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Phase separation of polymer solutions and interactions of globules
A. Grosberg, D. Kuznetsov
To cite this version:
A. Grosberg, D. Kuznetsov. Phase separation of polymer solutions and interactions of globules. Jour- nal de Physique II, EDP Sciences, 1992, 2 (6), pp.1327-1339. �10.1051/jp2:1992201�. �jpa-00247730�
Classification Physics Abstracts
36.20 64.70
Phase separation of polymer solutions and interactions of
globules
A. Yu. Grosberg and D. V. Kuznetsov
Institute of Chemical Physics, Russian Academy of Sciences, 117977 Moscow, Russia (Received 9 october I991, revised 9 March 1992, accepted12 March 1992)
Abstract. A new method is proposed for a quantitative investigation of interchain interactions in a poor solvent within the framework of a Lifshits type mean-field approximation. Density
distributions and interaction free energy for two globules equilibrated at an arbitrary distance ( from each other are calculated. The corresponding pair virial coefficient and coexistence binodal
curve in the second virial approximation are determined, and spinodal curve is estimated as well.
The theoretical phase diagram without any fitting parameters is compared with experiments
some interpretations and questions related to the experiments are formulated.
1. Introduction.
The phenomenon of precipitation of polymer solutions has been the subject of many theoretical studies since the classical ones of Flory and Huggins (see, for example, the books ill and [2]). The Flory-Huggins theory provides a qualitatively adequate description of most
experiments and, at the same time, requires some quantitative corrections.
First of all, being a mean-field-type theory, it needs some corrections in the vicinity of critical point these corrections were analyzed recently by Sanchez [3] and for a wider region by Chu and Wang [4]. They have generalized the usual scaling theory of liquid-gas critical
point for long polymer chains with N » I. The results of these works were fornlulated in
tennis of scaling combination of the type e.N~, where b=0.313 ±0.004, while
s m (T~~ T)/T~~ is the dimensionless deviation from the critical point T~~, but not from the
one ; and these results are assumed to be valid up to s ~
0.3/N°.~.
In the present paper, we will consider the range of essentially greater values of s, where another restriction of applicability of the usual Flory-Huggins theory arises. Indeed, the dilute
as well as the concentrated phase, is usually supposed to be uniform therefore, the usual
neglect of volume interactions in the dilute phase means that this phase is assumed to be a
suspension of ideal coils. But a coil-to-globule phase transition should take place in individual chains in the dilute phase not very far from the b-point (and, consequently, from the critical one) [1, 5]. To describe quantitatively the precipitation of polymer solutions, it is necessary to
take into account globular structure of chains in the dilute phase. This is exactly the subject of
the present paper.
The theory of globular state of individual polymer chains is well developed at present, main ideas are those of Flory and Lifshitz (see, for example, Refs. [1, 6]). The detailed quantitative
version of the theory has been proposed earlier [7-10]. To analyze the phenomenon of
precipitation, the single chain theory is not sufficient and it is necessary to consider interactions between chains, that is, between globules in our case.
An attempt to analyze globule-to-globule pair interactions using a Flory-type theory has been realized in reference [11]. This approach is based on the use of a single scalar order parameter like expansion factor a for the description of two globule complexes. Some non-
controlled and non-obvious assumptions seem to be inevitable in these frameworks. The
problem of precipitation has not been analyzed in reference Ii Il.
In the present paper we have calculated quantitatively the binodal (coexistence) and
spinodal curves of polymer solutions in the globular region near the globule-to-coil transition
point. Previously the pair free energy of interacting macromolecules (with detailed density distributions) versus internlolecular distance and their second virial coefficient have been described using a Lifshits type mean-field globule theory. The results presented below are
compared with experiments.
2. Effective potential of chain-chain interactions in a poor solvent.
The second virial coefficient of chain-chain interactions through the solvent medium can be evaluated by definition Ill as follows :
A~
=
l
exp(
~ ~~~~~~ ~~~~~~~
d~(. (I)
Here F~~~~ie and F~~i~ (( ) are free energies, of single polymer chain and of two chains placed at
a distance ( from each other, correspondingly, in the same solvent and under the same
conditions ; T stands for the temperature in energetic units. (A slightly different definition of
A~ is typically used in the chemical literature, namely, Aj~~~~~ = ~~A~, where M is the
M~
molecular mass and N~ is Avogadro's number.)
To calculate the value of pair virial coefficient (Eq. (I)) the effective potential energy u~ilf)
m F~~(f) 2 F~i~~j~ has to be deternlined.
Let us consider the system of two polymer chains of the same chemical nature in a
sufficiently poor solvent, where each chain should collapse to the globular state. In the most
interesting and non-trivial case the distance ( between the mass centers of globules is
comparable with their own sizes. Due to the attraction of monomers in this case both globules
should be stretched to each other. Such a drawn out structure cannot be described rigorously
in tennis of a single scalar order parameter like the expansion factor a.
Here we will use a Lifshits-type theory [1, 5, 6]. The spatial density distribution, which is the order parameter in this approach, can, of course, describe the structure of stressed non-
spherical globule. However, there exists the following problem. Lifshits theory is of the self-
consistent type and, therefore, includes with the free energy minimization procedure.
Simultaneously the two-chain free energy F~~i~(~) has no minimum at any non-zero distance (.
The free energy minimum corresponds only to the globules merging with each other, I.e, to
(
=
0.
Our solution of this problem can be explained in physical and mathematical ways. From the
physical point of view, the idea is to introduce two fictitious uniform fields of opposite signs,
that act on two chains correspondingly. These two fields pull the globules in opposite
directions. Being unifornl, these fields do not perturb the globules' inner structures.
Meantime, these fields fix the distance between globules and, therefore, this distance can be
regulated by values of the fields.
Let us consider our approach in more detail. As usual, the free energy of our system in the framework of mean-field theory should be written as follows :
Fpair "
Fill + F#$
,
(2)
where F)($~ describes volume interactions between monomers and F)$( = TS~~ corresponds
to confornlational entropy of chains.
The interaction energy F~~~~~ depends on the total density of monomers only, since all
monomers of both chains are identical. If we denote the spatial density distribution of
monomers of the chain number I (I
= 1, 2) by ni(x), then the total density will be n(x) = n j(x) + n~(x)
and, therefore, the interaction energy can be written in the fornl
F~~~~~
=
f*(n(x)) d ~x (3)
Here f* is the interaction energy of the link system, a virial expansion of the type f*(n(x)) m TBn ~ + TCn ~ + is valid for the system under description [5] ; B and C are the
interaction constants of quasimonomers.
The elastic energy (I,e, polymer confornlational entropy), contrary to the interaction energy, is the sum of single chain contributions and each chain contribution depends on its
own density only :
~pdr ~ ~single(~l(X)) ~ ~single(~2(X)) (4)
Here the standard Lifshits expression [5] is valid for the single chain entropies :
S~i~~j~(n;(x) =
j n;(x) In
~~"
d~x
m
~~j
~,(x) A~;(x) d~x, (5)
4~; 6
where ~,(x) m /$, #
m I + (a~/6) A stands for a linear memory operator, A is a Laplacian operator, a is the link size for the standard Gaussian model of chain.
Three conditions should be taken into account for the free energy minimization: in
addition to the ordinary requirements of fixation of the total numbers of monomers for each chain
n,(x) d~x
= N, (I = 1, 2)
,
(6)
the distance between the mass centers of two chains must also be fixed :
lxnj(x) d~x xn~(x) d~x
N
j N~ ~~~
Therefore, three Lagrangian multipliers (A~, A~ and E) should be introduced and the
following effective potential should be minimized :
G
=
F~'~~~ TS TS~ A
j n~(x) d~x N j
A~ n~(x) d~x N ~l-
xn~(x) d~x xn~(x) d~x
~~ N~ N~ ~~~
The last tern1plays the role of the energy of chains in the extemal unifornl fields ± E and this is the mathematical nature of our approach.
Minimization of the effective potential (Eq. (8)) with respect to the density distributions
n~(x) and n~(x) leads to the following equations
~ A /~
= exp
~'
' l fi (I
= 1, 2), (9)
~T ~
it should be solved together with conditions (6) and (7). Here
l~i *
= l~ *(ni + n2) (E; x) (IO)
are the self-consistent fields, acting on monomers of chain number I. They include the usual
ternl ~ *(n~ + n~), which is deternlined by the volume interactions between monomers and,
additionally, the extemal unifornl fields which fix the distance ( between mass centers of two chains.
The minimal value (which is the equilibrium value) of the free energy of interest is equal to
~pair~'~l~l~'~2~2~ jP*(~l~~2)d~X, (11)
where p *(n)
= n~ *(n) f*(n) m TBn ~ + 2 TCn ~ is the pressure of an equivalent system of separate polymer links without the ideal gas ternl. Of course, at the ( - oJ limit equation (I I) gives just the sum of the free energies of two single macromolecules.
We will now restrict our consideration to the simplest case of two chains of same lengths, N~
= N~. In this case the spatial density distributions for these two chains are under mirror symmetry to each other and, as a result, our equations are simplified enough for numerical
solution.
The results can be summarized as follows :
a) The theory, just as for the single chain case, remains a three-parametric theory. It involves the values of (I) the Gaussian-coil state size (root-mean-square radius of gyration) R~
~
aN ~~( (ii) the reduced temperature
BN ~~~ T ~
N ~~~ ~~~~
t =
2 '
C~~( ~~~
and (iii) the chain stiffness parameter $la~ (it is proportional to (dli)~/~, where d and I are the diameter and effective segment of the chain).
b) One of the calculated density distributions for two same interacting globules is shown, as
an example, in figure I for a certain non-zero distance ( and the reduced temperature
t > 30.
---~- --i- -~--- i
i i
-1- ~
---~- --i- -~---
i i
Fig. I. Density distribution in the axial plane section of two polymer globules placed at a fixed non-
zero distance between their mass centers for the reduced temperature 1= 30, presented by means of both 3D diagram and level curves.
c) The two-chain free energy F~~~(~), just like the single chain energy F~i~~j~, scales as a~/ $ with the chain stiffness and
can be written in the fornl
Fpmrl'> = T w~3 Fpa«(t, 3>, (13>
where li is
a dimensionless function which depends on dimensionless values of the reduced temperature t and the distance 3
m
~ ~~ ~~~
only. (In our previous paper [7] we have
RG;~ $
~3
shown that F~i~~j~ = T -F~i~~j~(t).)
$
d~ The calculated profiles of the effective chain-chain interaction potential are shown in
figure 2 for two values of reduced temperature : t = 24 and 34. (These temperatures lie both in the globular range, since the coil-to-globule transition point corresponds [7] to t = t~ m -10.2.)
3. Pair virial coefficient of chain-chain interaction.
In the framework of three-parametric theory the expression (I) for the chain coupling
constant A~ can be rewritten in the following way
A~
=
N ~Bh (t, $la~)
= t
j ~~ 6~"R(
~
h (t, $la~) (14)
a '
JOURNAL DE PHYSIQUE n -T 2. N'6, JUNE1992 48
0
-1
m m
~~ ~ i
-2
? 2
~ m ,
M
j -3
,
-4
-5
o-O O.5 1-o 1.5 2.O
(C'"la~)~"~
~~ta
Fig. 2. Effective potential of chain-chain interactions vs. the dimensionless distance between chains
mass centers in the globular range for two values of the reduced temperature t = 24 ( Ii and 34 (2).
In the neighborhood of the b-point the value of the dimensionless coefficient h was calculated earlier using perturbation theory (see Ref. [Ii) :
fi 1/2
h
= 0.4931 t( j (14a)
a
For the globular range we have calculated this value using the definition (I) and the results discussed above on F~~~(() (Eq. (13)). Our results are shown in figure 3 in semilogarithmic
scale log (- A~/R(~) versus t for a few realistic values of the chain stiffness.
4. Binodal and spinodal curves of polymer solution.
The standard conditions of phase equilibrium require the coincidence of polymer osmotic pressures (w~j) and chemical potentials (p~~j) in both coexisting phases :
"P°~ ~
~ ~P°i fP°i
c c~,
"
~ ~P°i fP°'
c = ~~~~
' ~~~~
»pot - N ~li°' ~
~~
= N ~li°'
~~~
(16)
Here the polymer solvent free energy per unit volume is denoted by f~~j ; c and c/N are the average numbers of polymer links and of chains, respectively, per unit volume in dilute (c~t;j) and concentrated (c~~~~) phases.
30
25
- 20
m~~~
©$
' 15
~m ~
~ lo '
o 3
"
- '
~
'
5 ~
,
"
', ',
~
',
~Tiz
''
O
-40 -30 -20 -lo O
Reduced temperature (t)
Fig. 3. Pair virial coefficient A~ of chain-chain interactions, normalized to the ideal Gaussian-coil volume,( vs. the reduced temperature t for the following realistic values of chain stiffness parameter ;
= 0.05 (1), 0,10 (2), 0,15 (3), 0.20 (4). The perturbation theory results (Eq. (14a)) for the coil
a
region (where it is small enough) as well as the results of our theory for the range of well formed
globule (t
< 20) are shown using the solid lines. Dashed parts of the curves are the interpolations
between these regions.
The concentrated phase can be considered as a « very large globule », so that
where
F)#il~~~~(C) =
~ f*(C)
m T ~
(BC~ + CC~)
C C
is the single globule free energy in the so-called volume approximation [5]. This approxi-
mation neglects the globule surface energy, and, therefore, it is just valid for macroscopic precipitate. Of course, expression (17) is practically equivalent to the well-known Flory- Huggins formula [2].
The dilute phase is a nearly ideal « gas » of globules, therefore
fpoi(C)
c c~
~ ~~~~~~ ~ ~~ ~ ~~
~
~ ~~~~~~~~~~~ ~~
A novelty of our approach is the use of quantitative globule theory results [7] for the single
chain free energy F~;~~j~ as well as for the pair virial coefficient A~.
The coexisting curve, I.e, binodal, on the phase diagram can be expressed in dimensionless form as follows
l~~~l,d ~j/~~l~d ~~~ ~ ~~~~l~~~~(~) ~S<ngle(~)
~ '~~~~ '~ ~~' ~~~~' ~ ~~~
~il~ l~d) ~
~ ~~~ ~~~~~~l~~~~~~~~ ~s'ngle~~~
j
'
(19)
l~il~ l~d)
~ j9/4
~ ~~~~ ~ ~ ~ ~~~ ~
~~ ~j~~~~
'
~~~~
where F)((i~~PP~.~(t) = F)]Q(i~~PP~.~(c~~~~(t)) m T
~~ ~~~~,
the function F~j~~j~(t) was calcu-
$ 4. 6
lated in a previous article [7], A~ or h is described above the value of y
m c~t,i/c~~~~ « I is
negligible in the globular range under consideration. It is quite natural that
l~(~~ R(,~) m ~~~
j ~~~
TN ~/~
t ~ ec 2. 6 a
B
°~ ~C°nCIt
~ ec
* @
As a rule, however, it is difficult to establish experimentally whether equilibrium or not
quite equilibrium phases are observed. This is why the spinodal curve is also of interest as well
as the binodal curve. The general spinodal equation, as is well-known, is of the type
~~°(~~~
= 0 (21)
~
ac
Of course, the second virial approximation (Eq. (18)) for f~~j(c) is not sufficient for a spinodal
curve quantitative analysis. Nevertheless, it can be used as the simplest estimate and this
overestimate is the only available now. Substitution of expression (18) for f~j into spinodal equation (21) gives
c((f~.I < N/2 A
~, (22)
I-e- in dimensionless variables of the present paper
~~~~ R(
w
6~"
2 j ~~~
t) h (t, $la~)j (23)
N '~
a
The phase diagram in the dilute region of polymer solutions is shown in/fure4 in
semilogarithmic scale for a few typical values of the chain stiffness parameter Cla~. The
solid and dashed lines correspond to binodal curves (Eq. (19)) and to spinodal curves
(Eq. (23)) respectively.
5. Comparison of theoretical and experimental results.
Let us remind that our theory includes three parameters Na( t and $la~. The value of
Na~ does not play any role in our problems, since it determines absolute values of polymer
chain characteristics only, but not relative values that are under consideration. The next
parameter t (Eq. (12)) depends on two variables measured experimentally (temperature T
or r m (T o)/o and chain molecular mass M) as t
=
QrM~/( where Q
=
Cst, for each
polymer-solvent system. Therefore $la~ and Q should be considered
as two adjustable
-J o
4~
~~
l~ _io
~ 4~
Ch '
~ -20 / / j
4~ / / j
'~ / /
'~~ -30 2 (1) (2)
~
''(3) ,'
~ (4)
Q _/ / , /
~ l' 3
4 /
,
/ 'l3
4~
~ ~~3O -25 -20 -15 -lo -5 O
~°~(~~~~)
Fig. 4. Phase diagram in dilute region of polymer solution for the following realistic values of chain stiffness parameter :
j
= 0.05 (1), 0,10 (2), 0.15 (3), 0.20 (4). The solid curves are binodals and the
a
dashed ones are spinodals. The coil-to-globule transition region is shown by dotted line at
t~ m -10.2.
parameters. One of them (Q) can be simply determined as the scale of TM ~/~ axis. Thus, the main empirical parameter of our approach is the chain stiffness $la~.
We would like to emphasize, however, that here we do not fit any parameter at all : the
values of $la~ and Q
were both determined earlier [7, 8] by means of comparison of our
theory with experiments [12-14] on the coil-to-globule transition in single chains in extremely
dilute solutions. According to the results of the articles [7, 8], we have for the polystyrene- cyclohexane system
$la~
=
0,15 ± 0.04
,
Q
= 0.28 ± 0.08 (mol~R/g~R) (24)
and for the polystyrene-dioctyl phthalate system
$la~
= 0.06 ± 0.02
,
Q = 0.06 ± 0.02 (mol~/~/g~/~) (25)
Therefore in the present paper we do not use any adjustable parameter at all.
There are a few experiments on PS/CYH solution phase separation in the dilute region. The
results of the CEN-Saday group [15] for coexistence curve of this solution and the
corresponding theoretical curves (see Eqs. (19) and (23)) without any adjustable parameters
are presented in figure 5. The part of these experimental data with the lowest molecular mass of polystyrene is in a good quantitative fit with our binodal results. For the longer polymers
the experimental points are very close to the theoretical spinodal curve and, moreover, the
empirical coexistence curve equation of reference [15] coincides practically with our spinodal
formula (22). In our opinion it means that for very long polymer chains the precipitation time is essentially longer than for shorter chains in the solution of the same weight concentration
o
4J
fj~o
Q~ ~
~ ~~ ~
_J a D ~
tl$ u
# -15
i2w£
*
~
~
-20
'tl 4J
@
~
~~ -30
Ctl
~~o ~~
C
-
R
N $d
33
32
~
g~ 30
~
i 29
~tl$
----
~w~~
8°~
~
26
c
-R~
N id
