Defect-diffusion models of relaxation

Journal of Molecular Liquids, 36 (1987) 37-46

37

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

DEFECT-DIFFUSION MODELS OF RELAXATION

John T. Bendler Polymer Physics and Engineering Branch General Electric Corporate Research and Development Schenectady, New York 12301 and Michael F. Shlesinger Physics Division Office of Naval Research 800 North Quincy Street Arlington, Virginia 22217 (Received 26 November ]986) Abstract

A model of molecular relaxation in glassy materials is presented to describe configurational fluctuations which decay via the mediation of mobile defects. A mild distribution of free-energy barriers impeding the defect motion can generate a wide distribution of waiting times between defect displacements. When the mean waiting time is longer than the time of an experiment, no characteristic time scale exists. This case directly yields the Kohlrausch- Williams-Watts (KWW) stretched exponential relaxation law. Correlations between defects described by the Ornsteln-Zernicke theory of critical fluctuations cause aggregation of defects and produce a Vogel-Fulcher (VF) type law for the disappearance of singlet defects, and therefore a VF law for the dipole relaxation time. The model is also applied to predict the molecular weight dependence of mechanical relaxation times (physical aging) in glassy polymeric systems.

I.

INTRODUCTION

Robert Cole I pioneered the field of dielectrics as an technique to probe fundamental properties of matter. In 1960, Sivert Glarum, a Cole student, introduced a model where a frozen-in dipole was relaxed upon its encounter with a mobile defect. 2 dipole in one-dimension,

While Glarum considered a single defect and a single in 1975 Bordewijk 3 generalized this to include a fin-

ite concentration of mobile defects in three dimensions. In both cases the mobile defects were undergoing Brownlan motion. Bordewijk's basic result was that, in three dimensions, the dlpole-dlpole correlation function decays exponentially which is equivalent to a seml-clrcular Cole-Cole plot. This result seemed to rule-out defect diffusion mechanisms in viscous disordered materials inasmuch as asymmetric Cole-Cole plots and stretched exponential

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© 1987 Elsevier Science Publishers B.V.

38

Kohlrausch-Williams-Watts 4 (KWW) dipole-dipole

correlation

functions

are hall-

marks of the glassy state. 5 In this Cole Festschrift we present a model of dielectric

relaxation which leads directly to the KWW law. The relaxation

is

still assumed to be induced by mobile defects, but the defect motion is characterized

by a wide distribution

of waiting times rather than by Brownian

motion. We then show how an attraction between defects,

leading to a condensa-

tion of the defect "gas" at T O below Tg, can produce the non-Arrhenius Tamman-Fulcher T .

g

(VTF) temperature behavior of the dipole relaxation

In the last section,

the defect model is applied to mechanical

Vogel-

time above relaxa-

tion.

II. Dielectric Relaxation

A. Mobile Defect Formulation

Consider a polarizable electric

field.

orientation

material with individual

dipole moments ~(t) in an

When the field is turned off at t-0 the decay of the dipole

correlation

function

4(t)

-

< p(t)

~(o)

>

(i)

< ~2(0) >

is related to the frequency-dependent

dielectric

constant

~(~) of the material

by

~(~) - ~

~ --

~0 " ~

-

t ~ t d~(t) dt e

(2)

dt

0

Following Glarum, 2 we assume that a frozen-ln dipole relaxes upon its encounter with a mobile defect.

Let us start with a system at t-0

with one frozen-in dipole and N mobile defects distributed sites.

instantaneously

among V lattice

For simplicity we treat these V sites as forming a perfect 3D lattice

and place the dipole at the origin of this lattice. represents

the probability

Then 4(t) in eqn (I)

that none of the N mobile defects has reached the

dipole by time t. It is given by

39

N

4(t)-

the term in brackets

[i - V ' I ~

~ F(~0,~)dr

%0

is one minus the probability

(3)

]

that a given defect did

reach the origin by time t, i.e. this is the probability

t~at the defect in

question did not reach the origin by time t. If the defect started at ~n, F(~0,T ) is the probability at t ~ ~. We integrate positions

of the defect which has a probability

particular

I/V of initially being at a

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

the probability the origin. defects,

density that it reached ~ - 0 for the first time

over all times less than t and average over all initial

that none of the N defects

In the limit N,V

~

(non-interacting)

has yet reached

~ with N/V - c, a constant concentration

of

eqn (3) becomes

expl-c} F(% ,)d,

(4)

i0 0

which c a n b e

shown

to b e equal to

4(t)

-

(5)

exp[-cS(t)]

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

B. DISTRIBUTION

in a time t.

OF JUMP TIMES

We have not yet specified what a defect is or how it moves.

The crux of

the physics lles in these issues. Let us first assume that a defect jumps randomly on the lattice and concentrate probability

on the defect motion.

Let #(t) be the

density that a defect remains at a lattice site for a time t. If

the defect must Jump over an activation barrier of height A O , then

#(t) - A 0 exp(-A0t )

(6a)

40

with

A0

-

-A0 u ° exp[ -~-

(6b)

]

For a distribution of barrier heights f(A) (but not prefactors) eqn (6a) is replaced by

@(t) - ~ Ae "Atp(A)dA 0

(7a)

where

p ( ~ ) - f(a)l~ I

(7b)

- u exp(-A/kT)

(Sa)

The choice

and

(8b)

f(A) - (kTo)-lexp[ -A/kT o ]

yields

(8c) o

which when substituted into eqn (7a) gives

~(t)

- ? r(?+l)

4

t-l-?

as

t

-~ ®

(9)

41

with $ - T/T ° . The quantity S(t) can be calculated from #(t) to give 5'6

÷(t)

-

e

-const c t$

(io)

where the constant will depend upon ~. This expression is valid for ~ ~ i . If > 1 exponential decay will result.

The complete time dependence of 4(t),

which is somewhat more complicated than eqn (I0), has been given by the authors.6 Essentially,

the decay is initially exponential (due to defects

leaving shallow traps ) before it crosses over to stretched exponential behavior. The range of validity of eqn (i0) is shown to cover the range of experimental data. Note that to derive eqn (i0) the mean waiting time < t > m ~ t#(t)dt - ~, i.e., there is no characteristic time scale in the hop0 ping process. The mild distribution of barrier heights induces a wide distribution of hopping times. If < t > were finite, exponential decay, e.g., exp (-const t/), would result. The form

4(t) - e

is called the Kohlrausch-Williams-Watts

, ~ < I

(ii)

(KWW) or stretched exponential law.

It was first introduced by R. Kohlrausch 7 in 1854 for the analysis of charge decay in Leyden Jars. His son later employed the same model to describe creep in glass and rubber fibers. 8 Several other examples of creep and stress relaxation were reported, 9-II but this law only became widely known after 1970 when Williams and Watts 4 discovered its ubiquitous appearance in the dielectric relaxation in polymeric melts near the glass transition. The stretched 12 13 exponential law has since been found in NMR, remanent magnetization, luminescence s t u d i e s , 1 4 o p tlcal scatterlng ,15 and mechanical relaxatlon. 16

Several theories besides defect diffusion, such as hierarchically constrained dynamics and direct transfer, have been proposed to derive the stretched exponential law. The common mathematical basis of these theories is given in ref [17].

42 III. DEFECT CORRELATIONS

AND THE VOGEL-TAMMANN-FULCHER

The model leading to the KWW law can be generalized First 6 one may use free energies, (8a). This preserves dence of 8. Second, temperature.

voids

the linear temperature

one can allow the concentration in glasses

is lowered,

ness the case when the defect-defect

in eqn depen-

of defects to vary with

if the defects are vacancies

as the temperature

(lowering the system energy and entropy).

condensation

in several ways.

instead of just activation energies

the KWW form but modifys

For example,

attract each other,

(VTF) LAW

which

they can coalesce

into

We consider here for definite-

attraction

is strong enough to lead to a

of the defect gas at some finite temperature

To, which is, how-

ever,

lower than T . If the phase transition is second order or weakly first g one may use the familiar formalism of the 3D lattice gas or Ising model 18 to describe the defect-defect fluctuations in the high-temperature region.

order,

The net pair correlation Ornstein-Zernicke

function between defects above T

i

where c is the total defect density, function,

between defects.

then has the

e -~r

G(r) - 4~cR 2

correlation

g

form 19

R 2 is the second moment of the direct

and ~ is the reciprocal

As the temperature

falls,

of the correlation

the correlation

increases

in a manner determined by the temperature

(osmotic)

compressibility,

-i.

(i2)

r

length

length ~-I

dependence

of the

It may be shown that ~ and ~ are related by

A

(13)

At a second order phase transition ~ vanlshes 19

(14)

- ~°(T-To)7

where a Landau or mean-field

treatment of the phase transition gives 7 - i.

Combining eqns (12)- (14) gives the temperature dependence correlation length [-I(T);

of the defect

43

f ' l ( T ) - f~l(T.To)-~T

(is)

Thus in the mean-fleld limit (i.e., ~ - I), eqn (15) shows that the correlation length between defects diverges as (T-To)'h. lowered,

As the temperature is

the average size of a defect cluster grows and the probability of

finding an isolated single defect vanishes rapidly.

If we assume that only

isolated defects are mobile enough to bring about dipole relaxation, and that therefore it is the concentration of isolated defects which appears as c in eqn (i0), then we can understand the non-Arrhenius VTF behavior as due to the aggregation of defects.

Adopting a Poisson form for the defect-cluster proba-

bility density, and using the correlation length ~-i to define the average cluster radius, we find 18 that the concentration of defect singlets has the temperature behavior

cI(T)

- c exp[

-B (T-To)3/2]

(16)

where c is the total defect density ( including singlets, doublets, triplets, etc.) in the system.

When the singlet density of eqn (16) is substituted into

eqn (i0), the dipole relaxation time is found to have the (dominant) temperature dependence

~dlpole - ~ exp[

(17) (T_To)3/2]

Eqn 17 is similar to the Vogel-Tamman-Fulcher

TVT F - A e x p [ ~ ]

form

B

the difference being that (T-To) appears to the 3/2 power in eqn (17). In Table I we compare dielectric relaxation times in glycerol determined by Davidson and Cole 20 with fits using eqns (17) and (18).

(18)

44 TABLE I

Dielectric

Temperature

°C

Relaxation Times for Glycerol using eqns (17) and (18).

in ~ (exp) 20

In ~ (eqn (17)) a

In • (eqn (18)) b

-ii.373

-Ii.382

-11.399

-40.0 -50.0

-8.987

-8.989

-9.010

-60.0

-5.991

-5.986

-6.033

-65.5

-3.985

-3.991

-4.059

-70.0

-2.137

-2.131

-2.218

-74.6

-0.041

-0.026

-0.079

Using A--12.43,

.4343 ~ - 9867, T -I12.7°K o

From ref 20, A--14.41,

As is seen from Table I, the temperature in glycerol

.4343 B - 957, T -132.°K o

dependence

of dielectric

relaxation

is actually better described by eqn (17) than by the VTF form, eqn

(18).(We have compared eqn (17) to VTF for other glass-formers equally satisfactory

fits. 18) The principle

difference

and have found

is that the temperature

T O is about 20°K lower than that found using the VTF equation,

or about 60°K

below T . Our interpretation of T in eqn (17) is that it is a critical temg o perature for condensation of the defect gas. The system never reaches this limit however,

since the rapid disappearance

lattice to freeze at Tg, preventing display VTF are presumably tendency to aggregate. low transition

those with high concentrations

temperature

A likely candidate conformation

Glass-formers of defects,

which

with a

Non-VTF glasses may have few defects and therefore

a

T . o

IV. MECHANICAL RELAXATION

cis-trans

of slnglet defects causes the

further aggregation.

("Physical AzinK")

for a defect in polymeric ) propagating

glasses

is a kink (e,g.,

along a polymer chain. Relaxation

occur when a kink leaves a chain. For a chain with M units,

a random walk makes an average of M 2 Jumps to reach the end of a chain. kink is within a disordered environment

and its probability

can

a kink undergoing If the

density follows

eqn (9) where #(t) - t "I'~ , ~ < i, then the average number of jumps N(t)

45 which occur within a time interval t is 21

N(t) - t~

(19)

and ~ would equal unity if were finite. Thus the time to perform M 2 jumps is

2 ~chain

-

(20)

M~

If the polymer chain has ideal statistics, its end-to-end distance varies as M h. Thus each chain has a sphere of interaction of radius scaling as M h. This same kink now walking through a network of chains will leave this "interaction sphere" (i.e., lose contact with the original chain environment) after only M steps, as opposed to the M 2 steps if it had to follow the chain path itself. The average time to take these M steps is

1 Tnetwor k - M ~

(21)

Glass relaxation studies on various polymers suggest that the kinetics is governed by KWW with ~ ~ 0.15 to 0.2 16. This implies a relaxation time (for defect escape out chain ends) scaling

~ M5 ~ M7

(22)

in reasonable agreement with experiment. 22

We should emphasize that this mechanism of defect "evaporation" out chain ends is practically irreversible on the time-scale for polymer glasses.

It

could therefore he the origin of plastic flow, as well as physical aging and preliminary experiments suggest that the kinetics are strongly dependent on temperature and molecular weight. 23 These phenomena are to be contrasted with the anelastic deformations involving defect rearrangements between entanglement junctions, which involve defect kinetics which are weakly sensitive to temperature, molecular weight, aging, etc. 16

46 REFERENCES

i. Robert H. Cole 70th Birthday Symposium, IEEE Transactions on Electrical Insulation, Vol. EI-20, No.6, Dec 1985. 2. S.H. Glarum, J. Chem. Phys. 33, 1371 (1960). 3. P. Bordewijk, Chem. Phys. Lett. 32, 592 (1975). 4. G. Williams and D.C. Watts, Trans. Faraday Soc. 66, 80 (1970). 5. M.F. Shlesinger and E.W. Montroll, Prec. Nat. Acad. Scl. 81, 1280 (1984) 6. J.T. Bendler and M.F. Shlesinger, Maeromolecules 18, 591 (1985). 7. R. Kohlrausch, Pogg. Ann. Physlk 91, 198 (1854). 8. F. Kehlrauseh, Pogg. Ann. Physik 119, 352 (1863). 9. F.T. Pierce, J. Text. Inst. Vol 14, T390 (1923). I0. C.R. Kurkjian, Phys. Chem. Glasses, Vol. 4, 128 (1963). Ii. J. de Bast and P. Gilard, Phys. Chem. Glasses, Vol. 4 , 117 (1963). 12. A.A. Jones, J.F. OGara, P.T. Inglefleld, J.T. Bendler, A.F. Yee, and K.L. Ngai, Macromolecules 16,658 (1983). 13. R.V. Chamberlin, G. Mozurkewich and R. Orhach, Phys. Rev. Lett. 52, 867 (19847. 14. U. Even, K. Kademmann, J. Jortner, N. Manor and R. Reisfeld, Phys. Rev. Lett. 52, 2164 (1984). 15. G.D. Patterson, J. Chem. Phys. (1983). 16. J.T. Bendler, D.G. LeGrand, and

W.V. Olszewski,

Polym. Preprints (ACS) 26 (2),90 (1985). 17. J. Klafter and M.F. Shlesinger, Prec. Nat. Acad. Sci. (USA) 83, 848 (1986). 18. J.T. Bendler and

M.F. Shleslnger, unpublished.

19. M.E. Fisher, J. Math. Phys. 5, 944 (1964). 20. D.W. Davldson and R.H. Cole, J. Chem. Phys. 19, 1484 (1951). 21. M.F. Shlesin~er, J. Stat. Phys. I0, 421 (1974). 22. D.G. LeGrand, J. Applied Polym. Sci. 13, 2129 (1969). 23. D.G. LeGrand, W.V. Olszewski, and J.T. Bendler, Bull. Amer. Phys. Soc,,submitted for March (1987).