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

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5.4 Hydrodynamics of a solid 251

(Cohen et al., 1976; Fleming and Cohen, 1976) as a measure for the defect density ρ

(x, t)

in the solid,

(x, t) = δρ + ρ

∇ · u(x). (5.4.32)

In the next section we present the deduction of a set of nonlinear ﬂuctuating hydrodynamic

equations for the extended set of slow modes {ρ,g, u}. We drop the energy density from

the set to keep the discussion simple.

Linear dynamics

A uniﬁed uniﬁed hydrodynamic approach to many-particle systems was developed (Martin

et al., 1972; Cohen et al., 1976; Fleming and Cohen, 1976) to describe the linear transport

in a crystal as well as in an amorphous solid. For a crystal the number of slow modes

is extended to take into account the corresponding Nambu–Goldstone modes due to the

breaking of the isotropic symmetry. As was pointed out earlier, the extra slow modes in

the solid are the displacement variables u around a corresponding set of points forming

the rigid lattice structure. The latter in the case of the crystal is a structure with long-

range order representing various space groups. Thus, in a cubic crystal the total number

of slow modes goes up to eight, which is the combination of the ﬁve conservation laws of

an isotropic ﬂuid and the three displacement variables u. The resulting formulation of the

hydrodynamics of the cubic crystal gives rise to transverse and longitudinal sound modes

(three each) as well as the energy diffusion and the very slow diffusion of vacancies in

the crystal. For an amorphous solid the lattice structure mentioned above is a random set

of points without any long-range order. An approach similar to that for the crystal is also

adopted here, albeit with some obvious limitations. This description of the amorphous solid

holds over time scales shorter than that for structural relaxation. The amorphous solid is

treated as being similar to the crystal over length scales related to local structures, although

the frozen state is isotropic over long distances. This approximation is further supported

by the fact that in an experiment transverse sound modes similar to those in a crystal are

observed in the glass. The hydrodynamic description of the glassy state is formulated in

terms of a displacement ﬁeld u(x) about the local metastable positions of the atoms, which

remain unaltered for a long time in the glassy state.

The dynamical equations for the ﬂuctuating variables are obtained with the standard

recipe outlined in Section 5.3.1. We discuss the equations of linearized dynamics here

in terms of a set of Langevin equations obtained in the form (5.3.56) for a given set of

collective modes {

}≡{ρ,g, u}. The construction of the dynamical equations has three

main ingredients.

I. The Gaussian free-energy functional

In order to compute the nonlinear ﬂuctuating hydrodynamic equations we need the

explicit form of the effective Hamiltonian F

for the solid. In addition to the usual

terms that appear in the free-energy functional for an isotropic liquid, we include

252 Dynamics of collective modes

the energy cost due to distortion in the elastic solid. The free-energy functional is

expressed as a sum of two contributions,

= F + F

, (5.4.33)

where F is the free energy of the isotropic liquid. The elastic part F

of the free energy

of the solid state (Mazenko, 2002a, 2002b) is written as



dx C

ijkl

(x)u

(x), (5.4.34)

where C

ijkl

is the fourth-rank elastic tensor and u

is the strain tensor expressed in

terms of the symmetrized derivative of the displacement ﬁeld u,

∇

+∇

. (5.4.35)

depends only on the derivative of the u ﬁeld. A simple translation of the sys-

tem does not cost any extra free energy. The free-energy functional is kept Gaussian

(quadratic) in the hydrodynamic ﬁelds {ρ,g, u}. Coupling of the density ﬂuctuations

to the displacement ﬁeld u is being ignored here and will be considered later in the

context of nonlinear dynamics.

In general, the rank-four matrix C

ijkl

in three dimensions has 81 components. How-

ever, the number of independent components of the elastic tensor is determined by

symmetry considerations. Since u

is symmetric by construction, it follows from the

expression (5.4.34) for the free energy that the elastic tensor C

ijkl

remains

unchanged under the interchange of i ⇔ j as well as k ⇔ l, i.e., C

ijkl

= C

jikl

ijlk

.IntheVoigt(Weiner, 1983) notation the pair of indices ij is treated as a single

index α (say). In three dimensions α is equated to any of six different combina-

tions, namely α ≡{11, 22, 33, 12, 23, 31}. Furthermore, the elastic tensor remains

unchanged under the transformation {ij}⇔{kl}, i.e.,

ijkl

= C

kli j

= C

lkij

. (5.4.36)

Denoting C

ijkl

≡ C

αβ

, the above symmetries are equivalent to representing the elastic

tensor with a 6 × 6 symmetric matrix C

αβ

= C

βα

. Thus C

ijkl

has in general 21

components. For a cubic crystal the number of independent elastic constants reduces

to three,

ijkl

= C

+ δ

− δ

+ C

ijkl

, (5.4.37)

where δ

is the cubic invariant tensor which is equal to 1 if i = j = k = l and zero

otherwise. C

, C

, and C

are related to the corresponding elastic constants deﬁned in

the Voigt notation (Landau and Lifshitz, 1975b; Weiner, 1983)as

xxxx

≡ C

= C

, (5.4.38)

xxxy

≡ C

= C

−C

, (5.4.39)

xyxy

≡ C

= C

. (5.4.40)

5.4 Hydrodynamics of a solid 253

For an isotropic system this reduces to two independent elastic constants. We discuss

this case separately below.

II. The reversible part

The reversible terms in the linear Langevin equation (5.3.56) in the generalized hydro-

dynamic model involve the Poisson brackets Q

={φ

,φ

} between the different

slow variables of the set {φ

}. To evaluate these Poisson brackets one requires the

microscopic deﬁnitions for the variables

(x) corresponding to ﬁelds φ

(x).Inthe

present context we are dealing with a system of N classical particles, each of mass m,

and the position and momentum of the αth particle at time t are denoted by r

(t)

and p

(t), respectively. For the density ˆρ and the momentum density

g the stan-

dard prescriptions are stated in (5.3.34). The expression for the ﬂuctuating variable

u(x, t) is obtained in terms of the displacements of the individual particles u

(t)

(α = 1,...,N ) from their respective mean positions denoted by R

(see eqn. (5.4.28)),

ˆρ(x, t) ˆu

(x, t) = m



α=1

(t)δ(x − r

(t)). (5.4.41)

The above deﬁnition of the Poisson bracket follows from a generalization of the rela-

tion (5.4.30) (we take m = 1) so as to include the presence of vacancies. Using the

canonical Poisson-bracket relation

, p

= δ

αβ

, (5.4.42)

we obtain the following results for the various elements of the matrix Q

involving

the ﬁeld u:

≡{ˆg

(x), ˆu



)}=−δ(x − x



)



−∇



)

≡{ˆu

(x), ˆu



)}=0,

≡{ˆu

(x), ˆρ(x



)}=0. (5.4.43)

The other elements of the Q

matrix are obtained from eqn. (5.3.65). The equilibrium

averages of the above expressions for the various Poisson brackets are the constants

and are useful inputs Q

 in eqn. (5.3.56).

III. The dissipative part

The dissipative part of the dynamics of the slow variable is given by the third term on

the LHS of the Langevin equation (5.3.56). It involves the bare-transport-coefﬁcient

matrix L

. The equations of motion for the Nambu–Goldstone modes now include

new elements in the bare-transport-coefﬁcient matrix in addition to the bare viscosities

for the isotropic liquid. The dissipative term in the equation for the u ﬁeld is assumed

to have the simple time-dependent Ginzburg–Landau (TDGL) form,

≡ ϒ

= ϒ

, (5.4.44)

254 Dynamics of collective modes

and we require that all other L

= 0. The transport coefﬁcient ϒ

relates to the very

slow vacancy-diffusion process in the solid. For the momentum density equation, the

irreversible part has the standard form with the dissipative coefﬁcient L

≡ L

It is straightforward to link the expression L

δF/δg

for the irreversible term of the

Langevin equation to the generalized viscosity tensor η

ijkl

. The dissipative part of the

stress tensor σ

is expressed in terms of the constitutive relation

= η

ijkl

∇

(5.4.45)

involving the gradient of the velocity ﬁeld v thermodynamically conjugate to the

momentum density g. The dissipative term of the Langevin equation (5.3.56) is

expressed in terms of the viscosity tensor η

ijkl

as follows:

∇

≡ L

δF

δg

. (5.4.46)

As we have already discussed in Section 5.1.3, the fourth-rank viscosity tensor η

ijkl

has the same symmetries as the elasticity tensors and hence they have the same number

of independent coefﬁcients. The g-dependent part F

of the free energy F is obtained

as in eqn. (5.3.73) for the case of an isotropic liquid. Assuming a purely quadratic

form of the free-energy functional, ρ in the denominator of the expression for F

replaced by ρ

at the lowest order. Therefore, using the relation v

= δF/δg

= g

/ρ

and (5.4.46), the bare matrix L

is obtained in terms of the viscosity tensor as

= η

iklj

∇

. (5.4.47)

The number of independent elements of the fourth-rank viscosity tensor η

ijkl

for the

crystal is greater than that for an isotropic liquid. The actual number depends on the

symmetry of the crystal. For example, in a uniaxial crystal the number of independent

viscosity coefﬁcients is ﬁve, with the following form for η

ijkl

= η

+ η

(δ

+ δ

)

+ η

{δ

(δ

+ δ

) + δ

(δ

+ δ

)}

+ η

(δ

+ δ

). (5.4.48)

The isotropic solid has two independent coefﬁcients and we discuss this case sepa-

rately below.

5.4 Hydrodynamics of a solid 255

For the general case of a system with broken symmetry we obtain the following equa-

tions for the linearized dynamics of the slow modes:

∂ρ

∂t

+∇· g = 0, (5.4.49)

∂g

∂t



∇

= θ

, (5.4.50)

∂u

∂t

−

+ ϒ

δF

δu

= ζ

. (5.4.51)

The random parts in these equations are Gaussian noises and are related to the bare trans-

port coefﬁcients by

θ

(x, t)θ



, t



)=2ρ

δ(x − x



)δ(t − t



), (5.4.52)

ζ

(x, t)ζ



, t



)=2k

T δ

δ(x − x



)δ(t − t



)ϒ

, (5.4.53)

ζ

(x, t)θ



, t



)=0. (5.4.54)

The different matrices of bare transport coefﬁcients L

, ϒ

etc. for the inhomogeneous

crystalline state can be obtained by formulating a theory for the solid similar to the Enskog

theory for liquids. For the hard-sphere system such a calculation has been done

(Kirkpatrick et al., 1990). The stress-energy tensor σ

in eqn. (5.4.50) has reversible and

dissipative (irreversible) parts, denoted by σ

and σ

, respectively, such that

= σ

+ σ

. (5.4.55)

Using the expression (5.4.33) for the free-energy functional, we obtain its functional deriva-

tive with respect to u

as,

δF

δu

= C

ijkl

∇

. (5.4.56)

The reversible part of the stress-energy tensor is now given by



−1

ijkl

, (5.4.57)

where the strain tensor u

is deﬁned in eqn. (5.4.35). The dissipative part is obtained in

terms of the strain tensor as

=−η

ijkl

. (5.4.58)

Using the above result (5.4.56) for the functional derivative of the free energy, eqn. (5.4.51)

for the displacement ﬁeld reduces to the form

∂u

∂t

− ϒ

ijkl

∇

+ ζ

. (5.4.59)

The set of ﬂuctuating equations obtained above describes the dynamics of the slow

modes for the elastic solid. It is generally applicable to a crystal in which long-range

256 Dynamics of collective modes

translational symmetry is broken. The corresponding decay of hydrodynamic ﬂuctuations

conforms to the existence of transverse sound modes in addition to the longitudinal sound

modes in a solid. These equations have also been extended to study the dynamics of amor-

phous solids. In this case we treat the solid as a system in which freezing has occurred at

the scale of local structure but overall translational invariance is maintained over longer

distances.

The isotropic solid

For the isotropic solid the rank-four elasticity tensor C

ijkl

is determined in terms of only

two independent elastic constants,

ijkl

= λ

+ μ

(δ

+ δ

), (5.4.60)

where λ

and μ

are generally termed Lamé constants. They are, respectively, the bulk and

shear modulus of the amorphous solid. The longitudinal modulus is given by

= λ

+ μ

. (5.4.61)

The expression (5.4.34) for the free energy now reduces to the form





(∇ · u)

+ 2μ

(∇

)

. (5.4.62)

Similarly, for the isotropic solid the rank-four viscosity tensor η

ijkl

is determined in terms

of only two independent bare viscosities, ζ

(bulk viscosity) and η

(shear viscosity):

ijkl

=−ζ

− η



+ δ

−



. (5.4.63)

The second term on the RHS represents the traceless component. For the quantity in square

brackets in this term, with any ﬁxed i and j (k and l), if we set k = l (i = j) and sum over

the common index k (i), the trace is zero. The longitudinal viscosity is deﬁned as

= ζ

4η

. (5.4.64)

Using (5.4.63) in the relation (5.4.47), the dissipative matrix L

reduces to the form

(5.3.71) discussed earlier for the isotropic case. In the ﬂuctuating-hydrodynamics descrip-

tion, for the isotropic amorphous solid the bare transport coefﬁcients ζ

, η

, and D

act as

external input parameters in the theory.

The equation of motion for the momentum density is given by (5.4.50), in which the

stress-energy tensor

= σ

+ σ

(5.4.65)

is a sum of a reversible part and an irreversible part. The reversible part is a sum of a

diagonal term and an off-diagonal term,

= Pδ

+ μ

∇

, (5.4.66)

5.4 Hydrodynamics of a solid 257

where the pressure P is expressed as a functional of density and the displacement ﬁeld as

P = c

δρ + ϑ

∇ · u, (5.4.67)

with c

= ρ

−1

being the speed of sound in the isotropic ﬂuid. The dissipative part of the

stress tensor in this case is obtained in the same form as for the isotropic ﬂuid. In terms of

the gradients of the momentum density ﬁeld we write the σ

in terms of the corresponding

kinetic viscosities as

= δ

(

)

∇ · g +

∇

. (5.4.68)

We have deﬁned the kinetic bulk and shear viscosities of the solid as ς

= ζ

/ρ

and ν

/ρ

, respectively. The corresponding sound attenuation is deﬁned as 

= ς

+ 4ν

/3.

Finally, the equation of motion for the displacement ﬁeld u

now takes the form

∂u

∂t

− ϒ

(

)

∇

∇ · u +μ

∇

+ ζ

. (5.4.69)

For the isotropic solid the vacancy density ρ

is deﬁned above in eqn. (5.4.32) and the

dynamical equation for ρ

(x, t) is obtained using eqns. (5.4.49)–(5.4.51) for the slow

modes,



∂

∂t

−

∇



∇

ρ = f

, (5.4.70)

where we deﬁne

= ϑ

and the noise f

= ρ

∇ · f. The variable ρ

(x, t) is related

to the longitudinal part of u through the quantity ∇ · u. In the context of amorphous solids

has been interpreted as a density of free volumes in the amorphous solid (Cohen et al.,

1976). The transverse part of u is related to the transverse sound modes in the amorphous

solid.

In order to demonstrate the hydrodynamic modes in the isotropic solid, we analyze the

pole structure of the hydrodynamic correlation functions as was done in the earlier section

for the isotropic ﬂuid. Using the isotropic symmetry (in the present case of an isotropic

solid), we split the correlation functions with the vector ﬁelds into longitudinal and trans-

verse parts. Elements of the memory-function matrix K

for the extended set {ρ,g, u}

are computed using the relations (5.3.41) and (5.3.50) for the static and dynamic parts of

, respectively. The static correlation function matrix S

is diagonal, with its diagonal

elements the same as before. Of course, in this case we have an extra element S

.To

compute S

we note the identity

(x)

δF

δu



)

= δ

δ(x − x



). (5.4.71)

258 Dynamics of collective modes

Using the deﬁnition (5.4.62) for the free energy of the isotropic solid, the longitudinal and

transverse components of the correlation of the u ﬁeld are obtained as

(q) =

, (5.4.72)

(q) =

. (5.4.73)

The density ﬁeld couples to the longitudinal components g

and u

, reducing (5.3.23) to the

following explicit form:

⎡

⎣

zq 0

(q) z + i 

iϑ

0 −i/ρ

z + i

⎤

⎦

⎡

⎢

⎣

ρρ

ρg

ρu

gρ

uρ

⎤

⎥

⎦

=−β

−1

⎡

⎢

⎣

χ 00

0 ρ

00(ϑ

)

−1

⎤

⎥

⎦

(5.4.74)

To analyze the hydrodynamic poles we consider the determinant of the matrix on the LHS

of (5.4.74),

= (z + i

)



−q

+iz

+ic

, (5.4.75)

where c

= ϑ

/ρ

and c

= c

+ϑ

/ρ

≡ c

. The zero of D

gives rise to two poles

(up to leading order in q), at z =±qc

−(iq

/2)



, representing the two decay modes of

the density ﬂuctuation. These correspond to the propagating sound modes in the isotropic

ﬂuid at speed c

in directions ±q. The attenuation of the longitudinal sound wave (in either

direction) is given by



= 

+ α

, (5.4.76)

where α

= c



+ c



. In addition to the longitudinal sound modes, there is an extra

diffusive hydrodynamic mode given by the pole at z =−i

. This is often interpreted

as the diffusion of vacancies represented in eqn. (5.4.70) above. This is a characteristic

mode present in the solid state. The notion of voids, however, is less ambiguous for the

crystalline state with long-range order than for the amorphous glassy state.

The transverse part of the matrix of the correlation functions is obtained as



z + i ν

iμq

−i/ρ

z + i ϒ







=−β

−1



0 (μ

)

−1



. (5.4.77)

The corresponding pole in the transverse correlation functions is therefore obtained from

the determinant of the matrix on the LHS of eqn. (5.4.77),

= (z + i

)



z + ν

− c

, (5.4.78)

where c

= μ/ρ

. By setting the determinant equal to zero we obtain two poles, z =

±qc

− (iq

/2)˜ν

, with ˜ν

= ν

. These poles correspond to the propagating sound

5.4 Hydrodynamics of a solid 259

modes in the solid state. In the solid state the two diffusive transverse shear modes change

into transverse sound modes with speed and attenuation given by c

and ˜ν

, respectively.

The nature of the vibrational modes in the amorphous glassy state and that of those

in the crystalline state are different, which shows up in a number of anomalous prop-

erties of disordered solids at low temperature. The speciﬁc heat c

below 1 K does not

follow the Debye T

law, but instead increases linearly with the temperature T . This has

been explained phenomenologically in terms of two-level states (Anderson et al., 1972;

Phillips, 1972, 1981). In the somewhat higher temperature range, there is a peak in the T

dependence of c

. A universal feature of disordered glasses is the so-called boson peak

observed in neutron- and Raman-scattering experiments (Frick and Richter, 1995; Angell,

1995). These properties are generally ascribed to an excess density of vibrational states

(Alba-siminoesco and Krauzman, 1995; Schirmachar et al., 1998, 2007; Grigera et al.,

2003) in the amorphous solid over the predictions of the Debye model usually applied

to the crystalline state. The density of vibrational modes scaled by the square of the fre-

quency, i.e., g(ω)ω

, goes through a peak. This anomalous excitation has been observed in

disordered systems at terahertz (THz) frequencies using various techniques, e.g., neutron

scattering (Buchenau et al., 1984), Raman scattering (Tao et al., 1991) dielectric-relaxation

experiments (Lunkenheimer et al., 2000), X-ray techniques (Rufﬂe et al., 2003), and light

scattering (Surovtsev et al., 2004). This THz peak in the density of states is termed the

boson peak (Kanaya et al., 1994; Sokolov et al., 1994; Angell et al., 2003).

We have ignored in the above discussion the energy-conservation equation, which adds another diffusive hydrodynamic mode

(Martin et al., 1972).

Appendix to Chapter 5

A5.1 The microscopic-balance equations

We obtain the equations of motion for the collective densities of mass, momentum, and

energy as deﬁned in eqns. (5.1.1) and (5.1.3). The time dependence in the conserved den-

sities enters implicitly through the position and momentum coordinates of the phase-space

variables and thus in the following we omit them unless they are speciﬁcally required.

For the ﬁve conserved densities {ˆρ(r),

g(r), ˆe(r)}, the corresponding currents are the mass

current density

g(r), the momentum current density or stress tensor

¯σ(r), and the energy

current

(r). Microscopic expressions for the currents

in terms of the phase-space den-

sity are obtained using the equations of Hamiltonian dynamics. We begin with the equation

for the mass density, for which we obtain

∂ ˆρ

∂t



m{∇

δ(r − r

)}·

=−∇



δ(r − r

=−∇·

g. (A5.1.1)

We have used the property ∇

δ(r − r

) =−∇

δ(r − r

). Hence the current

for mass

density ˆρ is obtained as

(r) =



δ(r − r

) =

g(r). (A5.1.2)

Similarly, for the momentum density ˆg

we obtain

∂ ˆg

∂t





{∇

δ(r − r

)}·

+ δ(r − r

) ˙p

=−∇



δ(r − r

)





αβ

δ(r − r

αβ

=−



∇



δ(r − r

) −





αβ

{δ(r − r

) − δ(r −r

)}F

αβ

, (A5.1.3)

260