- Email: [email protected]
Slow dynamics in condensed matter: spinodal decomposition and glass transition
PHYSICA ELSEVIER
Physica A 224 (1996) 1-8
Slow dynamics in condensed matter: spinodal decomposition and glass transition Kyozi Kawasaki a, Tsuyoshi Koga b'l a Department of Natural Science and Mathematics Chubu University, Kasugai, Aichi 487, Japan b Department of Physics, Faculty of Science, Kyushu University, Fukuoka 812, Japan
Abstract
A brief review is presented about our recent work on computer studies of a continuum model describing spinodal decomposition of binary fluid, which quantitatively accounts for corresponding experimental results. We then switch over to discussing problems of glass transitions. First we consider the ultraslow mode observed in fragile glass-formers, where the diffusive behavior of this mode is explained in terms of phase dynamics describing slow relaxation of spatial heterogeneous structure resulting from spontaneous break-down of continuous translational symmetry. Next we present a stochastic model for density fluctuation in supercooled liquids, which is mapped onto a certain kinetic lattice gas model which is convenient for Monte Carlo simulation studies.
1. Introduction
Interests in slow dynamics in condensed matter arise from the fact that the processes inevitably involve cooperative motions of many particles, which has always provided challenging problems in statistical physics. Here we shall discuss two such examples, with which we have been concerned recently. The next section takes up spinodal decomposition in binary fluid and the subsequent two sections are devoted to more recent and ongoing works related to glass transitions.
1 Present address: Hashimoto Polymer Phasing Project, ERATO, JRDC, Morimoto-cho, Shimogamo, Sakyoku, Kyoto 606, Japan. 0378-4371/96/$15.00 (~) 1996 Elsevier Science B.V. All fights reserved SSD1037 8-437 1 ( 9 5 ) 0 0 3 1 0 - X
2
K. Kawasaki, T. Koga/Physica A 224 (1996) 1-8
2. Spinodal decomposition in binary fluids 2.1. TDGL equation f o r binary fluids
The model used in our computer simulation [ 1-3] is the following time-dependent Ginzburg-Landau (TDGL)-type equation for the order parameter S ( r ) [4]: S(r, t) = LV2/z(r) - V S ( r ) .
f T(r-
r ' ) . V ' S ( r t ) i z ( r ' ) d r ',
(2.1)
where L is the Onsager kinetic coefficient, /~(r) -- 6 F { S } / 6 S ( r ) is the chemical potential, F { S } is the Ginzburg-Landau type free energy functional and T ( r ) is the following Oseen tensor: T(r) = ~
4-
,
(2.2)
where 7/is the shear viscosity and 1 is the unit tensor. The second term on the right-hand side of Eq. (2.1) represents the hydrodynamic interaction between the order parameter fluctuations mediated by the velocity field. In the derivation of Eq. (2.1), we assume that the relaxation of the local velocity is more rapid compared to that of the local order parameter [ 2,4]. By rescaling the variables in Eq. (2.1), we find that there is only one dimensionless parameter in Eq. (2.1) [2]. We call this the dimensionless shear viscosity, which measures the inverse of the strength of the hydrodynamic interaction. In the computer simulation, we employ the cell-dynamic method [5]. In the cell-dynamic version of Eq. (2.1), the parameter which corresponds to the dimensionless shear viscosity is denoted by 7/D. The system used in the computer simulation is a cubic lattice of size 1283 with periodic boundary conditions. 2.2. Results and discussions
At first, we show the time dependence of the characteristic length scale I(t) and discuss the growth exponent z: l(t) ~ t z . We calculate the spherically averaged structure factor Ik(t) of the order parameter. We use the first moment of Ik(t) denoted as kl (t) and the peak position of lk(t) denoted as km(t) as the characteristic wave number. The time dependence of kl (t) for fluids 070 = 1 and 7/o = 5) and "solid" is presented in Fig. 1. We call the conserved system without the hydrodynamic effect "solid" in this paper. From Fig. 1, we find that in the low viscosity case (~TD = 1), the effective exponent Zeff(t) takes the value 1 in the late time region. This indicates that the domain growth in this time region is dominated by the flow induced by the surface tension [6]. On the other hand, the value of z~ff(t) approaches 1/3 in the late stage in the solid model, which is the Lifshitz-Slyozov type growth law [7]. In the high viscosity case (r/D = 5), ZerO(t) increases with time but has not yet reached 1 even in the late time region of our simulation. It is expected that it takes a long time to reach the z = 1 region
K. Kawasaki, T. Koga/Physica A 224 (1996) 1-8
3
Slope=-1
10-1'102
103
04
Fig. 1. Time dependence of kl(t) for fluids with ~/o = 1 (o) and 7/D = 5 ( ~ ) and for solid (E]) on double-logarithmic scales.
- - simulation (fluid) -- simulation (solid) o PB/PI
x ' ',
o o
o
/ x Fig. 2. Porod plots of the scaling functions obtained by the computer simulation and the experiments of PB/PI. In the computer simulation, we use the transformation: S(r) ~ Se sgn[S(r)], to removethe effects of the interface thickness. Here Se is the equilibrium value of IS(r)I.
in the high viscosity case. These results are qualitatively consistent with the results using the interface equation of motion [2,8]. The branching of the time dependence of the characteristic wave number in Fig. 1 is due to the difference of the strength of the hydrodynamic interaction and corresponds to the so-called N branching in polymer blends [2,9]. Next we take up the scaling function F ( x ) = k m ( t ) 3 1 k ( t ) , where x =_ k / k i n ( t ) . Porod plots of the scaling functions for fluid and solid are presented in Fig. 2 together with the experimental results of a critical mixture of polybutadiene (PB) and polyisoprene (PI) by Takenaka and Hashimoto [ 10]. In this experiment, the scaling function is obtained in
K. Kawasaki, T. Koga/Physica A 224 (1996) 1-8
the hydrodynamic growth region. We find that the scaling function for fluid is different from that for solid [2] and the scaling function obtained by the experiment is more close to the scaling function for fluid than that for solid [3]. We note that the same problem was subsequently studied in great detail by Shinozaki and Oono [ 11 ].
3. Uitraslow modes in glass-forming systems In the preceding section we described a typical example of the usual phase ordering where the nature of the ordered phases is simple. A more challenging class of problems would be the cases with ill-defined ordered phases. A typical example is the problem of glasses. Here the "ordered phase" contains lots of disorder and is still poorly understood. Recently much attention is being directed to non-equilibrium dynamics following the quench below the glass transition temperature, especially in the spin glass community. This is known as the aging problem of glass [ 12,13]. We shall not touch upon this fascinating problem at this time since this is beyond our competence, but we shall discuss a few other problems related to glass. The first topic taken up is the ultraslow modes which are associated with the excess scattering which often appears in the so-called fragile glass-formers [ 14]. This excess scattering is caused by fluctuations whose spatial lengths can reach a few thousand angstroms and whose origin is not well understood. Such excess scattering was earlier observed for some polymer solutions in the semi-dilute regime [ 15] and also for polyelectolyte solutions with low salt or salt free solvents [ 16]. The whole subject is full of controversies. We do not go into the controversies here but rather point out a close relationship between ultraslow modes and the static heterogeneous structure inherent in the system [ 17]. Here, however, we assume that our system possesses translational invariance and the spatial heterogeneities can appear as a result of the spontaneous break-down of the continuous translational symmetry. Such spatial structure is undergoing random deformations by thermal noise whose amplitude increases as the wavelength of deformation increases. That is, the restoration force of deformation decreases towards zero as the wavelength of deformation tends to infinity. The fact that the relaxation rate of ultraslow modes which is identified here with the relaxation of this deformation is of the diffusion type is a direct consequence of the spontaneous break-down of translational symmetry. The relationship mentioned here was put forward first for regular periodic spatial structures like those appearing in the Rayleigh-Brnard convection and is known under the name of phase dynamics [ 18]. We applied this idea to the irregular spatial structures which are more likely to be associated with glass-forming systems. We now explain this for a simple continuum model system with a scalar order parameter field S ( r ) . In contrast to the case of phase ordering of the preceding section, S ( r ) can contain components which vary rapidly over a microscopic distance. Hence the associated free energy functional H{S} cannot be expanded in gradients of S(r). For dynamics we assume the following equation of motion for S(r, t) (hereafter we shall suppress t from the argument of S and other quantities for simplicity):
K. Kawasaki,T. Koga/PhysicaA 224 (1996)1-8 0
t~H{S} S(r) = -1-1(r, {S}) aS(r) '
5 (3.1)
where ~2 is a self-adjoint positive definite operator. The static spatial structure is obtained as a spatially inhomogenous stationary solution of (3.1), namely, t$H
- -
8S(r)
=0,
(3.2)
which is denoted as So(r). Translational invariance of H{S} implies that So(r + u) is also a stationary solution for arbitrary constant vector u. On the other hand, with u a slowly-varying function of r instead of a constant, So(r + u(r)) is no longer an exact solution, but is still an approximate solution of (3.2). This fact suggests that substitution of So(r + u(r)) into (3.1) permits us to extract an equation of motion for u(r) after coarse-graining. A simple equation for u(r) results if S is not conserved and So(r) changes rapidly on the length scale of variation of u(r). We shall refer all the technical details to [ 17] and [ 19] and only the final results are presented here. u is decomposed into the longitudinal component Ull which takes care of compressional deformation of So(r) and the transverse components u±(r) which describe shear deformations of So(r). We then obtain Ou[I (r) = DIIV2Ull ( r ) , --ui(r)
Ot
(3.3)
= Di X72u±(r),
DII = (El +
(3.4)
62)/W,
(3.5)
D± = 61/w.
(3.6)
Here 61 and 62 are the moduli of deformation of So(r), which appear as coefficients when H{S(r)} is expressed in terms of u(r) as follows:
H= ½f dr { 61 ~ ~ aul~(r) Or~ Ou~(r) Or~ + 6 2 [ V
" u ( r ) ] 2 } +constant.
(3.7)
And also
w=-51 f dr[VSo(r) ] • 0 -1 (r, {So})VSo(r),
(3.8)
where ,(2-~ is the inverse operator of/'2. We see that the softer the spatial structure, that is, the smaller 61 and 62 in (3.7), the smaller are the diffusion constants D[I and D± and the stronger become the intensities of scattering which are inversely proportional to the e's. This apparently is what is experimentally observed as the temperature being lowered [ 14]. Although the origin of these ultraslow modes is not known, it is of some interest to note that another class of mysterious findings was reported on freezing transition. Bilgram and coworkers [20] have reported on anomalous excess light scattering of diffusive type by supercooled liquids near the solidification front. The correlation length
6
K. Kawasaki, T. Koga/Physica A 224 (1996) 1-8
of fluctuations reaches a few thousand angstroms, which is similar to those of the ultraslow modes [ 14]. We do not want to speculate on a possible relationship between the two sets of findings at this time, but all these findings indicate unexpected richness surrounding the problem of freezing and melting. In general, a diffusive mode is possible either where there is a conserved quantity or when continuous symmetry is spontaneously broken. Since the ultraslow mode apparently cannot be explained by the known conservation law of the fluid system alone, we are left with the second possibility, i.e. the Nambu-Goldstone mode. We also note that the defect diffusion whose importance in solids was emphasized in [21] does not follow solely from the conservation laws, but is intimately connected with the system being a solid, i.e., in an ordered phase.
4. On dynamical density functional theory of glass transitions The next topic to be discussed is an alternative approach to the well-known mode coupling theory (MCT) of the glass transition [22]. Instead of directly dealing with the density auto-correlation function, we first consider a model stochastic equation for the density probability distribution functional P ({p}, t) as follows [ 23 ] :
~P({p},t)=-L f ~p~Vp(r).V[~p~+fl~JP({p},t),
(4.1)
where L is a kinetic coefficient and H{p} is a free-energy functional. Here we are concerned with small-scale density fluctuations and H{p} is conveniently chosen to be of the following Ramakrishnan-Yusouff form:
H{p}=kBT/drp(r)lnlp(r)-I po
1]
-l kBT/ / drdr'C(r- r')[p(r) - po][p(r') - po],
(4.2)
where p0 and C(r) are the average density and the direct correlation function of the reference fluid. From (4.1) one can derive with further factorization approximation of four-body correlations the well-known MCT self-consistent equation for the density autocorrelation function [22]. The direct analysis of (4.1) has the advantages that (i) the above-mentioned factorization can be avoided and (ii) the so-called hopping process which is missing in the original version of the MCT equation [22] comes in a rather natural manner. However, the analysis of (4.1) is difficult since it is more complicated than the similar time dependent Ginzburg-Landau equation, especially because of the presence of the extra factor p(r) between the two xT. This extra factor, in fact, is important here to recover the correct MCT equation [ 23 ]. Furthermore, in the absence of the interaction term in (4.2) and the thermal noise term (the first term) in (4.1), (4.1) gives a diffusion process
K. Kawasaki,T. Koga/PhysicaA 224 (1996)1-8
7
~ - - ~ V pP({p},t),
(4.3)
which is what we expect for the short-distance behavior of high-density fluid. One way to analyze (4.1) numerically would be first to map it onto a stochastic equation for lattice gas (or equivalently a spin-exchange kinetic Ising model [24] ). Thus we consider a lattice gas described by the variables n = nl, n2 . . . . . nM, where M is the total number of lattice points and ni = 0 or 1 depending on the absence or presence of an atom on the lattice point i, respectively. The probability distribution function P ( n , t) obeys the following master equation (the time t is suppressed for simplicity):
-~P(n) = ~_~ {Wo(ntn')P(n') - Wo(n'ln)_P(n) } ,
(4.4)
nI
where Wo(nln') is the transition probability n I ---, n. If we require the detailed balance condition, we have
Wo(nln') = To(n[n') exp { ½fl[E0(n') -
E0(n) ] },
where To(nln') is symmetric with respect to n and n' and configuration n given by
Eo( n) = - l kl3T~ ~ i
C ( ri -
(4.5)
Eo(n)
is the energy in the
rj )ninj,
(4.6)
j~i
where ri is the position vector of the lattice point i. We can now employ a procedure similar to those of [25] to coarse-grain (4.4), where we assume that Wo(nln') only describes jumping processes of individual atoms to their nearest neighbor vacant lattice points. Denoting the coarse-grained number density of atoms by p(r) we would have after taking the continuum limit, 0 ~P({p}, t)
=-Ll f dr ~ p ~ V p ( r ) [ P m - p(r)] " V [ S p ~ + 1 3 ~ ] P ( { P } , t ) , (4.7) where Pm is the maximum number density where all the lattice points are fully occupied. L1 is some kinetic coefficient and H'{p} is the coarse-grained free energy functional. We choose the average p(r) to be much smaller than Pm so that Pm p(r) in (4.7) can be replaced by Pm and the coarse-graining length and hence the lattice constant to be much shorter than the characteristic length of the direct correlation function C (r) so that the form of the interaction energy (4.6) will not be altered by coarse-graining. As long as each coarse-graining cell contains a sufficiently large number of atoms, the coarse-graining with the local equilibrium assumption introduces an entropy term of the usual form in H'{p}. Thus finally H'{p} becomes identical to the RamakrishnanYusouff form (4.2). In this way (4.7) is reduced to (4.1) with Llpm identified as L. -
-
K. Kawasaki, T. Koga/Physica A 224 (1996) 1-8
P r o v i d e d the assumptions entering the coarse-graining procedure are correct, it w o u l d be m o r e c o n v e n i e n t to simulate the kinetic lattice gas m o d e l by e.g, a M o n t e Carlo method, rather than to solve (4.1) directly.
5. Concluding remarks H e r e we described the two s e e m i n g l y disparate topics on slow dynamics. However, w e think that the relationship will b e c o m e closer as increasing attention is being paid to aging p r o b l e m s in glasses w h i c h has many c o m m o n features with the phase ordering in usual phase transitions but is simply m u c h m o r e difficult and challenging.
References [ 1] [2] [3] 14] [5] 16] 171 181 [9] 110] [ 11 I 1121 [ 13]
[141 1151 [ 16] [ 171 [18] 119] [ 20] [21] 122] [231 124] [251
T. Koga and K. Kawasaki, Phys. Rev. A 44 (1991) R817. T. Koga and K. Kawasaki, Physica A 196 (1993) 389. T. Koga, K. Kawasaki, M. Takenaka and T. Hashimoto, Physica A 198 (1993) 473. K. Kawasaki, Prog. Theor. Phys. 57 (1977) 826. Y. Oono and S. Puff, Phys. Rev. A 38 (1988) 434. E.D. Siggia, Phys. Rev. A 20 (1979) 595. I.M. Lifshitz and V.V. Slyozov, J. Phys. Chem. Solid 19 (1961) 35. K. Kawasaki and T. Ohta, Physica A 118 (1983) 175. A. Onuki, J. Chem. Phys. 85 (1986) 1122. M. Takenaka and T. Hashimoto, J. Chem. Phys. 96 (1992) 6177. A. Shinozaki and Y. Oono, Phys. Rev. E 48 (1993) 2622. L.C.E. Struik, Physical Aging in Amorphous Polymers and Other Materials (North-Holland, Amsterdam, 1978). L. Lundgren, in: Slow Dynamics in Condensed Matter, K. Kawasaki, M. Tokuyama and T. Kawakatsu, eds. (American Physical Society, New York, 1992); E. Vincent, J. Hammann and M. Ocio, in: Recent Progress in Random Magnets, D.H. Ryan, ed. (World Scientific, Singapore, 1992). E.W. Fischer, Physica A 201 (1993) t83. H. Benoit and C. Picot, Pure Appl. Chem. 12 (1966) 545. K.S. Schmitz, Biopolymers 33 (1993) 953, and the references quoted therein. K. Kawasaki, Physica A (in press). Y. Pemeau and P. ManneviUe, J. Phys. Lett. (Paris) 40(1979) L-609. K. Kawasaki, in: Proceedings of Statphys 19, Hao Bai-lin, ed. (World Scientific, Singapore, 1996). S. Di Nardo and J.H. Biigram, Phys. Rev. B 51 (1995) 8012, and the references quoted therein. P.C. Martin, O. Parodi and P.S. Pershan, Phys. Rev. A 6 (1972) 2401. S. Yip, ed., Theme Issue on Relaxation Kinetics in Supercooled Liquids - Mode Coupling Theory and Its Experimental Tests, Transp. Theor. Stat. Phys. 24(6-8) (1995). K. Kawasaki, Physica A 208 (1994) 35; see also T. Munakata, J. Phys. Soc. Jpn. 43 (1977) 1723, and B. Bagchi, Physica A 145 (1987) 273. K. Kawasaki, Phys. Rev. 145 (1966) 224. J.S. Langer, Ann. Phys. (N.Y.) 54 (1969) 258; 65 (1971) 53; K. Kitahara and M. Imada, Suppl. Progr. Theor. Phys. 64 (1978) 65; K. Kawasaki, in: Pattern Recognition and Pattern Formation, H. Haken, ed. (Springer, Heidelberg, 1980).
Physica A 224 (1996) 1-8
Slow dynamics in condensed matter: spinodal decomposition and glass transition Kyozi Kawasaki a, Tsuyoshi Koga b'l a Department of Natural Science and Mathematics Chubu University, Kasugai, Aichi 487, Japan b Department of Physics, Faculty of Science, Kyushu University, Fukuoka 812, Japan
Abstract
A brief review is presented about our recent work on computer studies of a continuum model describing spinodal decomposition of binary fluid, which quantitatively accounts for corresponding experimental results. We then switch over to discussing problems of glass transitions. First we consider the ultraslow mode observed in fragile glass-formers, where the diffusive behavior of this mode is explained in terms of phase dynamics describing slow relaxation of spatial heterogeneous structure resulting from spontaneous break-down of continuous translational symmetry. Next we present a stochastic model for density fluctuation in supercooled liquids, which is mapped onto a certain kinetic lattice gas model which is convenient for Monte Carlo simulation studies.
1. Introduction
Interests in slow dynamics in condensed matter arise from the fact that the processes inevitably involve cooperative motions of many particles, which has always provided challenging problems in statistical physics. Here we shall discuss two such examples, with which we have been concerned recently. The next section takes up spinodal decomposition in binary fluid and the subsequent two sections are devoted to more recent and ongoing works related to glass transitions.
1 Present address: Hashimoto Polymer Phasing Project, ERATO, JRDC, Morimoto-cho, Shimogamo, Sakyoku, Kyoto 606, Japan. 0378-4371/96/$15.00 (~) 1996 Elsevier Science B.V. All fights reserved SSD1037 8-437 1 ( 9 5 ) 0 0 3 1 0 - X
2
K. Kawasaki, T. Koga/Physica A 224 (1996) 1-8
2. Spinodal decomposition in binary fluids 2.1. TDGL equation f o r binary fluids
The model used in our computer simulation [ 1-3] is the following time-dependent Ginzburg-Landau (TDGL)-type equation for the order parameter S ( r ) [4]: S(r, t) = LV2/z(r) - V S ( r ) .
f T(r-
r ' ) . V ' S ( r t ) i z ( r ' ) d r ',
(2.1)
where L is the Onsager kinetic coefficient, /~(r) -- 6 F { S } / 6 S ( r ) is the chemical potential, F { S } is the Ginzburg-Landau type free energy functional and T ( r ) is the following Oseen tensor: T(r) = ~
4-
,
(2.2)
where 7/is the shear viscosity and 1 is the unit tensor. The second term on the right-hand side of Eq. (2.1) represents the hydrodynamic interaction between the order parameter fluctuations mediated by the velocity field. In the derivation of Eq. (2.1), we assume that the relaxation of the local velocity is more rapid compared to that of the local order parameter [ 2,4]. By rescaling the variables in Eq. (2.1), we find that there is only one dimensionless parameter in Eq. (2.1) [2]. We call this the dimensionless shear viscosity, which measures the inverse of the strength of the hydrodynamic interaction. In the computer simulation, we employ the cell-dynamic method [5]. In the cell-dynamic version of Eq. (2.1), the parameter which corresponds to the dimensionless shear viscosity is denoted by 7/D. The system used in the computer simulation is a cubic lattice of size 1283 with periodic boundary conditions. 2.2. Results and discussions
At first, we show the time dependence of the characteristic length scale I(t) and discuss the growth exponent z: l(t) ~ t z . We calculate the spherically averaged structure factor Ik(t) of the order parameter. We use the first moment of Ik(t) denoted as kl (t) and the peak position of lk(t) denoted as km(t) as the characteristic wave number. The time dependence of kl (t) for fluids 070 = 1 and 7/o = 5) and "solid" is presented in Fig. 1. We call the conserved system without the hydrodynamic effect "solid" in this paper. From Fig. 1, we find that in the low viscosity case (~TD = 1), the effective exponent Zeff(t) takes the value 1 in the late time region. This indicates that the domain growth in this time region is dominated by the flow induced by the surface tension [6]. On the other hand, the value of z~ff(t) approaches 1/3 in the late stage in the solid model, which is the Lifshitz-Slyozov type growth law [7]. In the high viscosity case (r/D = 5), ZerO(t) increases with time but has not yet reached 1 even in the late time region of our simulation. It is expected that it takes a long time to reach the z = 1 region
K. Kawasaki, T. Koga/Physica A 224 (1996) 1-8
3
Slope=-1
10-1'102
103
04
Fig. 1. Time dependence of kl(t) for fluids with ~/o = 1 (o) and 7/D = 5 ( ~ ) and for solid (E]) on double-logarithmic scales.
- - simulation (fluid) -- simulation (solid) o PB/PI
x ' ',
o o
o
/ x Fig. 2. Porod plots of the scaling functions obtained by the computer simulation and the experiments of PB/PI. In the computer simulation, we use the transformation: S(r) ~ Se sgn[S(r)], to removethe effects of the interface thickness. Here Se is the equilibrium value of IS(r)I.
in the high viscosity case. These results are qualitatively consistent with the results using the interface equation of motion [2,8]. The branching of the time dependence of the characteristic wave number in Fig. 1 is due to the difference of the strength of the hydrodynamic interaction and corresponds to the so-called N branching in polymer blends [2,9]. Next we take up the scaling function F ( x ) = k m ( t ) 3 1 k ( t ) , where x =_ k / k i n ( t ) . Porod plots of the scaling functions for fluid and solid are presented in Fig. 2 together with the experimental results of a critical mixture of polybutadiene (PB) and polyisoprene (PI) by Takenaka and Hashimoto [ 10]. In this experiment, the scaling function is obtained in
K. Kawasaki, T. Koga/Physica A 224 (1996) 1-8
the hydrodynamic growth region. We find that the scaling function for fluid is different from that for solid [2] and the scaling function obtained by the experiment is more close to the scaling function for fluid than that for solid [3]. We note that the same problem was subsequently studied in great detail by Shinozaki and Oono [ 11 ].
3. Uitraslow modes in glass-forming systems In the preceding section we described a typical example of the usual phase ordering where the nature of the ordered phases is simple. A more challenging class of problems would be the cases with ill-defined ordered phases. A typical example is the problem of glasses. Here the "ordered phase" contains lots of disorder and is still poorly understood. Recently much attention is being directed to non-equilibrium dynamics following the quench below the glass transition temperature, especially in the spin glass community. This is known as the aging problem of glass [ 12,13]. We shall not touch upon this fascinating problem at this time since this is beyond our competence, but we shall discuss a few other problems related to glass. The first topic taken up is the ultraslow modes which are associated with the excess scattering which often appears in the so-called fragile glass-formers [ 14]. This excess scattering is caused by fluctuations whose spatial lengths can reach a few thousand angstroms and whose origin is not well understood. Such excess scattering was earlier observed for some polymer solutions in the semi-dilute regime [ 15] and also for polyelectolyte solutions with low salt or salt free solvents [ 16]. The whole subject is full of controversies. We do not go into the controversies here but rather point out a close relationship between ultraslow modes and the static heterogeneous structure inherent in the system [ 17]. Here, however, we assume that our system possesses translational invariance and the spatial heterogeneities can appear as a result of the spontaneous break-down of the continuous translational symmetry. Such spatial structure is undergoing random deformations by thermal noise whose amplitude increases as the wavelength of deformation increases. That is, the restoration force of deformation decreases towards zero as the wavelength of deformation tends to infinity. The fact that the relaxation rate of ultraslow modes which is identified here with the relaxation of this deformation is of the diffusion type is a direct consequence of the spontaneous break-down of translational symmetry. The relationship mentioned here was put forward first for regular periodic spatial structures like those appearing in the Rayleigh-Brnard convection and is known under the name of phase dynamics [ 18]. We applied this idea to the irregular spatial structures which are more likely to be associated with glass-forming systems. We now explain this for a simple continuum model system with a scalar order parameter field S ( r ) . In contrast to the case of phase ordering of the preceding section, S ( r ) can contain components which vary rapidly over a microscopic distance. Hence the associated free energy functional H{S} cannot be expanded in gradients of S(r). For dynamics we assume the following equation of motion for S(r, t) (hereafter we shall suppress t from the argument of S and other quantities for simplicity):
K. Kawasaki,T. Koga/PhysicaA 224 (1996)1-8 0
t~H{S} S(r) = -1-1(r, {S}) aS(r) '
5 (3.1)
where ~2 is a self-adjoint positive definite operator. The static spatial structure is obtained as a spatially inhomogenous stationary solution of (3.1), namely, t$H
- -
8S(r)
=0,
(3.2)
which is denoted as So(r). Translational invariance of H{S} implies that So(r + u) is also a stationary solution for arbitrary constant vector u. On the other hand, with u a slowly-varying function of r instead of a constant, So(r + u(r)) is no longer an exact solution, but is still an approximate solution of (3.2). This fact suggests that substitution of So(r + u(r)) into (3.1) permits us to extract an equation of motion for u(r) after coarse-graining. A simple equation for u(r) results if S is not conserved and So(r) changes rapidly on the length scale of variation of u(r). We shall refer all the technical details to [ 17] and [ 19] and only the final results are presented here. u is decomposed into the longitudinal component Ull which takes care of compressional deformation of So(r) and the transverse components u±(r) which describe shear deformations of So(r). We then obtain Ou[I (r) = DIIV2Ull ( r ) , --ui(r)
Ot
(3.3)
= Di X72u±(r),
DII = (El +
(3.4)
62)/W,
(3.5)
D± = 61/w.
(3.6)
Here 61 and 62 are the moduli of deformation of So(r), which appear as coefficients when H{S(r)} is expressed in terms of u(r) as follows:
H= ½f dr { 61 ~ ~ aul~(r) Or~ Ou~(r) Or~ + 6 2 [ V
" u ( r ) ] 2 } +constant.
(3.7)
And also
w=-51 f dr[VSo(r) ] • 0 -1 (r, {So})VSo(r),
(3.8)
where ,(2-~ is the inverse operator of/'2. We see that the softer the spatial structure, that is, the smaller 61 and 62 in (3.7), the smaller are the diffusion constants D[I and D± and the stronger become the intensities of scattering which are inversely proportional to the e's. This apparently is what is experimentally observed as the temperature being lowered [ 14]. Although the origin of these ultraslow modes is not known, it is of some interest to note that another class of mysterious findings was reported on freezing transition. Bilgram and coworkers [20] have reported on anomalous excess light scattering of diffusive type by supercooled liquids near the solidification front. The correlation length
6
K. Kawasaki, T. Koga/Physica A 224 (1996) 1-8
of fluctuations reaches a few thousand angstroms, which is similar to those of the ultraslow modes [ 14]. We do not want to speculate on a possible relationship between the two sets of findings at this time, but all these findings indicate unexpected richness surrounding the problem of freezing and melting. In general, a diffusive mode is possible either where there is a conserved quantity or when continuous symmetry is spontaneously broken. Since the ultraslow mode apparently cannot be explained by the known conservation law of the fluid system alone, we are left with the second possibility, i.e. the Nambu-Goldstone mode. We also note that the defect diffusion whose importance in solids was emphasized in [21] does not follow solely from the conservation laws, but is intimately connected with the system being a solid, i.e., in an ordered phase.
4. On dynamical density functional theory of glass transitions The next topic to be discussed is an alternative approach to the well-known mode coupling theory (MCT) of the glass transition [22]. Instead of directly dealing with the density auto-correlation function, we first consider a model stochastic equation for the density probability distribution functional P ({p}, t) as follows [ 23 ] :
~P({p},t)=-L f ~p~Vp(r).V[~p~+fl~JP({p},t),
(4.1)
where L is a kinetic coefficient and H{p} is a free-energy functional. Here we are concerned with small-scale density fluctuations and H{p} is conveniently chosen to be of the following Ramakrishnan-Yusouff form:
H{p}=kBT/drp(r)lnlp(r)-I po
1]
-l kBT/ / drdr'C(r- r')[p(r) - po][p(r') - po],
(4.2)
where p0 and C(r) are the average density and the direct correlation function of the reference fluid. From (4.1) one can derive with further factorization approximation of four-body correlations the well-known MCT self-consistent equation for the density autocorrelation function [22]. The direct analysis of (4.1) has the advantages that (i) the above-mentioned factorization can be avoided and (ii) the so-called hopping process which is missing in the original version of the MCT equation [22] comes in a rather natural manner. However, the analysis of (4.1) is difficult since it is more complicated than the similar time dependent Ginzburg-Landau equation, especially because of the presence of the extra factor p(r) between the two xT. This extra factor, in fact, is important here to recover the correct MCT equation [ 23 ]. Furthermore, in the absence of the interaction term in (4.2) and the thermal noise term (the first term) in (4.1), (4.1) gives a diffusion process
K. Kawasaki,T. Koga/PhysicaA 224 (1996)1-8
7
~ - - ~ V pP({p},t),
(4.3)
which is what we expect for the short-distance behavior of high-density fluid. One way to analyze (4.1) numerically would be first to map it onto a stochastic equation for lattice gas (or equivalently a spin-exchange kinetic Ising model [24] ). Thus we consider a lattice gas described by the variables n = nl, n2 . . . . . nM, where M is the total number of lattice points and ni = 0 or 1 depending on the absence or presence of an atom on the lattice point i, respectively. The probability distribution function P ( n , t) obeys the following master equation (the time t is suppressed for simplicity):
-~P(n) = ~_~ {Wo(ntn')P(n') - Wo(n'ln)_P(n) } ,
(4.4)
nI
where Wo(nln') is the transition probability n I ---, n. If we require the detailed balance condition, we have
Wo(nln') = To(n[n') exp { ½fl[E0(n') -
E0(n) ] },
where To(nln') is symmetric with respect to n and n' and configuration n given by
Eo( n) = - l kl3T~ ~ i
C ( ri -
(4.5)
Eo(n)
is the energy in the
rj )ninj,
(4.6)
j~i
where ri is the position vector of the lattice point i. We can now employ a procedure similar to those of [25] to coarse-grain (4.4), where we assume that Wo(nln') only describes jumping processes of individual atoms to their nearest neighbor vacant lattice points. Denoting the coarse-grained number density of atoms by p(r) we would have after taking the continuum limit, 0 ~P({p}, t)
=-Ll f dr ~ p ~ V p ( r ) [ P m - p(r)] " V [ S p ~ + 1 3 ~ ] P ( { P } , t ) , (4.7) where Pm is the maximum number density where all the lattice points are fully occupied. L1 is some kinetic coefficient and H'{p} is the coarse-grained free energy functional. We choose the average p(r) to be much smaller than Pm so that Pm p(r) in (4.7) can be replaced by Pm and the coarse-graining length and hence the lattice constant to be much shorter than the characteristic length of the direct correlation function C (r) so that the form of the interaction energy (4.6) will not be altered by coarse-graining. As long as each coarse-graining cell contains a sufficiently large number of atoms, the coarse-graining with the local equilibrium assumption introduces an entropy term of the usual form in H'{p}. Thus finally H'{p} becomes identical to the RamakrishnanYusouff form (4.2). In this way (4.7) is reduced to (4.1) with Llpm identified as L. -
-
K. Kawasaki, T. Koga/Physica A 224 (1996) 1-8
P r o v i d e d the assumptions entering the coarse-graining procedure are correct, it w o u l d be m o r e c o n v e n i e n t to simulate the kinetic lattice gas m o d e l by e.g, a M o n t e Carlo method, rather than to solve (4.1) directly.
5. Concluding remarks H e r e we described the two s e e m i n g l y disparate topics on slow dynamics. However, w e think that the relationship will b e c o m e closer as increasing attention is being paid to aging p r o b l e m s in glasses w h i c h has many c o m m o n features with the phase ordering in usual phase transitions but is simply m u c h m o r e difficult and challenging.
References [ 1] [2] [3] 14] [5] 16] 171 181 [9] 110] [ 11 I 1121 [ 13]
[141 1151 [ 16] [ 171 [18] 119] [ 20] [21] 122] [231 124] [251
T. Koga and K. Kawasaki, Phys. Rev. A 44 (1991) R817. T. Koga and K. Kawasaki, Physica A 196 (1993) 389. T. Koga, K. Kawasaki, M. Takenaka and T. Hashimoto, Physica A 198 (1993) 473. K. Kawasaki, Prog. Theor. Phys. 57 (1977) 826. Y. Oono and S. Puff, Phys. Rev. A 38 (1988) 434. E.D. Siggia, Phys. Rev. A 20 (1979) 595. I.M. Lifshitz and V.V. Slyozov, J. Phys. Chem. Solid 19 (1961) 35. K. Kawasaki and T. Ohta, Physica A 118 (1983) 175. A. Onuki, J. Chem. Phys. 85 (1986) 1122. M. Takenaka and T. Hashimoto, J. Chem. Phys. 96 (1992) 6177. A. Shinozaki and Y. Oono, Phys. Rev. E 48 (1993) 2622. L.C.E. Struik, Physical Aging in Amorphous Polymers and Other Materials (North-Holland, Amsterdam, 1978). L. Lundgren, in: Slow Dynamics in Condensed Matter, K. Kawasaki, M. Tokuyama and T. Kawakatsu, eds. (American Physical Society, New York, 1992); E. Vincent, J. Hammann and M. Ocio, in: Recent Progress in Random Magnets, D.H. Ryan, ed. (World Scientific, Singapore, 1992). E.W. Fischer, Physica A 201 (1993) t83. H. Benoit and C. Picot, Pure Appl. Chem. 12 (1966) 545. K.S. Schmitz, Biopolymers 33 (1993) 953, and the references quoted therein. K. Kawasaki, Physica A (in press). Y. Pemeau and P. ManneviUe, J. Phys. Lett. (Paris) 40(1979) L-609. K. Kawasaki, in: Proceedings of Statphys 19, Hao Bai-lin, ed. (World Scientific, Singapore, 1996). S. Di Nardo and J.H. Biigram, Phys. Rev. B 51 (1995) 8012, and the references quoted therein. P.C. Martin, O. Parodi and P.S. Pershan, Phys. Rev. A 6 (1972) 2401. S. Yip, ed., Theme Issue on Relaxation Kinetics in Supercooled Liquids - Mode Coupling Theory and Its Experimental Tests, Transp. Theor. Stat. Phys. 24(6-8) (1995). K. Kawasaki, Physica A 208 (1994) 35; see also T. Munakata, J. Phys. Soc. Jpn. 43 (1977) 1723, and B. Bagchi, Physica A 145 (1987) 273. K. Kawasaki, Phys. Rev. 145 (1966) 224. J.S. Langer, Ann. Phys. (N.Y.) 54 (1969) 258; 65 (1971) 53; K. Kitahara and M. Imada, Suppl. Progr. Theor. Phys. 64 (1978) 65; K. Kawasaki, in: Pattern Recognition and Pattern Formation, H. Haken, ed. (Springer, Heidelberg, 1980).