Integral equation for nonequilibrium chemical potential and the Kirkwood diffusion equation: derivation from the generalized Boltzmann equation

HAL Id: jpa-00212342

https://hal.archives-ouvertes.fr/jpa-00212342

Submitted on 1 Jan 1989

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Integral equation for nonequilibrium chemical potential

and the Kirkwood diffusion equation: derivation from

the generalized Boltzmann equation

Byung Chan Eu

To cite this version:

Integral equation

for

_{nonequilibrium}

chemical

_potential

and

the Kirkwood diffusion

_equation:

derivation from the

generalized

Boltzmann

_equation

Byung

Chan Eu

_(*)

Department

of

_Chemistry

and

_Department

of

_Physics,

McGill

_{University, Montreal,}

_PQ, Canada H3A 2K6

(Reçu

le 13 avril 1989,

accepté

le 8

_{septembre 1989)}

Résumé. 2014 Dans cet

article,

l’équation cinétique

des fluides

_simples

denses est

_{généralisée}

et

appliquée

à l’établissement d’une

_{équation intégrale}

pour le

potentiel

chimique

hors

_équilibre

et

de

l’équation

de diffusion de Kirkwood des fluides

_{polyatomiques}

denses

_(polymères,

par

exemple).

La déduction demande la

généralisation

de

_{l’équation intégrale}

de Kirkwood à des fonctions de distribution de

_{configuration, l’équation intégrale}

du

_{potentiel chimique}

local au cas

des fluides

polyatomiques

hors

_équilibre,

et un ensemble

_{d’ équations}

d’ évolution des variables

macroscopiques.

Les

_équations

d’évolution des variables

_{macroscopiques}

et la

_{thermodynamique}

des processus irréversibles ont les mêmes structures

_{mathématiques}

_{que dans le} cas des fluides

simples

denses. Les

_{liquides plus}

ou moins

_simples

sont traités

_séparément

dans cette théorie, par différentes

_intégrales

de collision

_qui

interviennent dans les

_équations

d’évolution

_(équations

constitutives)

pour divers flux tels que les tenseurs de contraintes, les flux de chaleur et les

_flux

diffusifs.

_{L’équation}

de diffusion de Kirkwood est _{obtenue pour la fonction de distribution de}

configurations

d’une molécule

_{polyatomique,}

en utilisant un ensemble

_{d’approximations}

sur les

équations

des fonctions de distribution et de diffusion du flux de masse. Comme

exemple

d’application

des

_équations

d’évolution aux variables

_{macroscopiques,}

on considère la viscosité d’une solution binaire de fluide

_polyatomique

et

_monoatomique

et on obtient une formule pour la

viscosité

_intrinsèque

en termes

_{d’intégrales}

de collision. Cette formule foumit une

_expression

de

mécanique statistique

pour la viscosité

intrinsèque.

Abstract. 2014 In

this paper the kinetic

equation

for dense

simple

fluids

_{reported previously}

is

generalized

and

_applied

to derive an

_{integral equation}

for

_{nonequilibrium}

chemical

_potential

and the Kirkwood diffusion

_equation

for dense

_polyatomic

fluids

_{(e.g., polymers).}

The derivation

requires

a

_{generalization}

of the Kirkwood

_{integral equation}

for

_{configuration}

distribution

function, the

_{integral equation}

for local chemical

_potential

to the case of

nonequilibrium

polyatomic

fluids, and a set of evolution

_equations

for

_macroscopic

variables. The evolution

equations

for

_macroscopic

variables and irreversible

_{thermodynamics}

are found to have the same

mathematical structures as for dense

_simple

fluids. The

simple

and

_nonsimple

fluids are

distinguished

in the present

theory by

the different collision bracket

_{integrals appearing}

in the evolution

_equations

_{(constitutive equations)}

for various fluxes such as stress tensors, heat fluxes Classification

Physics

Abstracts 05.00 - 51.00 - 66.00

(*)

Mailing

address :

Department

of

_Chemistry,

McGill

University,

801 Sherbrook Street West, Montreal, Quebec, Canada H3A 2K6.

and diffusion fluxes. The Kirkwood diffusion

_equation

is obtained for the

configuration

distribution function of a

_polyatomic

molecule

_{by using}

a set of

_{approximations}

on the distribution function and the mass flux diffusion

_equations.

As an illustration of

_application

of the evolution

equations

for

macroscopic

variables, the

_viscosity

of a

_binary

solution of

_polyatomic

and monatomic fluids is considered and an intrinsic

_viscosity

formula is obtained for it in terms of collision bracket

integrals.

This formula

_provides

a statistical mechanical formula for intrinsic

viscosity.

1. Introduction.

Statistical mechanical theories

_[1-5]

of dense fluids

_consisting

of

_complex

molecules

_(e.g.,

polymers

and

_{hydrocarbons)}

often

_proceed

on the basis of intuitive models and

approxi-mations that

_produce

tractable but

_{only qualitatively}

correct results. In such theories

_[1-5]

and also in

_equilibrium

theories

_[6-8]

of dense

_simple

fluids

_{approximations}

of various kinds are made in essence for chemical

_potentials

for the fluids of interest on the basis of models. As often shown in

_{equilibrium theory}

_[6-8]

of dense

_simple

_fluids,

chemical

_{potentials play}

an

important

role in

_{calculating thermodynamic}

functions,

and there is a

_great

deal of

_insight

one can

_{gain by studying approximation}

methods for chemical

_potentials.

The same would be true

for

_{nonequilibrium polyatomic}

_{fluids. In this paper} we

develop

a formal

_theory

for

nonequilibrium

chemical

_potentials

and derive the Kirkwood diffusion

_equation

for

configur-ation distribution function

_closely

related to the

former,

which would also

_help

us

_develop

some

_approximate

kinetic theories of

_complex

molecular fluids.

Kinetic

_theory

of

_transport

_{processes in dense monatomic fluids and the attendant}

_theory

of irreversible

_{thermodynamics}

have been studied in recent _papers

_[9-11]

in which we have made use of the

_generalized

Boltzmann

_equation

for dense

_simple

fluids. The formalisms

_developed

therein have been

_{successfully applied}

to

_{rheological problems}

_[10,

_{llb, c]}

and

_generalized

hydrodynamics

of non-Newtonian fluids

_[llb-d].

_They

are also

_suggestive

of

_{generalization}

to

nonsimple

fluids and condensed matter in

_general.

For the _{stated aims in this paper} we first

present

a formal

generalization

of the

_theory

to a fluid mixture

_consisting

of

_polyatomic

molecules or a mixture of monatomic and

_polyatomic

molecules. We will show that the evolution

_equations

for various

_macroscopic

variables such as the

_{density, velocity,}

internal energy, mass

_fluxes,

heat

fluxes,

stress tensors, etc. remain

_formally

similar to those obtained for dense

_simple

fluids

_except

_{for the necessary}

_{changes arising}

_{from the appearance of the} internal

_degrees

of freedom absent in the case of

_simple

(monatomic)

fluids. The

_similarity

we observe between the

_{macroscopic equations}

for

_nonsimple

and

_simple

fluids is not

_surprising

since

_nonsimple

and

_simple

fluids are, from the continuum

_theory

_standpoint,

_{distinguishable}

only

in terms of

_transport

coefficients and related molecular

_parameters.

The situation is therefore similar to the one in the

_equilibrium

_{thermodynamics}

_where,

for

_example,

the virial

equations

of state are similar for both

_simple

and

_nonsimple

fluids

_except

for the details of the virial coefficients which reflect the nature of interactions and the

_molecularity

of the fluid in

question.

By

using

the formalism

_developed,

we show that when

_polyatomic

molecules are

interpreted

to include

_polymers,

the

_present

kinetic

_theory

contains in it the well known

theory

of Kirkwood

_[12]

on

_polymer

solutions,

since the Kirkwood diffusion

equation

for

configuration

distribution function can be obtained for a

_polymer

solution if a set of

approximations

is made. We will list the

_{approximations}

in the text. The Kirkwood diffusion

equation

holds for dilute

_polymer

solutions,

but its extension into the

_higher

concentration

concentrated solutions if

_pair

correlation functions for

_polymers

were included in the calculation of conditional diffusion fluxes

_appearing

in the

_{general theory.}

_(Here

we mean

_by

a conditional diffusion flux the mass flux of a

_{particle, given}

a constrained

_{configuration}

of other

_particles).

_{It appears that}

_although

this kind of

_{generalization}

is feasible

_by

_proceeding

one

_step

further than the Kirkwood

_{theory requires}

in

_calculating

the effects of

_pair

correlations,

the

_{resulting theory}

would become

_unwieldy

and cumbersome. We find it

simpler

to use the stress tensor and other evolution

_equations

to calculate

_viscosity

and other

transport

properties

of a

_polyatomic

fluid. The formalism for

_{calculating viscosity presented}

in this paper

provides

an alternative route distinctive from the Kirkwood line of attack.

With the evolution

_{equations developed,}

we then

_investigate

_transport

_{processes of} a

_binary

dense

mixture,

with a

_particular

attention

_paid

to viscous

_phenomena

for their obvious

_utility

in connection with

_rheology

of

_polyatomic,

_e.g.,

_{polymeric, liquids.}

In section 2 the kinetic

_equation

and the evolution

_equations

are

_presented

for

_polyatomic

fluids. The latter

_equations

are to be

_compared

with those for

_simple

dense fluids with the

emphasis placed

on the

_points

of

difference,

if there is any.

By using

the

_macroscopic

evolution

_equations,

we

_{develop theory}

of irreversible

_{thermodynamics}

whose structure also remains the same as that for

_simple

dense fluids. In section 3 we

_present

a « derivation » of

the Kirkwood diffusion

_equation

for

_{configuration}

distribution function which was

_originally

obtained with the Brownian motion model for motion of

_particles

in a

_polyatomic

or

_polymer

molecule. The

_equation

can be obtained without a Brownian motion model.

_Moreover,

statistical mechanical formulas for

parameters

in the

_original

Kirkwood diffusion

_equation

are identified. We also

_present

the Kirkwood

_{integral equation}

for correlation

functions,

e.g.,

configuration

distribution functions and

_pair

correlation

functions,

for

_{nonequilibrium,}

and an

_{integral equation}

for

nonequilibrium

chemical

potentials.

These two

_equations

would facilitate

_study

of

_{nonequilibrium}

effects on correlation functions and

_{thermodynamic}

quantities

in

_general.

In section 4 formulas for

transport

coefficients are

_obtained

in terms of collision bracket

_integrals

for dense

_polyatomic

fluids. These results reduce to the

Chapman-Enskog

first

_{approximation}

for

_transport

coefficients for a dilute fluid as the

_density

of the fluid decreases.

_Specializing

to a

_{binary polymeric}

_solution,

we obtain the intrinsic

_viscosity

formula in terms of the collision bracket

_integrals.

This formula

_provides

an alternative method of calculation for intrinsic

_viscosity

which is

_basically

different from the

_existing

methods. Section 5 is for discussion and conclusion.

2. Kinetic

_equation

and

_macroscopic

evolution

_equations.

2.1 PRELIMINARY. - In view of the

_complexity

_{of notation necessary for}

_description

of

polyatomic

molecular

systems

it is useful to

_systematize

the

_symbols

and _{their usage. We will} reserve

_{subscripts a,}

b,

c, ... for

species, subscripts i, j,

k, f,

... for

molecules,

subscripts q, s, t,

... for atoms or groups in a molecule. These

subscripts

will often appear

consecutively.

For

example,

maq

means the mass of

_particle

or _{group q in molecular}

_species

a, and

raiq

means the coordinate vector of

_{particle q}

in the i-th molecule of

_species

a. With this code for usage of the

subscripts pertaining

to

_particles

in a molecule of a

_species,

we now define

symbols :

Maq =

mass of the

_{q-th particle}

_{(or group)}

in

_species

a.

raiq =

position

vector, relative to a fixed coordinate

_origin,

of the

_{q-th particle}

in the i-th

composite particle

(molecule)

of

species

a.

Rai =

center of mass vector of the i-th

_particle

of

_species

a.

aiq

= _{distance of the}

q-th particle

in

_{species a}

from its center of mass located at

=

velocity

vector of the

_{q-th particle}

in molecule i of

_species

a.

= _{center of mass}

velocity

vector of molecule i of

_{species a}

relative to the coordinate

origin.

=

maq

Valq

= momentum

conjugate

to

raiq.

=

ma

V ai

= momentum

conjugate

to

Rai .

- E

Maq,

the total mass of molecular

_species

a.

qEa

-

maq

gaiq,

the momentum

conjugate

to

taiq-The dot over a

_symbol

means the time derivative :

e.g., g

=

de/dt.

In the above

_system

of coordinates there

_clearly

holds the relation

for every

position

vectors

_{of q}

E i and

_consequently

This

_identity

_{is easy} to

_verify

if the definition of the center of mass of a molecule is used

_along

with

_(2.1).

This

_implies,

of _course, the relation

If the

_system

consists of r

_species

_{which may be}

_polyatomic

or monatomic

molecules,

the Hamiltonian may be written

or, with the center of mass kinetic energy and the internal energy of the molecules made

explict,

Here

where

_Vai

is the

_potential

_{energy of} an isolated molecule i E a. The

_potential

_energy

where

For dense

_polyatomic

fluids in which we are

_interested,

_expressing

the Hamiltonian in the

center of mass _{energy, the energy for internal}

_degrees

of freedom and the intermolecular interaction energy is not

_{particularly advantageous}

mode of

_{representation}

_{for energy from} the mathematical

_standpoint.

The

_phase

of

_{Na, Nb,}

...

particles

will be abbreviated with

X(N)=X(Na,Nb’’’.):

where {Paiq}

and

_{raiq}

_respectively

stand for the momenta and coordinates of a set of

particles

in N molecules of

_species

a, etc., and

2.2 KINETIC EQUATION AND MACROSCOPIC EQUATIONS. - We consider

an

_r-component

fluid mixture

_{containing Na, Nb, N c’}

... molecules for

species a,

b,

c, ... that may be

polyatomic,

diatomic or monatomic. The mixture contains at least one

_{polyatomic species.}

The volume of the

_system

is V. The evolution of the

_system

is

_{statistically}

described

_by

the distribution function

_obeying

a suitable kinetic

_equation.

In the case of monatomic

fluids,

either of a

single

_component

or of

_multiple

_components,

we have shown that the kinetic

equation

may be assumed to follow a

_generalized

Boltzmann

_equation

_[9].

This

_generalized

Boltzmann

_equation

has been deduced

_{by using}

the

_{similarity principle}

_[9e]

_{which may be} stated as follows : there is a set

_of

_equations

_of

motion

_for

distribution

_functions

at

_different

levels

_of

correlation which are similar in their basic mathematical structures and

_properties

and

particularly

in their broken time-reversal _symmetry. These

_equations

are

_structurally

similar to

the Boltzmann

_equation

_{for dilute gases. This}

_{principle together}

with the Boltzmann

_equation

for dilute gases has led us to a kinetic

_equation

for distribution function

_{¡(s) (x (s) ;}

_{t )}

for cluster

a

_consisting

of s monatomic

_particles

where

_Ls

is the Liouville

_operator,

_{TSI S2 ... SN}

_{(z )}

is the collision

_operator

for N clusters each of which consists of s

_{particles (thus}

_{sl, s2,} etc.

_meaning

the

first,

second,

etc. clusters of s

particles),

ctl is the renormalized

_{(i. e. ,}

scaled

_up)

volume of the

_system

lu = NV with N

standing

for the number of

_clusters,

and X is the number of

_particles,

X = sN. This manner

of renormalization leaves the bulk

_density

invariant : n =

lim s /V

= lim

X 1 crI .

In

(2.8)

the

limit z - + 0 must be understood. This

_parameter

_{z i e (e :::. 0)}

_gives

a measure of duration of molecular collisions which take the

system

of molecules from an initial to a final state. This collision time is assumed to be much _{shorter than the time span}

_(kinetic

time

_scale)

over which the distribution functions evolve.

Therefore,

the kinetic

_{equation (2.8)}

must be

_regarded

as a

time-coarse-grained

equation

for distribution functions that are _{averages of}

fine-grained

distribution functions over the duration of collision. _{The collision time span is difficult} to

The distribution functions

_i,,(’)

are normalized as follows :

and if there is a molecular

quantity

A (x (’) ;

t )

such that

where Aj

are

_{single-particle}

variables and

_Ak

are

two-particle

variables,

then the average for A at r is

_{given by}

We will henceforth use

_angular

brackets to _{denote the average of mechanical}

_quantity

A(x(’»

over the

_phase

_{space with the statistical}

_weight

In

_(2.10)

the second line follows from the first

_by

virtue of the normalization condition

_(2.9),

and the third line from the second

_by

the

_identity

and

_symmetry

of the clusters

_{(subsystems).}

The last line is

_simply

the definition of

_{F (X)(x (X) ;}

_{t ),}

but it may be also construed as the usual definition of A which is

_commonly

encountered in the dense fluid kinetic

_theory

based on the Liouville

_{equation :}

see

_Irving

and Kirkwood

_[13].

The kinetic

_{equation (2.8)}

with the rule of

_averaging

_(2.10)

_provides

a means to calculate necessary

macroscopic

variables to

_study

_{irreversible processes in dense}

_simple

fluids. The

argument

[9e]

used to obtain the kinetic

_equation

_(2.8)

and the rule of

_averaging

_(2.10)

are

general

since no reference is made to the

_molecularity

of the substance

_involved,

and hence can be so

_generalized

as to _{describe irreversible processes in dense}

_polyatomic

fluids.

r

We

_imagine

N clusters of s

_{= L sa}

_particles

_{where Sa}

stands for the number of molecules of

a = 1

species

a. Then with the

_meaning

of the

phase

for

particles

extended as in

(2.7)

we can

The collision

_operator

_TS(l),

s(2), ...,

s(N)(z) = TS1,

_S2, ..., _SN

(z)

for X

particles

broken up into N

clusters of s molecules is determined

_by

the classical

_{Lippmann-Schwinger equation}

_[9a]

where

_{c(N)== {s(l),}

_{s (2 ),}

_{..., s(N)}}

stands for N clusters of s

_particles,

_£C’(N)

denotes the

intercluster interaction Liouville

_operator

and

_{R(O ) (z)}

is the resolvent

_operator

for N

independent

correlated clusters

Here

_Ls(a)

is the Liouville

_operator

for the o’-th cluster and has the form as

given

in

_(2.11).

Then

_{by using}

the Boltzmann

_equation

for a

_polyatomic

_{gas mixture and the}

_similarity

principle

it is

_possible

to arrive at the kinetic

_equation

where

Here

_{sa ( a )}

stands for the number of

_particles

of

_{species a}

in cluster a. The kinetic

_equation

is

a

_postulate

for the

_polyatomic

fluid in

_question.

It

_clearly

remains

_structurally

the same as the dense

_simple

fluid kinetic

_equation

_(2.8)

_except

for the increased dimension of the

_phase

space to accommodate the internal

_degrees

of freedom for

_polyatomic

molecules. The

significance

of the

_parameter

z in

(2.12a), (2.12b)

and

(2.13)

is the same as discussed in connection with

_(2.8).

In

_(2.13)

the limit £ -. + 0 must be understood as taken. We thus must

regard

the distribution functions in

_(2.13)

as

_{coarse-grained}

ones which evolve on a kinetic

time scale much

_longer

than the collision time scale on which the collision

_operator

TS(l), s(2), ..., s(N)

is defined.

_Appearance

of the collision

_operator

in the kinetic

_equation

presumes that there are well

_separated

time scales of molecular collisions and kinetic

evolution of distribution functions. Existence of such time scales is

_generally

assumed in modern works

_[14]

on kinetic

_theory

and dense fluids in

_particular.

As a matter of

_fact,

a

binary

collision

_operator,

a

_{particular example}

of such collision

_operator,

can be used to write the Boltzmann collision

_operator

as is often done in the derivation of the Boltzmann

equation ;

see references

_[9a]

and

_[15].

In this case it is

_possible

to show

_{unambiguously}

that

£-1

1

coarse-grained

in space so that

_they

do not

_change

over the collision volume. We remark that the

Laplace

transform

_[9a]

associated with the definition of collision

_operators

must be

interpreted

as a time coarse

_graining

over the collisional time scale.

Because of the

_similarity

of the kinetic

_equation

_(2.13)

to

_(2.8)

we obtain

macroscopic

equations

which are also

_structurally

similar to those derived from

_(2.8).

Since

_they

_{may be} derived from

_(2.13)

in the same manner as from

_(2.8)

we

_simply

list the

_equations

below. We will use

_angular

brackets to denote the

_averaging

over the

_phase

_{space of} a

_polyatomic

system :

It is

_important

to recall the rule of

_averaging

described in

_(2.10)

when we derive the evolution

equations given

below. We first define various statistical formulas for

_macroscopic

observables

appearing

in the

_present

_theory.

To _{this end it is necessary} to define the

_following

molecular

expressions

for various moments :

where with the notation

It is useful to note that the definitions of

_haB)

_given

above are correct to the

_{approximation}

in which the terms of

_{0(À )}

or

_higher

in the

expansion

of the

_operator

exp (-

_{’k raiq ;}

_bks

V r)

are

neglected.

This latter

_operator

_{appears in the} terms

_containing

the

_{forces gaiq;}

_bles. Inclusion of the

_neglected

terms would be

_{mathematically}

correct, but very difficult to handle in

_practice.

Moreover,

they

appear to be

_nonphysical

since

_they

_{arise from the delta functions which may}

have too

_sharp

a distribution. For

p a, ha

and pa

see

_{(2.17), (2.33)}

and

(2.49b)

below. With the definitions of

_{h (’)}

we now calculate various

_macroscopic

variables as follows :

We construct an

_{antisymmetric}

tensor S =

p5

with

angular

momentum vector S as follows :

These

_macroscopic

variables

_obey

the

_following

conservation laws and evolution

_{equations :}

In the evolution

_equations

_above,

i.e.,

the

_enthalpy

_{per unit} mass

_{of a, ta}

_{is the internal energy}

_density

_{of a, p,, is the}

_hydrostatic

pressure and

where

_/ïJ

are the molecular formulas for the stress _tensor, heat

fluxes,

mass

_fluxes,

etc.

defined in table I. We call

_A«)

_dissipative

terms since

_they

are

_intimately

related to the

entropy

production

due to

_dissipative

_{processes such} as viscous

flow,

heat conduction and

mass diffusions

_occurring

in the

system.

The conservation

_equations

and the evolution

_equations

listed

_above,

with the

_exception

of the

_angular

momentum conservation

_equation,

have the same structure as those

_appearing

in the dense

_simple

fluid kinetic

_theory

_[9b-g]

_except

for the fact that the

_macroscopic

variables consist of the center of mass and internal contributions and the

_dissipative

terms must reflect

the involvement of the internal

_degrees

of freedom in the molecular collision processes in the

system.

Whatever the molecular collisional

_details,

the

_dissipative

terms _{may be}

_expressed

in terms of stress tensors, heat

_fluxes,

mass

_fluxes,

etc. which

_incidentally

are henceforth called

simply

fluxes,

and the flux

_dependence

of the

_dissipative

terms is similar to the

_simple

fluid

counterpart except

that the coefficients reflect the

_molecularity

of the fluid in

_question.

These

coefficients consists of collision bracket

_integrals

which indicates the details of collisions between the

_particles

in the

_system.

Therefore

_they

are the source of information on molecular

_properties

of the

_system

in the

present

theory.

It is

important

to note that the evolution

_equations

_{(2.29)-(2.32)}

are in a fixed

_frame,

but

_they

can be converted to the

corotating

frame version

_{by using}

the rules

_reported

in reference

_[9g].

(*)

These definitions of

_f/Ja)

and

_ya)

are correct to the

_{approximation neglecting}

the terms of

o (À)

or

_higher

in the _operator

_{exp (- À r aiq ; bks . V r )}

_{which also appears in the} definitions of

Table 1

_(continued).

{ABC}

= sum of nonreduntant

_{symmetrized products}

of the vectors or tensors.

2.3 ENTROPY AND THE EQUILIBRIUM SOLUTION TO THE KINETIC EQUATION. - The kinetic

equation

(2.13)

satisfies the H-theorem if the

solutions,

distribution

functions,

belong

to the zero class functions

_[9a]

for which the total

_eigenvalue

of the Liouville

_operator

N

£0

_{= Y}

_Ls(a)

equals

zero. The

_equilibrium

distribution

function,

for

_example,

has this

a = 1

property.

The

_{nonequilibrium}

distribution function must be chosen in such a _{way that the} H-theorem is

satisfied,

and the modified moment method

_[16]

we make use of in this series of work ensures that the

requirement

be met.

The

_entropy

of the

_system

is defined

_by

the formula

and for zero class functions

[9a, e]

there holds the

inequality

equilibrium

distribution function in fact is

_{uniquely given by}

the

_vanishing

collision

_integral

of

(2.13) :

We note here

_again

that the limit s -. + 0 should be understood in

_(2.42).

_Equation

_(2.42)

means that the

_operand

is a collisional invariant. It can be

shown,

by

following

the

_procedure

described elsewhere

_[9a],

that the solution to

_equation

_(2.42)

_subject

to the conservation of energy and momentum is the canonical distribution function

where

_Hs)

is the Hamiltonian of s

_particles

_{and is}

_proportional

to the

_reciprocal

temperature.

Therefore we obtain

with q

denoting

the normalization factor

. the number of internal

_degrees

of

freedom ,

and h is the Planck constant. It is

_put

into the

_integral

to make the latter dimensionless. The factor N! is also

_put

in there to

_compensate

for the

_overcounting

of the

_frequency

of states.

The

_{nonequilibrium}

distribution function is now looked for in a form to conform to the H-theorem

_{by taking}

an

_exponential

form

and the normalization factor

.ae (X)

is defined

_by

the

_expression

with

_/3

=

1 IkB

T. The

_temperature

T is defined

by

the statistical formula

That

is,

the

_temperature

is defined

_by

_{the average kinetic energy. It is consistent with the} notion of

_temperature

_being

a measure of motions of

_particles

in the

_system

and also with the definition of T in the case of dense

_simple

fluids. The reader is referred to reference

_[9e]

for a more detailed discussion on

_temperature.

The factors N! and

h f in

_(2.48)

_appear for the same reason as for the

_equilibrium

normalization factor

_(2.45).

_{The pressure of the}

_system

is defined in a manner similar to that in the dense

_simple

fluid

_{theory :}

where

It is to be noted that the

_temperature

is taken not with the

_equilibrium

_temperature,

but with the

_temperature

as defined in

_(2.49a).

Therefore

_FI(Z)

is different from

_F(x)

_by

the

temperature

factor. Often in kinetic

_theory,

the

_hydrostatic

_{pressure is defined}

_by

_[18]

the

trace of stress tensor : _pa = Tr

Pa/3.

Since the stress tensor is

generally

time

dependent

for a

nonequilibrium

system,

this definition makes

_hydrostatic

_pressure a

_{nonequilibrium}

_quantity.

Moreover,

it leads to the conclusion that

_da

=

pâ,,

in

(2.36)

vanishes and _{as a consequence}

the dilatational

_part

of the

_entropy

_{production accompanying}

the

_expansion

or

_compression

of the fluid vanishes

_together

with the bulk

_viscosity.

Therefore,

the author believes that it is not

appropriate

for dense fluids to define

_hydrostatic

_{pressure with the} trace of stress tensor as mentioned earlier. The definition in

_(2.49b)

does not have the

_difficulty

mentioned and

_easily

reduces to the

_equilibrium

statistical mechanical formula for

_hydrostatic

_pressure

_given

in

terms of the virial tensor.

If

_H{N’)

is

_decomposed

into

_components

related to various fluxes in the

_system

which is

_positive

for all values of

_xJa}:

The

_entropy

flux

_J,

is

_{given by}

the formula

where

_la

is the chemical

_potential

_{of a per unit} mass

and

The unknown

_xJa}

are determined

_by

the

_consistency

condition

which may be looked upon as a differential

_equation

for the

_entropy

_density

since

Hère

and

_Z

are defined in

_{(2.29)-(2.32).}

Attendant to

_(2.56),

there holds the extended Gibbs relation

Equation

(2.57)

is a _{consequence of}

_(2.58).

The

entropy

density,

entropy

flux and

_entropy

production

form a balance

_equation

where

This

_inequality

is

_equivalent

to the second law of

_{thermodynamics}

as was shown elsewhere

The

_macroscopic

evolution

_equations

_{(2.24)-(2.32), (2.58)}

and

_(2.59)

have the same form as those

_appearing

in the dense

_simple

fluid

_theory

_[9]

_except

for the statistical definitions of

macroscopic

variables and the details of the

_dissipative

terms which reflect the

_molecularity

of the

system

in

_question.

This is

_quite

understandable since these

_macroscopic

_equations

are

meant _{for gross}

_description

of behavior

_universally

exhibited

_{by macroscopic}

_systems

and,

in this

_particular

case,

_by

a

_polyatomic

fluid.

3. The Kirkwood diffusion

_equation.

Kirkwood

_[12]

_proposed

a Brownian motion model for

_polymeric

solutions which has been

extensively

used

_by

Kirkwood himself and later workers

_[19-22]

for

_study

of

_{viscoelasticity}

of

polymeric

solutions. In the model a diffusion

_equation

is obtained for the

_{configuration}

distribution function of a

_polymer

when a set of

_{approximations}

and

_assumptions

is made. The Brownian motion of beads

_making

_up a

_polymer

chain is one of the

_assumptions.

In this section we show that the Kirkwood diffusion

_equation

can be obtained within the framework of the

_present

_polyatomic

fluid kinetic

_theory.

The result itself is not new, but its derivation from the

_generalized

Boltzmann

_equation

_{demonstrates the power of the latter for kinetic}

phenomena.

Furthermore,

there are a

_couple

_{of intermediate results necessary for the} derivation which turn out to be new for

_{nonequilibrium polyatomic}

fluids but also can be useful for

_calculating

some of their

_{nonequilibrium}

_properties.

The intermediate results in

question

are the

_generalized

Kirkwood

_{integral equation}

for correlation functions and chemical

_potentials

when there exist

_{nonequilibrium}

fluxes in the

_system.

These

_equations,

when

_{appropriately}

_solved,

would

_yield

the

_magnitude

of the effects on correlation functions of

_{nonequilibrium}

_{processes, e.g., shear-induced distortion of}

_pair

correlation

functions,

etc.

They

have, therefore,

a

_potential

for

_application

to studies of

_{nonequilibrium}

fluid

_properties.

3.1 GENERALIZED KIRKWOOD INTEGRAL EQUATION. - To carry out this

_part

of discussion we assume _{that the interaction energy of the}

_system

consists of

_pairwise

additive

_potentials.

We therefore may write

Now

_following

Kirkwood

_[23],

we introduce

_charging

_parameters

[6]

which indicate the

strengths

of interactions between

_particles

in the

_system.

It is sufficient for our _purpose to

designate

a

_particular

_molecule,

_{say, al and consider its correlation with the} rest of the

system.

Thus

_denoting

the

_charging

_parameter

_{by e,}

we _{may write the}

_potential

V for the

system

in the form

Here the

_{charging parameter e}

_ranges

from e

= 0

to e

= 1. Thus

at e

= 0 the

particle

al is

_{completely decoupled}

from the rest of the

_system

and

_{at e}

= 1 it is

fully coupled

with all

The

_potential

energy

part

h (’)

of the molecular formulas for fluxes

h,,q

can be also written

similarly :

These

_{decompositions}

_(3.2)

and

_(3.3)

will be

_presently

used in the derivation of the

generalized

Kirkwood

_{integral equation.}

,

We now define the

configuration

distribution function

1JI’a(lR, t)

where R stands for the set

of coordinates for

_particles

in a

_polyatomic

_{(e.g., polymer)}

_molecule,

IIB =

{Raiq} :

This distribution function

_gives

the

_probability

of

_{finding particles q}

E a at

_Raq

for all q E a at time t. If the

_charging

_{parameter e}

is not

_equal

to

_unity,

then

for the

_{configuration}

distribution function at

_arbitrary

The

_{configuration}

distribution function

_{1/’ a}

is a

_{mass-weighted}

reduced distribution function

_describing

the evolution of the

configuration

of a

_{single polyatomic}

molecule

_interacting

with the rest of the

_system.

In

_(3.5)

we have inserted the

_{parameter e}

in F (oN’) to indicate that

_particle

al is

_partially

«

_charged

».

We may write

It is also useful to define

_pair

correlation functions

and the

_symbol

where

The reduced distribution function

_V(2)

describes the correlation between a molecule of

species a

and

_{particle s}

of

_species

c. This is the

_counterpart

of the

pair

correlation function for a

_simple

fluid.

Differentiation of

_(3.5)

with

_{respect to e}

and use of

_(3.6)

_together

with the

_appropriate

definitions of

_{H(N)(e), H1(x)(e)}

and

_{A (x)(e)}

_yield,

after some

_{algebraic manipulations,}

the

for which we made use of the

_identity

and the definition

Equation

(3.9)

is the

_generalized

Kirkwood

_{integral equation}

for

_{1/’a(lR, t ).}

If

_(3.7)

is differentiated

_{with e}

and a similar calculation is

made,

there follow

_{integral equations}

for

â 2 cs

and

so on, that

is,

essentially

a

nonequilibrium

Bogoliubov-Born-Green-Kirkwood-Yvon

_(BBGKY)

_hierarchy

for correlation functions. It is of

_{nonequilibrium}

since there are

nonequilibrium

fluxes

_appearing

in the

_equations.

These

_equations

reduce to those in the

equilibrium

BBGKY

_hierarchy

_[14a-d]

as the fluxes or

xJa)

vanish for all a and a The terms

containing

xJa)

are the new features in the

_generalized

Kirkwood

_{integral equation. They}

should in

_principle

be able to account for

_{nonequilibrium}

effects on the

_{configuration}

distribution of

_polyatomic

_{(e.g., polymer)}

molecules in

_{nonequilibrium}

flux fields. Since

_(3.9)

is sufficient for the purpose of the

present

work we will not work out the

_equation

for

1/’ Js

here. It is

_{straightforward}

to derive the

_{integral equation}

for

_{P (2 )}

and the

_subsequent

members of the

_hierarchy.

3.2 LOCAL CHEMICAL POTENTIALS. - The

_equation

for a local chemical

_potential

can be

obtained

_{by following}

the same

_procedure

as for dense monoatomic fluids when

(3.2)

and

(3.3)

are used

_along

with the definition of A

(e).

Since it will be

repetitive

if we _go

_through

the

derivation in

detail,

we will

_simply

_present

the result below. When the volume and the

temperature

are

_kept

constant the chemical

_potential

1£a may be written as

_{[6, 23]}

The

_one-particle

chemical

_potentiel

_{F£o a}

is defined

_by

the formula

for

_{particle a1, na}

=

X a/CU’,

and

xi(n’ ’)

_collectively

stands for the

phase

of the internal

degrees

of freedom for molecule al. With this definition of

_.°

we _{may write}

and thus

By using

this

_identity

we find

_in

the form

Eliminating

the second term on the

_right

hand side of

_(3.15)

_{by using}

_(3.9),

we obtain the

equation

for local chemical

_potential

_{J.L a :}

where qa is such that

If the

_{nonequilibrium}

contribution to _{qa is}

_neglected,

then we find

where

_{qâint )}

is the internal molecular

_partition

function for the

_polyatomic

molecule in hand. In the last term in

_(3.16)

the sum over s is restricted to _{those of s = q if} c = _a. That

is,

P 121

is the «

pair

correlation function » for a molecule of

_{species a}

and

_{particle s}

in another

species

if c =/= a or another

_polyatomic

molecules of the same

_species

if c = a.

Let us assume that a stands for a

_{polyatomic species}

and the concentrations of

_polyatomics

are

_sufficiently

low. Then since the «

pair

correlation function »

IP 121

_{may be}

_neglected,

we may

approximate

ILa to the lowest order as follows :

This is the

_approximate

local chemical

_potential

which we will use in the derivation of Kirkwood’s diffusion

_equation.

This

_{approximation}

limits the Kirkwood diffusion

_equation

to

the

_regime

of

_sufficiently

dilute solutions.

3.3 KIRKWOOD DIFFUSION EQUATION. - With the

_preparation

above,

we are now

_ready

to

partition

function _{qa with}

_(3.17’)

we find

where

and thus

_{F .}

is the force on

_{particle y}

in al

_by

other

_particles

in the molecule

_{- 1. e. ,}

it is the so-called

_spring

force if we borrow the

_commonly

used

_terminology

in

_polymer

solution theories.

In the case _{of uniform pressure the}

_{thermodynamic}

_force

X y4

may

then be written as

The evolution

_equation

for diffusion flux

_Jaq

_{may be} obtained from

_(2.30)

_by

_decomposing

the latter

_{into q}

_components

where

To the lowest order

_{approximation}

_taking

a linear term in the

_{thermodynamic}

force or the fluxes and also to a linear

_{approximation}

for

_{LBz >,}

where Àaq; bs

are collision bracket

_integrals

for diffusion fluxes which are

_{given by}

the formula

where

with m,

standing

for a mass

_(e.g.,

reduced

_mass)

and d for a size

_parameter

of a molecule

_(or

a

monomer unit in the case of a

_polymer),

and

faster time scale than conserved

_hydrodynamic

variables.

Therefore,

it is reasonable to set

That

_is,

Substituting

(3.21)

and

_(3.22)

into

_(3.23),

we find

Inverting

this

_relation,

we

_finally

obtain the linear

_{flux-thermodynamic}

force relations

It is now convenient to define the

_transport

coefficients

These are diffusion coefficients. Substitution of

_(3.19)

into

_(3.25)

_yields

the diffusion flux in the form

We have shown that

_1JI’a

_obeys

the

_generalized

Kirkwood

_{integral equation.}

The same function also

_obeys

the

_following

differential

_equation

which can be derived from the definition for

_1JI’a,

_(3.4),

and the kinetic

_equation

_{(2.13) :}

where

This flux factorizes

_{approximately}

to the form

This can be shown

_{by following}

the

_procedure

described in a

_previous

_paper

[9f]

on

_electrolyte

solutions. This

_{approximation}

is valid if the heat flux

_part

of the

_{nonequilibrium}

contribution

to the distribution function is assumed to consist of the _{kinetic energy}

part

depending

on the

where

the self-diffusion coefficient. This is related to the friction coefficient

_appearing

in the Brownian motion model. When

_AJj

is

_neglected

and the

_approximate

_Jaq

so obtained is substituted into

_(3.28),

there follows the

_equation

which is the Kirkwood diffusion

_equation

for

_{configuration}

distribution function

_{qf,, (R, t )}

for a

single polyatomic

(e.g., polymer)

molecule in a flow field u.

_Neglect

of

OJâq

is

_justifiable

if the diffusion of a bead is uninfluenced

_by

other beads since the

_{approximation}

amounts to the

assumption

that

_{Cùaq; as}

= 0

if q

s. Therefore the

_{approximation}

should

_yield

reasonable results if the chain is not

_{tangled by}

itself and others. This fact and the manner in which

_(3.32)

is derived indicate

_clearly

that it is an

_{equation holding}

for a dilute solution.

It is useful to collect the

_assumptions

and

_{approximations}

made to obtain the Kirkwood diffusion

_equation

in the

_present

kinetic

_theory.

1. Factorization

_ofj(2)

This is due to the

_assumption

on the heat flux term

_appearing

in the distribution function

_by

which the

_potential

_{energy contribution} to the heat flux is

_neglected.

If the processes do not include heat flows in the

_system

this

_assumption

is not

_required

and the factorization of

_Jâq

becomes exact within the thirteen moment

_{approximation}

which is

generally satisfactory

for the usual

_{hydrodynamic description}

of a

_{fluid ;}

cf. reference

[9f].

2.

_{Neglect of}

_{« pair}

correlation

_{function »}

_ll,(2)

- This

_{approximation}

holds if the solute concentration is

_sufficiently

low,

and

_permits

an

_approximate

calculation of local chemical

potentials

for solutes and

_{corresponding thermodynamic}

forces.

3.

_Steady

state

Îaq

in the substantial

_{frame of reference.}

- This

_assumption

is

_justifiable

since diffusion fluxes should relax faster than conserved variables. Since the evolution

_equation

(3.20)

for

_Jaq

_{is the average of Newton’s}

_equation

of motion for

_{particle q}

e a, this

assumption

is

_equivalent

to

_neglecting

the inertia term in the

_{Langevin equation}

for

_particle

q e a in the Brownian motion model

and,

in

addition,

also

_assuming

the

_homogeneity

of diffusion fluxes in space,

i. e. ,

If the diffusion occurs in a flow

field,

this

_assumption

is

_generally

violated,

and one must take the effect of flow into account. _{This may be done if} an

_approximate

formula for u is

_employed.

For

_example,

if the shear is not

_large,

then we _{may take}

where y is the shear rate tensor assumed to be constant.

_(It

is useful to note here that the

even further and take the Oseen tensor

_{approximation generally}

taken in

_polymer

solution

theories

_{[12, 19-22].}

To be more

_rigorous,

_{it is necessary} to solve the diffusion flux evolution

equation

together

with the momentum balance

_equation

and the stress tensor evolution

equation.

This

_approach,

_however,

is rather difficult to

_implement

in

_practice.

Neither is it the

most convenient of the methods for

_obtaining

_transport

_properties

of

_polyatomic

_(e.g.,

polymer)

solutions.

4.

_Neglect

_of

_{àj (2)}

This

_assumption

is

_justifiable

if the

_{configuration}

of the chain is

relatively simple,

i.e.,

untangled,

and if the concentration of

_polymers

is low. Therefore the retention of this term

_probably

would

_improve

_{the accuracy for}

_1JI’a

when the solute concentration is not

_sufficiently

low.

Finally,

a note on the flux evolution

_equation

_(3.20).

As mentioned

before,

it is

_equivalent

to the

_{Langevin equation}

for motion of a

_particle

immersed in the medium of other

_particles.

However,

there are some

_important

differences which endow

_(3.20)

with a more

_powerful

capability.

First of

all,

it can deal with nonlinear situations since the

_dissipative

term

llaq

is

_generally

_{nonlinear for processes removed far from}

_equilibrium.

The

_dissipative

terms

can be

_explicitly

calculated in terms of molecular

_properties

if the

_present

molecular

_approach

is

_pursued

to its ultimate end.

_Equation

_(3.20)

does not

_require

an

_assumption

on diluteness of the solution.

Therefore,

it can handle concentrated solutions as well. All these features are vested in the

_dissipative

terms in the

_present

_theory.

4.

_Viscosity

of a

_binary

mixture.

The kinetic

_equation

used in this work

_provides

a formalism with which to calculate various

transport

coefficients for

_polyatomic

fluids and we take this

_opportunity

to show some formal results for shear

_viscosity.

The

_procedure

to

_employ

_{for the purpose is}

_basically

the same as the one used for

_simple

dense fluid mixtures

_{reported previously}

[9e, 24].

Therefore we shall

not _{go into the}

_subject

in

_depth

here.

Instead,

we _{shall pay a close attention} to

_{viscosity only,}

since it is better studied

_{experimentally especially}

in connection with

_polymer

solutions which are relevant to the

_present

kinetic

_theory

and also related to the friction coefficients

À aq ; bs

presented

in section 3.

4.1 STEADY STATE SOLUTION OF STRESS EVOLUTION EQUATIONS. - To be more

_specific,

in our

_{investigation}

we shall confine the discussion to a

_binary

mixture in which a

_polyatomic

fluid is dissolved in a solvent

_consisting

of a

_spherical

molecular

_(simple)

fluid. If there is no

oscillatory

shear

_acting

on the

_system

the stress relaxes

_generally

on a faster time scale than the fluid

_velocity

of the

_density.

It is then sufficient to consider the

_steady

state stress

evolution

_equation.

Furthermore we will assume that the stress is

_homogeneous

so that it does

not

_depend

on

_position.

We also assume that the shear rate is

_sufficiently

small so that it is sufficient to consider the linear shear rate

_dependence

of the stress tensor. That

is,

we assume

that the flow is Newtonian. We will later

_develop

a

_theory

for non-Newtonian flow

along

the line taken in the

_previous

_paper

_[9]

on dense

_simple

fluids.

Under these

_assumptions

the stress tensor evolution

_equations

become linear

_algebraic

equations

for

_Rc,

_{(c = a, b ; a}

=

solute,

b =

solvent) :

term

_{A a (1)}

for

the solute

_species

as follows :

and

_similarly

for the solvent

_component

b. In

_{(4.1a, b)}

the coefficients

_Raa,

_Rab,

etc. are the collision bracket

_integrals

related to the stress tensors. Their

_explicit

forms are

_{given by}

the formula

where the

_subscripts

c and d are either

_{a (solute )}

or b

(solvent ).

These coefficients are related

to the Newtonian viscosities of the solvent and solute

_species.

The linear

_algebraic

set

_{(4.1a, b)}

can be solved for

_ee=

_{p P c,}

_(c

= _a,

b )

to

yield

where

These solutions can be used to calculate the viscosities of the solvent and the solute.

4.2 INTRINSIC VISCOSITY OF THE SOLUTION. - Since the viscosities

are defined

_by

the relations

we

_identify

71 g

and

q§,

_{respectively,}

with the coefficients in

(4.3a, b) :

The formulas in

_{(4.6a, b)}

_give

the Newtonian viscosities in terms of statistical formulas

which,

Since in

_practice

the

_viscosity

of a mixture is measured in the

laboratory,

the actual observable is the total

_{viscosity 17 0 :}

where

The intrinsic

_viscosity

of a solution is defined

_by

the

_limiting

formula

where c stands for the concentration of the solute in the units of the

_weight

_{per volume}

(e.g.,

g/deciliter).

Therefore the intrinsic

_viscosity

is the relative

_viscosity

of the solute in the infinite dilution limit and has the dimension of volume/mass. It is of interest to us here.

In order to have a better idea of the intrinsic

_viscosity

we use

_(4.4a-d)

and

_{(4.6a, b)}

and examine their mass fraction

_dependence

a

_step

_beyond

the formal structure. _{For this purpose}

we introduce a

_parameter

_Te of dimension time defined

_by

the formula

where mr is a mean mass and d is a mean size

parameter

of the molecules

_making

up the

system.

This

_parameter

_rc is

_roughly

a measure of collision time between two

_particles.

We then scale the collision bracket

_{integrals Raa,}

etc. in the

_following

manner :

These

_scalings

leave

_{Rbb, Raa,}

etc. dimensionless. The choice for the

_density

factors is made on the basis of

_analysis

for the collision bracket

_integrals

and the collision

_operator

in

_particular.

The

_analysis

in

_question

is made

_by

means of a cluster

_expansion

for the collision _operator

appearing

in the collision bracket

_integrals,

for details of which the reader is referred to