Das Sh.P. Statistical Physics of Liquids at Freezing and Beyond

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A7.4 The memory-function approach 361

which follows from the orthogonality condition (A7.4.19). Using the initial value

≡ R

(0) = QLA

, (A7.4.22)

the memory function M

in eqn. (A7.4.12) is obtained as an autocorrelation of the noise R

=R

(t)R



∗

−1

. (A7.4.23)

This is a generalized ﬂuctuation–dissipation relation whose validity does not require the

system to be close to the equilibrium state but is dependent on the orthogonality condition

(A7.4.19).

The dynamical equation for the time evolution of the correlation functions C

(t) deﬁned

in (A7.4.2)) is reached in a matrix form by multiplying eqn. (A7.4.12) from the right by

∗

· A

∗

A

−1

and averaging to obtain

∂C

∂t

−i C(t) +



M(t − s) · C(s)ds = 0, (A7.4.24)

where we have used the fact that AR(t )=0. The one-sided Laplace transform of the

memory-function equation (A7.4.24) reduces to the form

[zI + + M]C(z) = χ. (A7.4.25)

The equation χ ≡ C(t = 0) represents the initial conditions for (A7.4.24). Note that the

equation is linear in correlation functions with Green–Kubo transport coefﬁcients.

A7.4.2 The mode-coupling approximation

The dynamical equations (A7.4.12) linear in the ﬂuctuations A

involve the matrix M(t) of

non-Markovian transport coefﬁcients, which are also expressed in the Green–Kubo form

in (A7.4.23). The collective effects that inﬂuence the long-time dynamic behavior are now

included in the memory-function matrix M. In the mode-coupling theories the frequency-

and wave-number-dependent functions in M are obtained in terms of the products of time

correlation functions (Götze and Lücke, 1975; Munakata and Igarashi, 1977, 1978). The

memory effects are thus computed in terms of correlation functions themselves rather than

being approximated through phenomenological time-dependent functions (Boon and Yip,

1991). The analysis of the memory function in the projected space of an extended set of

slow modes (involving coupling of slow modes) gives the frequency-dependent part in

terms of hydrodynamic correlation functions. We analyze here the expression (A7.4.23)

for the non-Markovian transport coefﬁcient in the Mori equations (A7.4.12) corresponding

to a speciﬁc set of collective modes. The generalized force corresponding to density,

= QLg

, (A7.4.26)

362 Appendix to Chapter 7

is expressed (Götze, 1991) in terms of its projection on the coupled hydrodynamic modes

C(kp ...)(say) following deﬁnition (A7.4.3) as





K

|C(k



)χ

−1

(kp, k



)C(kp), (A7.4.27)

where χ

−1

is the inverse of the correlation matrix

C(kp)C(k



). (A7.4.28)

On substituting (A7.4.27) into (A7.4.23), the memory function is obtained in terms of time

correlation functions like

C(kp) exp(−iQLQ)C(k



) and C(kp)Lg

. (A7.4.29)

These are further simpliﬁed with some crucial approximations described below.

(a) Near the glass transition, the density ﬂuctuations are assumed to be most dominant,

leading to the simple choice C(kp) = δρ (k)δρ(p). The corresponding equal-time

correlation matrix is obtained as

−1



) = δ



/[β N

S(k)S( p)]. (A7.4.30)

(b) The next important approximation involves equating four-point correlations of den-

sity ﬂuctuations evolving in time with generator QLQ to corresponding two-point

correlations evolving in time with L:

δρ (k)δρ(p)e

−iQLQ

δρ (k



)δρ(p



)δρ (k, t)

∗

δρ (k



)δρ (p, t)

∗

δρ (p



).

(A7.4.31)

This reduction has been termed the Gaussian approximation or Kawasaki approxima-

tion in the MCT literature. The memory function is now obtained in the form

(q, t)  (1/n

N )



|(

q · K)C(kk

)|

S(k)S(k

)φ(k, t)φ(k

, t), (A7.4.32)

where k

= q − k. The equilibrium averages

K

C(kk

)≡ρ(k)ρ(k

)Lg

 (A7.4.33)

are evaluated using classical Poisson-bracket relations between ρ and g to obtain the

mode-coupling kernel (7.3.66) and (7.3.67).

The ergodic–nonergodic transition

We now discuss a model for the slow dynamics of supercooled liquids that follows from the

analysis of the renormalized theory of liquid dynamics discussed in the previous chapter.

This theoretical development has been termed in the literature the mode-coupling theory

(MCT) of supercooled liquids.

8.1 Mode-coupling theory

Let us ﬁrst describe two basic steps in the formulation of the MCT starting from the model

for the dynamics of the liquid developed in the last chapter.

I. The liquid dynamics is formulated in terms of the nonlinear equations of the gen-

eralized hydrodynamics for a set of slow modes. Renormalized expressions for the

correlation functions of the slow modes involving generalized transport coefﬁcients

that are wave-number- and frequency-dependent generalizations of the hydrodynamic

transport coefﬁcients are obtained. The Laplace transform of the density–density corre-

lation function ψ(q, z) is obtained in terms of the renormalized longitudinal viscosity

L(q, z) as

ψ(q, z) =



z −



z + iq

L(q, z)



−1

, (8.1.1)

where 

represents a microscopic frequency that is characteristic of the liquid state.

II. As a consequence of the nonlinear coupling of the slow modes the bare transport coef-

ﬁcient 

(say) that appears in the equations of linearized dynamics is renormalized to

L(q, z). The MSR ﬁeld theory outlined in the previous chapter provides a systematic

scheme for computing these corrections from a set of self-energies, which are expressed

in terms of the correlation functions and response functions. A renormalized perturba-

tion expansion for the transport coefﬁcients at arbitrary order in the nonlinearities can

thus be constructed in principle. For example, the renormalized longitudinal viscos-

ity L(q, z) at one-loop order is given by eqns. (7.3.66) and (7.3.67). In general, the

363

364 The ergodic–nonergodic transition

renormalized transport coefﬁcient L(q, z) is expressed as a self-consistent functional

of the density correlation function of the {ψ}:

L(q, z) ≡ M

[{ψ}]. (8.1.2)

Therefore, with the coupled set of equations (8.1.1) and (8.1.2) obtained in steps I and

II above, the time correlation function of the density ﬂuctuations is expressed in terms

of renormalized transport coefﬁcients, which themselves are expressed in terms of corre-

lation functions. This gives rise to a self-consistent feedback mechanism for the density

correlation function with characteristic slow dynamics.

It is useful to note at this point that originally non-self-consistent mode-coupling models

were used in the study of relaxation in systems near second-order phase-transition points

(Fixman, 1962; Kadanoff and Swift, 1968; Kawasaki, 1970a, 1970b). In this approach the

renormalizing contributions to the transport coefﬁcients are approximated as a functional

[{C

}] of the correlation functions C

between the different hydrodynamic modes in

the linear theory which involve only the bare transport coefﬁcients. In the case of critical

phenomena, the static correlations grow indeﬁnitely near the transition point and, through

the mode-coupling terms, the cooperative effects over different length and time scales are

probed. For the normal liquid state, mode-coupling contributions to transport coefﬁcients

(Alley et al., 1983) and diffusivity (Curkier and Mehaffey, 1978) were computed at low-

est order to account for the substantial difference between computer-simulation data and

corresponding results obtained from the Enskog theory (Résibois and de Leener, 1977).

The latter takes into account only short-time uncorrelated dynamics of the particles and

is applicable for low-density ﬂuids. More exotic results from inclusion of mode-coupling

effects indicated breakdown of conventional hydrodynamics, e.g., the long-time tails in the

dynamic correlation functions (Forster et al., 1977), or divergence of components of the

dynamic viscosity tensor in smectic-A liquid crystals (Mazenko et al., 1983; Milner and

Martin, 1986).

The relevance of the MCT to the theory of the glass transition came with the idea of

expressing the mode-coupling terms, i.e., the functional M

[{C}]for the generalized trans-

port coefﬁcient, solely in terms of the full density–density correlation function ψ itself.

The functional M

[{C}] and the deﬁning relation for ψ in terms of the generalized trans-

port coefﬁcient gave a closed equation for ψ (Geszti, 1983). From expression (7.3.66) we

see that, if the density correlation freezes in the long-time limit, L(q, z) has a 1/z pole.

Such a pole in the viscosity, when substituted into (7.3.68), appears to be self-consistent

with the 1/z pole for the correlation function. The mode-coupling model that we discuss

in this chapter is based on this crucial dynamic feedback mechanism. We note here in

this regard that the model equations follow from a generalized-hydrodynamics approach

extended to the ﬁnite-wave-vector or corresponding short-length-scale regime. The glass

transition is not intrinsically related to the long distance and long time scales which con-

cern hydrodynamics in the traditional sense. The transition to an amorphous solid-like

state is characterized by the freezing process occurring ﬁrst over small or intermediate

8.1 Mode-coupling theory 365

length scales. However, the effects of the jamming process should ﬁnally propagate to

hydrodynamic regimes.

We present an outline of the present chapter. As a result of the nonlinear feedback

mechanism outlined above, a dynamic transition to a nonergodic state is predicted in

the simple MCT. The long-time limit of the density correlation ψ freezes at a nonzero

value beyond the transition point, while the thermodynamic properties of the liquid vary

smoothly through the transition. The viscosity follows a power-law divergence approach-

ing the dynamic transition point, accompanied by a ﬁnite shear modulus in the glassy state.

Subsequent works (Das and Mazenko, 1986) demonstrated that the sharp dynamic transi-

tion to an ideal glassy state predicted in the preliminary works by Leutheusser (1984),

Bengtzelius et al. (1984), and Das et al. (1985a) is ﬁnally cut off and ergodicity is main-

tained at all densities. However, the above-described feedback mechanism from the cou-

pling of density ﬂuctuations causes a substantial enhancement of the viscosity. Thus,

although the sharp transition is cut off, there are strong remnants of it, causing a qualitative

change in the supercooled liquid dynamics around a temperature T

that is higher than the

usual glass-transition temperature T

. Identiﬁcation of the new temperature T

> T

)

for liquids, generally of the fragile type, has been one of the main outcomes from self-

consistent MCT. Subsequent theoretical works as well as experimental works have focused

on ﬁnding the signatures of the mode-coupling transition at T

8.1.1 The schematic model

The complex relaxation scenario that results from the nonlinear feedback mechanism of

density ﬂuctuations is conveniently demonstrated in terms of a schematic model in which

all wave-vector dependences are dropped. The q-independent density correlation function,

normalized with respect to its equal-time value, is obtained as

ψ(q, t) =

ρρ

(q, t)

S(q)

≡ ψ(t). (8.1.3)

The Laplace transform ψ(z) of the the density correaltion function ψ(t) satisﬁes the

equation

ψ(z) =



z −

z + im(z)



−1

, (8.1.4)

where time has been rescaled to t

, with 

≡ cq treated as a characteristic microscopic

frequency in the model, dropping all wave-vector dependence. The renormalized viscosity

L(q, z) has been written here in a rescaled form in terms of m(z). In the schematic models

the one-loop contribution to the viscosity is obtained as

m(z) = m



∞

dt e

izt

(t), (8.1.5)

366 The ergodic–nonergodic transition

where m

=

is the scaled bare transport coefﬁcient. From the ﬁeld-theoretic model

presented in the previous chapter, the constant c

is determined in terms of the vertex func-

tions of cubic and higher order in the action functional (see Section 7.3.3, eqns. (7.3.63),

(7.3.64), (7.3.66), and (7.3.67)). The actual dependence of c

on the thermodynamic prop-

erties of the ﬂuid can be determined by taking into account the proper wave-vector depen-

dence in the model, to be considered in the next section.

In order to keep the analysis at a more general level we will in fact consider renormal-

ization of the transport coefﬁcient beyond one-loop order and generalize the integral in the

second term on the RHS of eqn. (8.1.5) to

m(z) = m



∞

dt e

izt

H[ψ(t)], (8.1.6)

where H[ψ(t)] is a local functional of the density correlation function ψ(t) (Das et al.,

1985a; Götze, 1991; Kim and Mazenko, 1992),

H[ψ(t)]=



n=1

(t), (8.1.7)

with the c

being determined by the thermodynamic state of the ﬂuid. Thus the one-loop

result considered above corresponds to H [φ]=c

. Equations (8.1.4) and (8.1.5) form

a coupled set of equations for the density autocorrelation function. In the time space they

can be simply expressed as one closed nonlinear integral equation for ψ(t),

ψ(t) + ν

ψ(t) + ψ(t) +



ds H[ψ(t − s)]

ψ(s) = 0, (8.1.8)

where ν

is the bare transport coefﬁcient.

Initially, eqn. (8.1.8) was derived (Leutheusser, 1984) from a physical picture for the

supercooled liquid dynamics as follows. The deeply supercooled state is characterized by

strong density ﬂuctuations. The basic quantity describing the dynamic properties of density

ﬂuctuations is the dynamic structure factor, whose decay is controlled by the longitudinal

viscosity. The dynamic transport coefﬁcients at high density are strongly inﬂuenced by cor-

related motion of the ﬂuid particles and are expressed mainly through the mode-coupling

terms involving products of strong density ﬂuctuations. Therefore, through this feedback

process, the relaxation time of density ﬂuctuations is enhanced by the slow decay of the

density ﬂuctuations themselves. A similar enhancement was argued also for the shear vis-

cosity (Geszti, 1983). Bengtzelius et al. (1984) discussed a model in which the feedback

mechanism for the liquid was considered in greater detail, and in their model the proper

structure factor of a real liquid was included.

The basic assumption in the analysis of eqn. (8.1.4) is that, depending on the the kernel

H[ψ], i.e., the coefﬁcients c

, there is a time range over which ψ(t) is approximately

time-independent and has the form

ψ(t) = f + (1 − f )ψ

(t), (8.1.9)

8.1 Mode-coupling theory 367

where f is the value of ψ(t ) in some metastable state that is yet to be speciﬁed. We con-

sider time scales on which the inequality |zψ

(z)|1 is valid for the Laplace transform

[ψ

(t)].Usingeqn. (8.1.9) in eqn. (8.1.4) and keeping terms up to O

(z)

, we obtain



+ 

(z) −(1 − f )H



( f )zψ

(z) +

(1 − f )



( f )L



(t)

= 0, (8.1.10)

with



= H( f ) −

1 − f



∂

∂ f

(1 − f )H ( f ) − f

. (8.1.11)

In eqn. (8.1.10) the primes indicate derivatives with respect to f . An ideal metastable state

is obtained when both 

and 

are zero, giving a solution for the decaying function

(t). We can determine f by setting 

=0. This is obtained for

H( f ) + 1 =

1 − f

, (8.1.12)

where C

is a constant not depending on f . Under this condition 

is given by



− 1

1 − f

. (8.1.13)

An ideal nonergodic state is obtained when, for a given set of c

≡ c

∗

,wehave



∗



=0.

Thus the ideal transition takes place for C

=1, giving 

=0.

The decay of the density–density correlation ψ(t) becomes extremely slow in a certain

part of the parameter space (involving the c

) and eventually it freezes at a nonzero value

f , indicating a dynamic transition of the ﬂuid to a nonergodic state. For the Leutheusser

model we obtain the nonergodicity parameter f =

when the coupling c

has a critical

value of c

∗

=4. For the model that includes both c

and c

(Götze, 1991) having a linear

term in the kernel, i.e.,

H[ψ]=c

ψ + c

, (8.1.14)

the critical values are parametrized in terms of a single quantity λ

∗

2λ

− 1

∗

, (8.1.15)

and correspond to a line in the c

− c

space given by

∗

= 2

√

∗

− c

∗

(8.1.16)

368 The ergodic–nonergodic transition

representing the ergodic–nonergodic transition. For higher-order models there are critical

surfaces. Note that, with 

= 0, eqn. (8.1.10) is given by



−

1 − f

zψ

(z) +

(1 − f )



( f )L



(t)

= 0. (8.1.17)

For the high-frequency region where the term proportional to 

can be neglected, i.e.,

(|

1/2

|zψ

(z)|, (8.1.18)

the above equation reduces to the simple form

zψ

(z) −λ



(t)

= 0, (8.1.19)

with λ

at the transition point determined from the relation

(1 − f )



( f ) ≡



( f )



( f )]

3/2

. (8.1.20)

This equation is satisﬁed by the function

(t) = A(t

)

−a

, (8.1.21)

where the exponent a is given in terms of the parameter λ

by the equation



(1 − a)

(1 − 2a)

= λ

. (8.1.22)

The inequality (8.1.18) and the solution (8.1.21) obviously hold in the time region

 t  τ

≡ t

|

−1/(2a)

, (8.1.23)

where t

=

−1

represents a microscopic time scale and τ

diverges at the ideal transition

point, i.e., 

=0. Therefore ψ(t) decays algebraically towards the metastable value f .

For 

= 0, i.e., points away from the transition, the dynamics is quite different for the

cases 

> 0 and 

< 0. For 

> 0, eqn. (8.1.17) has the solution ψ

(z) ∼ 

1/2

/z as

z → 0 and

f = f

+ c(

)

1/2

+ O(

), (8.1.24)

where f

is the value of f at the ideal transition point and c is a constant. For 

< 0we

have from eqn. (8.1.17) a solution for ψ

(z) that is more singular than 1/z as z → 0 and

has the following form in the time regime:

ψ(t) = f − B(t/τ

)

, (8.1.25)

where B is a positive constant. Similarly to a in eqn. (8.1.21), b satisﬁes the transcendental

equation



(1 + b)

(1 + 2b)

= λ

. (8.1.26)

8.1 Mode-coupling theory 369

Equation (8.1.25) is referred to as the von Schweidler relaxation law. However, this relax-

ation eventually violates the inequality |zψ

(z)|1 and this decay mechanism is valid in

the region

 t  τ

, (8.1.27)

where τ

=|

−[1/(2a)+1/(2b)]

represents the time scale for the power law or the

β-process. For t ≥τ

, the system enters into the α-relaxation regime. In the α-relaxation

regime no fully analytic solution of the mode-coupling equations is known. Numerical

solution in this regime is well ﬁt by a stretched exponential. Assuming the form

ψ(t) = fe

−(t/τ

)

(8.1.28)

to be the solution of eqn. (8.1.8), we obtain an approximate expression for the stretching

exponent β in terms of the parameters c



n=1

−1/β

= 1. (8.1.29)

For the Leutheusser model at the ideal transition c

∗

=4 and f =

we get β

ln 2/ ln(c

f ) =1, i.e., relaxation is exponential. For the model with c

and c

, referred

to as the φ

model (Sjögren, 1986; Götze, 1991),

ln 2

ln [c

f /(1 − c

)]

=−

ln 2

ln (1 − λ

)

. (8.1.30)

Therefore β

varies along the transition line given by (8.1.16).

Let us summarize here the complex relaxation scenario that characterizes the dynamics

driven by the feedback mechanism of the schematic model discussed above. For 

< 0,

ﬁrst there is power-law decay, followed by the von Schweidler relaxation and ﬁnally the

stretched exponential relaxation. This is the ergodic behavior. At the transition point,



=0, the power-law decay extends to the longest time scale. In the nonergodic state,



> 0, the density correlation function freezes in a state of structural arrest, after the initial

power-law decay. The sequence of relaxations in the ergodic state is shown schematically

in Fig. 8.1.

8.1.2 Effects of structure on dynamics

The discussion of the schematic model outlined above demonstrates that the feedback

mechanism central to MCT is insensitive to the wave-vector dependence. The transition

predicted is essentially dynamic in the sense that the divergence of the viscosity at the

critical point is not accompanied by any sharp change in the thermodynamic properties of

the liquid. Certain mean-ﬁeld models with intrinsic (quenched) disorder, namely equations

similar to that of the schematic model, also provide a useful description of the dynamics

(see Section 8.4). However, the MCT for the glass transition is not just the self-consistent

370 The ergodic–nonergodic transition

Fig. 8.1 The relaxation behaviors over different time scales as predicted for the liquid in the ergodic

state below the ideal dynamic transition point of the schematic mode-coupling model.

treatment of an integral equation with one degree of freedom. The actual impact of the

feedback mechanism becomes clear when we consider the wave-vector-dependent model

(Kirkpatrick, 1985a). Using the static structure factor as an input, the predictions of the

theory are directly linked to thermodynamic properties such as temperature and density.

In the wave-vector-dependent model, even with the simplest form of quadratic coupling

of density ﬂuctuations, many different time scales (corresponding to different wave num-

bers) are coupled. This as a result produces stretched exponential relaxation behavior.

In the present section we consider the effects of including structure on the dynamics

predicted in MCT.

In the wave-vector-dependent model the Laplace transform of the density correlation

function φ(q, z) is now expressed with a generalization of the memory function m

(q, z),

in the form

φ(q, z)

zφ(q, z) − 1

= z + im

(q, z), (8.1.31)

where we have rescaled time to a dimensionless form with a characteristic microscopic

frequency 

≡ cq. The memory function is expressed as a sum of two parts,

(q, z) = m

(q) +



∞

dt e

izt

˜m

(q, t), (8.1.32)

where m

(q, t) is the mode-coupling contribution to the viscosity and m

corresponds

to the bare or short-time contribution to the viscosity. For practical calculations of the

self-consistent model the renormalizations are computed from the relevant self-energies at

one-loop order (see eqns. (7.3.66) and (7.3.67) discussed earlier),

˜m

(q, t) =

2mβ



(2π)

(q, k)]

S(k)S(k

)φ(k

, t)φ(k, t), (8.1.33)