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Comments on the mode coupling theory for structural relaxation
Chemical Physics ELSEVIER
Chemical Physics 212 (1996) 47-59
Comments on the mode coupling theory for structural relaxation W. G 6 t z e a'b, L. S j 6 g r e n c a Physik-Department, Technische UniversiRit Miinchen, D-85747 Garching, Germany b Max-Planck-lnsfitutfiir Physik, Werner-Heisenberg-lnstitut, P.O. Box 401212, D-80805 Miinchen, Germany c Institutionen f6r teoretiskfysik. Giiteborgs Universitet, S-41296 Giiteborg, Sweden
Received 22 February 1996
Abstract In condensed matter the coupling of fluctuating forces to the infinite set of density fluctuation pairs leads to non-Markovian equations for the time evolution. Treating the dynamics of the density pairs with a factorization approximation yields closed equations of motion which provide a mathematically well-defined treatment of the cage effect for particle motion in liquids. The equations imply a bifurcation singularity connected with the appearance of a spontaneous arrest of the particle positions in disordered arrays. The evolution of structural relaxation on cooling or compression of liquids is obtained when the temperature or density approach critical values, which characterize the singularity. Von Schweidler's law is obtained as a generic reason for a-relaxation stretching. There appears a dynamical window where structural relaxation is described by a universal law, which deals with two time fractals. The relation of the mode coupling theory with Mountain's theory of structural relaxation is discussed and the interplay of relaxations and oscillations in supercooled liquids is demonstrated.
1. Introduction It is well known that cooled liquids exhibit low lying excitations whose relaxation times r,~ vary drastically with temperature T or density n. It is anticipated that these excitations are caused by the motion of clusters of molecules. Therefore they are referred to as structural relaxations. The processes with the largest time scales r~ are called a-relaxation. Upon supercooling, r~ can become of macroscopic size, i.e. 10-14 orders of magnitude larger than the femto second scale to, characterizing normal liquid dynamics. The scale r,, can become larger than the time scale texp, characterizing the experiment which was designed to probe the system's properties. If r,, << texp the sample exhibits a typical liquid response. However, if r,~ >> texp the sample behaves effectively nonergodically and responds like an amorphous solid, i.e.
it is a glass. In a cooling experiment there occurs a transformation from a liquid to a glass near a temperature Tg, called the calorimetric glass transition temperature, where r~ (T = Tg) = rex p. An understanding of structural relaxation is a prerequisite for a microscopic explanation of the glass transition. Structural relaxation is generic for condensed disordered matter. It occurs whenever crystallization can be bypassed in systems as different as molten salt mixtures, such as Ca(NO3)2. KNO3 ( C K N ) , and van der Waals liquids, e.g. salol. It is observed in systems as complicated as polymers and as simple as hard sphere colloidal suspensions. In this paper we will briefly review how proper extensions of earlier treatments of the cage effect in liquids leads to an ab initio theory of structural relaxation. Originally this approach, referred to as mode coupling theory (MCT), grew out of theories for simple liquids
0301-0104/96/$15.00 Copyright (~) 1996 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 0 1 - 0 1 0 4 ( 9 6 ) 0 0 0 9 4 - 8
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W. G6tze, L. Sj6gren/Chemical Physics 212 (1996) 47-59
[ 1-3 ]. It will be explained, why some results of MCT are of general significance and can also be used to analyze experiments for very complex liquids. We will consider the relation beetween the visco elastic theory of liquids and the MCT. Furthermore, the evolution of structural relaxation out of the oscillatory motion of cooled condensed matter will be demonstrated.
2. Some major problems Three major problems can be specified with which a theory of structural relaxation has to cope.
2.1. Spontaneous arrest There is consensus that phonon assisted hopping of particles in an almost frozen potential landscape is the elementary process which leads to relaxation within the strongly supercooled state, say for T near Tg [4]. The landscape is produced by the liquid molecules which are arrested in some random array. Explaining relaxation requires the study of complicated percolation problems. Such problems have been analyzed in the theory of amorphous semi-conductors [5]. In these systems the molecules form a disordered static matrix and the electrons exhibit the dynamics of interest. In liquids there is no ad hoc separation of particles which produce free energy barriers and particles which percolate. The same interactions which give spontaneous arrest create also the potential barriers for the percolating particles. The microscopic equations of motion must lead to a separation of time scales: the duration of a hopping event must be orders of magnitude shorter than the lifetime of some free energy barrier. The separation of time scales mentioned above does not exist in the normal liquid state, say for parameters T and n near the triple point. Hence there must be a characteristic temperature Tc such that the whole picture of hopping transport loses its meaning as T increases seriously above Te.
2.2. The evolution of structural relaxation Simple liquids in their normal state do not exhibit structural relaxation. Thus there appears the question: how do structural relaxation phenomena evolve upon condensing the system by lowering T or increasing n?
The normal state dynamics of simple liquids is governed by binary collisions between the molecules, excluded volume and molecular field effects. The dynamics can be described by generalizing Boltzmann's kinetic equation. On condensing the liquid there appear several new features of the dynamics. They can be understood as result of correlated binary collision events or, equivalently, as non-linear interaction effects between excitation modes [6]. A perturbation calculation of these effects can explain for example the gradual decrease of the diffusivity D by up to one order of magnitude. Obviously, there must exist a critical temperature such that the above view of liquid dynamics becomes insufficient if T decreases seriously below it. It is plausible that this is the same temperature Tc already mentioned. Thus there should exist a cross over temperature Tc separating regimes of different dynamical behaviour [7]. In the strongly supercooled state, T << Tc, activated processes cause relaxation. In the normal and moderately supercooled state, T >> To, well understood interaction effects govern the dynamics. How can Tc be identified?
2.3. Relaxation stretching The dynamics of a many particle system is described by correlation functions or correlators ~b(t) or by their Fourier-Laplace transforms ~b(to) = ~b' ( ~o) + i~b"(to). Here the correlator of variable A is defined by ~b(t) = (A(t)*a)/(I A 12). The evolution in time, t, is given by the microscopic equations of motion and the averages are defined canonically. The correlation spectrum ~b"(to) for frequency to is related trivially to the susceptibility spectrum g " (w) = to~b" (to) [ 6 ]. Decay curves ~b(t) or spectra g " (to) for structural relaxation are stretched over huge windows of time or frequency respectively. For the a-process t has to be increased by two or more decades to observe a decrease of ~b(t) from 90% to 10% of its initial value. This window is at least 5 times larger than that for a Debye process ~bD(t) = f e x p - - ( t / r ~ ) . The Debye susceptibility spectrum is a Lorentzian with a half width of 1.14 decades: X~(to) = f(tor,~)/[1 + (toT"a)2]. However, a-peaks for the susceptibility spectra usually have a half width of two or more decades. Relaxation stretching is a paradigm for the dynamics of condensed disordered matter.
W. G6tze, L. Sj6gren/Chemical Physics 212 (1996) 47-59 Several fitting formulae have been proposed in the past, which often describe a-relaxation curves reasonably well• The oldest ones are due to Kohlrausch [8] and von Schweidler [9]. The Kohlrausch function modifies the Debye law by a stretching exponent /3 which is smaller than unity: ¢(t) = fexp[-(t/Ta)/3].
(1)
Here f2q is the mentioned phonon frequency. Parameter Vq = ~q')/q denotes a Newtonian friction• The non-trivial part is the memory kernel mq which is the constant of proportionality between a friction force at time t and the 'velocity' at the preceding time t'. The equation of motion (4) can be solved by introducing Fourier-Laplace transforms Cq (co) and mq (to) for the correlator and the kernel respectively:
Von Schweidler's law can be considered as the short time part of ( l ): ¢ ( t << ra) - f o¢ - ( t / r a ) b q- O(t2b).
(2)
It is a great challenge to explain how regular microscopic equations of motion can yield the power laws specified by the anomalous exponents/3 or b. Data analysis shows, that usually/3 5t b. This suggests that ( 1) is not an exact law but merely some data interpolation formula. The simplest function dealing with structure dynamics is the density correlator qbq(t). It is the correlator for density fluctuations of wave vector q: A = pq, q =l q ]. For the short time transient one gets [6]
(bq(t << to) = 1 - l(~Qqt)2 -q- . . . .
(3)
Here f2q = qv/V/-~q is the effective phonon frequency; v is the thermal velocity and Sq = (I Pq 12) is the structure factor• Thus (1) or (2) cannot be correct if t is small enough. There must exist some time scale, say rfl, such that the a-process describes dynamics only for t/> r E. Explaining r E is a necessary part of an explanation of a-relaxation.
3. Relaxation kernels
3.1• The equations of motion MCT starts with equations of motion which express the correlators ¢ in terms of relaxation kernels M. For the density correlator ¢q(t) a generalized oscillator equation is used [2]:
02q~q( t) d- J~q ((bq( t) -~- )'qtgt(bq( t ) t
+ fmq. o
(4)
49
2 L ( tO) tO -k d"2qMq cq(tO) =
tO2_
+ tOa qM,q(tO) ,
MLq( tO) = i'yq -q- mq( tO) .
(5)
This formula shows that ML(to) plays the same role in liquid theory as the polarization kernel in the theory of anharmonic phonons or of electromagnetic fields in matter. Shear dynamics is described by the correlator CqT for transversal currents A = jq.T [ 6 ] • The correlator is also expressed in terms of some kernel M T, which is a generalized (q, to)-dependent shear diffusivity or viscosity:
¢T (to) =
1 CO q- q2v2M]'q ( t o ) '
MT(to) --_ . l y qT + mqT (co) .
(6)
The zero wave vector limits of the kernels MoL'T(w) are, up to trivial manipulations, the longitudinal and transversal elastic modulus respectively. Within the Zwanzig-Mori formalism [6] one can define fluctuating forces Fq(t) and express the relaxation kernels for the current correlators in terms of the force correlators (F~ (t)* F~); a, fl = x, y, z. Because of rotational and inflection symmetry there enter only two independent functions and these are ME,X(t). Restricting our attention to frequencies below the band of microscopic excitations, tot0 << l, we can split off f r o m Mq(to) a white noise background iyq as a b o v e . The difficult part dealing with low frequency anomalies is mq(to). Eqs. ( 4 ) - ( 6 ) are exact reformulations of the microscopic theory; possible structural relaxation phenomena are hidden in the kernels mq(to) and mT(to).
W. Ggtze, L. Sjggren/Chemical Physics 212 (1996) 47-59
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3.2. Force relaxation and density relaxation Let us consider a liquid where a single mode causes some low lying excitation. The Zwanzig-Mori reduction can be continued and the kernel written as fraction mq(Og) = - F q / [ t o + ~q(Og)]. Here the new kernel IXq(W) does not contain the mentioned mode dynamics anymore. Thus the frequency dependence of tZq(W) can be ignored for tot0 << 1 and it is possible to write tZq(W) = i/7"q. In this way a Debye model is obtained for the force relaxation: mq( t) = Fq exp(--t/'rq).
(7a)
This model approximates the structural relaxation peak for the force spectrum mq~(W) by a Lorentzian of area Fq and width 1/rq. Furthermore, if the discussion is restricted to small q, the wave vector dependencies of Fq, rq, Yq, Sq can be ignored. Then ( 4 ) - ( 7 a ) are equivalent to Maxwell's visco-elastic theory of liquids [6]. One implication of Maxwell's theory should be recalled. If the structural relaxation spectrum for the force correlator is well separated from the other liquid excitations, the density correlator exhibits a quasielastic resonance, called Mountain's peak [ 10] :
~q(~) = - - f q / [ ~ + iFq]; ~, 1/Tq<<~q, l / t o .
(7b)
The area fq of this peak in the density spectrum is given by the area Fq in the force spectrum:
fq = G / ( l + Fq).
(8)
The width parameter Fq of the structural relaxation peak for the density fluctuations is proportional to the rate parameter 1/7"q for the force fluctuations:
Fq = Cql'l"q,
Cq = 1/( 1 + Fq) .
(9)
The parameters fq, Fq approach finite non-zero constants for q --, 0. However, Maxwell's theory is not adequate for the discussion of structural relaxation, since it misses the stretching phenomenon. Apparently, structural relaxation is not caused by a single mode or by only a few low lying modes. It is important to understand that some of Mountain's results [ 10] are strict implications of the exact equations (4) and (5) which do not
depend on the approximation (7a). To identify these results we give first a precise mathematical meaning to the concepts 'well separated' and '<<' in (7b). To proceed we write mq(t) = fnq(t') with t" = ts. This is equivalent to mq(tO) = fnq((O)/S with ~ = to/s and rhq (~) denoting the Fourier-Laplace transform of rhq (t'). Now we consider the limit s tending to zero for fixed rescaled time ~', rescaled frequency ~ and shape function rhq. This limit procedure achieves the desired separation of the c~-process from the other processes. From (5) one derives (bq(~O)S --~ qbq(&) for s --~ 0, where q~q(~) = - 1 / [ & - 1 / & q ( ~ ) ] . This result is equivalent to (bq(t) --~ qbq(t-), where ~q(t") is the Fourier-Laplace back transform of 6q(W). Equivalently one gets d 6q(t3 =,~q(t) - = [ r n q ( 7 dt J 0
T'),~q(?)di'.
(10)
If one uses Maxwell's model (Ta) for rhq, Eq. (10) is solved by Mountain's results ( 7 b ) - ( 9 ) for 6q. However, independently of any specific model one concludes from (10): a quasi-elastic peak for the force spectrum of shape rhq~(~) causes a corresponding peak of shape ~qt(t~) for the density spectrum and vice versa. The areas Fq and fq of the a-peaks of force spectrum rhqt(~) and density spectrum t~q~(~) respectively are given by Fq = rhq(~"--~ 0) and fq = qbq(~ --, 0). The limit ? ~ 0 in (10) yields Mountain's result (8). There are different ways to define a relaxation time rq for the force correlator or a relaxation rate Fq for the density correlator. In any case, 7"q cx 1Is and Fq oz s, where the constants of proportionality are given by the shape functions rhq and t~q respectively. Eq. (9) holds with vanishing I'q and 1/7"qand a model dependent constant Cq. Let us examine the ansatz for the small i"dynamics: l?lq( t-) = Fq -- a q P + g q t 2b + . . . .
6q(t') = fq - hqr ~ + kqf 2b + . . . .
(lla)
This is an extension of yon Schweidler's law (2) to an asymptotic expansion of rh and q~. In a generalization of (8) one derives from (10), that the coefficients of the rhq-expansion determine recursively those for the 6q-expansion and vice versa. For example:
W. GiJtze, L. Sj6gren/Chemical Physics 212 (1996) 47-59 hq = Hq/(1 q- Fq) 2,
kq=[Kq(l+Fq)-AH2q]/(l+Fq)
3.
(lib)
Here parameter A abbreviates ,t = F ( 1 + b)2/F( 1 + 2b) and F is the gamma function. For b < 1 one gets A > 1/2. If ~q(?) or (~q(~') obey (1), one gets the identity 2Kq = H q/ or 2kq = h2 / f q respectively. The simultaneous validity of both these identities is compatible with (8) and ( l l b ) only if ~ = 1/2. The following result may therefore be concluded. If the a-process for the force relaxation follows the Kohlrausch formula (1) with exponent fl < 1, the a-relaxation for the density correlator is not a Kohlrausch process and vice versa. The microscopic equations of motion establish exact relations between certain structural relaxation processes. Such a relation between the elastic modulus spectrum, mq - " (~), and the dynamical structure factor q~q'(&) is given by (10). These relations exclude the possibility that the Kohlrausch function is a general law describing all a-processes. It is not possible to characterize a glass forming liquid by a single stretching exponent fl, contrary to what is anticipated occasionally [ 11 ]. On the other hand, there appears no obvious problem if one assumes a yon Schweidler expansion (2) such, that all a-processes of a given liquid are characterized by a common exponent b. In such case, the microscopic equations of motion imply relations among the expansion coefficients as exemplified by Eqs. (8) and (11).
Let us recall that the Maxwell-Mountain theory is based on the assumption that there is only a single slow mode which causes the low frequency dynamics of the force correlator mq(t). This assumption is not valid for a treatment of density fluctuations in supercooled liquids. Rather the infinite set of modes Pq-kPk couples to the forces Fq, and it is this set which causes the structural relaxation contribution to the fluctuating force correlators Mq. Approximations for correlators of products of slow modes like pip2 were studied in detail first by Kawasaki [ 12]. The essence of his procedure is to replace averages of products by products of averages: (Pl (t)p2(t)p3P4) "~ (Pl (t)p3)(P2(t)p4) -ff (3 ~ 4). Following this procedure we express the force kernels as polynomials of the density correlators: mq(t) = E V( q, kp)fbk( t)dpp ( t) = ~q( dpk ( t) ) , k+p=q (13)
mqT(t) = ~
v+(q, kp)4,k(t)4,,(t)
k +p=q = FT(q~k(t)) .
V(q, kp) = Sq&Sp{q[ k( l - S~ l ) (15)
Thus the vertices V,VT are equilibrium functions. Like Sq they vary smoothly with T and n, and they can be evaluated within established liquid structure theories
4.1. The mode coupling approximation MCT is based on an approximate expression of the relaxation kernels in terms of the density correlators. The approximation is motivated by the following observation. If a density fluctuation Pk relaxes with rate Fk, product modes like PkPp relax in the first approximation with rate Fk + Fp. Therefore all product modes relax within the same window as the simple variable pq. For zero wave vector the fluctuating force is a combination of pair modes:
Fo = E U k p - k p k . k
(14)
The coefficient Uk in (12) is given by the bare interaction potential. However, this potential is renormalized and replaced by combinations of the static correlation function Sq in the vertices V and VT. For example [2,3]:
+ p(1 - s p l ) ] }2/(2q3n).
4. Mode coupling functionals
51
(12)
[6]. Let us summarize the preceding results. The correlation functions are expressed first in terms of correlators of those fluctuating forces which couple directly to density fluctuation products. Force correlators are then approximated by polynomials of ~b~(t) like .Tq, .TqT. These polynomials are called mode coupling functionals since they express the coupling of force fluctuations and density fluctuations. In some cases one can express the desired quantity in terms of the ~bk(t); and then its structural relaxation anomalies are obtained as corollary of those for the density
W. G6tze, L. Sj6gren/Chemical Physics 212 (1996) 47-59
52
correlators. This is the case for the shear correlator, in particular for the shear viscosity, as follows from (6) and (14). The density correlator is reduced to the kernel mq(t) and the latter to dpq(t). One does not get an explicit expression for dpq(t) but the closed set of equations (4) and (13).
4.2. The cage effect Consider a tagged particle. If the other liquid particles were fixed, the tagged one could not move very far. For normal densities it would be trapped in a cage formed by its neighbours. Diffusion in normal liquids is possible only because the neighbouring particles are not fixed but move out of the way. The neighbours can move because their neighbours move, etc. Normal liquid dynamics, like diffusion, is possible only because complicated streaming patterns, called backflow, are formed. The rattling motion of the particles in cages does not contribute to the diffusivity D. Because of the cage effect, D is suppressed below the value Do expected within a kinetic equation picture. It was possible to evaluate the decrease of D/Do with increasing n with formulae equivalent to (13) and (15) and using the truncation of (13) to mq(t) [13]. Here ~bk~°) is some first order approximation for ~bk. But for large n, ~bk differs drastically from ~b~°) and the cited approach becomes inconsistent. The cage forming particles exhibit the same dynamics as those which are trapped in the cages. All fluctuations have to be treated on the same level and this is the aim of MCT. '~
.~q(¢~0))
4.3. The simplified MCT The closed set of equations (4), (13) and (15) defines the basic version of the MCT, which deals only with the cage effect. This version is referred to as simplified MCT in order to distinguish it from the extended theory, which includes also phonon assisted hopping effects. It is also referred to as the idealized MCT since it deals with a sharp transition to an ideal, i.e. truly non-ergodic, glass state. In most cases it is obvious how hopping effects alter qualitatively the results of the simplified theory. There are subtleties due to the interplay of hopping events with the cage effect; but these will not be considered in this paper.
The MCT equations are regular. There are no input parameters which depend strongly on T and n. All resuits have to be calculated by solving (4), (13) and (15) without inventing additional hypotheses. Temperature T enters essentially only indirectly via Sq in (15). I f T is lowered from the melting temperature T,, to Tg, Sq changes by about 20%. This small change causes all the known drastic changes of the MCT solutions. The equations of motion (4) and (13) define unique solutions ~bq(t), which have all the standard properties of correlators. For every fixed finite time interval 0 ~< t ~< tmax < ~ , the solutions fbq(t) depend smoothly on the parameters 12q,Vq and V(q, kp), characterizing the equations. Thus MCT provides a mathematically well defined model for a dynamics. There is no input in the MCT equations of motion which is directly related to structural relaxation or glass formation. In order to stress this point a schematic model for the equations will be formulated. This model ignores the wave vector q, and so it deals with a single correlator ~b only. The equation of motion becomes t
f mft-
~(t) + v(b(t) + 122c~(t) + $"22
t')qb(t') dt'
o = O.
(16a)
The kernel m is a polynomial such as
m( t) = vl~b(t) + v2~b(t) 2 .
(16b)
The interesting parameters of this model are the two coupling constants Vl /> 0, v2 /> 0. This schematic model, occasionally referred to as F12-model, has been used before in order to demonstrate some mathematical features of MCT [ 14]. The non-trivial results of the theory have two origins. One is given by the non-linearities: kernel m in (4) or (16a) depends on ~b and the mode coupling functional in (13) or (16b) is non-linear. The other is the retardation effect formulated by the integrals in (4) or (16a). The retardation interval (t - tt), where m(t - t r) is appreciable, is not given but depends on the solution. It is the interplay of the non-linearities with diverging retardation effects, which renders MCT so different from other dynamics theories.
W. GOtze,L. Sjligren/ChemicalPhysics 212 (1996)47-59 I
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53
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Fig. 1. Correlators 40as functionof time t for the schematic model specified in Eqs. (16a) and (16b). The couplingconstantsdepend on the separation parameter e = -4-1/4" as specified in Eq. (1 6c). The time unit is chosen such, that/2 = 1. The damping constant is u = I. The dashed line is the solution at the critical point E = 0. The dashed dotted line is the critical law 40 = ( t o ~ t ) a with a = 0.327. The dotted line is the Debye curve 0.3 exp(--t/~ra).
-8
-6
-4
-2
0
logo Fig. 3. Correlationspectra 40" = X"/w as function of frequency ca for some correlators shown in Fig. 1.
vl,2 = v~,2( 1 + e ) , e=±l/4",
n=0,1 .....
(16c)
d
C<0
10~
n=
7
6
5
4
3
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54
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Fig. 2. Susceptibilityspectra X" as function of frequency ~o for the correlators shown in Fig. 1.
5. A simplified view of structural relaxation Figs. 1, 2 and 3 show the correlators ~b(t), susceptibility spectra g"(o~) and correlator spectra (b"(~o) calculated for the schematic model which is defined by equ. (16). A damping constant v = 1.0/2 is taken and units are chosen such that /'2 = 1. The coupling constants in the vl-v2 parameter plane are shifted on a straight line through the point v~ = ( 2 a - 1 ) / A 2 , v~ = 1/A 2, A = 0 . 7 :
The correlators for e < 0 and n > 3 exhibit a decay process from the plateau fc = 1 - • to zero, whose time scale ~-,~ increases dramatically with decreasing e. This decay produces the low frequency susceptibility peaks in Fig. 2 and the quasielastic peaks for the spectra ~b"(w), shown in Fig. 3 for n = 3 and 5. A Debye peak g ~ cx (o)7"a)/[ 1 Jr- ((.o'/'a) 2] is added as dotted line in Fig. 2 with r,~ chosen so, that the peak maximum for n = 7 is matched. The corresponding Debye decay, fc e x p ( - t / T ~ ) , is also shown as the dotted curve in Fig. 1. The slow decay process can be fitted very well by the Kohlrausch law (1) with [3 = 0.58. Thus the schematic model in (16) exhibits the main feature of structural relaxation: a slow control parameter sensitive o~-process, which is stretched. In this section some comments on the physical picture of structural relaxation as obtained from the simplified MCT will be made.
5.1. Ideal glass states There is a weak coupling regime of small vertices
V(q, kp) in (15), where the correlators relax to the equilibrium value zero: dpq(t ~ o~) = 0. These solutions describe ergodic liquid dynamics as exempli-
W. GOtze, L. SjOgren/ Chemical Physics 212 (1996) 47-59
54
fled by the • < 0 curves in Figs. 1-3. In the complementary strong coupling regime density fluctuations do not disappear but arrest (bq(t --~ cxz) = f q > 0, as exemplified by the e > 0 curves in Fig. 1. For the strong coupling solutions the density spectra exhibit elastic peaks of intensity fq : &q'(W ~ O) = 7rfq(3((o). This is a signature for a solid with DebyeWaller factor fq. Contrary to what is obtained for a crystalline solid, fq is a smooth function for all wave vectors q, i.e. the solid found is amorphous. From (14) a non-zero long time limit is obtained for the transver> sal force fluctuations: mT ( t -+ 00 ) -T= 0. This yields for the transversal current correlator (6): (bT (to) oc l/[to2--c2q2 +O(wq2) ] , Cq2 = v2fTq. Long wave length low frequency shear disturbances propagate as isotropic transverse sound waves with speed co. One concludes that the strong coupling solutions describe ideal glasses. For low T and large n, the coupling constants V(q, kp) are large and then the cage effect yields a spontaneous arrest of the density fluctuations. MCT explains the appearance of a frozen potential landscape as result of cooperative effects among density fluctuations. A tagged particle cannot percolate because its neighbours cannot move out of the way. The latter cannot move because their neighbours are blocked etc. The Debye-Waller factor of the glass obeys the equation
fq/(1-fq)
=5Fq(fk),
0<~ f q < 1 .
(17)
This result is equivalent to Mountain's formula (8), where Fq = Uq ( f k ) . However, the area Fq of the structural relaxation peak is not implemented within our theory as an ad hoc parameter, but it has to be evaluated from the equilibrium structure by solving the coupled equations (13) and (17).
5.2. Glass transition singularities On lowering T or increasing n, the vertices V(q, kp) in (15) increase. Critical values Tc or nc appear, where the simplified MCT predicts a bifurcation. The low frequency spectra or long time correlators exhibit a singular dependence for small separation parameters o- = C•, where • = (Tc - T)/Tc or • = (n - nc)Inc. The critical parameters are referred
to as glass transition singularities (GTS). Within the simplified MCT, the critical values Tc or nc mark transitions from liquid to glass dynamics. If hopping effects are incorporated the system cannot be driven through a GTS, but it can be close to it. In this case Tc or nc mark the cross overs between regions of different dynamical behaviour as discussed in section 2.2 For parameters near a GTS, there appear relaxation processes outside the transient regime: t >> to. These cause spectra below the band of microscopic liquid excitations. This dynamics which can neither be understood within normal liquid pictures nor within standard solid state models is structural relaxation. Its origin is the GTS. From Figs. 1-3 one infers to ,-~ 1. For this schematic model structural relaxation is observed for t > 10 or w < 0.1 for all parameters in (16c) w i t h [ • I< 1/4.
5.3. The critical dynamics for to << t << r~ At the GTS, ~bq exhibits power law decay towards the critical Debye-Waller factor f~q: (bq( t) = f~q + hq( t o / t ) a + O ( 1 / t 2 a ) .
(18)
The corresponding susceptibility spectrum varies sublinearly: Xq'(W) cx hq(wto) a. This process is called critical dynamics. There is a closed but involved set of formulae to calculate the critical exponent 0 < a < 1/2 and the critical amplitude hq > 0 from f'q [14]. The time to is obtained by matching (18) to the short time transient. For the model (16) one gets a = 0.327. The solutions at the critical point • = 0 are shown by the dashed lines in the figures and the leading terms (to~t) a and (wt0) a, respectively, by dashed dotted lines. If parameters are chosen away from the GTS, be it because T 5i To, n 5/ nc or because of hopping effects, a finite time scale rB appears such that the critical dynamics is observed for to << t << rt~ ; 1/r~ << to << l/to. Within the specified window the correlators and spectra vary only weakly with the control parameters. However, the scale rB depends sensitively on T and n; r~ diverges if the GTS is approached. For to << t << ra the particles are trapped in cages. The increase of rt~ with increasing T, for T < To, is a precursor phenomenon signalling the instability of the frozen
W. G6tze, L. Sj6gren/ Chemical Physics 212 (1996) 47-59
structure at Tc. The increase of ~'t~ on cooling, for T > To, is a precursor of the freezing at To. The critical spectra have first been measured by neutron scattering [ 15 ] and by polarized as well as depolarized light scattering [ 16] for the above mentioned glass former CKN.
55
cay from ]q proportional to t b (2). As suggested by Levy's generalization of the central limit theorem, (10) and (13) are solved for q ~ ~ by O~q(t-) = f~exp(-Fq? b) [18]. Thus MCT has established a derivation of the Kohlrausch law from the equilibrium structure, albeit in a very special limit only. In general ( 1) is not valid, but often it is a good fitting formula.
5.4. MCT a-relaxation for ~'t~<< t 5.5. MCTfl-relaxationfor to << t << ~', For ideal glass states, the time z# marks the cross over to arrest: (bq(t ~ Tfl) ,~ fq. The corresponding elastic peak in the density spectrum can be viewed as an a-peak of width zero: 1/76 = 0. For T > Tc or n < nc the time 7t~ marks the beginning of the decay below f~. The decay of q~q(t) from f~ to the equilibrium value is the MCT a-process in the proper sense. The time ~'t~specifies the a-relaxation window: r# << t ; w << lira. Both r,~ and r# diverge upon approaching the GTS; and the ratio r,/7a also diverges. Structural relaxation is a process occurring in several steps, specified by scales like r,~ and ~-#. Since z~ diverges, (10) is valid as an asymptotic law near the GTS. However, the kernel rhq(t-) must not be used as an ad hoc input but rather it is related to the a-correlator via specialization of (13) to the critical parameters: /nq (t-) ----o~";( ~k (t-) ). Thereby closed equations are obtained allowing the evaluation of the a-spectra from the equilibrium structure [ 17]. Von Schweidler's law (2) is obtained as an exact implication of (10) and (13) [ 14]. There is a straightforward but involved procedure to evaluate from 5t-q the so-called exponent parameter .h, 0.5 ~< ,h < 1. This parameter fixes both exponents a and b:
The essence of structural relaxation is the partial arrest of density fluctuation near the critical DebyeWaller factor, fq. There is the mesoscopic dynamical window, to << t << z~, where t~qbq(t) = (bq(t) -- f~ is small. The dynamics within this window is referred to as the fl-process. This process begins with the critical decay and it ends with the von Schweidler process [ 14]. The beginning of the a-process is the end of the fl-decay. Within the fl-relaxation window one finds the factorization: ~(bq(t) = h q G ( t ) + O(G2). One factor is the above mentioned critical amplitude, hq. The other factor is called the/3-correlator G; it depends sensitively on t and on the control parameters. Hence there are no correlations between the propagation of density disturbances in space with those in time. In other words, fl-dynamics deals with localized modes, a clear manifestation of the cage effect. If one substitutes Cq (t) = f q + hqG + 0 ( G 2) e.g. in (14), one gets my(t) = ~'Tq( fk) + h~G( t) + O(G2), c where hqT = Ek[O.T'~T (fk)/Ofk]hk. A similar result holds also for other variables A, which can be expressed as polynomials of density correlators:
F( 1 - a)2/I'( 1 - 2a) = A = F ( 1 + b)2/F( 1 + 2b).
dpA ( t) = f cA + h A G ( t / t o ) -4- OA( G 2) .
(19) For special choices of V(q, kp), it is possible to obtain ,h = 1/2, i.e. b = 1. For the model (16) this happens only for v~ = 0, v~ = 4, and (10) and (13) are then solved by Maxwell's model. Generically b < 1, and von Schweidler's law is identified as the general reason for stretching. For large wave vector q the phase space for the decay of pq into pair modes Pq-kPk becomes large. Simultaneously, the a-peak intensity, f~q,becomes small. Thus the kernel mq in (13) becomes a sum of many small terms, and each term exhibits the initial de-
(20)
All such functions, the density correlator (A = pq), the longitudinal and transverse elastic moduli (obtained from A = F0), depolarized light scattering functions which are caused by dipole induced dipole interactions, the dielectric loss spectrum etc., exhibit the same fl-dynamics. Different variables merely project different contributions of the same local modes. The projection is quantified by the equilibrium parameter hA. The problem of solving the coupled equations (4) and (13) for the many functions qbq(t) is reduced to solving a single equation for G [ 14],
56
O"
W. GOtze, L. Sj6gren/Chemical Physics 212 (1996) 47-59
+ hG(t') 2 -
d /, ^ / G ( t - ~ ) G ( F ) dF = 0 J
(21)
with the initial condition G ( f ~ 0)• ~ 1. Here to, h and tr have the same meaning as before: matching scale, exponent parameter and separation parameter respectively. The solution of (21 ) depends smoothly on A and formula (19) is an implication of (21). 5.6. Universality features
Eqs. (20) and (21) establish universality features or laws of corresponding states. Systems with the same a exhibit the same fl-spectra: X ~ ( t o ) / h a = & G " ( & ) . This holds if the spectra are measured with scale ha and frequencies with scale 1~to, & = toto. In addition, the evolution of the spectra with changes of a control parameter such as T is the same, provided these are entered in the rescaled form o- = C(Tc - T)/Tc. The results are robust. The details enter merely as the set of parameters ha, to, ,~, C and Tc. The parameters change smoothly if e.g. the pressure is altered. With the appropriate specification of scales, the fldynamics shown in the figures for the schematic model (16) is the same as that for salol. For the figures A -0.7 was chosen, this being the measured value for salol [191. MCT equations have also been derived for binary mixtures. Eqs. ( 4 ) - ( 6 ) , ( 13)-(15) and (17) are generalized so that all quantities such as /2q, fq etc. are replaced by matrices. The same liquid to glass bifurcation singularity has been identified as for simple systems. Since the universal results ( 1 8 ) - ( 2 1 ) merely reflect the topology of the bifurcation manifold, they remain valid without any modification. Even for polymers, mode coupling equations have been derived, which contain the cage effect functional (13) [20]. Hence Eqs. (20) and (21) hold also in this case. The theory provides a scenario for structural relaxation and the general results can be used without explicit reference to the microscopic details. Microscopic calculations are necessary if one wants to determine numerical values for parameters such as T c, hq or .A. They are also necessary to predict the sizes of the O(G 2) terms in (20). It cannot be predicted without detailed calculations, whether a system can be driven so closely to a GTS, that the discussed
asymptotic laws (2), (10) and (20) are of relevance for the interpretation of data. The specified universality refers to the fl-dynamics, including the initial part of the a-process. The full aprocess depends on the detailed structure, as follows from (10) and (13). Different systems can exhibit quite different a-peak shapes, even if they have the same von Schweidler exponent b. The solutions of the MCT equations for a and fl processes are summarized elsewhere [21,22]. Comparisons of the asymptotic formulae of the MCT with experiments are reviewed in Refs. [23,24] and in a series of articles in Ref. [25].
6. Structure and structural relaxation
The glassy dynamics, which was discussed above, is due to large values of the kernels mq(tO). Within the dynamical window, where w//2q2 and yq can be neglected in comparison with mq(¢.o), Eq. (5) simplifies to [23] ~/)q(O)) -~ mq(tO)/[ 1 -- tOmq(tO) ] .
(22)
This equation is independent of the parameters /2q and yq, which characterize the microscopic dynamics. Therefore one can conclude the following. The long time or low frequency dynamics is specified completely by the vertices (15). The latter are given by the equilibrium structure. Glassy dynamics deals merely with the statistics of the orbits of the many particle system in configuration space. This statistics is determined by the potential landscape, i.e. by the Boltzmann factor e x p ( - U / T ) with U denoting the interaction potential. The microscopic dynamics merely sets the time scale to for the exploration of the potential landscape. If q~q(t) solves the coupled equations (13) and (22), the same is true for ~bl (tx) for every x > 0. The scale to for the glassy dynamics is set by matching the solution of (22) to the microscopic transient. Thereby MCT gives a precise meaning and a justification of the statement that glassy dynamics is structure dynamics. The results of the preceding paragraph can be demonstrated for the schematic model, defined by (16a) and (16b). While Figs. 1-3 exhibit the results for a microscopic dynamics with/2 = u = 1, Figs. 4-6 exhibit solutions for/2 = 1 and r, = 0. Corresponding
W. GStze, L. Sjt~gren/Chemical Physics 212 (1996) 47-59 I
I
I
I
i~
I
0
n=
o.8
57 I
I
I
I
10 6
1
0.6
2
c>O
e
10 4
0.4 E>O 0.2
10 2
" " " , , ' ~
5
"
-
-
•
3
. . . . .
lo
. . . . . . .
n=
0.0
e
10o I
,
-2
L 2
I
0
I 4
I 6
I 8
I 10
Iogt Fig. 4. Correlators for the same model described in Fig. 1 but for ~,=0.
i
-8
-6
-4
-2
0
logo) •
'
I
e<0 n=
I
I
7
6
I
5
I
4
I
3
I
Fig. 6. Correlation spectra for some of the correlators shown in Fig. 4.
2 1 ~ L//llll|
can show [26], that the correlators are completely monotone functions; i.e. there are distributions of rates pq( [') ~ 0 such that P
10 .2 1
(bq(t) =/exp(-Ft)pq(F) dF.
(23)
o h
i
-12
-10
-8
-6
-4
-2
0
log0~ Fig. 5. Susceptibility spectra for the correlators shown in Fig. 4.
solutions refer to identical mode coupling functionals, specified by (16c). The short time dynamics in Fig. 1 differs from that shown in Fig. 4. For the latter the short time oscillations are more pronounced. However, the decay curves for tO > 200 of Fig. 1 coincide with the corresponding ones of Fig. 4 for tO > 100, provided the time scale for the latter is changed by a factor 4.00. This rescaling amounts to a shift of the curves parallel to the logarithmic time abscissa. A similar statement can be inferred by comparing the susceptibility spectra of Fig. 2 with the shifted ones of Fig. 5. A special model for a microscopic dynamics is obtained by dropping the first term o3?qbq(t)in Eq. (4). In this case the microscopic dynamics is specified by the set of relaxation rates "yq > 0. For this model one
This model exhibits identical structure dynamics as discussed above in connection with (22). Therefore one concludes that structure dynamics is a superposition of Debye relaxation processes. In particular the aprocesses and the fl-processes can be written as such a superposition (23). The proof of (23) provides the justification to view structure dynamics as relaxation. A relaxation function like (23) starts with a linear short time expansion, ~b(t) = 1 - (t/T) + .--, provided the distribution pq(I') has a finite first moment. However, the MCT solutions exhibit the exact short time expansion (3) as any N-particle system with regular interaction potentials. A superposition of Debye processes (23) decreases monotonically with increasing time, while MCT solutions can exhibit oscillations as shown for tO < l0 in Figs. 1 and 4. The spectra ~bq'(to) for a relaxation process (23) decrease monotonically with increasing frequency to. But the MCT spectra can have bumps as is shown for frequencies to near ,O in Figs. 3 and 6. The results of Fig. 6 for e > 0 and n = 0 - 3 are also shown on linear scales in Fig. 7.
W. GOtze, L. SjOgren/Chemical Physics 212 (1996) 47-59
58
_-
v/-y+ -
1 (24b)
0.6 n~
0.4 v
0.2
I
•
I
\
I
2
3
Fig. 7. Correlation spectra from Fig. 6 for e > 0 and n = 0 - 3 plotted on linear scales. The dashed curve is the asymptotic result (24b) for n = 0.
These bumps are more pronounced for lower temperatures, as one can notice by comparing the results for • > 0 and n = 0 with that for n = 1. These bumps are due to inhomogeneously broadened oscillations. Deep in the glass state, i.e. for T considerably below Tc, the particles are well localized in cages. They will perform essentially harmonic oscillations. The distribution of sizes and shapes of the cages implies a distribution of oscillator frequencies. Hence one gets for the disordered solid a bump in the spectrum instead of a sharp line which one would get for a crystalline solid. In the strong coupling limit, i.e. for temperatures T far below the critical value Tc, the MCT equations simplify considerably. One finds as solution for the dynamical susceptibility X ( w ) a superposition of oscillator functions
X(w) =
f
--J'22
dx p ( x ) o~(o~ + i~) - ~ ( x ) 2 "
(24a)
The squares of the oscillator frequencies/'2(x)2/~ 2 = tzj + 2 x v / - ~ are distributed with density p ( x ) = 2~/1 - xZ/Ir. The parameters/*l and ,L~2 can be expressed in terms of the mode coupling functional. From (16b) one finds/Zl = 1 + v l +v2,/*2 = Vl +2v2 [ 14]. Figs. 4-7 demonstrate the example v = 0. In this case (24a) implies the spectral density for 0<~2
Here the edges/2+ of the bump are given by/'2~:/J22 = /Zl 4- 2v/-~-~. The dashed curve in Fig. 7 is the result (24b) calculated for parameters/zl,/x2 for the n = 0 coupling constants. Thus the MCT predicts deep in the glass a semi-ellipse for the shape of the bump as one would expect within Wigner's theory of random matrices. Let us emphasize that the MCT results for the short time oscillations and spectral bumps, as opposed to the structural relaxation phenomena in the strict sense, depend on the details of the short time dynamics. Spectral bumps are often observed for glass forming liquids and they are occasionally referred to as boson peaks. Tao et al. [27] pointed out that the cited MCT results provide a natural explanation of the boson peak phenomenon. Alba-Simionesco et al. [28] have shown that results for simple schematic MCT models can fit reasonably well light scattering spectra for toluidine, salol and CKN. The fits [ 28 ] hold for the whole Raman band including the boson peak region; in addition they describe the evolution of the structural relaxation. From equations (24) one concludes the following. With decreasing coupling the boson peak position shifts to lower frequencies and the peak intensity increases. These phenomena are demonstrated in Figs. 6 and 7. Thus MCT implies for the simplest models a vibrational softening and an increase of vibrational spectral densities in the boson peak region on heating. These predictions have been verified in a recent neutron scattering study [29].
7. Conclusions The MCT deals with a mathematical model for a dynamics which explains the evolution of structural relaxation as result of the appearance of a bifurcation singularity. In this paper we have considered the simplest example for such singularity. In all cases a singularity establishes a universality class or a scenario, where the correlation functions within certain windows depend on a finite number of parameters such as Tc, hq and ,t only. The connection between these parameters and the microscopic details is robust. Thereby a qualitative understanding of the solutions can be ob-
v~ G6tze, L. Sj6gren/Chemical Physics 212 (1996) 47-59 tained and a variety o f predictions for relations between m e a s u r a b l e quantities, like ( 1 9 ) , can be made. T h e M C T is based on the m i c r o s c o p i c theory o f the cage effect in s i m p l e liquids. Quantitative details for fq, hq, nc, A, and a - r e l a x a t i o n shape functions have been w o r k e d out for the hard sphere system, LennardJones systems and s o m e mixtures. The w o r k for m o r e c o m p l e x liquids remains to be done. The observation o f the evolution o f structural relaxation requires spectroscopic techniques, which bec a m e available o n l y recently. The first detailed measurements have been m a d e by depolarized light scattering for C K N by Li et al. [30] and by photon correlation spectroscopy for hard sphere colloidal suspensions by van M e g e n and U n d e r w o o d [ 31 ]. These authors [ 30,31 ] have c o m p a r e d their data c o m p r e hensively w i t h the available M C T results. Their w o r k p r o v i d e s a solid basis for a rational assessment o f the m o d e c o u p l i n g theory for structural relaxation.
Acknowledgement We thank H.Z. C u m m i n s for helpful and stimulating discussions and M. Fuchs for the c o m p u t e r p r o g r a m used to evaluate the figures.
References I 1 I E. Leutheusser, Phys. Rev. A 29 (1984) 2765. [21 U. Bengtzelius, W. GOtze and A. Sj(51ander, J. Phys. C 17 (1984) 5915. [31 W. G6tze and L. Sj5gren, Z. Physik B 65 (1987) 415. [41 J. Wong and C.A. Angell, Glass: structure by spectroscopy (Dekker, Basel, 1976); S. Brawer, Relaxation in viscous liquids and glasses (The American Ceramics Society, Columbus, 1985). 151 H. Scher, in: Slow dynamics in condensed matter, eds. K. Kawasaki, T. Kawakatsu and M. Tokuyama, AIP Conference Proceedings 256 (1992) 485, and references therein. [6] J.P. Boon and S. Yip, Molecular hydrodynamics (McGrawHill, New York, 1980); J.P. Hansen and I. McDonald, Theory of simple liquids (Academic Press, London, 1986). [71 M. Goldstein, J. Chem. Phys. 51 (1969) 3728.
59
[8] R. Kohlrausch, Pogg. Ann. Phys. 51 (1854) 56. [9] E. von Schweidler, Ann. Phys. 24 (1907) 711. [101 R.D. Mountain, J. Res. Nat. Bur. Standards 70 A (1966) 207. I 11 ] J.C. Phillips, J. Non-Cryst. Solids 192/193 (1995) 393. [121 K. Kawasaki, Phys. Rev. 150 (1966) 291. [131 R.I. Curkier and J.R. Mehaffey, Phys. Rev. A 18 (1978) 1202. [ 14] W. G6tze, Liquids, freezing and the glass transition, eds. J.P. Hansen, D. Levesque and J. Zinn-Justin (Noah-Holland, Amsterdam, 1991) p. 287. [15] W. Knaak, E Mezei and B. Farago, Europhys. Letters 7 (1988) 529. 116] N.J. Tao, G. Li and H.Z. Cummins, Phys. Rev. Letters 66 (1991) 1334. I17] M. Fuchs, W. GOtze, I. Hofacker and A. Latz, J. Phys. Condens. Matter 3 ( 199t ) 5047. [181 M. Fuchs, J. Non-Cryst. Solids 172-174 (1994) 241. [19] G. Li, W.M. Du, A. Sakai and H.Z. Cummins, Phys. Rev. A 46 (1992) 3343. [201 W. Hess, Macromolecules 19 (1986) 1395; 20 (1987) 2587; G.V. Rostiashvili, Soviet Phys. JETP 70 (1990) 563; K.S. Schweizer, Physica Scripta T 49 (1993) 99. 1211 W. G6tze and L. Sj6gren, J. Non-Cryst. Solids 172-174 (1994) 16. [22] L. Sj6gren and W. G6tze, J. Non-Cryst. Solids 172-174 (1994) 7. [231 W. G6tze and L. Sjtigren, Rep. Prog. Phys. 55 (1992) 241. [24] H.Z. Cummins, G. Li, W.M. Du and J. Hernandez, Physica A 204 (1994) 169. [25] S. Yip, ed., Mode coupling theory and its experimental tests, Special Issue devoted to relaxation kinetics in supercooled liquids, Transport Theory Star. Phys. 24 (1995) pp. 9811267. [261 W. G6tze and L. Sj6gren, J. Math. Anal. Appl. 195 (1995) 230. [271 N.J. Tao, G. Li, X. Chen, W.M. Du and H.Z. Cummins, Phys. Rev. A 44 (1991) 6665. [281 C. Alba-Simionesco and M. Krauzman, J. Chem. Phys. 102 (1995) 6574; V. Krakoviak, C. Alba-Simionesco and M. Krauzman, preprint (1995). [29] U. Buchenau, C. Schtinfeld, D. Richter, T. Kanaya, K. Kaji and R. Wehrmann, Phys. Rev. Letters 73 (1994) 2344. [30] G. Li, W.M. Du, X.K. Chen, H.Z. Cummins and N.J. Tao, Phys. Rev. A 45 (1992) 3867; H.Z. Cummins, W.M. Du, M. Fuchs, W. G6tze, S. Hildebrand, A. Latz, G. Li and N.J. Tao, Phys. Rev. E 47 (1993) 4223. [ 31 ] W. van Megen and S.M. Underwood, Phys. Rev. Letters 70 (1993) 2766; Phys. Rev. E 49 (1994) 4206.
Chemical Physics 212 (1996) 47-59
Comments on the mode coupling theory for structural relaxation W. G 6 t z e a'b, L. S j 6 g r e n c a Physik-Department, Technische UniversiRit Miinchen, D-85747 Garching, Germany b Max-Planck-lnsfitutfiir Physik, Werner-Heisenberg-lnstitut, P.O. Box 401212, D-80805 Miinchen, Germany c Institutionen f6r teoretiskfysik. Giiteborgs Universitet, S-41296 Giiteborg, Sweden
Received 22 February 1996
Abstract In condensed matter the coupling of fluctuating forces to the infinite set of density fluctuation pairs leads to non-Markovian equations for the time evolution. Treating the dynamics of the density pairs with a factorization approximation yields closed equations of motion which provide a mathematically well-defined treatment of the cage effect for particle motion in liquids. The equations imply a bifurcation singularity connected with the appearance of a spontaneous arrest of the particle positions in disordered arrays. The evolution of structural relaxation on cooling or compression of liquids is obtained when the temperature or density approach critical values, which characterize the singularity. Von Schweidler's law is obtained as a generic reason for a-relaxation stretching. There appears a dynamical window where structural relaxation is described by a universal law, which deals with two time fractals. The relation of the mode coupling theory with Mountain's theory of structural relaxation is discussed and the interplay of relaxations and oscillations in supercooled liquids is demonstrated.
1. Introduction It is well known that cooled liquids exhibit low lying excitations whose relaxation times r,~ vary drastically with temperature T or density n. It is anticipated that these excitations are caused by the motion of clusters of molecules. Therefore they are referred to as structural relaxations. The processes with the largest time scales r~ are called a-relaxation. Upon supercooling, r~ can become of macroscopic size, i.e. 10-14 orders of magnitude larger than the femto second scale to, characterizing normal liquid dynamics. The scale r,, can become larger than the time scale texp, characterizing the experiment which was designed to probe the system's properties. If r,, << texp the sample exhibits a typical liquid response. However, if r,~ >> texp the sample behaves effectively nonergodically and responds like an amorphous solid, i.e.
it is a glass. In a cooling experiment there occurs a transformation from a liquid to a glass near a temperature Tg, called the calorimetric glass transition temperature, where r~ (T = Tg) = rex p. An understanding of structural relaxation is a prerequisite for a microscopic explanation of the glass transition. Structural relaxation is generic for condensed disordered matter. It occurs whenever crystallization can be bypassed in systems as different as molten salt mixtures, such as Ca(NO3)2. KNO3 ( C K N ) , and van der Waals liquids, e.g. salol. It is observed in systems as complicated as polymers and as simple as hard sphere colloidal suspensions. In this paper we will briefly review how proper extensions of earlier treatments of the cage effect in liquids leads to an ab initio theory of structural relaxation. Originally this approach, referred to as mode coupling theory (MCT), grew out of theories for simple liquids
0301-0104/96/$15.00 Copyright (~) 1996 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 0 1 - 0 1 0 4 ( 9 6 ) 0 0 0 9 4 - 8
48
W. G6tze, L. Sj6gren/Chemical Physics 212 (1996) 47-59
[ 1-3 ]. It will be explained, why some results of MCT are of general significance and can also be used to analyze experiments for very complex liquids. We will consider the relation beetween the visco elastic theory of liquids and the MCT. Furthermore, the evolution of structural relaxation out of the oscillatory motion of cooled condensed matter will be demonstrated.
2. Some major problems Three major problems can be specified with which a theory of structural relaxation has to cope.
2.1. Spontaneous arrest There is consensus that phonon assisted hopping of particles in an almost frozen potential landscape is the elementary process which leads to relaxation within the strongly supercooled state, say for T near Tg [4]. The landscape is produced by the liquid molecules which are arrested in some random array. Explaining relaxation requires the study of complicated percolation problems. Such problems have been analyzed in the theory of amorphous semi-conductors [5]. In these systems the molecules form a disordered static matrix and the electrons exhibit the dynamics of interest. In liquids there is no ad hoc separation of particles which produce free energy barriers and particles which percolate. The same interactions which give spontaneous arrest create also the potential barriers for the percolating particles. The microscopic equations of motion must lead to a separation of time scales: the duration of a hopping event must be orders of magnitude shorter than the lifetime of some free energy barrier. The separation of time scales mentioned above does not exist in the normal liquid state, say for parameters T and n near the triple point. Hence there must be a characteristic temperature Tc such that the whole picture of hopping transport loses its meaning as T increases seriously above Te.
2.2. The evolution of structural relaxation Simple liquids in their normal state do not exhibit structural relaxation. Thus there appears the question: how do structural relaxation phenomena evolve upon condensing the system by lowering T or increasing n?
The normal state dynamics of simple liquids is governed by binary collisions between the molecules, excluded volume and molecular field effects. The dynamics can be described by generalizing Boltzmann's kinetic equation. On condensing the liquid there appear several new features of the dynamics. They can be understood as result of correlated binary collision events or, equivalently, as non-linear interaction effects between excitation modes [6]. A perturbation calculation of these effects can explain for example the gradual decrease of the diffusivity D by up to one order of magnitude. Obviously, there must exist a critical temperature such that the above view of liquid dynamics becomes insufficient if T decreases seriously below it. It is plausible that this is the same temperature Tc already mentioned. Thus there should exist a cross over temperature Tc separating regimes of different dynamical behaviour [7]. In the strongly supercooled state, T << Tc, activated processes cause relaxation. In the normal and moderately supercooled state, T >> To, well understood interaction effects govern the dynamics. How can Tc be identified?
2.3. Relaxation stretching The dynamics of a many particle system is described by correlation functions or correlators ~b(t) or by their Fourier-Laplace transforms ~b(to) = ~b' ( ~o) + i~b"(to). Here the correlator of variable A is defined by ~b(t) = (A(t)*a)/(I A 12). The evolution in time, t, is given by the microscopic equations of motion and the averages are defined canonically. The correlation spectrum ~b"(to) for frequency to is related trivially to the susceptibility spectrum g " (w) = to~b" (to) [ 6 ]. Decay curves ~b(t) or spectra g " (to) for structural relaxation are stretched over huge windows of time or frequency respectively. For the a-process t has to be increased by two or more decades to observe a decrease of ~b(t) from 90% to 10% of its initial value. This window is at least 5 times larger than that for a Debye process ~bD(t) = f e x p - - ( t / r ~ ) . The Debye susceptibility spectrum is a Lorentzian with a half width of 1.14 decades: X~(to) = f(tor,~)/[1 + (toT"a)2]. However, a-peaks for the susceptibility spectra usually have a half width of two or more decades. Relaxation stretching is a paradigm for the dynamics of condensed disordered matter.
W. G6tze, L. Sj6gren/Chemical Physics 212 (1996) 47-59 Several fitting formulae have been proposed in the past, which often describe a-relaxation curves reasonably well• The oldest ones are due to Kohlrausch [8] and von Schweidler [9]. The Kohlrausch function modifies the Debye law by a stretching exponent /3 which is smaller than unity: ¢(t) = fexp[-(t/Ta)/3].
(1)
Here f2q is the mentioned phonon frequency. Parameter Vq = ~q')/q denotes a Newtonian friction• The non-trivial part is the memory kernel mq which is the constant of proportionality between a friction force at time t and the 'velocity' at the preceding time t'. The equation of motion (4) can be solved by introducing Fourier-Laplace transforms Cq (co) and mq (to) for the correlator and the kernel respectively:
Von Schweidler's law can be considered as the short time part of ( l ): ¢ ( t << ra) - f o¢ - ( t / r a ) b q- O(t2b).
(2)
It is a great challenge to explain how regular microscopic equations of motion can yield the power laws specified by the anomalous exponents/3 or b. Data analysis shows, that usually/3 5t b. This suggests that ( 1) is not an exact law but merely some data interpolation formula. The simplest function dealing with structure dynamics is the density correlator qbq(t). It is the correlator for density fluctuations of wave vector q: A = pq, q =l q ]. For the short time transient one gets [6]
(bq(t << to) = 1 - l(~Qqt)2 -q- . . . .
(3)
Here f2q = qv/V/-~q is the effective phonon frequency; v is the thermal velocity and Sq = (I Pq 12) is the structure factor• Thus (1) or (2) cannot be correct if t is small enough. There must exist some time scale, say rfl, such that the a-process describes dynamics only for t/> r E. Explaining r E is a necessary part of an explanation of a-relaxation.
3. Relaxation kernels
3.1• The equations of motion MCT starts with equations of motion which express the correlators ¢ in terms of relaxation kernels M. For the density correlator ¢q(t) a generalized oscillator equation is used [2]:
02q~q( t) d- J~q ((bq( t) -~- )'qtgt(bq( t ) t
+ fmq. o
(4)
49
2 L ( tO) tO -k d"2qMq cq(tO) =
tO2_
+ tOa qM,q(tO) ,
MLq( tO) = i'yq -q- mq( tO) .
(5)
This formula shows that ML(to) plays the same role in liquid theory as the polarization kernel in the theory of anharmonic phonons or of electromagnetic fields in matter. Shear dynamics is described by the correlator CqT for transversal currents A = jq.T [ 6 ] • The correlator is also expressed in terms of some kernel M T, which is a generalized (q, to)-dependent shear diffusivity or viscosity:
¢T (to) =
1 CO q- q2v2M]'q ( t o ) '
MT(to) --_ . l y qT + mqT (co) .
(6)
The zero wave vector limits of the kernels MoL'T(w) are, up to trivial manipulations, the longitudinal and transversal elastic modulus respectively. Within the Zwanzig-Mori formalism [6] one can define fluctuating forces Fq(t) and express the relaxation kernels for the current correlators in terms of the force correlators (F~ (t)* F~); a, fl = x, y, z. Because of rotational and inflection symmetry there enter only two independent functions and these are ME,X(t). Restricting our attention to frequencies below the band of microscopic excitations, tot0 << l, we can split off f r o m Mq(to) a white noise background iyq as a b o v e . The difficult part dealing with low frequency anomalies is mq(to). Eqs. ( 4 ) - ( 6 ) are exact reformulations of the microscopic theory; possible structural relaxation phenomena are hidden in the kernels mq(to) and mT(to).
W. Ggtze, L. Sjggren/Chemical Physics 212 (1996) 47-59
50
3.2. Force relaxation and density relaxation Let us consider a liquid where a single mode causes some low lying excitation. The Zwanzig-Mori reduction can be continued and the kernel written as fraction mq(Og) = - F q / [ t o + ~q(Og)]. Here the new kernel IXq(W) does not contain the mentioned mode dynamics anymore. Thus the frequency dependence of tZq(W) can be ignored for tot0 << 1 and it is possible to write tZq(W) = i/7"q. In this way a Debye model is obtained for the force relaxation: mq( t) = Fq exp(--t/'rq).
(7a)
This model approximates the structural relaxation peak for the force spectrum mq~(W) by a Lorentzian of area Fq and width 1/rq. Furthermore, if the discussion is restricted to small q, the wave vector dependencies of Fq, rq, Yq, Sq can be ignored. Then ( 4 ) - ( 7 a ) are equivalent to Maxwell's visco-elastic theory of liquids [6]. One implication of Maxwell's theory should be recalled. If the structural relaxation spectrum for the force correlator is well separated from the other liquid excitations, the density correlator exhibits a quasielastic resonance, called Mountain's peak [ 10] :
~q(~) = - - f q / [ ~ + iFq]; ~, 1/Tq<<~q, l / t o .
(7b)
The area fq of this peak in the density spectrum is given by the area Fq in the force spectrum:
fq = G / ( l + Fq).
(8)
The width parameter Fq of the structural relaxation peak for the density fluctuations is proportional to the rate parameter 1/7"q for the force fluctuations:
Fq = Cql'l"q,
Cq = 1/( 1 + Fq) .
(9)
The parameters fq, Fq approach finite non-zero constants for q --, 0. However, Maxwell's theory is not adequate for the discussion of structural relaxation, since it misses the stretching phenomenon. Apparently, structural relaxation is not caused by a single mode or by only a few low lying modes. It is important to understand that some of Mountain's results [ 10] are strict implications of the exact equations (4) and (5) which do not
depend on the approximation (7a). To identify these results we give first a precise mathematical meaning to the concepts 'well separated' and '<<' in (7b). To proceed we write mq(t) = fnq(t') with t" = ts. This is equivalent to mq(tO) = fnq((O)/S with ~ = to/s and rhq (~) denoting the Fourier-Laplace transform of rhq (t'). Now we consider the limit s tending to zero for fixed rescaled time ~', rescaled frequency ~ and shape function rhq. This limit procedure achieves the desired separation of the c~-process from the other processes. From (5) one derives (bq(~O)S --~ qbq(&) for s --~ 0, where q~q(~) = - 1 / [ & - 1 / & q ( ~ ) ] . This result is equivalent to (bq(t) --~ qbq(t-), where ~q(t") is the Fourier-Laplace back transform of 6q(W). Equivalently one gets d 6q(t3 =,~q(t) - = [ r n q ( 7 dt J 0
T'),~q(?)di'.
(10)
If one uses Maxwell's model (Ta) for rhq, Eq. (10) is solved by Mountain's results ( 7 b ) - ( 9 ) for 6q. However, independently of any specific model one concludes from (10): a quasi-elastic peak for the force spectrum of shape rhq~(~) causes a corresponding peak of shape ~qt(t~) for the density spectrum and vice versa. The areas Fq and fq of the a-peaks of force spectrum rhqt(~) and density spectrum t~q~(~) respectively are given by Fq = rhq(~"--~ 0) and fq = qbq(~ --, 0). The limit ? ~ 0 in (10) yields Mountain's result (8). There are different ways to define a relaxation time rq for the force correlator or a relaxation rate Fq for the density correlator. In any case, 7"q cx 1Is and Fq oz s, where the constants of proportionality are given by the shape functions rhq and t~q respectively. Eq. (9) holds with vanishing I'q and 1/7"qand a model dependent constant Cq. Let us examine the ansatz for the small i"dynamics: l?lq( t-) = Fq -- a q P + g q t 2b + . . . .
6q(t') = fq - hqr ~ + kqf 2b + . . . .
(lla)
This is an extension of yon Schweidler's law (2) to an asymptotic expansion of rh and q~. In a generalization of (8) one derives from (10), that the coefficients of the rhq-expansion determine recursively those for the 6q-expansion and vice versa. For example:
W. GiJtze, L. Sj6gren/Chemical Physics 212 (1996) 47-59 hq = Hq/(1 q- Fq) 2,
kq=[Kq(l+Fq)-AH2q]/(l+Fq)
3.
(lib)
Here parameter A abbreviates ,t = F ( 1 + b)2/F( 1 + 2b) and F is the gamma function. For b < 1 one gets A > 1/2. If ~q(?) or (~q(~') obey (1), one gets the identity 2Kq = H q/ or 2kq = h2 / f q respectively. The simultaneous validity of both these identities is compatible with (8) and ( l l b ) only if ~ = 1/2. The following result may therefore be concluded. If the a-process for the force relaxation follows the Kohlrausch formula (1) with exponent fl < 1, the a-relaxation for the density correlator is not a Kohlrausch process and vice versa. The microscopic equations of motion establish exact relations between certain structural relaxation processes. Such a relation between the elastic modulus spectrum, mq - " (~), and the dynamical structure factor q~q'(&) is given by (10). These relations exclude the possibility that the Kohlrausch function is a general law describing all a-processes. It is not possible to characterize a glass forming liquid by a single stretching exponent fl, contrary to what is anticipated occasionally [ 11 ]. On the other hand, there appears no obvious problem if one assumes a yon Schweidler expansion (2) such, that all a-processes of a given liquid are characterized by a common exponent b. In such case, the microscopic equations of motion imply relations among the expansion coefficients as exemplified by Eqs. (8) and (11).
Let us recall that the Maxwell-Mountain theory is based on the assumption that there is only a single slow mode which causes the low frequency dynamics of the force correlator mq(t). This assumption is not valid for a treatment of density fluctuations in supercooled liquids. Rather the infinite set of modes Pq-kPk couples to the forces Fq, and it is this set which causes the structural relaxation contribution to the fluctuating force correlators Mq. Approximations for correlators of products of slow modes like pip2 were studied in detail first by Kawasaki [ 12]. The essence of his procedure is to replace averages of products by products of averages: (Pl (t)p2(t)p3P4) "~ (Pl (t)p3)(P2(t)p4) -ff (3 ~ 4). Following this procedure we express the force kernels as polynomials of the density correlators: mq(t) = E V( q, kp)fbk( t)dpp ( t) = ~q( dpk ( t) ) , k+p=q (13)
mqT(t) = ~
v+(q, kp)4,k(t)4,,(t)
k +p=q = FT(q~k(t)) .
V(q, kp) = Sq&Sp{q[ k( l - S~ l ) (15)
Thus the vertices V,VT are equilibrium functions. Like Sq they vary smoothly with T and n, and they can be evaluated within established liquid structure theories
4.1. The mode coupling approximation MCT is based on an approximate expression of the relaxation kernels in terms of the density correlators. The approximation is motivated by the following observation. If a density fluctuation Pk relaxes with rate Fk, product modes like PkPp relax in the first approximation with rate Fk + Fp. Therefore all product modes relax within the same window as the simple variable pq. For zero wave vector the fluctuating force is a combination of pair modes:
Fo = E U k p - k p k . k
(14)
The coefficient Uk in (12) is given by the bare interaction potential. However, this potential is renormalized and replaced by combinations of the static correlation function Sq in the vertices V and VT. For example [2,3]:
+ p(1 - s p l ) ] }2/(2q3n).
4. Mode coupling functionals
51
(12)
[6]. Let us summarize the preceding results. The correlation functions are expressed first in terms of correlators of those fluctuating forces which couple directly to density fluctuation products. Force correlators are then approximated by polynomials of ~b~(t) like .Tq, .TqT. These polynomials are called mode coupling functionals since they express the coupling of force fluctuations and density fluctuations. In some cases one can express the desired quantity in terms of the ~bk(t); and then its structural relaxation anomalies are obtained as corollary of those for the density
W. G6tze, L. Sj6gren/Chemical Physics 212 (1996) 47-59
52
correlators. This is the case for the shear correlator, in particular for the shear viscosity, as follows from (6) and (14). The density correlator is reduced to the kernel mq(t) and the latter to dpq(t). One does not get an explicit expression for dpq(t) but the closed set of equations (4) and (13).
4.2. The cage effect Consider a tagged particle. If the other liquid particles were fixed, the tagged one could not move very far. For normal densities it would be trapped in a cage formed by its neighbours. Diffusion in normal liquids is possible only because the neighbouring particles are not fixed but move out of the way. The neighbours can move because their neighbours move, etc. Normal liquid dynamics, like diffusion, is possible only because complicated streaming patterns, called backflow, are formed. The rattling motion of the particles in cages does not contribute to the diffusivity D. Because of the cage effect, D is suppressed below the value Do expected within a kinetic equation picture. It was possible to evaluate the decrease of D/Do with increasing n with formulae equivalent to (13) and (15) and using the truncation of (13) to mq(t) [13]. Here ~bk~°) is some first order approximation for ~bk. But for large n, ~bk differs drastically from ~b~°) and the cited approach becomes inconsistent. The cage forming particles exhibit the same dynamics as those which are trapped in the cages. All fluctuations have to be treated on the same level and this is the aim of MCT. '~
.~q(¢~0))
4.3. The simplified MCT The closed set of equations (4), (13) and (15) defines the basic version of the MCT, which deals only with the cage effect. This version is referred to as simplified MCT in order to distinguish it from the extended theory, which includes also phonon assisted hopping effects. It is also referred to as the idealized MCT since it deals with a sharp transition to an ideal, i.e. truly non-ergodic, glass state. In most cases it is obvious how hopping effects alter qualitatively the results of the simplified theory. There are subtleties due to the interplay of hopping events with the cage effect; but these will not be considered in this paper.
The MCT equations are regular. There are no input parameters which depend strongly on T and n. All resuits have to be calculated by solving (4), (13) and (15) without inventing additional hypotheses. Temperature T enters essentially only indirectly via Sq in (15). I f T is lowered from the melting temperature T,, to Tg, Sq changes by about 20%. This small change causes all the known drastic changes of the MCT solutions. The equations of motion (4) and (13) define unique solutions ~bq(t), which have all the standard properties of correlators. For every fixed finite time interval 0 ~< t ~< tmax < ~ , the solutions fbq(t) depend smoothly on the parameters 12q,Vq and V(q, kp), characterizing the equations. Thus MCT provides a mathematically well defined model for a dynamics. There is no input in the MCT equations of motion which is directly related to structural relaxation or glass formation. In order to stress this point a schematic model for the equations will be formulated. This model ignores the wave vector q, and so it deals with a single correlator ~b only. The equation of motion becomes t
f mft-
~(t) + v(b(t) + 122c~(t) + $"22
t')qb(t') dt'
o = O.
(16a)
The kernel m is a polynomial such as
m( t) = vl~b(t) + v2~b(t) 2 .
(16b)
The interesting parameters of this model are the two coupling constants Vl /> 0, v2 /> 0. This schematic model, occasionally referred to as F12-model, has been used before in order to demonstrate some mathematical features of MCT [ 14]. The non-trivial results of the theory have two origins. One is given by the non-linearities: kernel m in (4) or (16a) depends on ~b and the mode coupling functional in (13) or (16b) is non-linear. The other is the retardation effect formulated by the integrals in (4) or (16a). The retardation interval (t - tt), where m(t - t r) is appreciable, is not given but depends on the solution. It is the interplay of the non-linearities with diverging retardation effects, which renders MCT so different from other dynamics theories.
W. GOtze,L. Sjligren/ChemicalPhysics 212 (1996)47-59 I
I
I
I
I
53
I
I
I
I
I
n=
.:o
0.8
~
0.6
.E
10 6
1 ~
2
E<0
E>O
0.4
\ \\3
10'
0.2
n= f 10 2
o.0
5
" ~.
e<0 I
I
I
-2
~
2
I
I
I
4
6
8
,
|
10
10 0
.
.
.
.
.
.
logt
Fig. 1. Correlators 40as functionof time t for the schematic model specified in Eqs. (16a) and (16b). The couplingconstantsdepend on the separation parameter e = -4-1/4" as specified in Eq. (1 6c). The time unit is chosen such, that/2 = 1. The damping constant is u = I. The dashed line is the solution at the critical point E = 0. The dashed dotted line is the critical law 40 = ( t o ~ t ) a with a = 0.327. The dotted line is the Debye curve 0.3 exp(--t/~ra).
-8
-6
-4
-2
0
logo Fig. 3. Correlationspectra 40" = X"/w as function of frequency ca for some correlators shown in Fig. 1.
vl,2 = v~,2( 1 + e ) , e=±l/4",
n=0,1 .....
(16c)
d
C<0
10~
n=
7
6
5
4
3
10 -2
I
-12
-10
-8
54
-6
-4
-2
0
logo
Fig. 2. Susceptibilityspectra X" as function of frequency ~o for the correlators shown in Fig. 1.
5. A simplified view of structural relaxation Figs. 1, 2 and 3 show the correlators ~b(t), susceptibility spectra g"(o~) and correlator spectra (b"(~o) calculated for the schematic model which is defined by equ. (16). A damping constant v = 1.0/2 is taken and units are chosen such that /'2 = 1. The coupling constants in the vl-v2 parameter plane are shifted on a straight line through the point v~ = ( 2 a - 1 ) / A 2 , v~ = 1/A 2, A = 0 . 7 :
The correlators for e < 0 and n > 3 exhibit a decay process from the plateau fc = 1 - • to zero, whose time scale ~-,~ increases dramatically with decreasing e. This decay produces the low frequency susceptibility peaks in Fig. 2 and the quasielastic peaks for the spectra ~b"(w), shown in Fig. 3 for n = 3 and 5. A Debye peak g ~ cx (o)7"a)/[ 1 Jr- ((.o'/'a) 2] is added as dotted line in Fig. 2 with r,~ chosen so, that the peak maximum for n = 7 is matched. The corresponding Debye decay, fc e x p ( - t / T ~ ) , is also shown as the dotted curve in Fig. 1. The slow decay process can be fitted very well by the Kohlrausch law (1) with [3 = 0.58. Thus the schematic model in (16) exhibits the main feature of structural relaxation: a slow control parameter sensitive o~-process, which is stretched. In this section some comments on the physical picture of structural relaxation as obtained from the simplified MCT will be made.
5.1. Ideal glass states There is a weak coupling regime of small vertices
V(q, kp) in (15), where the correlators relax to the equilibrium value zero: dpq(t ~ o~) = 0. These solutions describe ergodic liquid dynamics as exempli-
W. GOtze, L. SjOgren/ Chemical Physics 212 (1996) 47-59
54
fled by the • < 0 curves in Figs. 1-3. In the complementary strong coupling regime density fluctuations do not disappear but arrest (bq(t --~ cxz) = f q > 0, as exemplified by the e > 0 curves in Fig. 1. For the strong coupling solutions the density spectra exhibit elastic peaks of intensity fq : &q'(W ~ O) = 7rfq(3((o). This is a signature for a solid with DebyeWaller factor fq. Contrary to what is obtained for a crystalline solid, fq is a smooth function for all wave vectors q, i.e. the solid found is amorphous. From (14) a non-zero long time limit is obtained for the transver> sal force fluctuations: mT ( t -+ 00 ) -T= 0. This yields for the transversal current correlator (6): (bT (to) oc l/[to2--c2q2 +O(wq2) ] , Cq2 = v2fTq. Long wave length low frequency shear disturbances propagate as isotropic transverse sound waves with speed co. One concludes that the strong coupling solutions describe ideal glasses. For low T and large n, the coupling constants V(q, kp) are large and then the cage effect yields a spontaneous arrest of the density fluctuations. MCT explains the appearance of a frozen potential landscape as result of cooperative effects among density fluctuations. A tagged particle cannot percolate because its neighbours cannot move out of the way. The latter cannot move because their neighbours are blocked etc. The Debye-Waller factor of the glass obeys the equation
fq/(1-fq)
=5Fq(fk),
0<~ f q < 1 .
(17)
This result is equivalent to Mountain's formula (8), where Fq = Uq ( f k ) . However, the area Fq of the structural relaxation peak is not implemented within our theory as an ad hoc parameter, but it has to be evaluated from the equilibrium structure by solving the coupled equations (13) and (17).
5.2. Glass transition singularities On lowering T or increasing n, the vertices V(q, kp) in (15) increase. Critical values Tc or nc appear, where the simplified MCT predicts a bifurcation. The low frequency spectra or long time correlators exhibit a singular dependence for small separation parameters o- = C•, where • = (Tc - T)/Tc or • = (n - nc)Inc. The critical parameters are referred
to as glass transition singularities (GTS). Within the simplified MCT, the critical values Tc or nc mark transitions from liquid to glass dynamics. If hopping effects are incorporated the system cannot be driven through a GTS, but it can be close to it. In this case Tc or nc mark the cross overs between regions of different dynamical behaviour as discussed in section 2.2 For parameters near a GTS, there appear relaxation processes outside the transient regime: t >> to. These cause spectra below the band of microscopic liquid excitations. This dynamics which can neither be understood within normal liquid pictures nor within standard solid state models is structural relaxation. Its origin is the GTS. From Figs. 1-3 one infers to ,-~ 1. For this schematic model structural relaxation is observed for t > 10 or w < 0.1 for all parameters in (16c) w i t h [ • I< 1/4.
5.3. The critical dynamics for to << t << r~ At the GTS, ~bq exhibits power law decay towards the critical Debye-Waller factor f~q: (bq( t) = f~q + hq( t o / t ) a + O ( 1 / t 2 a ) .
(18)
The corresponding susceptibility spectrum varies sublinearly: Xq'(W) cx hq(wto) a. This process is called critical dynamics. There is a closed but involved set of formulae to calculate the critical exponent 0 < a < 1/2 and the critical amplitude hq > 0 from f'q [14]. The time to is obtained by matching (18) to the short time transient. For the model (16) one gets a = 0.327. The solutions at the critical point • = 0 are shown by the dashed lines in the figures and the leading terms (to~t) a and (wt0) a, respectively, by dashed dotted lines. If parameters are chosen away from the GTS, be it because T 5i To, n 5/ nc or because of hopping effects, a finite time scale rB appears such that the critical dynamics is observed for to << t << rt~ ; 1/r~ << to << l/to. Within the specified window the correlators and spectra vary only weakly with the control parameters. However, the scale rB depends sensitively on T and n; r~ diverges if the GTS is approached. For to << t << ra the particles are trapped in cages. The increase of rt~ with increasing T, for T < To, is a precursor phenomenon signalling the instability of the frozen
W. G6tze, L. Sj6gren/ Chemical Physics 212 (1996) 47-59
structure at Tc. The increase of ~'t~ on cooling, for T > To, is a precursor of the freezing at To. The critical spectra have first been measured by neutron scattering [ 15 ] and by polarized as well as depolarized light scattering [ 16] for the above mentioned glass former CKN.
55
cay from ]q proportional to t b (2). As suggested by Levy's generalization of the central limit theorem, (10) and (13) are solved for q ~ ~ by O~q(t-) = f~exp(-Fq? b) [18]. Thus MCT has established a derivation of the Kohlrausch law from the equilibrium structure, albeit in a very special limit only. In general ( 1) is not valid, but often it is a good fitting formula.
5.4. MCT a-relaxation for ~'t~<< t 5.5. MCTfl-relaxationfor to << t << ~', For ideal glass states, the time z# marks the cross over to arrest: (bq(t ~ Tfl) ,~ fq. The corresponding elastic peak in the density spectrum can be viewed as an a-peak of width zero: 1/76 = 0. For T > Tc or n < nc the time 7t~ marks the beginning of the decay below f~. The decay of q~q(t) from f~ to the equilibrium value is the MCT a-process in the proper sense. The time ~'t~specifies the a-relaxation window: r# << t ; w << lira. Both r,~ and r# diverge upon approaching the GTS; and the ratio r,/7a also diverges. Structural relaxation is a process occurring in several steps, specified by scales like r,~ and ~-#. Since z~ diverges, (10) is valid as an asymptotic law near the GTS. However, the kernel rhq(t-) must not be used as an ad hoc input but rather it is related to the a-correlator via specialization of (13) to the critical parameters: /nq (t-) ----o~";( ~k (t-) ). Thereby closed equations are obtained allowing the evaluation of the a-spectra from the equilibrium structure [ 17]. Von Schweidler's law (2) is obtained as an exact implication of (10) and (13) [ 14]. There is a straightforward but involved procedure to evaluate from 5t-q the so-called exponent parameter .h, 0.5 ~< ,h < 1. This parameter fixes both exponents a and b:
The essence of structural relaxation is the partial arrest of density fluctuation near the critical DebyeWaller factor, fq. There is the mesoscopic dynamical window, to << t << z~, where t~qbq(t) = (bq(t) -- f~ is small. The dynamics within this window is referred to as the fl-process. This process begins with the critical decay and it ends with the von Schweidler process [ 14]. The beginning of the a-process is the end of the fl-decay. Within the fl-relaxation window one finds the factorization: ~(bq(t) = h q G ( t ) + O(G2). One factor is the above mentioned critical amplitude, hq. The other factor is called the/3-correlator G; it depends sensitively on t and on the control parameters. Hence there are no correlations between the propagation of density disturbances in space with those in time. In other words, fl-dynamics deals with localized modes, a clear manifestation of the cage effect. If one substitutes Cq (t) = f q + hqG + 0 ( G 2) e.g. in (14), one gets my(t) = ~'Tq( fk) + h~G( t) + O(G2), c where hqT = Ek[O.T'~T (fk)/Ofk]hk. A similar result holds also for other variables A, which can be expressed as polynomials of density correlators:
F( 1 - a)2/I'( 1 - 2a) = A = F ( 1 + b)2/F( 1 + 2b).
dpA ( t) = f cA + h A G ( t / t o ) -4- OA( G 2) .
(19) For special choices of V(q, kp), it is possible to obtain ,h = 1/2, i.e. b = 1. For the model (16) this happens only for v~ = 0, v~ = 4, and (10) and (13) are then solved by Maxwell's model. Generically b < 1, and von Schweidler's law is identified as the general reason for stretching. For large wave vector q the phase space for the decay of pq into pair modes Pq-kPk becomes large. Simultaneously, the a-peak intensity, f~q,becomes small. Thus the kernel mq in (13) becomes a sum of many small terms, and each term exhibits the initial de-
(20)
All such functions, the density correlator (A = pq), the longitudinal and transverse elastic moduli (obtained from A = F0), depolarized light scattering functions which are caused by dipole induced dipole interactions, the dielectric loss spectrum etc., exhibit the same fl-dynamics. Different variables merely project different contributions of the same local modes. The projection is quantified by the equilibrium parameter hA. The problem of solving the coupled equations (4) and (13) for the many functions qbq(t) is reduced to solving a single equation for G [ 14],
56
O"
W. GOtze, L. Sj6gren/Chemical Physics 212 (1996) 47-59
+ hG(t') 2 -
d /, ^ / G ( t - ~ ) G ( F ) dF = 0 J
(21)
with the initial condition G ( f ~ 0)• ~ 1. Here to, h and tr have the same meaning as before: matching scale, exponent parameter and separation parameter respectively. The solution of (21 ) depends smoothly on A and formula (19) is an implication of (21). 5.6. Universality features
Eqs. (20) and (21) establish universality features or laws of corresponding states. Systems with the same a exhibit the same fl-spectra: X ~ ( t o ) / h a = & G " ( & ) . This holds if the spectra are measured with scale ha and frequencies with scale 1~to, & = toto. In addition, the evolution of the spectra with changes of a control parameter such as T is the same, provided these are entered in the rescaled form o- = C(Tc - T)/Tc. The results are robust. The details enter merely as the set of parameters ha, to, ,~, C and Tc. The parameters change smoothly if e.g. the pressure is altered. With the appropriate specification of scales, the fldynamics shown in the figures for the schematic model (16) is the same as that for salol. For the figures A -0.7 was chosen, this being the measured value for salol [191. MCT equations have also been derived for binary mixtures. Eqs. ( 4 ) - ( 6 ) , ( 13)-(15) and (17) are generalized so that all quantities such as /2q, fq etc. are replaced by matrices. The same liquid to glass bifurcation singularity has been identified as for simple systems. Since the universal results ( 1 8 ) - ( 2 1 ) merely reflect the topology of the bifurcation manifold, they remain valid without any modification. Even for polymers, mode coupling equations have been derived, which contain the cage effect functional (13) [20]. Hence Eqs. (20) and (21) hold also in this case. The theory provides a scenario for structural relaxation and the general results can be used without explicit reference to the microscopic details. Microscopic calculations are necessary if one wants to determine numerical values for parameters such as T c, hq or .A. They are also necessary to predict the sizes of the O(G 2) terms in (20). It cannot be predicted without detailed calculations, whether a system can be driven so closely to a GTS, that the discussed
asymptotic laws (2), (10) and (20) are of relevance for the interpretation of data. The specified universality refers to the fl-dynamics, including the initial part of the a-process. The full aprocess depends on the detailed structure, as follows from (10) and (13). Different systems can exhibit quite different a-peak shapes, even if they have the same von Schweidler exponent b. The solutions of the MCT equations for a and fl processes are summarized elsewhere [21,22]. Comparisons of the asymptotic formulae of the MCT with experiments are reviewed in Refs. [23,24] and in a series of articles in Ref. [25].
6. Structure and structural relaxation
The glassy dynamics, which was discussed above, is due to large values of the kernels mq(tO). Within the dynamical window, where w//2q2 and yq can be neglected in comparison with mq(¢.o), Eq. (5) simplifies to [23] ~/)q(O)) -~ mq(tO)/[ 1 -- tOmq(tO) ] .
(22)
This equation is independent of the parameters /2q and yq, which characterize the microscopic dynamics. Therefore one can conclude the following. The long time or low frequency dynamics is specified completely by the vertices (15). The latter are given by the equilibrium structure. Glassy dynamics deals merely with the statistics of the orbits of the many particle system in configuration space. This statistics is determined by the potential landscape, i.e. by the Boltzmann factor e x p ( - U / T ) with U denoting the interaction potential. The microscopic dynamics merely sets the time scale to for the exploration of the potential landscape. If q~q(t) solves the coupled equations (13) and (22), the same is true for ~bl (tx) for every x > 0. The scale to for the glassy dynamics is set by matching the solution of (22) to the microscopic transient. Thereby MCT gives a precise meaning and a justification of the statement that glassy dynamics is structure dynamics. The results of the preceding paragraph can be demonstrated for the schematic model, defined by (16a) and (16b). While Figs. 1-3 exhibit the results for a microscopic dynamics with/2 = u = 1, Figs. 4-6 exhibit solutions for/2 = 1 and r, = 0. Corresponding
W. GStze, L. Sjt~gren/Chemical Physics 212 (1996) 47-59 I
I
I
I
i~
I
0
n=
o.8
57 I
I
I
I
10 6
1
0.6
2
c>O
e
10 4
0.4 E>O 0.2
10 2
" " " , , ' ~
5
"
-
-
•
3
. . . . .
lo
. . . . . . .
n=
0.0
e
10o I
,
-2
L 2
I
0
I 4
I 6
I 8
I 10
Iogt Fig. 4. Correlators for the same model described in Fig. 1 but for ~,=0.
i
-8
-6
-4
-2
0
logo) •
'
I
e<0 n=
I
I
7
6
I
5
I
4
I
3
I
Fig. 6. Correlation spectra for some of the correlators shown in Fig. 4.
2 1 ~ L//llll|
can show [26], that the correlators are completely monotone functions; i.e. there are distributions of rates pq( [') ~ 0 such that P
10 .2 1
(bq(t) =/exp(-Ft)pq(F) dF.
(23)
o h
i
-12
-10
-8
-6
-4
-2
0
log0~ Fig. 5. Susceptibility spectra for the correlators shown in Fig. 4.
solutions refer to identical mode coupling functionals, specified by (16c). The short time dynamics in Fig. 1 differs from that shown in Fig. 4. For the latter the short time oscillations are more pronounced. However, the decay curves for tO > 200 of Fig. 1 coincide with the corresponding ones of Fig. 4 for tO > 100, provided the time scale for the latter is changed by a factor 4.00. This rescaling amounts to a shift of the curves parallel to the logarithmic time abscissa. A similar statement can be inferred by comparing the susceptibility spectra of Fig. 2 with the shifted ones of Fig. 5. A special model for a microscopic dynamics is obtained by dropping the first term o3?qbq(t)in Eq. (4). In this case the microscopic dynamics is specified by the set of relaxation rates "yq > 0. For this model one
This model exhibits identical structure dynamics as discussed above in connection with (22). Therefore one concludes that structure dynamics is a superposition of Debye relaxation processes. In particular the aprocesses and the fl-processes can be written as such a superposition (23). The proof of (23) provides the justification to view structure dynamics as relaxation. A relaxation function like (23) starts with a linear short time expansion, ~b(t) = 1 - (t/T) + .--, provided the distribution pq(I') has a finite first moment. However, the MCT solutions exhibit the exact short time expansion (3) as any N-particle system with regular interaction potentials. A superposition of Debye processes (23) decreases monotonically with increasing time, while MCT solutions can exhibit oscillations as shown for tO < l0 in Figs. 1 and 4. The spectra ~bq'(to) for a relaxation process (23) decrease monotonically with increasing frequency to. But the MCT spectra can have bumps as is shown for frequencies to near ,O in Figs. 3 and 6. The results of Fig. 6 for e > 0 and n = 0 - 3 are also shown on linear scales in Fig. 7.
W. GOtze, L. SjOgren/Chemical Physics 212 (1996) 47-59
58
_-
v/-y+ -
1 (24b)
0.6 n~
0.4 v
0.2
I
•
I
\
I
2
3
Fig. 7. Correlation spectra from Fig. 6 for e > 0 and n = 0 - 3 plotted on linear scales. The dashed curve is the asymptotic result (24b) for n = 0.
These bumps are more pronounced for lower temperatures, as one can notice by comparing the results for • > 0 and n = 0 with that for n = 1. These bumps are due to inhomogeneously broadened oscillations. Deep in the glass state, i.e. for T considerably below Tc, the particles are well localized in cages. They will perform essentially harmonic oscillations. The distribution of sizes and shapes of the cages implies a distribution of oscillator frequencies. Hence one gets for the disordered solid a bump in the spectrum instead of a sharp line which one would get for a crystalline solid. In the strong coupling limit, i.e. for temperatures T far below the critical value Tc, the MCT equations simplify considerably. One finds as solution for the dynamical susceptibility X ( w ) a superposition of oscillator functions
X(w) =
f
--J'22
dx p ( x ) o~(o~ + i~) - ~ ( x ) 2 "
(24a)
The squares of the oscillator frequencies/'2(x)2/~ 2 = tzj + 2 x v / - ~ are distributed with density p ( x ) = 2~/1 - xZ/Ir. The parameters/*l and ,L~2 can be expressed in terms of the mode coupling functional. From (16b) one finds/Zl = 1 + v l +v2,/*2 = Vl +2v2 [ 14]. Figs. 4-7 demonstrate the example v = 0. In this case (24a) implies the spectral density for 0<~2
Here the edges/2+ of the bump are given by/'2~:/J22 = /Zl 4- 2v/-~-~. The dashed curve in Fig. 7 is the result (24b) calculated for parameters/zl,/x2 for the n = 0 coupling constants. Thus the MCT predicts deep in the glass a semi-ellipse for the shape of the bump as one would expect within Wigner's theory of random matrices. Let us emphasize that the MCT results for the short time oscillations and spectral bumps, as opposed to the structural relaxation phenomena in the strict sense, depend on the details of the short time dynamics. Spectral bumps are often observed for glass forming liquids and they are occasionally referred to as boson peaks. Tao et al. [27] pointed out that the cited MCT results provide a natural explanation of the boson peak phenomenon. Alba-Simionesco et al. [28] have shown that results for simple schematic MCT models can fit reasonably well light scattering spectra for toluidine, salol and CKN. The fits [ 28 ] hold for the whole Raman band including the boson peak region; in addition they describe the evolution of the structural relaxation. From equations (24) one concludes the following. With decreasing coupling the boson peak position shifts to lower frequencies and the peak intensity increases. These phenomena are demonstrated in Figs. 6 and 7. Thus MCT implies for the simplest models a vibrational softening and an increase of vibrational spectral densities in the boson peak region on heating. These predictions have been verified in a recent neutron scattering study [29].
7. Conclusions The MCT deals with a mathematical model for a dynamics which explains the evolution of structural relaxation as result of the appearance of a bifurcation singularity. In this paper we have considered the simplest example for such singularity. In all cases a singularity establishes a universality class or a scenario, where the correlation functions within certain windows depend on a finite number of parameters such as Tc, hq and ,t only. The connection between these parameters and the microscopic details is robust. Thereby a qualitative understanding of the solutions can be ob-
v~ G6tze, L. Sj6gren/Chemical Physics 212 (1996) 47-59 tained and a variety o f predictions for relations between m e a s u r a b l e quantities, like ( 1 9 ) , can be made. T h e M C T is based on the m i c r o s c o p i c theory o f the cage effect in s i m p l e liquids. Quantitative details for fq, hq, nc, A, and a - r e l a x a t i o n shape functions have been w o r k e d out for the hard sphere system, LennardJones systems and s o m e mixtures. The w o r k for m o r e c o m p l e x liquids remains to be done. The observation o f the evolution o f structural relaxation requires spectroscopic techniques, which bec a m e available o n l y recently. The first detailed measurements have been m a d e by depolarized light scattering for C K N by Li et al. [30] and by photon correlation spectroscopy for hard sphere colloidal suspensions by van M e g e n and U n d e r w o o d [ 31 ]. These authors [ 30,31 ] have c o m p a r e d their data c o m p r e hensively w i t h the available M C T results. Their w o r k p r o v i d e s a solid basis for a rational assessment o f the m o d e c o u p l i n g theory for structural relaxation.
Acknowledgement We thank H.Z. C u m m i n s for helpful and stimulating discussions and M. Fuchs for the c o m p u t e r p r o g r a m used to evaluate the figures.
References I 1 I E. Leutheusser, Phys. Rev. A 29 (1984) 2765. [21 U. Bengtzelius, W. GOtze and A. Sj(51ander, J. Phys. C 17 (1984) 5915. [31 W. G6tze and L. Sj5gren, Z. Physik B 65 (1987) 415. [41 J. Wong and C.A. Angell, Glass: structure by spectroscopy (Dekker, Basel, 1976); S. Brawer, Relaxation in viscous liquids and glasses (The American Ceramics Society, Columbus, 1985). 151 H. Scher, in: Slow dynamics in condensed matter, eds. K. Kawasaki, T. Kawakatsu and M. Tokuyama, AIP Conference Proceedings 256 (1992) 485, and references therein. [6] J.P. Boon and S. Yip, Molecular hydrodynamics (McGrawHill, New York, 1980); J.P. Hansen and I. McDonald, Theory of simple liquids (Academic Press, London, 1986). [71 M. Goldstein, J. Chem. Phys. 51 (1969) 3728.
59
[8] R. Kohlrausch, Pogg. Ann. Phys. 51 (1854) 56. [9] E. von Schweidler, Ann. Phys. 24 (1907) 711. [101 R.D. Mountain, J. Res. Nat. Bur. Standards 70 A (1966) 207. I 11 ] J.C. Phillips, J. Non-Cryst. Solids 192/193 (1995) 393. [121 K. Kawasaki, Phys. Rev. 150 (1966) 291. [131 R.I. Curkier and J.R. Mehaffey, Phys. Rev. A 18 (1978) 1202. [ 14] W. G6tze, Liquids, freezing and the glass transition, eds. J.P. Hansen, D. Levesque and J. Zinn-Justin (Noah-Holland, Amsterdam, 1991) p. 287. [15] W. Knaak, E Mezei and B. Farago, Europhys. Letters 7 (1988) 529. 116] N.J. Tao, G. Li and H.Z. Cummins, Phys. Rev. Letters 66 (1991) 1334. I17] M. Fuchs, W. GOtze, I. Hofacker and A. Latz, J. Phys. Condens. Matter 3 ( 199t ) 5047. [181 M. Fuchs, J. Non-Cryst. Solids 172-174 (1994) 241. [19] G. Li, W.M. Du, A. Sakai and H.Z. Cummins, Phys. Rev. A 46 (1992) 3343. [201 W. Hess, Macromolecules 19 (1986) 1395; 20 (1987) 2587; G.V. Rostiashvili, Soviet Phys. JETP 70 (1990) 563; K.S. Schweizer, Physica Scripta T 49 (1993) 99. 1211 W. G6tze and L. Sj6gren, J. Non-Cryst. Solids 172-174 (1994) 16. [22] L. Sj6gren and W. G6tze, J. Non-Cryst. Solids 172-174 (1994) 7. [231 W. G6tze and L. Sjtigren, Rep. Prog. Phys. 55 (1992) 241. [24] H.Z. Cummins, G. Li, W.M. Du and J. Hernandez, Physica A 204 (1994) 169. [25] S. Yip, ed., Mode coupling theory and its experimental tests, Special Issue devoted to relaxation kinetics in supercooled liquids, Transport Theory Star. Phys. 24 (1995) pp. 9811267. [261 W. G6tze and L. Sj6gren, J. Math. Anal. Appl. 195 (1995) 230. [271 N.J. Tao, G. Li, X. Chen, W.M. Du and H.Z. Cummins, Phys. Rev. A 44 (1991) 6665. [281 C. Alba-Simionesco and M. Krauzman, J. Chem. Phys. 102 (1995) 6574; V. Krakoviak, C. Alba-Simionesco and M. Krauzman, preprint (1995). [29] U. Buchenau, C. Schtinfeld, D. Richter, T. Kanaya, K. Kaji and R. Wehrmann, Phys. Rev. Letters 73 (1994) 2344. [30] G. Li, W.M. Du, X.K. Chen, H.Z. Cummins and N.J. Tao, Phys. Rev. A 45 (1992) 3867; H.Z. Cummins, W.M. Du, M. Fuchs, W. G6tze, S. Hildebrand, A. Latz, G. Li and N.J. Tao, Phys. Rev. E 47 (1993) 4223. [ 31 ] W. van Megen and S.M. Underwood, Phys. Rev. Letters 70 (1993) 2766; Phys. Rev. E 49 (1994) 4206.