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

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8.1 Mode-coupling theory 371

with the notation k

= q − k. The vertex function V

is deﬁned as

(q, k) = (

q · k)c(k) + (

q · k

)c(k

). (8.1.34)

The renormalization of the transport coefﬁcients in terms of density correlation functions

is expressed in a self-consistent form here.

In an intuitive picture this self-consistent treatment is justiﬁed by appealing to the nature

of the single-particle motion in the strongly correlated liquid: the tagged particle rattles

around in the cage formed by its surrounding particles, which are also trapped. The tagged

particle thus needs to be treated in the same manner as the ones forming the cage around it

and hence the renormalized transport coefﬁcient is taken to be a self-consistent functional

of the density correlation function. The feedback process is a manifestation of the fact that

the motion of a particle in a dense medium inﬂuences the surroundings, which will in turn

react and inﬂuence its subsequent motion. The nonergodic phase is deﬁned in terms of the

φ(q, t) having a nondecaying part f (q) in the long-time limit,

φ(q, t) = f (q) + (1 − f (q))φ

(q, t). (8.1.35)

(q, t) goes to zero for large t. The f (q), which are termed nonergodicity parameters

(NEPs), are determined from a set of coupled nonlinear integral equations,

1 − f



˜m

(q, t →∞) ≡ H

[ f

]. (8.1.36)

In writing the set of eqns. (8.1.36) we have treated the integral over the wave vector k in

the mode-coupling term as a sum over a discrete set of k values uniformly distributed over

a grid of size, say, M, and extending up to a suitable upper cutoff value. In the discrete

form we denote, for a q = 1,...,M grid,

f (q) → f

φ(q, t) →φ

(t).

The functional H

on the RHS of (8.1.36) is obtained from the long-time limit of the

expression for ˜m

(q, t) in eqn. (8.1.33) and is a functional of all the f

. Equation (8.1.36)

now represents a set of M coupled integral equations, which are numerically solved using

iterative methods. The necessary input for this is the static structure factor S(q) for the

liquid. For low densities, only the trivial solution set with all of the f

equal to zero

(corresponding to the ergodic liquid state) is obtained. The critical density at which all

of the f

simultaneously converge to a nonzero set of values marks a dynamic transi-

tion of the ﬂuid to a nonergodic state. For example, let us consider a system of hard

spheres of diameter σ with the packing fraction ϕ =π nσ

/6. The Percus–Yevick solu-

tion with the Verlet–Weiss correction (Verlet and Weiss, 1972; Barker and Henderson,

1976)forS(k) of a hard-sphere system is then used as an input in eqn. (8.1.36). The ﬂuid

undergoes an ergodic-to-nonergodic transition at the critical value of the packing fraction

ϕ =ϕ

=0.525. The contributions to the mode-coupling integral (8.1.36) from k values

above the upper cutoff value 50σ

−1

do not affect the transition point much.

372 The ergodic–nonergodic transition

1. The cusp behavior of the NEP. In the close vicinity of this dynamic transition point,

the order parameter of this transition, i.e., the f

, exhibits a cusp behavior similar to

eqn. (8.1.24) in the schematic model. Understanding this behavior in the wave-vector-

dependent model involves an asymptotic analysis of the integral equation (8.1.36) near

the ideal transition point (Götze, 1985). The starting point of such an exercise is the

stability matrix C

, which is deﬁned as

δH

δ f

(1 − f

)

, (8.1.37)

all elements of which are positive numbers by deﬁnition. Let e

and ˆe

, respectively,

denote the right and left eigenvectors of the matrix C

at the transition point corre-

sponding to the eigenvalue 1:



= e

, (8.1.38)



ˆe

=ˆe

. (8.1.39)

The nonergodicity parameters f

are related to the corresponding f

at the dynamic

transition point as

= f

+ g



1/2

+ O(), (8.1.40)

where the amplitude is given by

= (1 − λ)

−1/2

, (8.1.41)

(

1 − f

)

. (8.1.42)

The separation parameter 

is deﬁned in terms of , namely 

=C, where C =

ˆe

and  is the relative distance from the dynamic transition point in terms of a suit-

able control parameter. For a hard-sphere system the packing fraction is the controlling

thermodynamic variable,  =(ϕ −ϕ

)/ϕ

. The parameter λ in (8.1.40) is obtained as

λ =



qkp

ˆe

q,kp

, (8.1.43)

with



∂ H

∂n



n=n

, (8.1.44)

q,kp



(1 − f

)

(1 − f

)

δ f



n=n

. (8.1.45)

It should be noted that the square-root cusp behavior of the NEP is an asymptotic result.

Results extracted from experimental data for the Debye–Waller factor are used to locate

the existence of a possible mode-coupling transition point. Since the sharp dynamic

8.1 Mode-coupling theory 373

transition of the supercooled liquid gets smeared in a real system, the square-root cusp in

reality is an idealization and its existence can be concluded only indirectly. Furthermore,

even if we conﬁne the discussion to the ideal model and ignore the cutoff mechanism,

the cusp behavior still gets modiﬁed due to leading-order corrections to the asymptotic

result (8.1.40):

= f

+ g



1/2



1 + 

1/2

. (8.1.46)

The amplitude functions

have been computed by Franosch et al. (1997)forthe

simple form of the memory function (8.1.33).

2. Scaling in power-law relaxations. The decay of the density correlation function φ

(t)

has distinct relaxation regimes similar to those occurring in the case of the schematic

model described above. First, φ

(t) decays to a plateau value f

algebraically,

(t) = f

+ h

(τ

/t)

. (8.1.47)

As in the schematic case, (8.1.47) is valid in the time window t

 t  τ

.The

time scale τ

diverges algebraically with the separation parameter 

such that τ

|

−1/(2a)

. Above the critical density, in the nonergodic glassy phase φ

(t) decays

byapowerlawt

−a

to the corresponding value of the NEP f

. For densities less than

the critical value, the initial power-law relaxation crosses over to the von Schweidler

relaxation,

(t) = f

− h

(t/τ

)

. (8.1.48)

This occurs in the time window τ

 t  τ

(which is the same as in the schematic

case in (8.1.25)). The exponents a and b of the power-law relaxations follow from the

same transcendental equation as in the corresponding schematic case. The parameter λ

in the schematic case now corresponds to the exponent parameter λ and is computed

using the formula presented in eqn. (8.1.43). For hard-sphere systems with one-loop

vertex contributions, this is computed as λ =0.74 (Fuchs et al., 1992).

The power-law decays in different time windows as described above can be expressed

in the form of a single scaling function,

−1



(t) − f

;



(

t ), (8.1.49)

where g

(

t ) refer to the master functions in terms of the reduced time

t =t/τ

, above

and below the dynamic transition point, respectively. The LHS of eqn. (8.1.49) scaled

with a factor of

√



will follow one master curve independently of the wave number

q. The power-law relaxations then correspond to the asymptotic behavior of the scaling

functions. For short times, when t  τ

, to leading orders

(

t  1) =

−a

. (8.1.50)

374 The ergodic–nonergodic transition

In the other limit of t  τ

, the scaling functions are different above and below the

transition point. In the nonergodic state

→ g

= (1 − λ)

−1/2

for

t  1. (8.1.51)

In this time regime eqn. (8.1.49) reduces to the cusp behavior of the NEP described

by eqn. (8.1.40). A crossover time ˆτ

is sometimes deﬁned when the monotonically

decreasing function reaches within, say, 0.1% of its asymptotic value, i.e., g

( ˆτ

) =

1.001g

. For the liquid side g

−

for large

t approaches the von Schweidler law

−

(

t  1) =−B

, (8.1.52)

where B is a constant. In this case a corresponding crossover time ˆτ

−

can be deﬁned

where g

−

( ˆτ

−

) = 0.

3. The factorization property. Note that both of the exponents a and b in the time

regimes corresponding to the power-law relaxations are independent of temperature

and density. From (8.1.49) it is clear that the space and time dependence of the quan-

tity φ(q, t) − f (q) factorizes. This rather remarkable prediction of the MCT has been

termed the factorization property. This is an asymptotic result that holds in the close

vicinity of the dynamic transition point (Das, 1993). The higher-order corrections can

be presented in the form of a series with factorized terms (Franosch et al., 1997)in

powers of (τ

/t)

(t) − f

= h

(

)



1 + K

(1)

(a)

(

)

, (8.1.53)

and a similar expansion in powers of (t/τ

)

for the von Schweidler relaxation,

(t) − f

=−h







1 − K

(1)

(−b)







. (8.1.54)

The q-dependent amplitudes K

(1)

(x) are the so-called “corrections to scaling.” One will

need to keep adding higher-order terms as the distance from the transition increases.

The wave-vector independence implied in the factorization property is also affected

due to other reasons. For example, the microscopic time scale t

in the q-dependent

model now assumes special signiﬁcance (Das, 1993). The time t

quantiﬁes the match-

ing of the short-time decays for all the q values to the corresponding transients which

are obviously dependent on the wave number. The microscopic frequency for wave

number q is given by 

[βmS(q)]

−1

. The time window between the crossover

from the transient to the power-law behavior given by eqn. (8.1.53) is now strongly

dependent on wave number. Furthermore, within the range of q values over which the

amplitude of the correction term becomes small, the factorization property is naturally

valid over longer times. Thus the range of validity of the so-called factorization property

is strongly dependent on q.

8.1 Mode-coupling theory 375

With the next-to-asymptotic-order terms in eqn. (8.1.49) taken into account, the new

scaling functions consist of corresponding correction terms,

−1



(t) − f

;





(

t ) +

;



(1)

(q,

t )

, (8.1.55)

where

t =t/τ

is the rescaled time. The leading-order correction terms g

(1)

for the mas-

ter functions (Franosch et al., 1997) are dependent on the wave number q. As we move

away from the dynamic transition point, the q-independence of the RHS of eqn. (8.1.55)

becomes invalid. In short, closer to the dynamic transition point, the time window over

which the correlation function φ(q, t) remains near the plateau f

expands, and hence

the range of validity of the factorization property grows.

Finally, we note that the corrections to the scaling described here are for the sim-

ple MCT model in which all aspects of the ergodicity-restoring mechanism have been

ignored. In the presence of such physically relevant processes (to be described next),

which remove the sharp dynamic transition, it remains unclear whether the system will

ever get close enough to the transition point for it to clearly display the above behaviors.

4. α-Relaxation scaling. In the region asymptotically close to the dynamic transition at ϕ

with the time scale τ

tending to ∞,theα-relaxation in the MCT is described in terms

of a temperature-independent master function φ

(t) ≡



(t/τ

). All the relaxation

curves at different temperatures will merge into a single curve with a temperature-

dependent relaxation time τ

. This property is termed the time–temperature superpo-

sition principle. The α-relaxation is asymptotically deﬁned in the limit of the control

parameter ϕ

approaching the critical value ϕ → ϕ

, with z → 0, τ

→∞, so that

ˆz =zτ

remains ﬁnite. The mode-coupling prediction for the α-relaxation is obtained

through numerical solution of the model equations. The equation for the master func-

tion  is obtained from (8.1.31)–(8.1.33) in the scaling limit z → 0, τ

→∞, so that

ˆz =zτ

. The equation for the Laplace transform



(ˆz) =τ

−1

φ(q, z) is obtained as



(ˆz) =

ˆm

(ˆz)

1 +ˆz ˆm

(ˆz)

, (8.1.56)

where we deﬁne ˆm

(ˆz) =−im

(q, z)/(τ



). The mode-coupling integrals in ˆm are

evaluated at the transition point ϕ =ϕ

. The microscopic and power-law relaxations are

eliminated here, so the

t =0 value of the scaling function

 is equal to f

. Therefore

eqn. (8.1.36) for locating the dynamic transition now corresponds to the

t → 0 limit of

(8.1.56). The scaling function 

(

t ) is obtained very accurately by numerically solving

(Fuchs et al., 1992) eqn. (8.1.56) in the time space as



(

t ) =ˆm

(

t ) −



ds ˆm

(

t − s)



(s) (8.1.57)

and using the the Percus–Yevick solution for the hard-sphere-system input static struc-

ture factor. An asymmetrically stretched shape of the spectra in the frequency space is

obtained due to the overlap of the von Schweidler relaxation. The stretched-exponential

376 The ergodic–nonergodic transition

form is a good ﬁt for the master function at all wave numbers, with better quality of

agreement at large q. The stretching exponent β is wave-number-dependent and β

shows a minimum at the ﬁrst peak of the static structure factor. The corresponding

relaxation times τ

(q) also show a peak at this wave vector.

Similarly to the scaling function for the initial power-law regimes, the relaxation

behavior described by the master function  is an asymptotic result that is valid in the

close vicinity of the transition point. The time–temperature superposition is a prediction

of the simple MCT only near the dynamic transition point. As we move away from the

transition, the temperature-independent master function requires correction terms very

much in the same way as for the power-law regime,

(t) =



(

t ) +

(

1 − f

)



(1)

(

t ) + O

(



)

. (8.1.58)

The temperature independence of the α-relaxation master function is lost with the inclu-

sion of the correction term. In contrast to the case of power-law relaxations, the leading-

order correction to the α master function in eqn. (8.1.58) is of O(

). The correspond-

ing correction to the power-law scaling function is of O





1/2



. Indeed, the numeri-

cal solutions for φ

(t) from the simple mode-coupling model show a marked depar-

ture from time–temperature superposition except in the close vicinity of the dynamic

transition point. The solutions of the mode-coupling equations away from the transi-

tion follow power-law or stretched-exponential forms with effective exponents that are

dependent both on temperature and on the wave number.

8.1.3 Tagged-particle dynamics

The single-particle density ρ

(x, t) in a ﬂuid is a microscopically conserved mode very

much like the total density ρ(x, t) or the total momentum density g(x, t). This conserva-

tion law gives rise to a new slow mode in the ﬂuid. Over hydrodynamic length and time

scales (large compared with microscopic length and time scales, respectively) this mode

corresponds to the tagged-particle diffusion or self-diffusion process in the ﬂuid. In this

section we discuss the implications of slowly decaying density ﬂuctuations in a dense ﬂuid

on the tagged-particle motion.

The equation of motion

First the tagged-particle density is deﬁned microscopically through a relation similar to

eqn. (5.3.62) for the total density,

ˆρ

(x, t) = mδ(x −r

). (8.1.59)

The corresponding density correlation or the van Hove correlation function,

(|r − r



|, t − t



) =

ˆρ

(r, t) ˆρ



, t



)

(8.1.60)

8.1 Mode-coupling theory 377

represents the time-dependent spatial distribution of the tagged particle. The constraint



dr G

(r, t) = 1 (8.1.61)

ensures the particle conservation. In the thermodynamic limit the average





is zero and

hence δρ

≡ ρ

It is rather straightforward to see, by taking a time derivative of the RHS of eqn. (8.1.59),

that the conserved tagged-particle density ρ

satisﬁes the microscopic continuity equation,

∂ ˆρ

∂t

+∇·

= 0. (8.1.62)

The corresponding current density

is obtained as

(x, t) = p

δ(x − r

). (8.1.63)

The coarse-grained quantities ρ

and g

are related through a phenomenological constitu-

tive relation:

(x, t) =−D

∇

(x, t), (8.1.64)

linking the tagged-particle density and this current at large length scales. On combin-

ing eqns. (8.1.59) and (8.1.64) we obtain the diffusion equation for the tagged-particle

density ρ



∂

∂t

− D

∇



= 0, (8.1.65)

where D

is the tagged-particle diffusion constant.

However, unlike the total momentum density

g, the momentum density

of the tagged

particle is not a conserved property. It is also not a slow variable in the sense a heavy

Brownian particle is, since a tagged particle is in the sea of identical particles in the

ﬂuid. Obtaining a ﬂuctuating equation for the tagged-particle current density g

requires

a phenomenological approach. The equation of motion for the current g

is obtained by

following the standard procedure described above with respect to the dynamics of {ρ,g}.

The reversible part of the equation for g

is obtained from the usual Poisson brackets. From

the above microscopic deﬁnitions the Poisson brackets are obtained as

ˆg

(x), ˆρ



)

=∇

[δ(x − x



) ˆρ



)], (8.1.66)

ˆg

(x), ˆg



)

=−∇



δ(x − x



) ˆg

(x)

+∇





δ(x − x



) ˆg



)

. (8.1.67)

The other ingredient required for construction of the equations of ﬂuctuating nonlinear

hydrodynamics for the tagged particle is the driving free-energy functional in terms of the

coarse-grained densities. For this we consider a functional similar to the one used for the

one-component system,

= F

[ρ

]+F

[ρ

, g

], (8.1.68)

378 The ergodic–nonergodic transition

where F

is the so-called potential or density (ρ

)-dependent part of the free-energy func-

tional F

for the single-particle dynamics. F

is the kinetic or the momentum density

-dependent part of the free-energy functional and is assumed to take the form

[ρ

, g





(x, t)

2ρ

(x, t)



. (8.1.69)

This is very similar to the corresponding kinetic-energy part F

of the driving free energy

(see eqn. (6.2.7)) in our earlier discussion of the ﬂuctuating-hydrodynamics description

with respect to the total variables {ρ,g}. This form (8.1.69) gives for the equation of motion

for ρ

a continuity equation similar to (8.1.62), namely

∂ρ

∂t

+∇· g

= 0. (8.1.70)

Now, using the standard prescription outlined in the earlier chapter (see Section 6.3), we

obtain the generalized Langevin equation for the tagged-particle momentum density,

∂g

∂t

=−ρ

∇

δF

δρ

−



∇





+ 

δF

δg

+ ζ

. (8.1.71)

Here we have taken the nonzero elements in the bare transport matrix to correspond to



≡ 

. The bare matrix determines the correlation of the Gaussian noise ζ

through

the ﬂuctuation–dissipation relation,

(x, t)ζ



, t



)

= 2k

T 

δ(x − x



)δ(t − t



). (8.1.72)

Since g

is not a conserved density, 

does not involve any spatial gradient operator. The

corresponding Fourier transform is nonzero in the small-wave-number limit. The ﬁrst term

on the RHS represents the reversible part of the dynamics,

=−ρ

∇

δF

δρ

. (8.1.73)

With the above choice for F

, the dissipative part of the single-particle dynamics given by

the third term on the RHS of eqn. (8.1.71) is obtained as



δF

δg

= γ

, (8.1.74)

where we have deﬁned the dissipative coefﬁcients for the tagged-particle dynamics to be

proportional to the tagged-particle density,



≈ ρ

(8.1.75)

as the simplest approximation (Kirkpatrick and Nieuwoudt, 1986a). Equation (8.1.71) for

the tagged-particle momentum density is now obtained in the form

∂g

∂t

+ ρ

∇

δF

δρ

+ 

δF

δg

= ζ

(x, t). (8.1.76)

8.1 Mode-coupling theory 379

The above equation of motion is similar to the balance equation (6.3.31) which we obtained

in Section 6.3.2 of Chapter 6 for the microscopic current

(x, t) with Fokker–Planck

dynamics. This similarity also requires identifying the bare interaction potential βU with

the Ornstein–Zernike direct correlation function −c(r). The primary difference between

these two equations is that, with dissipative Fokker–Planck dynamics, the equations deal

with the microscopic densities ˆρ

and

, whereas in the Newtonian case presented above

we are dealing with the coarse-grained densities. With the above choice of (8.1.75) of

the dissipative coefﬁcients the noise becomes multiplicative in this case as in the case of

eqn. (6.3.31).

To complete the construction of the equation of motion for g

we need to evaluate the

functional derivative



δF

/δρ



which appears on the RHS of (8.1.73). Similarly to in the

case of the total density, we obtain F

[ρ

]=F

[ρ

]+F

[ρ

]. (8.1.77)

The ﬁrst term F

on the RHS comes from the ideal-gas part while the second term F

represents the contribution from the interaction. The ideal part F

is approximated from

the one-component limit

of the result (6.2.60) for the binary mixture,

β F

[ρ



dx ρ

(x)





(x)



− 1



, (8.1.78)

where the subscript signiﬁes the self-part here. Using the above expression for the ideal

gas part F

, we obtain the functional derivative as

δF

δρ

[

ln ρ

(x, t)

]

δF

δρ

. (8.1.79)

The ﬁrst term actually produces a linear term for the dynamics.

Linear dynamics

Let us ﬁrst consider the linear equation of motion for g

. The ideal-gas contribution (the

ﬁrst term on the RHS of eqn. (8.1.79)) contributes a linear term in the reversible current in

the equation for g

∂g

∂t

+ 

+ v

∇

= ζ

. (8.1.80)

We ignore the convective nonlinearity given by the second term on the RHS of eqn. (8.1.71)

in the treatment below. It should be noted here that, in our earlier discussion of the tagged–

particle momentum-density equation in Section 6.3.2 with Fokker–Planck dynamics, we

saw that in the adiabatic approximation a similar nonlinear convective term gives rise to a

term that is linear in the density ﬂuctuation. This is identical to the ideal-gas contribution or

the excluded-volume term in the dynamics involving ∇

, i.e., the third term on the LHS of

eqn. (8.1.80). The single-particle dynamics at the linear level is now completely decoupled

This is equivalent to making the two species in the mixture identical and setting N

and N

equal to N − 1 and 1, respectively.

380 The ergodic–nonergodic transition

from the collective dynamics involving {ρ,g}. Equations (8.1.70) and (8.1.80) are similar

to the corresponding equations (6.2.16) and (6.2.20), respectively, for ρ and g. To keep the

treatment simple, we take 

as a constant (independent of frequency) and replace it by

its value in the small-q limit. The time correlation functions of the tagged-particle density

for these two linear equations of motion are easily computed using the MSR technique

outlined above, or using directly a memory-function approach (Ma and Mazenko, 1975).

The Fourier–Laplace transform F

(q, z) of the van Hove correlation function G

(r, t) is

obtained in the form of a two-step continued fraction

(q, z) =



z −

z + i 



−1

. (8.1.81)

This result for F

(q, z) is similar to the expression (7.3.68) for the total density correla-

tion. Here v

is the average speed of the ﬂuid particle. In the long-time limit (small z) F

develops a diffusive pole

(q, z) =

z + iq

, (8.1.82)

with the bare self-diffusion coefﬁcient D

/

. For the hard-sphere system 

com-

puted from the Enskog model gives the Enskog result, 

=2/(3t

Mode-coupling effects

Since the average value of





∼ V

−1

, where V is the volume of the system, it is negli-

gible in the thermodynamic limit. The second term on the RHS of eqn. (8.1.79) involving

the interaction part of F

therefore makes a contribution nonlinear in ρ

. The crucial input

here is the functional derivative of F

with respect to ρ

. Using for the interaction part F

an expansion in terms of the density ﬂuctuations and involving the Ornstein–Zernike direct

correlation functions of the one-component ﬂuid, we obtain

δF

δρ

=−

βm



c(x −x



)δρ(x



, t)dx



, (8.1.83)

keeping only the lowest-order contributions involving the two-point direct correlation func-

tions. See the Ramakrishnan–Yussouff free-energy functional given by eqns. (2.1.38) and

also (6.2.14). This form was considered earlier with respect to both equilibrium density-

functional calculations and the mode-coupling dynamics of the collective variables in

Chapters 2 and 6, respectively. The above form (8.1.83) of the dynamic equation includes

in the single-particle dynamics the feedback effects resulting from coupling of the tagged-

particle density ρ

with the total density ﬂuctuations, i.e., δρ . Following standard proce-

dures (see Section 7.1.1 for how to compute the perturbative corrections to the transport

properties) we obtain the mode-coupling contribution to the kernel 

. Upon including