Exhausting of Fluctuating Free Volume of Dyna~nic Glass Transition at Low Te~nperatures
E. Donth (*)
Universitàt Halle, Fachbereich Physik, 06099 Halle (Saale), Germany
(Received 29 December1995, revised J4 March 1996, accepted 13 May1996)
PACS.64.70.Pf Glass transitions PACS.61.43.Fs Glasses
Abstract. The conventional free volume, usually externally given by uand, is substituted
by an internai fluctuating free volume. The temperature dependence of this free volume can be calculated from the transition to Levy stable distributions assuming dominance of fluctuations in functional a subsystems. An exhausting temperature is obtained (well above trie Vogel
temperature) where this free volume becomes practically zero. A cellular pattern fluctuating in
space and time is suggested for the dynamic glass transition. Partial freezing-in of the cells con
qualitatively explain several fundamental properties of glasses.
1. Introduction
It is not easy to define wuat is meant by tue term "cooperativity" (1-4j of molecular motions.
Firstly I shall refer to a formulation of Fredrickson. "A particular molecule is trapped by its
neighbors in a 'cage', which may persist for long periods of time. To destroy trie cage (required
for viscous flow), very cooperative dynamical events m trie vicmity of tue molecule are required.
Tuis follows because tue neiguboring molecules tuat constitute tue cage are tuemselves caged.
Tue spatial extent over wuicu cooperative rearrangements must occur to relax a cage very
likely increases as tue fluid is densified", or as tue temperature is lowered.
A ratuer successful translation of tuis picture in molecular computer models (see also [5]) is a facilated Ising model [2] or a Jàckle model [4j, wuere tue "cage" is defined by a certain
state or by certain possibilities of tue environment of tue particular molecule. For instances,
tue movement of t(is molecule is tue more facilated tue uigher its cage compressibility is. Tue
rearranging movement of trie other molecules needs not to occur simultaneously in tue strict meaning of tue word, but at least some causal succession is needed.
Modifying Fredrickson's passage ". persist for long periods of time" it seems tuat kinetic moiecuiar randomness is an important aspect for molecular cooperativity. Tue randomness results from tue uigu thermal velocity of tue molecules. Tue latter is of order 100 m/s wuicu results from vT
" (3kT/mo)~/~ witu mû tue mass of tue partiale, e.g. of a monomenc unit for
polymers. Trie rearrangements, however, are connected with a mean velocity va = fa/Ta, where
fa is a typical Îength of cooperativity (order nanometers), and Ta trie typical rearrangement
or relaxation lime. For (~
= l nm and T~
= 10~° s we would obtain va
=
10~~ m/s. Let be
(*) e-mail: donthilphysik.uni-halle.d400.de
© Les Éditions de Physique 1996
vT » va, e.g. vT/vo > 103. Tuen tue uigu and cuaotic tuermal velocities generate during tue rearrangement lime Ta a very large number of different local temporary configurations with very mucu local chances or possibilities for rearrangements wuicu may only occasionally aiid
randomly be used for trie slow a cooperative rearrangements. Tuis means that tue rearranging
patin of eàch molecule bas local random kinks of order
one molecule diameter, say, and trie
patins of dilferent partiales are dilferent, with randoni eveiits.
An attempt to include tuis randomness in a Fredrickson model is to choose randomly both barrier energies and interaction energies anew at eacu computational step (6j.
It is a consequence of tue random events tuat tue rearranging a movements are to be consid- ered as independent from otuer movements. Local fl relaxations or uigu-frequency vibrations
or puonons, for instance, cannot accelerate, decelerate, or even quench molecular a relaxations.
Trie term "cooperatively rearranging region" (CAR) was introduced by Adam and Gibbs [ii
as a spatiaiiy independent subsystem. Its volume will be denoted by i
= (( (cooperativity
volume containing Na partiales), and fa is called characteristic length (2j. Na obtained from molecular fluctuations is usually of order 100, and fa of order 3 nm [2]. Considering trie kinetic mdependence of dilferent dispersion zones mentioned above trie terni "a functional subsystem"
was introduced in [7j: it contains only trie rearranging motions of trie a dispersion zone, 1-e- of trie dynamic glass transition.
A priori trie concept of cuaracteristic lengtu is not trie same as trie concept of correlation
lengtu, since tue latter is usually defined by tue density-density correlation at a given time.
One can imagine [4j to find situations witu a certain cooperativity region altuougu tue position of tue partiales je-g. on a lattice) is random.
Assuming, uowever, tuat tue local rearranging mobility is controlled by tue local density (or,
in other words, by local concentration of free volume), tuen we bave some coupling between local density fluctuation and cooperativity. A puenomenological model of tuis free-volume
facilitation was developed in (7,8j and will be forwarded in tuis paper.
For tue description of tue dynamic glass transition tue free volume is conventionally given by uand, extemally, as
VÎ~~ " °f(~ ~co), Il
witu of a constant temperature coefficient and Tcc tue Vogel temperature. Putting it in a
Batcuinski Doolittle equation for tue viscosity one immediately obtains tue WLF equation (9,loi.
Tue aim of this paper is trie substitution of trie external free volume by an internal one tuat can self-consistently be calculated from tue facilities of a cooperativity. It is assumed tuat molecular fluctuations dominate tue description of tue glass transition. A fluctuating free
volume can puenomenologically be derived from tuis dommance. We suall,see tuat simple as- sumptions about fluctuating cooperativity lead to Levy stable limit distributions for stocuastic
processes representing tue a spectrum for volume fluctuations. Tue tueory of Levy stable dis- tributions does not only define tue hmit distribution, but also tue way to tue limit (Eq. (27), below). Tuis gives an equation for tue temperature dependence of fluctuating free volume tuat
shows its exuausting at an exuausting temperature TE far above Tcc. Exuausting practically stops tue original, uigu-temperature WLF equation.
Changes in tue WLF (or VFT) scaling at low temperature are well known for long times
[11,12]. Very precise dielectric investigations seem to show (13j tuat tuere is generally a temperature (called Fischer Stickel temperature uere, TFS) wuere a uigu-temperature WLF
smootuly changes to anotuer regime (probably a low-temperature WLF with smaller Tcc, or
an Arruenius regime).
Tue paper is organized as follows. Tue size of CRR'S is crucial for oui approacu. In Sec-
tion 2, tuerefore, an experimental metuod for tue estimation of fa is recalled. Tue concept of fluctuation dominance is described in Section 3. Tue results of our approacu (existence
of fluctuating free volume, WLF equation, temperature dependence of cuaracteristic lengtu, Koulrauscu function for isotuermal bulk compliance, and temperature dependence of tue fluc- tuating free volume) are described in Section 4. Tue exuausting puenomenon will be compared
witu tue experimental Fischer Stickel crossover for a in Section 5 (TE " TFS). Tue last section suortly compares our puenomenological fluctuation approacu to tue microscopic mode coupling tueory for tue glass transition. Two Appendices report on a fluctuating pattem tuat suould uelp to visuahze tue cuaracteristic lengtu and its consequences for tue frozen-in glassy state.
2. Estimation of trie Characteristic Length of Glass Transition, (~
Tue fluctuation dissipation tueorem FDT implies a mean temperature fluctuation of tue func- tional a subsystem, ôT (7,8,14j. Knowing tuis ôT, tue size of tue latter can be estimated from tue general fluctuation formula
ôT/T m 1llÎ ii)
Taking two tuermodynamic dimensions, tue volume of a CRR can be calculated (8, lsj by specifying of equation (2),
Va =
kT~/h(1/cv)/pôT~. (3)
Tuis is an exact equation provided tuat tue o dispersion is independent from otuer relaxations
(cf. tue introduction) and tuat fluctuations dominate (see tue following Sect. 3). Suortly, we
start from tue Landau Lifsuits fluctuation formula, /hT2 = kT~ /Cv, wuere Cv is tue extensive
ueat capacity, unit J/K. Dominance of fluctuations implies tuat V~ can solely determined from such a fluctuation formula. Using trie a independence 1/Cv must be substituted by trie a relaxation strength, 1-e- by trie step of a "temperature modulus", i-e- /h(1/Cv). Introducing tue specijic ueat capacity, cv, unit J/K g, expanding witu ôÎ
= number of CRR'S in trie sample
and m~ = mass of a CAR, and using cv " Cv/ôlma and p
= ma /V~, we obtain equation (3).
Tue ôT values can be estimated from tue Cj'(uJ) dispersion là lnuJ m 1/flKww [2j) of tue ueat spectroscopy peaks at tue glass transition, using tue local time-temperature equivalence
à in uJ/ôT = d inuJ/dT (aiong WLF (4)
Tuis equivalence corresponds to tue principle of local eqmlibrium [16j wuen ôlnuJ is mcluded in tue stock of extensive tuermodynamic variables, see below, equation (10). ôT can also be
estimated from DSC curves (2,17,18j.
3. Dominance of Fluctuation for trie Functional o Subsystem
Dur fluctuation approacu is based on tue principle of dominance of fluctuations wuicu means
that trie functional a subsystem is not too large and not too small. Not too large means that molecular fluctuations +~ 1llÇ, counted (14j by one single Boltzmann constant k land not by Nak) and not becoming critical, can play an essential rote i.e. tuat tuere is some uope tuat trie
a properties can solely be calculated from general, puenomenological properties of moiecuiar fluctuations witu no recall to response or memory wuicu latter are, mstead, tue atm of tue calculation. Counting by one k is a consequence of kinetic molecular randomness producing
a large number of energetically (order kTg) equivalent configurations (energy landscape (19j).
Not too small means so tuat usual tuermodynamic variables become some sense and tuat
general laws sucu as tue FDT and tue principle of local equilibrium [7,16] can be applied. Trie characteristic length fa estimated above (nanometers) seems to be in trie right scale to apply
this principle.
4. Phenomenological Fluctuation Approach to trie Dynamic Glass Transition 4.1. MINIMAL COUPLING. Let us now try to describe trie molecular cooperativity in an
average CRR (Î e Va
= fi by means of phenomenological terms. Construct a partition of
partial volumes mside a CRR, 1-e- divide trie number of partiales in a CRR (N~) m n equal parts, 1, each in
a partial volume l~, =1,2..., n le-g- n = 8 or 27), and assume that trie parts are also large enough to define a reasonable density in it,
p~ = ilf. (5)
In other words, f
= NaIn is trie number of partiales defining l~. Thus,
Î= ~Î l~, Na= ~Î N~=nN~, 1=1,2,.,n. (6)
Let us assume that, for given N~, each partial volume fit) is a stationary stochastic function,
~i(t),
K(~) " ~~(~), (7)
with an average value f
= TIR, and that a partial frequency uJ~ (that is assumed to depend only from its own partial volume f) can be associated to each partial volume,
bJ~ = MilK). 18)
Interprete trie partial frequencies as some probabilities uJ~ = AP~: dilferent events pet second, [Ai = Ils. Trie stochastic functions uJ~
= j(t) are assumed to be statistically independent (minimal coupiing [7,8j between trie parts), Fa
= Pi P~.. Pn. Rescaling by uJ~/"/À
~ uJ~
gives
QJa # WI 'QJ2 'QJn. (9)
Tue term minimal couphng is supposed to recall a property of gauge field tueory wuere an interaction is also reduced to
a space property. In otuer words, minimal coupling means tuat tue local mobility (uJ~) is monitored (facilitated) by tue local density p~ (or l~), and tuat tue cooperativity can puenomenologically be described solely by tue geometric connection equation (6) between tue l~ inside a CRR: enlarging tue density uere means its diminution tuere, and so on.
Remark. To escape from tue problem of too small CRR'S for partitioning we cari tuink about usmg additionally a certain number of neigubored partial volumes to define p~ for tue given
by equation (5) wituout violating equations (6, 8, 9). A quasi continuous description reacuing
down to small N~ values will be presented in reference [14j.
4.2. WLF EQUATION. From tuis approacu we can derive tue WLF equation [20,21j. We see from equation (6) witu given constant f, and from tue statistical independence equation (9),
tuat lnuJ is an "extensive" variable witu a dispersion increasing witu tue size of a CRR,
à lu Ma
+~
n~/~
+~
N]/~ (io)
On tue otuer uand, tue temperature fluctuation of tue functional
a subsystem is intensive ôT
+~
Nj~/~, (ii)
wuicu means ôT ôlnuJ
r~J
Ni~/~~~/~
r~J
N]. Taking tue Na dependence for governing tue
Inw(T) relation of tue dispersion curves we can ask: wuat are tue lnuJ(T) curves tuat are mvariant against a "parallel" shift of ôlnuJ ôT
= const. between neighboring curves? Trie
answer follows from dilferential geometry: excluding straight fines (with limite In uJ at T
= o)
trie only curve sets with ôT ô In w
= const. between curves are hyperbolas, 1-e- WLF curves,
(T Tco) In (uJ/Q) = (To Tco)În(uJo/Q), (12)
where Tcc and Q are trie asymptotic constants of trie set, and (Mo,To) labels one curve of trie set [8j. From equations (10-12) we obtain
ôln'uJ
r~J 1/(T Tm)
,
ôT
r~J
(T Tcc), (13)
and, in three dimensions,
Na r~J f$
r~J 1/(T Tcc)~. (14)
Since trie presumed dominance of fluctuation is based on median charactenstic lengths (not
too small and not too large, according to Sect. 3), we see from equation (14) that trie whole
approach, including trie WLF equation, is restricted to a certain, median loguJ and T interval;
both trie asymptotic regions near loge and Tcc cannot be described by our dominance of
fluctuations. It should be seen that trie low-temperature WLF region is heavily restricted by trie exhausting of Section 5.
4.3. FLUCTUATING FREE VOLUME. A second consequence of minimal cooperativity is trie
existence of a "fluctuating free volume", vf. Equations (6-9) can be considered as a functional equation for a general function uJ(V) with trie only solution
uJ/UJM = exp(V/vf), (là)
where (16)
~~ = (UJM~)~
is a reduction constant depending on tue partition of Va in tue n partial volumes l~ witu equal 1, and vf is a volume not depending on tue partition for given size of a CRR.
We see from equation (là) tuat tue fluctuations of lnuJ~ and j (Eq. (7)) are monitored by trie parameter vf = /hVllh In uJ from whicu property tue name was cuosen. Equation (là) tuus
describes trie local, fluctuating facilitation of trie mobilities, /h In uJ, by small local volume vari- ations /hV, and not for instance how a mode frequency depends on its mode length (dispersion law).
Trie inclusion of a dispersion law gives trie possibility to construct fluctuating density pattern that are sketched in trie two Appendices A and B.
4.4. ISOTHERMAL KOHLRAUSCH FUNCTION FOR BULK COMPLIANCE. Trie stretched ex-
ponential decay, ço(x) r~J
exp(-(t/To)~)
,
is called Kohlrausch or KWW [22j law, fl = flKww.
Shlesinger [23j wrote "why this law is so widespread is that it is a probability limit distribu- tion". This subsection is to show that, m an isothermal treatment, T = const., equations (9, là) lead to a Levy stable distribution, p(lr~uJ), uJ = Ma, for trie total CRR. Trie stationary
stochastic functions of trie partial volumes of a CRR are, according to equation (7), denoted
as
~~(t)
= v(t). (17)
From equation (là), applied to partial volumes, we bave InuJ~
= a~~ + (18)
with
a = 1/vf > 0 j19)
and
b = InuJj~, (20)
wuere a and b formally depend on tue size of a CRR
r~J n a = an, b
= bn, if CRR'S of dilferent size are cuaracterized by dilferent n for constant partial volumes f.
Let tue general volume coordinate be denoted by x, 1.e. j E x. Tue statistical independence
of w~, equation (9), imphes tue statistical independence of Inw~ and j for dilferent i. Tue general probability density for Inw~ is denoted by p. We obtain from equations (9, 18) tue
convolution to a total p(x),
fllZ) " fl~~l~nX + bn), 121)
depending on tue experimental frequency w E wa of tue CRR ma equations (8, 9), applied
to tue wuole CRR. Tuis convolution is part of a second, ratuer obvious principle of our pue- nomenological approacu witu dommance of fluctuation: naturai coupiing, wuicu means tuat tue stocuastic functions j(t) generate,
ma equation (21), sucu volume fluctuations tuat can
immediately be inserted in tue spectral density of tue FDT:
/hV~(w)
= kTB"(uJ)/~w. j22a)
B"(w) is tue loss part of tue dynamic bulk comphance, B*(w)
= B'(w) iB"(uJ). Tue concepts of minimal and natural coupling suould not be confounded: tue former is to describe tue
molecular cooperativity in puenomenological terms, tue latter is to describe uow sucu molecular fluctuations can by measured in a tuermodynamic experiment. Tue puysical entrance of tue fluctuation dominance principle (Sect. 3) is tuerefore witu equation (22a). Tuis means tuat
pj>)/~
+~ ~v2jw) j~~b)
because d~
r~J d lnuJ
+~ dw/w. Tuis includes, as mdicated above, tue reduction of tue n partial frequencies
uJ~ to one common frequency w m tue dynamic linear-response experiment on tue
functional a subsystem represented by one CRR, (w~) ~ w (w identity in terms of Ref. [7j).
Tue statistical independence as formulated by equation (21) is a suflicient condition tuat,
for large n. a Levy stable limit distribution is obtained [24-26j. Tue cuaracteristic function, j(t) =
e~~~p(x)dx.
123)
is tuen Caucuy scaled.
f(t)(
= exp(-c If"()
,
0 < a < 2. (24)
In general, a cuaracteristic function f(t) is not tue same as a correlation function q7(t). But, since tue Fourier transformation equation (23) must be considered as a translation of a spectral density, equation (22), witu p(x)
r~J B"(w), from frequency w domam (Eq. (18)) to time t domain resulting, of course, m a correlation function ço, and since tue cuaracteristic function
f, in particular its amount fi equation (24), and tue correlation function
ço bave trie same general properties (positive definiteness (27j) we can put
~2lt) r~~ exP(-lt/To)~) (25)
for tue correlation function of volume fluctuations
m a CAR.
