Flory-Huggins theory for the solubility of heterogeneously-modified polymers
Patrick B. Warren
Unilever R&D Port Sunlight, Bebington, Wirral, CH63 3JW, UK.
(Dated: April 5, 2007)
Many water soluble polymers are chemically modified versions of insoluble base materials such as
cellulose. A Flory-Huggins model is solved to determine the effects of heterogeneity in modification
on the solubility of such polymers. It is found that heterogeneity leads to decreased solubility, with
the effect increasing with increasing blockiness. In the limit of extreme blockiness, the nature of the
phase coexistence crosses over to a polymer-polymer demixing transition. Some consequences are
discussed for the synthesis of partially modified polymers, and the experimental characterisation of
such systems.
Many water-soluble polymers are made by chemically
modifying insoluble base materials such as starches and
gums, for example a wide class of water-soluble polymers
are obtained from cellulose [1, 2]. It is often possible to
vary the degree of modification of the base polymer to
obtain water soluble polymers with, in principle, con-
tinuously variable properties. A basic characteristic of
these polymers is their solubility, but given the essen-
tially stochastic nature of the chemical modification step,
what is the effect of heterogeneity in modification on the
solubility of the resulting materials?
In the present paper, this question is approached from
a theoretical point of view by setting up a Flory-Huggins
model for the phase behaviour of a polymer-solvent mix-
ture [3], where the polymers have a random degree of
modification. In this approach, the issue of solubility
is translated into the problem of determing the phase
coexistence between a dissolved aqueous phase and an
undissolved (water-poor) phase. The solubility is then
formally given by the polymer concentration in the aque-
ous phase. Determination of the full phase behaviour for
a multicomponent Flory-Huggins theory is an onerous
task though, and a simpler approach is to examine the
spinodal stability of the system, which can be taken to
be representative of the full phase behaviour. This is the
approach taken in the present paper. It is arguably more
insightful than a full calculation of the phase behaviour
since closed-form analytic expressions can be obtained for
the spinodal stability limit. The approach taken is sim-
ilar to models for the phase behaviour of random block
copolymer melts which have been developed in the past
[4, 5, 6]. There has been rather little work though on
random copolymers which also include a solvent, apart
from a brief example described by Sollich et al [7].
In the present model, it is supposed that the system
comprises a large number of species of polymers i with
differing degrees of modification 0 < αi < 1 and con-
centrations ρi. For simplicity, length polydispersity is
neglected, and all the polymers are assumed to have the
same number N of segments. The system is then de-
scribed by the following (mean field) Flory-Huggins free
energy density,
i ρi log ρi+(1−φ) log(1−φ)+χ(φ−η)(1−φ), (1)
where φ is the total polymer segment concentration and
η is the concentration of chemically modified segments,
given respectively by φ = N
i ρi and η = N
i ρiαi.
The first term in Eq. (1) is the ideal free energy of mix-
ing. The second term is the usual Flory-Huggins configu-
rational chain entropy. The third term is the free energy
cost of the unmodified polymer segments at a concentra-
tion φ− η coming into contact with solvent (water) at a
concentration 1 − φ. Typically one expects χ > 1/2 for
this interaction, to represent the repulsion between un-
modified segments and water which leads to phase sepa-
ration of unmodified polymers. To keep the model sim-
ple, this is the only χ-parameter that is retained in the
problem.
Eq. (1) has the structure of a moment free energy,
since the excess free energy, comprising the second and
third terms, only depends on φ and η which are mo-
ment densities. Such a system can be analysed using the
methods developed by Sollich and coworkers [7, 8, 9, 10].
In particular, Ref. [10] describes how the spinodal sta-
bility conditions for systems with an excess free energy
can be expressed in terms of moment densities, gen-
eralising various truncation theorems obtained by ear-
lier workers [11, 12]. I now summarise the relevant re-
sults, translated into terms suitable for the present prob-
lem. Let us consider such a system with a free energy
i ρi log ρi + f
(ex)(φ(1) . . . φ(n)), where the excess
free energy depends on moment densities of the form
φ(r) =
i ρiw
i (r = 1 . . . n), with the w
i being
species-dependent weights. The fundamental idea is that
the moment densities can be treated as effective species
concentrations. In particular, it can be proved that spin-
odal stability corresponds to the positive-definiteness of
the matrix M of second partial derivatives of the free en-
ergy with respect to the moment densities. In Ref. [10]
it is shown that M = Mid + Mex where (M
id )rs =
i ρiw
i and (Mex)rs = ∂
2f (ex)/∂φ(r)∂φ(s). The
limit of spinodal stability is given by detM = 0. This
condition usually corresponds to the vanishing of a single
eigenvalue of M, with an eigenvector ∆φ(s) that satisfies
s(M)rs∆φ
(s) = 0. It is shown in Ref. [10] that the
spinodal instability direction in the space of species con-
centrations is given by ∆ρi =
rs ρiw
i (Mid)rs∆φ
For the present problem, there are two moment densi-
ties φ and η, defined respectively with w
i = N (a con-
http://arxiv.org/abs/0704.0707v1
stant) and w
i = Nαi (the number of modified groups
on the ith species). Application of the above theory to
Eq. (1) leads to
id = N
i ρiαi
i ρiαi
i ρiα
Mex =
(1− φ)−1 − 2χ χ
. (3)
After some algebra the condition detM = 0 reduces to
−2χ(1−〈α〉)−χ2Nφ(〈α2〉−〈α〉) = 0, (4)
where
〈α〉 =
i ρiαi /
i ρi, 〈α
i ρiα
i ρi. (5)
I emphasise that, despite being remarkably simple,
Eq. (4) is exact.
One already reaches a significant conclusion from this.
The first three terms in Eq. (4) are what one would ex-
pect from standard Flory-Huggins theory [3], with an ef-
fective χ-parameter given by the product of the original
χ-parameter and the fraction 1− 〈α〉 of unmodified seg-
ments. These terms therefore take account of the mean
degree of modification. The final term in Eq. (4) is a
correction due to the heterogeneity. Since the variance
〈α2〉− 〈α〉2 is positive, this term is always negative. The
effect is that heterogeneity in modification reduces the
solubility, over and above what would be expected from
the mean degree of modification.
To make further progress, it is convenient to specify a
model for the distribution of the αi. In particular, such a
model can be used to examine the effect of blockiness in
modification which is expected to play an important role.
In previous work on random block copolymers [4, 5], a
Markov model was used to characterise the correlations
between different kinds of segments. Whilst such a model
may be appropriate for the stochastic nature of the syn-
thetic route for such random block copolymers, as dis-
cussed below it is probably not appropriate in the present
case. I therefore consider instead a very simple model for
the heterogeneity in which the modified segments occur
in blocks of size M , where 1 < M < N . In this model,
it is supposed that each block has an equal probability p
of being modified, and there are no further correlations.
Then, for any particular species, αi = (1/N)
j=1 Mǫij
where j labels the blocks, and ǫij is zero or one with prob-
ability 1 − p and p respectively. Thus the αi are drawn
from scaled binomial distribution, with
〈α〉 = p, 〈α2〉 − 〈α〉 = (M/N) p(1− p). (6)
Eq. (4) becomes
− 2χ(1− p)− χ2Mφp(1− p) = 0. (7)
This is a quadratic equation for χ and the appropriate
root is
)}1/2
. (8)
I now examine the consequences of this result.
The formal limit M → 0 corresponds to a vanishing
variance and a completely uniform distribution of mod-
ified segments, as though each monomer has undergone
an identical fractional modification by a fraction p, rather
than being modified or not with probability p and 1− p.
As noted already above, this limit corresponds to sim-
ple Flory-Huggins theory with an effective χ-parameter
equal to χ(1−p). For large N , this indicates the absence
of phase separation for χ(1− p) < 1/2 or p > 1− 1/(2χ).
Now let us consider Eq. (8) for block size M = 1.
In this case, individual segments are modified randomly
with no correlations. For M = 1 and large N in Eq. (8),
there are two behaviours depending on the value of p.
For p < 4/5, there is an absence of phase separation
for χ(1 − p) < 1/2, just as for the M → 0 limit. For
4/5 < p < 1, the behaviour is more complicated. To be
precise, the location of the minimum value of the χ(φ)
spinodal shifts from φmin ∼ N
−1/2 for p < 4/5 to a non-
vanishing 0 < φmin < 1 for p > 4/5 (it is the examination
of Eq. (8) in the limit φ ∼ N−1/2 that gives the cross over
point p = 4/5). The change in behaviour can be seen for
the M = 1 curves (dashed lines) in Fig. 1 and is shown
explicitly in the upper plot of Fig. 2.
Let us next consider the limit of extreme blockiness
M = N . This limit is strikingly different from the M = 1
case. For largeN and p > 0, one can show that there is an
absence of phase separation only for χ
Np(1− p) < 2.
In the large N limit, this inequality is always violated,
indicating that the system always has a tendency to un-
dergo phase separation in the limit of extreme blockiness.
Since the unmodified polymer system itself only phase
separates for χ > 1/2, this suggests that the phase sep-
aration has the nature of a polymer-polymer demixing
transition rather than a solvent-driven phase separation.
This insight is confirmed by analysis of the spinodal in-
stability direction below.
For large N and general M in Eq. (8), one would ex-
pect that the above two cases represent the two classes of
behaviour. In the first case M ≪ N and the behaviour
is similar to the M = 1 limit where individual segments
are randomly modified. In the second case, M ∝ N
and the behaviour is similar to the M = N limit of ex-
treme blockiness. Fig. 1 shows typical spinodal curves
calculated from Eq. (8) for various values of p and M .
The location of the minimum (φmin, χmin) of the spin-
odal curves can be numerically determined, and Fig. 2
shows how this depends on p.
The results show firstly that for M ≪ N , increasing p
leads to increasing solubility as the value of χ required to
reach the spinodal instability is increased. Moreover, a
decrease in solubility between a uniform model (M → 0)
with no heterogeneity, and a model with fine-grained
0.0 0.2 0.4 0.6 0.8 1.0
PSfrag replacements
p = 0.1
0.0 0.2 0.4 0.6 0.8 1.0
PSfrag replacements
p = 0.1
p = 0.5
0.0 0.2 0.4 0.6 0.8 1.0
PSfrag replacements
p = 0.1
p = 0.5
p = 0.9
FIG. 1: Spinodal curves calculated from Eq. (8) for polymers
of length N = 103, for three values of the mean degree of
modification p, and for block sizes M → 0 (uniform limit,
solid line), M = 1 (dashed line), M = 10 (dash-dot line),
M = 100 (dash-dot-dot line) and M = 103 (dash-dash-dot
line). The system is spinodally unstable above the indicated
curves. Note the change in shape of the M = 1 curves: for
p = 0.1 and 0.5 the minimum is at φ → 0, whereas for p = 0.9
the minimum is at φ ≈ 0.25.
blockiness (M = 1), is apparent. The major effect arises
as M → N though, where the tendency for phase sepa-
ration is greatly enhanced.
The above analysis is augmented considering the spin-
odal instability direction associated with the spinodal
stability limit which can provide a useful mechanistic in-
sight. As explained above, the spinodal instability direc-
tion is characterised by the eigenvector that corresponds
to the vanishing eigenvalue responsible for the vanish-
0.0 0.2 0.4 0.6 0.8 1.0
PSfrag replacements
0.0 0.2 0.4 0.6 0.8 1.0
PSfrag replacements
FIG. 2: The location of the numerically determined minimum
of the spinodal curves from Eq. (8) is plotted as a function
of p, for polymers of length N = 103 and block sizes M → 0
(uniform limit, solid line), M = 1 (dashed line), M = 10
(dash-dot line), M = 100 (dash-dot-dot line) and M = 103
(dash-dash-dot line). For M = 1 (dashed line) the upper
plot shows clearly that φmin ≈ N
≈ 0.03 only holds for
4/5 = 0.8.
ing spinodal determinant. For the present problem, from
Eqs. (2)–(3), one finds the instability direction is charac-
terised by
∆η /∆φ = 〈α〉 − χNφ(〈α2〉 − 〈α〉2) (9)
The corresponding spinodal instability direction in the
space of species concentrations is
〈α2〉∆φ − 〈α〉∆η + αi(∆η − 〈α〉∆φ)
φ(〈α2〉 − 〈α〉2)
1 + χNφ(〈α〉 − αi)
where the second line follows by inserting the result for
the ratio ∆η/∆φ. These results should be evaluated on
the spinodal. They are all exact, for an arbitrary distri-
bution of αi.
For the instability direction to lie along a pure dilution
line, one should have ∆ρi/ρi independent of species i.
One can conclude that this only happens if ∆η/∆φ =
〈α〉, in other words if the variance 〈α2〉 − 〈α〉2 vanishes.
In such a case, the phase transition is purely associative,
or solvent-driven, meaning that the compositions of the
coexisting phases remain the same (∆ρi/∆φ = ρi/φ).
If one specialises to the model of blockiness described
above by inserting the value of χ corresponding to the
spinodal stability limit, the instability direction becomes
)}1/2
This confirms that the spinodal instability lies along a
dilution line (∆η/∆φ = 〈α〉 = p) only in the limit
M → 0 which formally corresponds to a vanishing vari-
ance. For M = 1 (and M ≪ N in general) the phase
transition has a mixed character. The interesting case
occurs when M = N (or M ∝ N in general) for which
∆η/∆φ ∼ (−)N1/2 in the limit of largeN . One can write
this as ∆φ/∆η → 0 as N → ∞. This shows that the
phase transition tends towards being purely segregative,
meaning that the overall polymer concentration in coex-
isting phases remains the same (∆φ = 0). This confirms
the suggestion above, that in the limit of extreme block-
iness, the system tends towards a segregative polymer-
polymer demixing transition.
Let us now try to draw some conclusions. The main ef-
fect of randomness is to reduce the solubility of partially-
modified polymers beyond what would be expected from
the mean degree of modification. The extent to which
this occurs depends on the blockiness in substitution.
For fine-grained blockiness, the phase behaviour is ex-
pected to be similar to a system for which there is no ran-
domness, albeit with a somewhat reduced solubility. For
coarse-grained blockiness, where the block size is com-
parable to the polymer length, the nature of the phase
transition changes to a polymer-polymer demixing tran-
sition. In this situation, one expects that the modified
polymers (being almost fully modified) will partition into
the aqueous phase, leaving the unmodified polymers be-
hind.
The reason for considering the two extreme kinds of
blockiness is now clearer: namely one can envisage two
different mechanisms of chemical modification (this is the
reason why a Markov model for the distribution of mod-
ified segments has not been used). Fine-grained block-
iness would arise if monomers are equally accessible to
the modifying agent, irrespective of their surroudings. If
this cannot be achieved in a one-step process (for the
reason described below) it could perhaps be achieved in
a two-step process, by fully modifying the polymers then
removing a random fraction of the derivative groups. Ex-
treme blockiness on the scale of the polymer chain itself
would arise if the modifying agent was present only in the
aqueous phase, and as such only able to access polymer
which had already been solubilised. This would lead to a
mixture of polymers which were either fully modified, or
remained unmodified and insoluble. The process of mod-
ification of insoluble polymers could still be initiated be-
cause the modifying agent is able to access the tiny pro-
portion of the insoluble polymer segments which lie at
the interface between the insoluble and aqueous phases.
Experimentally, confirmation of the scenario of extreme
blockiness would be given by measuring the mean degree
of modification for the dissolved polymers. One should
find that this is much in excess of the apparent mean
degree of modification.
In the calculation, the major effect arises from inter-
chain rather than intra-chain heterogeneities. The model
is not sophisticated enough to take account of the solu-
tion structures such as micelles or mesophases that could
form for blocky polymers with block sizes M ≫ 1 but
still M < N (for example, diblock copolymers). Such
polymers would be expected to have greater solubilities
than would be predicted from the Flory-Huggins theory
since the hydrophobic groups can be buried in micelles
or other solution structures. The present theory could be
extended to discuss these inhomogeneous situations using
a Landau approach developed for random block copoly-
mers [4, 5, 6]. For the mechanistic routes discussed above
though, it is difficult to envisage that polymers with in-
termediate block sizes could arise very easily. I therefore
expect that the general conclusions will remain.
Finally I note that in principle the above model for
the phase behaviour could be combined with a model for
the chemical modification reaction, to obtain a theory for
reaction-induced solubility. However, one needs to take
great care to capture the kinetics correctly [13].
I thank Nigel Clarke for a critical reading of the
manuscript.
[1] R. L. Davidson, Handbook of water-soluble gums and
resins (McGraw-Hill, New York, 1980).
[2] J. Rueben, Macromol. 17, 156 (1984).
[3] P. J. Flory, Principles of polymer chemistry (Cornell Uni-
versity Press, Ithaca, New York, 1953).
[4] G. H. Fredrickson and S. T. Milner, Phys. Rev. Lett. 67,
835 (1991).
[5] G. H. Fredrickson, S. T. Milner, and L. Leibler, Macro-
molecules 25, 6341 (1992).
[6] A. Nesariker, M. Olvera de la Cruz, and B. Crist, J.
Chem. Phys. 98, 7385 (1993).
[7] P. Sollich, P. B. Warren, and M. E. Cates, Adv. Chem.
Phys. 116, 265 (2001).
[8] P. Sollich and M. E. Cates, Phys. Rev. Lett. 80, 1365
(1998).
[9] P. B. Warren, Phys. Rev. Lett. 80, 1369 (1998).
[10] P. B. Warren, Europhys. Lett. 46, 295 (1999).
[11] P. Irvine and M. Gordon, Proc. R. Soc. Lond. A 375,
397 (1981).
[12] E. M. Hendriks, Ind. Eng. Chem. Res. 27, 1728 (1988).
[13] G. A. Buxton and N. Clarke, Macromolecules 38, 8929
(2005).

  Many water soluble polymers are chemically modified versions of insoluble
base materials such as cellulose. A Flory-Huggins model is solved to determine
the effects of heterogeneity in modification on the solubility of such
polymers. It is found that heterogeneity leads to decreased solubility, with
the effect increasing with increasing blockiness. In the limit of extreme
blockiness, the nature of the phase coexistence crosses over to a
polymer-polymer demixing transition. Some consequences are discussed for the
synthesis of partially modified polymers, and the experimental characterisation
of such systems.

Flory-Huggins theory for the solubility of heterogeneously-modified polymers
Patrick B. Warren
Unilever R&D Port Sunlight, Bebington, Wirral, CH63 3JW, UK.
(Dated: April 5, 2007)
Many water soluble polymers are chemically modified versions of insoluble base materials such as
cellulose. A Flory-Huggins model is solved to determine the effects of heterogeneity in modification
on the solubility of such polymers. It is found that heterogeneity leads to decreased solubility, with
the effect increasing with increasing blockiness. In the limit of extreme blockiness, the nature of the
phase coexistence crosses over to a polymer-polymer demixing transition. Some consequences are
discussed for the synthesis of partially modified polymers, and the experimental characterisation of
such systems.
Many water-soluble polymers are made by chemically
modifying insoluble base materials such as starches and
gums, for example a wide class of water-soluble polymers
are obtained from cellulose [1, 2]. It is often possible to
vary the degree of modification of the base polymer to
obtain water soluble polymers with, in principle, con-
tinuously variable properties. A basic characteristic of
these polymers is their solubility, but given the essen-
tially stochastic nature of the chemical modification step,
what is the effect of heterogeneity in modification on the
solubility of the resulting materials?
In the present paper, this question is approached from
a theoretical point of view by setting up a Flory-Huggins
model for the phase behaviour of a polymer-solvent mix-
ture [3], where the polymers have a random degree of
modification. In this approach, the issue of solubility
is translated into the problem of determing the phase
coexistence between a dissolved aqueous phase and an
undissolved (water-poor) phase. The solubility is then
formally given by the polymer concentration in the aque-
ous phase. Determination of the full phase behaviour for
a multicomponent Flory-Huggins theory is an onerous
task though, and a simpler approach is to examine the
spinodal stability of the system, which can be taken to
be representative of the full phase behaviour. This is the
approach taken in the present paper. It is arguably more
insightful than a full calculation of the phase behaviour
since closed-form analytic expressions can be obtained for
the spinodal stability limit. The approach taken is sim-
ilar to models for the phase behaviour of random block
copolymer melts which have been developed in the past
[4, 5, 6]. There has been rather little work though on
random copolymers which also include a solvent, apart
from a brief example described by Sollich et al [7].
In the present model, it is supposed that the system
comprises a large number of species of polymers i with
differing degrees of modification 0 < αi < 1 and con-
centrations ρi. For simplicity, length polydispersity is
neglected, and all the polymers are assumed to have the
same number N of segments. The system is then de-
scribed by the following (mean field) Flory-Huggins free
energy density,
i ρi log ρi+(1−φ) log(1−φ)+χ(φ−η)(1−φ), (1)
where φ is the total polymer segment concentration and
η is the concentration of chemically modified segments,
given respectively by φ = N
i ρi and η = N
i ρiαi.
The first term in Eq. (1) is the ideal free energy of mix-
ing. The second term is the usual Flory-Huggins configu-
rational chain entropy. The third term is the free energy
cost of the unmodified polymer segments at a concentra-
tion φ− η coming into contact with solvent (water) at a
concentration 1 − φ. Typically one expects χ > 1/2 for
this interaction, to represent the repulsion between un-
modified segments and water which leads to phase sepa-
ration of unmodified polymers. To keep the model sim-
ple, this is the only χ-parameter that is retained in the
problem.
Eq. (1) has the structure of a moment free energy,
since the excess free energy, comprising the second and
third terms, only depends on φ and η which are mo-
ment densities. Such a system can be analysed using the
methods developed by Sollich and coworkers [7, 8, 9, 10].
In particular, Ref. [10] describes how the spinodal sta-
bility conditions for systems with an excess free energy
can be expressed in terms of moment densities, gen-
eralising various truncation theorems obtained by ear-
lier workers [11, 12]. I now summarise the relevant re-
sults, translated into terms suitable for the present prob-
lem. Let us consider such a system with a free energy
i ρi log ρi + f
(ex)(φ(1) . . . φ(n)), where the excess
free energy depends on moment densities of the form
φ(r) =
i ρiw
i (r = 1 . . . n), with the w
i being
species-dependent weights. The fundamental idea is that
the moment densities can be treated as effective species
concentrations. In particular, it can be proved that spin-
odal stability corresponds to the positive-definiteness of
the matrix M of second partial derivatives of the free en-
ergy with respect to the moment densities. In Ref. [10]
it is shown that M = Mid + Mex where (M
id )rs =
i ρiw
i and (Mex)rs = ∂
2f (ex)/∂φ(r)∂φ(s). The
limit of spinodal stability is given by detM = 0. This
condition usually corresponds to the vanishing of a single
eigenvalue of M, with an eigenvector ∆φ(s) that satisfies
s(M)rs∆φ
(s) = 0. It is shown in Ref. [10] that the
spinodal instability direction in the space of species con-
centrations is given by ∆ρi =
rs ρiw
i (Mid)rs∆φ
For the present problem, there are two moment densi-
ties φ and η, defined respectively with w
i = N (a con-
http://arxiv.org/abs/0704.0707v1
stant) and w
i = Nαi (the number of modified groups
on the ith species). Application of the above theory to
Eq. (1) leads to
id = N
i ρiαi
i ρiαi
i ρiα
Mex =
(1− φ)−1 − 2χ χ
. (3)
After some algebra the condition detM = 0 reduces to
−2χ(1−〈α〉)−χ2Nφ(〈α2〉−〈α〉) = 0, (4)
where
〈α〉 =
i ρiαi /
i ρi, 〈α
i ρiα
i ρi. (5)
I emphasise that, despite being remarkably simple,
Eq. (4) is exact.
One already reaches a significant conclusion from this.
The first three terms in Eq. (4) are what one would ex-
pect from standard Flory-Huggins theory [3], with an ef-
fective χ-parameter given by the product of the original
χ-parameter and the fraction 1− 〈α〉 of unmodified seg-
ments. These terms therefore take account of the mean
degree of modification. The final term in Eq. (4) is a
correction due to the heterogeneity. Since the variance
〈α2〉− 〈α〉2 is positive, this term is always negative. The
effect is that heterogeneity in modification reduces the
solubility, over and above what would be expected from
the mean degree of modification.
To make further progress, it is convenient to specify a
model for the distribution of the αi. In particular, such a
model can be used to examine the effect of blockiness in
modification which is expected to play an important role.
In previous work on random block copolymers [4, 5], a
Markov model was used to characterise the correlations
between different kinds of segments. Whilst such a model
may be appropriate for the stochastic nature of the syn-
thetic route for such random block copolymers, as dis-
cussed below it is probably not appropriate in the present
case. I therefore consider instead a very simple model for
the heterogeneity in which the modified segments occur
in blocks of size M , where 1 < M < N . In this model,
it is supposed that each block has an equal probability p
of being modified, and there are no further correlations.
Then, for any particular species, αi = (1/N)
j=1 Mǫij
where j labels the blocks, and ǫij is zero or one with prob-
ability 1 − p and p respectively. Thus the αi are drawn
from scaled binomial distribution, with
〈α〉 = p, 〈α2〉 − 〈α〉 = (M/N) p(1− p). (6)
Eq. (4) becomes
− 2χ(1− p)− χ2Mφp(1− p) = 0. (7)
This is a quadratic equation for χ and the appropriate
root is
)}1/2
. (8)
I now examine the consequences of this result.
The formal limit M → 0 corresponds to a vanishing
variance and a completely uniform distribution of mod-
ified segments, as though each monomer has undergone
an identical fractional modification by a fraction p, rather
than being modified or not with probability p and 1− p.
As noted already above, this limit corresponds to sim-
ple Flory-Huggins theory with an effective χ-parameter
equal to χ(1−p). For large N , this indicates the absence
of phase separation for χ(1− p) < 1/2 or p > 1− 1/(2χ).
Now let us consider Eq. (8) for block size M = 1.
In this case, individual segments are modified randomly
with no correlations. For M = 1 and large N in Eq. (8),
there are two behaviours depending on the value of p.
For p < 4/5, there is an absence of phase separation
for χ(1 − p) < 1/2, just as for the M → 0 limit. For
4/5 < p < 1, the behaviour is more complicated. To be
precise, the location of the minimum value of the χ(φ)
spinodal shifts from φmin ∼ N
−1/2 for p < 4/5 to a non-
vanishing 0 < φmin < 1 for p > 4/5 (it is the examination
of Eq. (8) in the limit φ ∼ N−1/2 that gives the cross over
point p = 4/5). The change in behaviour can be seen for
the M = 1 curves (dashed lines) in Fig. 1 and is shown
explicitly in the upper plot of Fig. 2.
Let us next consider the limit of extreme blockiness
M = N . This limit is strikingly different from the M = 1
case. For largeN and p > 0, one can show that there is an
absence of phase separation only for χ
Np(1− p) < 2.
In the large N limit, this inequality is always violated,
indicating that the system always has a tendency to un-
dergo phase separation in the limit of extreme blockiness.
Since the unmodified polymer system itself only phase
separates for χ > 1/2, this suggests that the phase sep-
aration has the nature of a polymer-polymer demixing
transition rather than a solvent-driven phase separation.
This insight is confirmed by analysis of the spinodal in-
stability direction below.
For large N and general M in Eq. (8), one would ex-
pect that the above two cases represent the two classes of
behaviour. In the first case M ≪ N and the behaviour
is similar to the M = 1 limit where individual segments
are randomly modified. In the second case, M ∝ N
and the behaviour is similar to the M = N limit of ex-
treme blockiness. Fig. 1 shows typical spinodal curves
calculated from Eq. (8) for various values of p and M .
The location of the minimum (φmin, χmin) of the spin-
odal curves can be numerically determined, and Fig. 2
shows how this depends on p.
The results show firstly that for M ≪ N , increasing p
leads to increasing solubility as the value of χ required to
reach the spinodal instability is increased. Moreover, a
decrease in solubility between a uniform model (M → 0)
with no heterogeneity, and a model with fine-grained
0.0 0.2 0.4 0.6 0.8 1.0
PSfrag replacements
p = 0.1
0.0 0.2 0.4 0.6 0.8 1.0
PSfrag replacements
p = 0.1
p = 0.5
0.0 0.2 0.4 0.6 0.8 1.0
PSfrag replacements
p = 0.1
p = 0.5
p = 0.9
FIG. 1: Spinodal curves calculated from Eq. (8) for polymers
of length N = 103, for three values of the mean degree of
modification p, and for block sizes M → 0 (uniform limit,
solid line), M = 1 (dashed line), M = 10 (dash-dot line),
M = 100 (dash-dot-dot line) and M = 103 (dash-dash-dot
line). The system is spinodally unstable above the indicated
curves. Note the change in shape of the M = 1 curves: for
p = 0.1 and 0.5 the minimum is at φ → 0, whereas for p = 0.9
the minimum is at φ ≈ 0.25.
blockiness (M = 1), is apparent. The major effect arises
as M → N though, where the tendency for phase sepa-
ration is greatly enhanced.
The above analysis is augmented considering the spin-
odal instability direction associated with the spinodal
stability limit which can provide a useful mechanistic in-
sight. As explained above, the spinodal instability direc-
tion is characterised by the eigenvector that corresponds
to the vanishing eigenvalue responsible for the vanish-
0.0 0.2 0.4 0.6 0.8 1.0
PSfrag replacements
0.0 0.2 0.4 0.6 0.8 1.0
PSfrag replacements
FIG. 2: The location of the numerically determined minimum
of the spinodal curves from Eq. (8) is plotted as a function
of p, for polymers of length N = 103 and block sizes M → 0
(uniform limit, solid line), M = 1 (dashed line), M = 10
(dash-dot line), M = 100 (dash-dot-dot line) and M = 103
(dash-dash-dot line). For M = 1 (dashed line) the upper
plot shows clearly that φmin ≈ N
≈ 0.03 only holds for
4/5 = 0.8.
ing spinodal determinant. For the present problem, from
Eqs. (2)–(3), one finds the instability direction is charac-
terised by
∆η /∆φ = 〈α〉 − χNφ(〈α2〉 − 〈α〉2) (9)
The corresponding spinodal instability direction in the
space of species concentrations is
〈α2〉∆φ − 〈α〉∆η + αi(∆η − 〈α〉∆φ)
φ(〈α2〉 − 〈α〉2)
1 + χNφ(〈α〉 − αi)
where the second line follows by inserting the result for
the ratio ∆η/∆φ. These results should be evaluated on
the spinodal. They are all exact, for an arbitrary distri-
bution of αi.
For the instability direction to lie along a pure dilution
line, one should have ∆ρi/ρi independent of species i.
One can conclude that this only happens if ∆η/∆φ =
〈α〉, in other words if the variance 〈α2〉 − 〈α〉2 vanishes.
In such a case, the phase transition is purely associative,
or solvent-driven, meaning that the compositions of the
coexisting phases remain the same (∆ρi/∆φ = ρi/φ).
If one specialises to the model of blockiness described
above by inserting the value of χ corresponding to the
spinodal stability limit, the instability direction becomes
)}1/2
This confirms that the spinodal instability lies along a
dilution line (∆η/∆φ = 〈α〉 = p) only in the limit
M → 0 which formally corresponds to a vanishing vari-
ance. For M = 1 (and M ≪ N in general) the phase
transition has a mixed character. The interesting case
occurs when M = N (or M ∝ N in general) for which
∆η/∆φ ∼ (−)N1/2 in the limit of largeN . One can write
this as ∆φ/∆η → 0 as N → ∞. This shows that the
phase transition tends towards being purely segregative,
meaning that the overall polymer concentration in coex-
isting phases remains the same (∆φ = 0). This confirms
the suggestion above, that in the limit of extreme block-
iness, the system tends towards a segregative polymer-
polymer demixing transition.
Let us now try to draw some conclusions. The main ef-
fect of randomness is to reduce the solubility of partially-
modified polymers beyond what would be expected from
the mean degree of modification. The extent to which
this occurs depends on the blockiness in substitution.
For fine-grained blockiness, the phase behaviour is ex-
pected to be similar to a system for which there is no ran-
domness, albeit with a somewhat reduced solubility. For
coarse-grained blockiness, where the block size is com-
parable to the polymer length, the nature of the phase
transition changes to a polymer-polymer demixing tran-
sition. In this situation, one expects that the modified
polymers (being almost fully modified) will partition into
the aqueous phase, leaving the unmodified polymers be-
hind.
The reason for considering the two extreme kinds of
blockiness is now clearer: namely one can envisage two
different mechanisms of chemical modification (this is the
reason why a Markov model for the distribution of mod-
ified segments has not been used). Fine-grained block-
iness would arise if monomers are equally accessible to
the modifying agent, irrespective of their surroudings. If
this cannot be achieved in a one-step process (for the
reason described below) it could perhaps be achieved in
a two-step process, by fully modifying the polymers then
removing a random fraction of the derivative groups. Ex-
treme blockiness on the scale of the polymer chain itself
would arise if the modifying agent was present only in the
aqueous phase, and as such only able to access polymer
which had already been solubilised. This would lead to a
mixture of polymers which were either fully modified, or
remained unmodified and insoluble. The process of mod-
ification of insoluble polymers could still be initiated be-
cause the modifying agent is able to access the tiny pro-
portion of the insoluble polymer segments which lie at
the interface between the insoluble and aqueous phases.
Experimentally, confirmation of the scenario of extreme
blockiness would be given by measuring the mean degree
of modification for the dissolved polymers. One should
find that this is much in excess of the apparent mean
degree of modification.
In the calculation, the major effect arises from inter-
chain rather than intra-chain heterogeneities. The model
is not sophisticated enough to take account of the solu-
tion structures such as micelles or mesophases that could
form for blocky polymers with block sizes M ≫ 1 but
still M < N (for example, diblock copolymers). Such
polymers would be expected to have greater solubilities
than would be predicted from the Flory-Huggins theory
since the hydrophobic groups can be buried in micelles
or other solution structures. The present theory could be
extended to discuss these inhomogeneous situations using
a Landau approach developed for random block copoly-
mers [4, 5, 6]. For the mechanistic routes discussed above
though, it is difficult to envisage that polymers with in-
termediate block sizes could arise very easily. I therefore
expect that the general conclusions will remain.
Finally I note that in principle the above model for
the phase behaviour could be combined with a model for
the chemical modification reaction, to obtain a theory for
reaction-induced solubility. However, one needs to take
great care to capture the kinetics correctly [13].
I thank Nigel Clarke for a critical reading of the
manuscript.
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