e-(+)t~ (12)

IS determined by the concentration of mobile defects, i e.

1 r - 1 (13)

c p

If we lower the temperature of our system then we must

lower the entropy. In our model, only the defects are

mobile so the only pathway for lowering the entropy is

for the defects to coalesce We further assume that only

smgk, defects of concentration c l are mobile and can

contribute to the relaxation. Eqn (13) is then modified to

read

1 r - 1 (14)

Cl p

The probability of finding a single defect is

Cl = cO-c )* (15)

where z is the number of neighboring (correlated) sites.

J T Bendler, M F Shlesmger / Fractal ttme symmeto' m the glass transmon 85

Using the mean-field formalism of a three dimensional

lattice gas system we find that a diverging correlation

length f develops between defects as the temperature is

lowered, such that

= (T .To) -1/2

The correlation volume z = f3 and using eqns (14) and

(15)

r = exp[ c°~mt~ ] (16)

L(r-r0) -r ]

which daffers from the Vogel law in that the temperature

3 difference is raised to the ~ power instead of the first

power. We have found eqn (16) to be a better fit to data

than the Vogel law, especially near the glass transition

temperature.

Note that T o is the temperature at which single defects

no longer exist. The glass transition temperature Tg is

the temperature at which rigidity percolates throughout

the system. The rigidity onset will occur when single

defects still exist. A few defects will not disrupt the

freezing of the liquid into a glass. Thus, To < Tg.

Both cases exist in nature. The former are called by

Angell 3 fragile liquids and the latter strong liquids. Both

cases are consistent with the stretched exponential as

described here.

REFERENCES

1 G. Williams and D.C. Watts, Trans. Faraday So¢. 66, 80 (1970).

2. (a) H. Vogel, Z. Phys. 22, 645 (1921); (b) G. Fulcher, J. Amer. Ceram. Soc. 8, 339 (1925); (c) G. Tammann and N. Hesse, Z. Anorg. Chem. 156, 245 (1926).

3. C.A. Angell, Strong and Fragile Liquids, in Relaxa- tion in Complex Systems , ed K. Ngai and G. Wright 1985, US Government Printing Office, Wash. DC. Available from National Technical Information Service, 5285 Port Royal Road, Springfield, VA 22161.

4. M.F. Shlesinger and E.W. Montroll, Proc. Nat. Acad. Sci. 81, 1280 (1984).

5. J T. Bendler and M.F. Shlesinger, J. Molecular Liquids 36, 37 (1987).

6. S.H. Glarum, J. Chem. Phys. 33, 1371 (1960).

7. M.F. Shlesinger, J. Stat. Phys. 10, 421 (1974).

5. Conclusions.

Our defect diffusion model derives the stretched

exponential law as a probability limit distribution. This

can account for the robustness of the law. The question

of temperature dependence of the time scale r in the

stretched exponential is an additional problem. We

relate it to the number of mobile defects, a highly tem-

perature dependent quantity. Depending on the nature

of the system it may be Vogel-like if a phase transition in

the number of single defects occurs, or it may be

Arrhenius if the number of defects is thermally activated.

8. J.T. Bendler and M.F. Shlesinger, Macromolecules 18, 591 (1985).

9. J.T. Bendler and M.F. Shlesinger, J. Star. Phys. (in press) Van Kampen Festschrift.