with the Gibbs“Duhem relation c1 (∇μ1 )p,T + c2 (∇)μ1 )p,T = 0, or alterna-

tively as

1

σdi¬ = ’ (u1 ’ u0 )c1 (∇μ1 )p,T . (10.7)

T

The linear response assumption is that a system that is not in overall equi-

librium will develop ¬‚ows J i proportional to driving forces X j (de¬ned such

that σ = i J i · X i ) according to the Onsager phenomenological relations

(16.109) and (16.111):

Ji = Lij X j ; Lij = Lji . (10.8)

j

On the mesoscopic level of theory the transport coe¬cients Lij are input

parameters for mesoscopic simulations; they can be derived from experiment

or from non-equilibrium simulations at the atomic level, but do not follow

from mesoscopic system simulation. One may adopt the simplifying but

poor assumption that there are only diagonal transport coe¬cients.

For a two-component system there is only one coe¬cient connecting the

relative particle ¬‚ux (i.e., di¬usional ¬‚ux) to the chemical potential gradi-

ents. This coe¬cient is related to the di¬usion constant in the following way.

For a dilute or ideal solution of component 1 in solvent 0 (i.e., small c1 ),

the thermodynamic potential (see Chapter 16, Section 16.6 on page 435) is

given by

μ1 = μ0 + RT ln(c1 /c0 ), (10.9)

1

and hence

RT

∇μ1 = ∇c1 , (10.10)

c1

while the di¬usional ¬‚ux equals the di¬usion constant D times the concen-

tration gradient:

J d = c1 (u1 ’ u0 ) = ’D∇c1 . (10.11)

1

Combined this implies that

D

u1 ’ u0 = ’ ∇μ1 . (10.12)

RT

The negative gradient of μ1 is the thermodynamic force that tries to move

component 1 with respect to component 0; in the steady state the thermo-

dynamic force is counterbalanced by an average frictional force ζ(u1 ’ u0 ),

where ζ is the friction coe¬cient. The friction coe¬cient is therefore related

300 Mesoscopic continuum dynamics

to the di¬usion coe¬cient by

RT

ζ=

. (10.13)

D

For n-component mixtures there are n ’ 1 independent concentrations and

1 2

2 n(n’1) di¬usion coe¬cients. In the local coupling approximation (LCA) it

is assumed that the transport coe¬cient is proportional to the local density

and the gradient of the thermodynamic potential.

Now consider the time evolution of the concentration ci of species i. In the

mesoscopic literature it is costumary to indicate this quantity by the density

ρi , expressed either in number of particles or in moles per unit volume, and

we shall adopt this convention. We shall focus on the structural rearrange-

ments in mixtures following material transport and therefore simplify the

system considerably by considering an isothermal/isobaric system, in which

there is no heat ¬‚ux, electric current, or bulk ¬‚ow. The continuity equation

for species i reads

‚ρi

= ’∇J i (10.14)

‚t

with the ¬‚ux in the local coupling approximation and including a random

term J rand due to thermal ¬‚uctuation:

i

J i = ’M ρi ∇μi + J rand , (10.15)

i

where we take for simplicity a single transport coe¬cient

D

= ζ ’1

M= (10.16)

RT

and where J rand is the random residual of the ¬‚ux which cannot be neglected

i

when the coarse-graining averages over a ¬nite number of particles. This

“noise” must satisfy the ¬‚uctuation“dissipation theorem and is intimately

linked with the friction term; it is considered in the next section.

Note The friction can be treated with considerably more detail, e.g., one may

distinguish the frictional contribution of di¬erent species (if there are more than

two species), in which case the ¬‚ux equation becomes a matrix equation. One

may also generalize the local coupling approximation inherent in (10.15) and use a

spread function for the local friction. So the general form is

J i (r) = ’ Λij (r; r )∇μj (r ) dr + J rand , (10.17)

i

V

j

2 The mutual di¬usion constants are complicated functions of the concentrations, but the depen-

dencies become much simpler in the Maxwell“Stefan description in terms of inverse di¬usion

constants or friction coe¬cients, because the frictional forces with respect to other components

add up to compensate the thermodynamic force. See Wesselingh and Krishna (1990) for an

educational introduction to the Maxwell“Stefan approach, as applied to chemical engineering.

10.3 The mean ¬eld approach to the chemical potential 301

with

Λij (r; r ) = M ρi δij δ(r ’ r ) (10.18)

in the local coupling approximation.

The equation for the evolution of the density of species i is given by the

continuity equation for each species, provided there are no chemical reactions

between species:

‚ρ1

= ’∇ · J i = M ∇ · (ρi ∇μi ) ’ ∇J rand . (10.19)

i

‚t

10.3 The mean ¬eld approach to the chemical potential

What we are still missing is a description of the position-dependent chemi-

cal potential given the density distribution. When we have such a relation

the gradients of the thermodynamic potentials are known and with a proper

choice of the mobility matrix the time evolution of a given density distribu-

tion can be simulated. Thus we can see how an arbitrary, for example homo-

geneous, density distribution of, e.g., the components of a block copolymer,

develops in time into an ordered structural arrangement.

The thermodynamic potential is in fact a functional of the density distri-

bution, and vice versa. In order to ¬nd the chemical potential, one needs

the total free energy A of the system, which follows in the usual way from

the partition function. The Hamiltonian can be approximated as the sum

of a local contribution, independent of the density distribution, based on a

local description of the unperturbed polymer, and a non-local contribution

resulting from the density distribution. Simple models like the Gaussian

chain model su¬ce for the local contribution. The non-local contribution

to the chemical potential due to the density distribution is in mesoscopic

continuum theory evaluated in the mean-¬eld approximation, essentially fol-

lowing Landau“Ginzburg theory.

If the free energy A, which is a functional of the density distribution, is

known, the position-dependent chemical potential is its functional derivative

to the density:

δA

μ(r) = . (10.20)

δρ(r)

When the system is in equilibrium, the density distribution is such that A is

a global minimum, and the chemical potential is a constant. By adding an

energy term U (r), which we call the “external ¬eld”, to the Hamiltonian,

the equilibrium density distribution will change; there is a bijective relation

between the density distribution and the external ¬eld U . The evaluation

302 Mesoscopic continuum dynamics

of the functionals is quite intricate and the reader is referred to the original

literature: Fraaije (1993) and Fraaije et al. (1997).