Thermodynamics and hydrodynamics of chemical gels. K: Sekimoto'. Defiartment of. Applied Physics, Nagoya. University,. Nagoya 464, Japan. (Repu le 5 juin ...
Phys.
J.
(1991)
II1
19~36
1991,
JANVIER
19
PAGE
Classification
Physics
Abstracts
82.70G
62.20
05.70L
/
Thermodynamics
hydrodynamics
and
of
gels
chemical
Sekimoto'
K:
Defiartment (Repu
of
juin
le 5
Applied
Physics,
idvisd
1990,
le 3
Nagoya
University,
sepiemhre
accepid
1990,
464,
Nagoya le 16
Japan
octohre
1990)
Abstract. have developed'a framework We of thermodynamics hydrodynamics of and chemically crosslinked framework of thermodynamics is analogous to that of binary gels. Our fluid mixtures, except that the former includes a variable describing the deformation of the solid (network) of gels. -The< framework portion of the hydrodynamics is based on the theory of hydrodynamics of systems with broken symmetry and on the assumption of local equilibrium. Inthe framework there is no plastic flow of solid portions of gels, an that present assume we assumption which leads to the identification field with the displacements of the order-parameter of the material composini the §olid portion. We also investigated the properties of the symmetry Olisager coeffiiients fir this system with an emphasis ok the relation to the metric tensor the'defirmation of the solid portion of iels. We investigated the limit of incompressib(describing lity'of both'the solid and the fluid portion of gels, as well as some other special cases of isotropic processes and the states with quasi~mechanical equilibribm. Our deformations, isothermal framework deformed introduces the presents a generalization of Darcy's law to the systems and osmotic
1.
.Introducdon.
Gels
are
properly.
reactions the
of
part
[i~crystals
gels there
through. the gel, plastic
important
bulk
call
Another,
of in
network
in
gels
and
of
gels
feature
gel
occurs
chemical
chemical
which
would
portion slowly than more gels usually occurs
of the
solid
gels simply gels
in
this
atoms
features
the
paper.
[2, 3], or colloidal distinguish them
their is
that
purely in
crystals [4] from
mechanical
those
gels) applicability
the
gels
made
deformation
of the
,
media
Porous
linear
other
modeling through of the
elasticity
if we allowed thq brea$ up and the Although the perrneatioi of solvents in the propagation of the acoustic waves slowly than permeation does. We more
occur
of gels. still
which
irreversible the
boun(.
without
move
interstitial
with
(so-called the chemical gels or the gel can become very large beyond
of the flow
can
however,
are,
plastic flows, the flow
without
reconnection
hereafter
gas
,
efficts, often are restricted jpplicability.
rather
theory
or
In
particles
which-§olvent
Thermal
chemical solid
,.
through fluid
with
this
systems has
identifies
three
a~ (pu~ )
(Ff
F(
k~
w
~Ji~
(3.I1)
0 ,
k)j
J
ax~
(3,12) ,
+ab~~~-~a~a,
+
Hi
Uf
Pi
(3,13)
r
~
identity is given in
PeV~
~t
local
+1~r a~vP
8~) +
RI,
(3.14>
where
have
we
(Ff
m
(Ff
il m
obtaining
second
the
line
decompose ((«~°~)$) by introducing
(3,15)
of
the
we
F(
V
have
we
following
)(T~~
V
(3,15)
Tf)
currents
mass
the
F(
k~
(T~ 8~~
=
Now
M
II.
defined 8~
In
PHYSIQUE
JOURNAL'DE
26
k~
(T~f
) a~
V
used
formula
(2.7).
p~&
and
piui
(3.16)
and
the
momentum
current
quantities,
it m
jr «
(~'~)S "
P
f(uf
)
(3. ii)
v)
(3.ig)
v
p~(x
Tf
(~'~~~JS
(3.19)
(or &) and ((tr~°~)$) in (3.14) using (3.17)-(3.19). As in the case of jf + j$ vanishes identically due [20] the sum of the diffusive currents definitions (3.7), (3.8), (3.17) and (3,18). Below, however, these diffusive to currents are treated separately, just as they were treated in the case of the binary fluid mixtures, since by doing so the results the symmetric forms which thdrefore would be easily possess more generalizable for the gels with multi-component In the resulting equation we solvents. substitute (2.ll) and (2.12), the latter being here interpreted under local equilibrium a assumption as
We the
eliminate
binary
pi ui,
fluid
p~
mixtures
0
We
now
define
dissipative
the
j thus
obtain
the
JE
aAMi
p~ aA
+
if
current
Hi
~v
2 a~
+
(T~f.
~MfJf
o
~~
~MmJm
(3.20)
as
~~
T))
J#~
Pm
(ji~
following a
pi
entropy
D~
"p
=
+
(T$ 8~~
l
~~
JS
We
aAr
ps
=
(ps)
~
fji~
(#m1
~
~
Pm
ji~j
(3.21)
v
equation,
a~(psv~
=
at
M
+
js~ ~)
a~r
~ js~
r
a~mi
~Jf
~ ~
a~MS ~
'
+
Aj'+ A~,
~ jm~
+
i~V~ (« ~ )S ~ (3.22)
M
THERMODYNAMICS
I
where
HYDRODYNAMICS
AND
OF
CHEMICAL
GELS
27
have
we
I
Aj
m
(-
a~Tf
f VU
=
T
+
(Ff a~ (T~fj
~J-
T$ a ~ aXP
a yP
ax"
0
~
~2
(3.23)
=
[
F( j#~)
(F~J/~~
"
ax
~'
[JT)
I
J tr T8
Jp~
r
l
evaluatiig
description
AI
used
we
ax
j
D~
(3.24)
~
~
(2.7) and
(A.7)).'In transforrhing
below
~
J#~
~
~
~
Pm
rdlation
the
~
~
,,
I tr T
l'T
T
In
JT)
(aliXP)(J aXP/ ax")
identity
the
expreision
the
of A~
have
we
0 =
(see
introduced
derivative,
covariant
Af,~ma~Af+Ff~A)-F)~Af, where
symbol F(~ is
Christofel's
defined
F(A
'~
and-
we
have
appendix and
entropy
the
used C.
that
Jp~ is
all of
these
fact
Gathering
expresjion
the
of
the
a
~~~~
entropy
3~T,,
jJs
(3.26)
ax
ax
(see (2.14)), The derivation of (3.24) is given finally obtain the equation of the balance of production as follows
constant
together
we
3~(psu~
=
"
~~j,
m
( (Ps) fbs
(3.25)
as
ax
in
3~ u~
+'j
~ ~
j
~~~ji~
+
I'
#m
~
j?~)
latter
respectively,
expression are
shows
that
dissipative
the
if, if, j$
=
PS
3vT
+
)$ ~
1
~
~,
contain not
so
this
(he
relation
we
covariant
Pi
surprising
:
reexpress The
can
derivative. When
the
ji~
(3.28)
A
tr~
defined in (3.21), (3,17), (3,18) and (3.19), corresponding "currents. In the next section we relating these dissipative with the local thercurrents that the variant of the Gibbs-Duhem relation (3.20) can be and
as
3vMi+
follows, Pm
'#m
T
I
+
Pm
Using
(3.27)
of the
parts
discuss the Onsager coefficients modynamic forces. We note here rewritten using the covariant derivative
°
Is
+
m
The
the
the
the
right
appearance deforrnation of
:hand
side
of the
gel
is
of
(3.28) in
covariant not
(3.29)
T), A
v;A
a
derivative
homogeneous,
form
in the
that
does
(3.24)-(3.29) inverse
of
not
is the
mapping On
X
x(X)
-
other
the
defines
hand
it is
curved
the
easy
that
means
-component
(~P At-
as
the
derivative
of A
expressed by of
-component
~'"
3x~
respect
the
the
to
curved
3x~
3x~
3xP
3x~
tr-th
coordinate
the
reference
coordinate,
curved
X-coordinate
Cartesian
the
x~, x~) in that
has
system Thus
system.
a
lax",
of
the
representation
the
that
say
can
we
space.
the
v
derivatives
covariant in
derivative case
3xP
with
the
(~
~
~
and
~~
~~
M
II
(xi, (3.25)
system
(3.24)
from
3x~ 3x~
3
? This
coordinate
verify
to
PHYSIQUE
DE
JOURNAL
28
of
first
the
gels the last since
terra
spatial
of
two
variation
space
terrns
spatial
the of
quantities
the
reference
on
is
mbre
appear natural
the
right
hand
therrnodynamic
other
side
of
in
real
as
TIP
+
p~
V~
k
)
where
8~~
is the
Kronecker
a~ipv~
delta.
Here
the
as
the
as
the
forrnalism
should
be
conservation
mass
converted
:
(3.30)
0 =
(~rt°t>~v
vv
as as
densities, (2.17), is
constraint have dynamical (3.30) used (3.2) and (3.3). The we constraint introduce component) constraint. order to satisfy this In we motion field [21] which is generated (3.4) the constraining force mixtures fi, as is the case for binary fluid
=
important important
the
in
~,
where
a,(pvv)
that
etc.
above
the
a~ (pi Vi uf
space.
generally are a~XPlax~ ax~, is
which
of add
We
(3.25)
describe how the remaining part of this section we the mass constraint modified in the incompressible The on case. variables using the equations of into that on the dynamical In
quantities
those
the
distortion, quantities such
of the
variation
for
when than
(tr~°~)~~ is in fact
+
not
is
a
(or equation
scalar
in
the
by
a
scalar
the
of
field
(3.31)
iii ~v
one-
,
total
stress
since it
excludes
the the incompressibility of the constituents. Next the relation of local equilibrium (3.10) is modified following the recipe given in (2.21). We then identify fl with p since the latter has been introduced incompressibility constraint. The to satisfy the derivation of the expression of the production tbs is completely in parallel with that entropy given in the compressible case, except for replacements according to (2.21). We obtain in place of (3.28) and (3.29) the following equations
contribution
yields
that
4s
a~r
jjs~
~
m
a~vv +
(«
~
3~(vf+ Pip) Jf
i
~~
p
~
0
Ps 3 vT +
=
#m +
Pm
Pi
)S
3v(vi
+
~/ +
T
Mm I
j
+
+
~
VmPi
A
~ Jm~
(3.32)
Pip) ~mP~
A
P~ )~,
(3.33)
derivative in (3.32) by using (3.33). As noted in the again we can eliminate the covariant of section including (2.22), if the therrnodynamic potential is given as a paragraph 3 last therrnodynamic variables of the incompressible systems (see (2.18)), function of independent Here
M
THERMODYNAMICS
I
HYDRODYNAMICS
AND
CHEMICAL
OF
GELS
29
have to (3.32) and (3.33) into those modified for such a case by applying the convert replacements (2.19) to equations (3.32) and (3.33).,In this case the Lagrange multiplier field p( fl), which has been introduced merely for generating the constraining force, has the real meaning of a field. (2,18) the From reversible of the H parts stress, pressure or we
=
H~,
calculated
are
volume,
Onsager
4. We
the
contribution
no
relations
linear
and
within
derivatives
have
restricted
of the
space
bulk
the
to
states
with
constant
pressure.
coefficients.
assume
gels,
the
as
therefore
and
we
thermodynamic coefficients L~~f
the
between
Onsager
the
introduce-
forces
and
and
Af/
the
irreversible
through
the
fluxes
in
following
equations
ji~
L,,hsfl
(tr~)t therrnodynamic
the
f(
forces
defined
are
f]
(4.i)
m
(4.2)
as
) ~
w
f/
(4.3) (4.4)
w
~#~l-
f(w-
or
=
All
(4. 16) ,
we
have
the
following
relations
Ajabi,p Bj(~ Bj(~
=
=
Bj(~ B((~
"
p, q p, q
(4.17)
Ajbai,p =
=
0, 0,
or
or
2
(4,18)
2.
(4.19)
M
THERMODYNAMICS
I
Since in this
paper
assume
we
can
the
following
do
we
that
the
allow
not
vorticity
internal,
for
the
v
does
A
ff (h )
rot
or
couple
not
GELS
CHEMICAL
degrees of freedom,
intrinsic
rotational
with
dissipative
any
31
flux.
implies
This
relation A
fj (h
(4. 20)
=
which
OF
HYDRODYNAMICS
AND
,
implies
~jj)
Bjj) ~
Bj(~
=
(4.18), (4,19)
From
(4,21)
and
Ari(h)
we
obtain
( ( ( (
+
where
introduced
have
we
expression
above
for
+
the
Afi(h),
components
(h~>~v(hP>~«i
Bj(~[(hP)(h~)]~~~~,
(4.22)
notation
[~'~)(Q))~vA« The
expression
the
(4.21)
2.
q-o
2p=oq=o
the
or
=
Bll~i(hP>~v(h~)~«
=
p=o
0,
p, q
~~A Qua
"
the
assures
+
~~«
QVA
+
Af/(h)
symmetry
folliwing relatioi,
~v« Q~A
=
A([(h),
+
Q~a.
~vA from
(~.~~)
which
we
(tr~)1.
(tr~)S
have
the
(4.24)
=
Concluding
5.
remarks.
therrnodynamics and hydrodynamics of gels both in framework described the of the describe how the this section In compressible and in the incompressible we cases, faIniliar reduced into sections forrnulae for incompressible gels given in the previous more are isothermal such as that of forms when we consider certain special cases process or of isotropic
We
have
the
and
deformation
on.
so
therrnodynamics in which we did not introdqce the Lagrange multiplier (see (2,18), (2,19) and the paragraph including these equations) it is very H H becomes WI, and w deformation. The the case of isotropic consider tensor easy to for isothermal have Especially of gel. identified with the osmotic be processes we pressure can relation, familiar the following the
In
first
type
forrnulation
of
of
=
d(E
longer
We
no
we
discuss
(dr (2,21)
0
),
discuss
the
the
second
above
becomes
d
(E
TS)
(~
.n~~~
w
=
(5,1)
dV
since its hydrodynamic counterpart is lacking. Next isotherrnal using the Lagrange multiplier. In the processes version of (2.4) obtained by applying the replacements given in of (dM~ 0) as follows amount monomers fornlalism
forrnalism
incompressible for a given
the
=
rS)
=
M M~
fixed
~
(M f
+
VfP ) d @
+
? [T~
JpF~
'
]( dF(
(5.2)
dM~
of
Because
and
0
PHYSIQUE
DE
JOURNAL
32
incompressibility
the
=
dmi
constraint-
have
we
M~
VI
=
M
II
(5.3)
dJ.
~
Pm
Using (5.3), (2.14)
(2,15)
and
d(E
view
In
(2.7)
of
M~
fixed
find
we
~
be
identified
as
notation
same
replacement
osmotic
as
stress
by (2.21),
isotropic
a
fact
deforrnation
of
VI
T +
which
incompressible gel, Note
with
Ml)
and
reduced
thus
is
stress
physical
our
further
Vi (vf
"
this
that
agrees
gel (5.5) is
WI
(5.5)
vi1
of the
(2.19).
in
(5.4)
vi
tensor
tensor
introduced
that
described
For
pressure.
the
ldfi
JF~
+
~
the
that
m
can
f
T~
~
H
the
as
~
TS)
(5.4)
and
(5.2)
rewrite
we
have
we
invariant
intuition of relation the
to
used
under
the
osmotic
(~'~~
1 ,
where
have
we
defined
incompressible subtracted.
As
it)
if (= Especially
from
solvent for
VI M/I.
Tm-
which
the
Ml
Here
from
potential
chemical
the
is
contribution
'
constraining
the
hydrodynamics the incompressible version of the by applying the replacements given in (2.21) isotherrnal dr 0, using (3.33) we obtain processes,
obtained
is
for
the
of
force
relation
linear to
pure
a
properly
is
(4.4)
for
(4.5).
and
=
ji ~
Lu(
p~
the
other
hand
mechanical
equilibrium
motion
of =
within
3~iPV~
of
center
mass
(3.31)
is
now
given
(«~)ti
Tt +P8
V~
(5.7)
T
as
(5.8)
~v
gel
the
is have
we
the
the
rapidly,
very
of
and
center
mass
condition
as
is the
tr~,
mechanical
of
a~(Tf -p8~v)
=
have
reached
velocity
the
0
we
local
of the
pi )[
a~(T
Vi~) Pm
neglect in (5.8) the terrns containing velocity gradient (see (4.2)). Then
Substituting (5.9) into (5.7) (nonisotropic) ge[s,
Li~( +
T
equation
the
3 If
(Hi
aA
=
Pm
On
Li~(
Pi
~
case
which
isually, is
driven
we
can
by
the
balance
(5.9)
following expression
for
Darcy's law of perrneation in
the
Vf(pm Lff~ jf
~fm~ )
Pi
~A(Mf
~~ "
Pm
Vi (P
+
~fP)
~
Lffl
P
m =
~
f
L furl
~~
~ «
A
(5. 10)
M
THERMODYNAMICS
I
where
H
defined
was
using
rewritten
(5.5).
in
derivatives
HYDRODYNAMICS
AND
We
with
add
j
This
formula
useful
is
for
submitting
After
the
first
33
B
tensor
BY
[JK
be
can
following
XP in the
way
:
(5,I I)
aXP
of the
calculation
the
GELS
arbitrary
an
coordinates
reference ~
=
divergence of
the
the
to
a~B[
(XP).
that
here
respect
CHEMICAL
OF
version
of the
using
models
the M.
present
reference
coordinate
informed
Doi
paper isotherrnal the
the
system author of
in which other things he studies slow perrneation of gels among incompressibility assumption. His results agree with (5.9) and (5.10) if we apply to these transforrnation equations a according to recipe (2,19) (see the discussion in the last paragraph of section 3).
his
[23]
paper
under
the
Acknowledgments. acknowledges
author
The
for his
valuable
and
Appendix A. homogeneous
A
author
The
comments.
publication,
to
Kawakatsu
T.
Maggs
A.
of
relation
in
Doi
M.
for
his
the
referee
work
prior
manuscript.
the
thermodynamics
the
thanks
also
He
communicating
binary
of
gels
under
isotropic
and
mixtures.
fluid
of gels. (Here the adjective homogeneous is in fact isotropic and non-homogeneous gels we realize cannot an deformation constraint.) The isotropic such that under the coherency deforrnation is defined J~'~ I with J VI (p[ M~), where V is the volume in the real sjace occupied by the gel F reference sample and pi M~ is its corresponding volume. If we regard the gel in this state as introduce the the binary fluid mixture of the and the it is natural solvent to monomers, consider
We
superfluous
the
since
=
isotropic
to
discussions.
fruitful thanks
reading
for
description
brief
deformations
for also
in
deforrnation
chemical
=
entropy
$~~
as
a
$~~(E, Mi, M~,
To
the
see
chemical
correspondence potentials and
easy
to
show
that
V
)
S =
V,
and
(E,
which
Mi, M~,
defined
is
v
F
m
Pm,
as
ij3
(Al)
l
Mm
ofbinary fluid mixtures we introduce therrnodynamics the through the following equation : hydrostatic pressure
with the
dS,~~
It is
Mi, M~
of E,
function
we
m
can
T
=
dE
b dmi
identify
vi in>
pi,
T
d dM~ (b2)
with
Pm ,
=
v
~ m
p
Mm
(A2)
p d V
defined
that
v
~ =
Pm
+
2j3
pi
the
in
the
text
and
that
(A3)
(A4)
Appendix
Derivation
B.
introduce
We
of
derivative
time
:
$ DA
W
(Bl)
at
~
3A
(X, t)
~~
~~'
at
~
m
~~
~~
V
+
at
(82)
3x~
have
we
Using
(x, t)
aA
aA
Thus
M
II
(3.12).
of
kinds
two
PHYSIQUE
DE
JOURNAL
34
following
the
a
F(
fi
J
D
F(
Dt
J
3
F(
ax~
J
~
~'
~~~~
relations
$
j
(84)
"
DJ
aXPa
ax~
%
ax~ fi
$
aXP
ak~
ax~
aXP
~
J
=
rewrite
can
we
(83)
~~~ ax~
(85) ,
as
~~
3
lk_ =
J
at
JaXP
~
l
ax~
J
F~gA) P
=~(iff-~F(j-V~P~, where
we
defined
have
~ ~P~
Since we
~P~
obtain
Appendix For
the
can
be
directly
shown
expression (3.12)
C.
(86)
J
J
ax~
Derivation
simplicity
of
of
in
to
the
m
vanish,
ax~
F)
(87)
J
which is
essentially
the
identity d(da
A
dh
m
o
[22],
text.
(3.24). the
equations
we
temporarily
introduce
a
vector
y
defined
as
M
THERMODYNAMICS
I
y~
(rJpn~)- j#~.
m
first
line
~~~~
JT] lY~ r(A
Y~
y~ (a~(JT))
=
y
=
(J7j
v
the
second
~~~
a~~~~ ~
r(A
+ Y
right
the
on
GELS
CHEMICAL
side
hand
(3.24)
of
35
derived
is
from
the
(JT])
~
y
~
~
Y
~
~~ ~~ ~~~~
~~~~
~~~~
~
F)~ JTf)
F(~ JT$
+
line
OF
:
~~~~
~~
"
follows
as
Then
HYDRODYNAMICS
AND
(JT$)
a~(JT~)
~
JT~ ))
(ci) ~
This
.leads
to
hand
side
of
the required expression in (3.24) we only need to use
JPm
~~~~~ ~'
In proceeding following relation
the
text.
the
arbitrary
an
Appendix D.
key
The
Derivation
identities
where
A,
tensor
(4,10)
of we
~~~~
the
used
following
the
to
L~bf(h)
are
and
defined
(D2)
of
Aff(h)
antisymmetric tensor L~~f(h ) and Ail (h)
can
dressed
lines
like
be
Ii
=
8p~
[16],
formula
I~1)~~ E~~~
h +
8
8
flu
yv
the
metric
expressed
the
and
as
three-point (bare) (h~)~y whose both
(D2) (D3)
8 yv
the
h~p.
tensor
diagram in
bounded
are
line
text
and
the
E~ p~
the
assume
the
of
terra
of
we
8~p,
delta
h~ p
Generically
vertices.
ends
Each
which
second
the
are
that
completely
expansion of represented,
these
graphs
by three-point
vertices
of
some
right
posess in the
way,
QIS~>
terms
~h~
(4.12)-(4.14) in the text. In going to already used (Dl). As noted in Kronecker expanded only by using the
E~p~ and
Using
~)aA ~A~v
in
are
two-
following
constant.
a
(Dl)
have
we
respectively, by internal
is
=0,
~3(~ ~
E«py E«~v
side
pi
=
;
"
hand
Jp~
Cayley-Hamilton
the
are
use
~afly ~fl~ ~yv
(, I~ and I~
A
that
fact
h~-Iih~+I~h-I~l and
right
(4.ll),
and
going
are
have
we
the
on
'
Pm
;
for
line
:
AS
lJAf PI
~
third
the
to
formulae which
do
(Dl-3) not
we
include
show
below
three-point
m
~«py(h~)yy
that
a
bare
quantity vertices
(D4)
~y«> like :
Qj$jj
From
(Dl)
can we
be
decomposed into the following
have
identity, 1=
(h~-Iih+I~l)h. /3
(D5)
Applying
identity
this
I
Qjj~p
Qt)&1
formula
Ij
~h~
h
Ii
to
I~
h +
i
yj
(h~ )~~
E~ p~
+121>~ja~ I(h~
~
(hn>~ ~
(h~)~y
II
on
~ ~
right
hand
pa
~~«f
~&if
+12 iYl
h
[~h~-Iih+I~l)~]~~ [(h~-
vertices,
pairs
the
terms
and
eliminate lead
to
(8~j8~j-8~j
(D6)
we
8~~),
have
(D7)
to
of
the
second
three-point
equation bare
have
we
vertices
used
connected
the
by
formula any
(D3). number
In of
this bare
way
we
can
two-point
bare three-point in each of the topologically is at most only one vertex topologically joined graphs with a single external leg, which correspond to Thus for vanish since h is a symmetric like E~p~(h~)p~, do, however, tensor. Ail we can eliminate all the three-point bare vertices. In each term thus obtained we Cayley-Hamilton formula, and then we are the higher powers of h's by using the
until
joined graphs. L~~f
side of
Ii
where, in going eliminate
h+I~l)~]p~
Ii
(D6>
p~(hn>~~ ~~a~
the
13 =
M
II
obtain
we
~
(D2) n-timis
~ ~
(D4)
(h2
Ii
=
Employing
in
n-times
PHYSIQUE
DE
JOURNAL
36
there
The
expressions
(4,12)
and
(4.13) in
the
text.
References
[1] BIOT M. A. and WILLIS D. G., J. Appl. Mech. 24 (1957) 594. [2] ALEXANDER J. I. D. and JOHNSON W. C., J. Appl. Phys. 58 (1985) 816. J. I. D., J. Appl. Phys. 59 (1986) 2735. ALEXANDER [3] JOHNSON W. C. and DuBoIs-VIOLETTE E., PANSU B. and ROTHEN F., J. Phys. France 49 (1988) [4] JORAND M., [5] TANAKA T., HOCKER L. O. and BENEDEK G. B., J. Chem. Phys. 59 (1973) 5151.
ll19.
GENNES P. G., Macromolecules 9 (1976) 587 and 594. [~ TANAKA T., ISHIWATA S. and ISHIMOTO C., Phys. Rev. Lent. 38 (1977) 771. D., J. Chem. Phys. 70 (1979) 1214. FILLMORE [8] TANAKA T. and [9] LI Y. and TANAKA T., J. Chem. Phys. 92 (1990) 1365. [10] MATsuo E. S., TANAKA T., J. Chem. Phys. 90 (1989) 5161. [ll] GEISLLER E. and HECHT A. M., J. Chem. Phys. 77 (1982) 1548. [12] SEKIMOTO K., SUEMATSU Ni and KAWASAKI K., Phys. Rev. A 39 (1989) 4912. [13] TANAKA T., Phys. Rev. Lent. 40 (1978) 820. [14] JOHNSON D. L., J. Chem. Phys. 77 (1982) 1531. [15] ONUKI A., Dynamics of Ordering Processes in Condensed and H. Mater, Y. Komura Furukawa Eds. ~Plenurn, New York, 1988). [16] RIVLIN R. S., Rheology, F. Eirich Ed. (Academic Press, New York, London, 1956) vol.1. Statistical Physics, English Ed. (Pergamon, 1958). [17] LANDAU L. D. and LIFSHITz E. M., [18] MARTIN P. C., PARODI O. and PERSHAN P. S., Phys. Rev. A 6 (1972) 2401. [19] FLORY P. J., Principles of Polymer Chemistry (Comell Univ. Press, Ithaca, 1966) Chaps.12 and
[6] DE
13.
(McGraw-Hill, 1962). Thermodynamics j20] See, for example, D. D. Fitts, Nonequilibrium (Addison-Wesley Pub., 1950).p. 40. Mechanics, j21] GOLDSTEtN H., Classical Mathematical Physics, (Cambridge Univ. Methods of Geometrical j22] See, for example, B. F. Schutz, Press, 1980). j23] DOI M., Dynamics and Patterns in Complex Fluids, A. Onuki and K. Kawasaki Eds. (SpringerVerlag, 1990) to be published.
