Effects of strongly selective additives on volume phase transition in gels
Yuki Uematsu and Takeaki Araki
Citation: J. Chem. Phys. 137, 024902 (2012); doi: 10.1063/1.4732857
View online: http://dx.doi.org/10.1063/1.4732857
View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v137/i2
Published by the American Institute of Physics.

Additional information on J. Chem. Phys.
Journal Homepage: http://jcp.aip.org/
Journal Information: http://jcp.aip.org/about/about_the_journal
Top downloads: http://jcp.aip.org/features/most_downloaded
Information for Authors: http://jcp.aip.org/authors

THE JOURNAL OF CHEMICAL PHYSICS 137, 024902 (2012)

Effects of strongly selective additives on volume phase transition in gels
Yuki Uematsu and Takeaki Araki
Department of Physics, Kyoto University, Kyoto 606-8502, Japan(Received 26 April 2012; accepted 19 June 2012; published online 10 July 2012)
We investigate volume phase transition in gels immersed in mixture solvents, on the basis of a threecomponent Flory-Rehner theory. When the selectivity of the minority solvent component to the polymer network is strong, the gel tends to shrink with an increasing concentration of the additive, regardless of whether it is good or poor. This behavior originates from the difference of the additive
concentration between inside and outside the gel. We also found the gap of the gel volume at the transition point can be controlled by adding the strongly selective solutes. By dissolving a strongly poor
additive, for instance, the discontinuous volume phase transition can be extinguished. Furthermore,
we observed that another volume phase transition occurs far from the original transition point. Thesebehaviors can be well explained by a simplified theory neglecting the nonlinearity of the additive
concentration. © 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4732857]
I. INTRODUCTION

Swelling behavior of a polymer network was first investigated by Flory and Rehner.1 On the basis of this work, discontinuous volume phase transition (VPT) of gels was predicted
theoretically.2 Hence, the VPT has received much attention
from both scientific and industrial viewpoints.3–5 In various
types of gels, addition of solutes into the solvent often plays
very important roles in their volume change.3–6 For example,
the VPT was first realized in a polyacrylamide (PAA) gel immersed in a mixture of water and acetone.3 In this experiment,
the solvent quality, or χ parameter, is controlled by changing the volume fraction of the acetone. It is also well known
that addition of salts dramatically affects the behaviors ofionic gels.7–9 Furthermore, a nonioninc hydrated gel such aspoly(N-isopropylacrylamide) gel (NIPA) changes its volume
discontinuously. In the hydrated gels, the VPT is attributed totemperature-dependence of hydrogen bondings of the polymer network.6, 10 It has been reported that its VPT is affectedby adding salts11–14 and other additives.15–21 Although somemicroscopic theories are devoted to explain the VPT,21–24
there remain unresolved problems. For instance, the transition
point is lowered by dissolving salts,12–14 whereas it is raisedwith surfactants.16 The difference between these additive effects remains unclear. Interestingly, is was reported that thediscontinuous transition of the NIPA gel disappears when NaI
is dissolved.14 We consider that a macroscopic description
will also help us to understand these unresolved phenomena.
It is often assumed that the composition of a mixture solvent is same in and out of a gel. This assumption is referred toas single liquid approximation (SLA). The SLA is experimentally confirmed in a NIPA gel in a dimethyl sulfoxide-water
mixture, for instance.25 However, the compositions of mixture solvents are not necessarily homogeneous.25–27 There, thedifference of the composition between the interior and exterior of the gel would lead to a dramatic effect on its volume
change. Some extended Flory-Huggins models dealing with apolymer network and two species of solvent molecules weredeveloped to investigate the volume change of a polymer net0021-9606/2012/137(2)/024902/8/$30.00

work in a mixture solvent.26–30 In these theories, the volume
change is characterized by the compositions of the three components and three χ parameters. They were studied systematically for fixed sets of the χ parameters. Usually, the χ parameters in polymer systems depend on environment parameterssuch as temperature. Since it is quite complicated to incorporate the temperature-dependences of the χ parameters, theswelling behavior in mixture solvents has not been fully understood.
Recently, the effects of solutes with strong selective
solvation on phase behaviors of a water-oil mixture wasinvestigated.31–33 By dissolving a very small amount ofstrongly hydrophobic solute into the water-oil mixture, theoil-rich phase containing the solute is precipitated even in
the one-phase region. This finding implies that such solutes
would affect the VPT in gels.
The aim of this study is to clarify the effect of the additives on the VPT in gels, based on a simple theoretical argument. In particular, we focus on the cases, in which oneof the solvent components has strong selectivity to the polymer network. So far, the VPT is realized in hydrophilic gels(polyelectrolyte and hydrated ones). Our model is based ona simple Flory-Huggins theory, so that it is also applicableto hydrophobic polymer networks. Then, we hope our model
will help one to find VPT in other new types of gels.

II. MODEL FREE ENERGY

Swelling behaviors of gels are well described by the
Flory-Rehner theory.1 The free energy F consists of a mixing part Fmix and an elastic part Fel as
F = Fmix + Fel .

(1)

For a gel immersed in a mixture solvent, the mixing free energy Fmix is given by26–28, 30, 32
Fmix = kB T v0−1 [Vg f (φ1g , φ2g , φ3 ) + Vs f (φ1s , φ2s , 0)],137, 024902-1(2)© 2012 American Institute of Physics

024902-2

Y. Uematsu and T. Araki

J. Chem. Phys. 137, 024902 (2012)

where kB is the Boltzmann constant and T is the temperature.
Vg and Vs are the volumes of the interior and exterior of thegel, respectively. In this model, we assume that the volumes ofeach monomeric unit of the polymer network and the solvent
molecules are equal to a characteristic volume v0 for simplicity. f (φ 1 , φ 2 , φ 3 ) is the Flory-Huggins type mixing free energyof the ternary system as34
f (φ1 , φ2 , φ3 ) = φ1 ln φ1 + φ2 ln φ2 +
χij φi φj . (3)Here, φ 1 and φ 2 are the volume fractions of the first andsecond components of the mixture solvent, respectively. In
Eq. (2), φ ig and φ is stand for the volume fraction of the ith
component (i = 1, 2) in and out of the gel. φ 3 is the volume
fraction of the polymer network. Since all the polymer chainsare chemically connected forming a single network, the polymers are not dissolved outside the gel, that is, φ 3s = 0. In
Eq. (3), the first two terms stem from the translational entropyof the solvent molecules. Here, the translational entropy of the
network is neglected. The last term is the interaction energy
among the solvent molecules and the monomeric units of the
network. χ ij is the interaction parameter between the ith andjth components.
We employ Flory’s rubber elasticity as35
⎫
⎧
⎬
⎨ 
1
Fel = kB T νVg0
γk2 − 3 − 2B ln(γx γy γz ) , (4)⎭⎩2k=x,y,zwhere Vg0 is the volume of the gel in a reference state, ν is thedensity of cross linking points in the reference state, and γ k
is the elongation ratio in the k-axis (k = x, y, z). B is a nonlinear elastic coefficient. In polyelectrolyte gels, the translational
entropy of the counter-ions is renormalized into B as B + b,
where b is the number of the dissociable monomeric unit pereffective chain. In this study, however, we do not consider theelectric charges explicitly. In the case of isotropic swelling,
the elongation ratio is coupled with the volume fraction of thepolymer network as
γk =

Vg
Vg0

φ30
φ3

=

1/3

(5)

,

where φ 30 is the volume fraction of the network in the reference state. Therefore, the elastic energy Eq. (4) is rewritten by
a function of φ 3 as,
Fel =

1
φ30
kB T νVg0 3
2
φ3

2/3

− 2B ln

∂
∂
=
= 0.
∂φ2g
∂φ2s

(10)

∂
∂
=
= 0,
∂Vg
∂Vs

(11)

Then, we obtained the equilibrium conditions

i<j

1/3

equilibrium state is characterized by minimizing ,

φ30
φ3

.

(6)

We impose the incompressible conditions for both in and outof the gel,
φ1g + φ2g + φ3 = 1,(7)

φ1s + φ2s = 1.

(8)Here, we define a grand potential as = Fmix + Fel − μ2 (Vg φ2g + Vs φ2s ) + κ(Vg + Vs ), (9)where μ2 and κ are Lagrange multipliers to conserve theamount of the second component and the total volume. Theμ̃(φ2g , φ3 ) = μ̃(φ2s , 0) =

v0 μ2
kB T

,

kB T ˜( 0 + ˜ ad + ˜ el ) = 0,
v0

(12)

(13)

where μ̃ is the reduced chemical potential given byμ̃(φ2 , φ3 ) = ln

φ2
1 − φ2 − φ3

+ χ12 (1 − 2φ2 − φ3 ) + (χ23 − χ31 )φ3 .

(14)

Equations (12) and (13) represent the balances of the chemical
potential for the second component and the osmotic pressure,respectively.
˜ 0 is a part of the osmotic pressure for the gel without
the second solvent component. It stems from the mixing freeenergy and given by˜ 0 = −φ3 − χ31 φ32 − ln(1 − φ3 ).

(15)

˜ el is the contribution of the elasticity as
˜ el = −νv0

φ3
φ30

1/3

−B

φ3
φ30

.

(16)

If the solvent does not contain the additive φ 2 = 0, the balanceof the osmotic pressure is expressed by ˜ 0 + ˜ el = 0.
˜ ad is the contribution of the second component and isgiven by
φ2g
+ ln(1 − φ2s )1 − φ3
2
2+ Gφ2g φ3 ,− χ12 φ2g
− φ2s

˜ ad = − ln 1 −(17)

where G = χ 23 − χ 31 − χ 12 is a parameter describing theaffinity of the second component to the polymer network. ˜ 0
and ˜ ad come from the mixing free energy Fmix . If G < 0 and|G|  1, the additive tends to be adsorbed selectively to thepolymer network. If G  1, on the other hand, the additive
would be expelled from the gel.
In this study, we assume that the binary solvent is completely mixed outside the gel. We note that the volume of theequilibrated gel does not change if we add more mixture solvent whose composition is equal to the equilibrated outer solvent. Hence, we take a limit of Vs /Vg → ∞ with fixing φ 2s ,so that φ 2s is uniquely determined for a fixed μ2 . Hereafter,
we use φ 2s as a control parameter.

024902-3

Y. Uematsu and T. Araki

J. Chem. Phys. 137, 024902 (2012)

0.06

0.06

(a)

(b)
0

0.05

5 ×10 −3 1×10 −2 1.5 ×10 −2

0.04

0.04

0.03

0.03

0.02

0.02

0.01
0.528

0.532

0.536

0.54

1.5 ×10 −2 1×10 −2 5 ×10 −3

0.05

0.01
0.518

0.522

0.526

0.53

0

0.534

FIG. 1. The swelling curves of a gel immersed in a mixture solvent. We set φ 30 = 1.0 × 10−2 , νv0 = 1.0 × 10−5 , B = 1.15, and χ 12 = 0.0. In the absence ofadditives, the gel undergoes volume phase transition with increasing χ 31 . We dissolve solutes of χ 23 = −1.0 (pro-gel) in (a) and χ 23 = 1.0 (anti-gel) in (b).
The black solid lines are calculated from Eq. (21).

III. RESULTS AND DISCUSSIONS
A. The first volume phase transition

We numerically study the effects of additives on volume
changes of gels. First, we focus on the cases, in which thegel can undergo VPT without additives. We set φ 30 = 1.0
× 10−2 , νv0 = 1.0 × 10−5 and B = 1.15. The effective chain
polymerization is N = φ30 /νv0  1 and φ 30 ∼ N−1/2 .26
We vary χ 31 continuously to induce the volume changewith fixing the other parameters. We obtain the swelling
curves of the gel by solving Eqs. (12) and (13) numerically.
Figure 1 shows the swelling curves for several values of φ 2s .
(0)
In the absence of additives, the VPT occurs at χ31 = χ31t(≈ 0.5327). In Fig. 1(a), we dissolve the additive ofχ 12 = 0.0 and χ 23 = −1.0 , which has affinity to the polymer
network (pro-gel). As its concentration φ 2s is increased, thetransition point shifts to higher χ 31 . In Fig. 1(b), the additiveof χ 12 = 0.0 and χ 23 = 1.0 is dissolved into the solvent. Thisadditive dislikes the polymer network (anti-gel). It is shownthat the transition point shifts to lower χ 31 with an increasing
φ 2s . For the cases as in Fig. 1, the SLA can explain the effectsof the solute on the volume changes qualitatively. In the SLA,
the volume fractions of the solvent components inside the gel
are assumed to be φ 1g = (1 − φ 3 )(1 − φ 2s ) and φ 2g = (1− φ 3 )φ 2s . Then, the interaction parameter between the polymer network and the mixture solvent is approximated asχ̃31 = χ31 (1 − φ2s ) + χ23 φ2s − χ12 φ2s (1 − φ2s )2= χ31 + Gφ2s + O φ2s.

(18)This means that the interaction parameter changes effectively
with φ 2s depending on G. For the plotted range in Fig. 1(a)(χ 31 ∈ [0.528, 0.542]), G remains negative, so that the solvent
becomes more good, swelling the gel with an increasing φ 2s .
As shown in Fig. 1(b), on the other hand, the solvent changes
to more poor and the gel shrinks with φ 2s for the solute of
positive G.
In Fig. 1, the absolute value of the resulting G israther small (|G|  2). Next, we study the effects of solutes of strong selectivity |G|  1. Figures 2(a) and 2(b)show the swelling curves in the mixture solvents of χ 23
= −10.0 and χ 23 = 10.0, respectively. The other parameters
0.06

0.06

(a)

(b)

0.05

0.05

0.04

0.04

0.03

0.03

0.02

0.02

0.01
0.524

0.528

0.532

0.536

0.01
0.515

0.525

0.535

FIG. 2. The swelling curves of a gel immersed in a mixture solvent. We set φ 30 = 1.0 × 10−2 , νv0 = 1.0 × 10−5 , B = 1.15, and χ 12 = 0.0. We dissolve solutesof χ 23 = −10.0 (pro-gel) in (a) and χ 23 = 10.0 (anti-gel) in (b). The black solid lines are calculated from Eq. (21). The additives have strong selectivities tothe polymer network.

024902-4

Y. Uematsu and T. Araki

0.004

(a)

J. Chem. Phys. 137, 024902 (2012)

10.0

7.5

5.0

1.0

-10.0

-7.5

-5.0

-1.0

0.001

(b)

0.000

0.000

-0.004-0.001-0.008-0.002-0.012-0.00300.00040.0008-0.004

-2

-1

0

1

2

(0)

FIG. 3. (a) The shifts of the transition point from χ31t are shown for several values of χ 23 . The shift is numerically obtained with the same parameters as inFigs. 1 and 2. For χ 23 = 7.5 and 10.0, the transition points are terminated since the discontinuous volume changes disappear. (b) The dependence of χ 31t on(0)χ 23 is plotted for φ 2s = 0.001. The broken curve represents g(φ3c , Ḡ) [Eq. (26)] with φ 3c = 0.0267 and χ31t = 0.5327.

are the same as those in Fig. 1. In the both cases, the transition
points are shifted to lower χ 31 , regardless of whether the solute is good or poor. This behavior is in contrast to the volume
changes in Fig. 1 and indicates the SLA does not work wellwhen the additives have the strong selectivity to the polymer
network.
Figure 3(a) shows the dependences of the transition
shift χ 31t on the additive concentration φ 2s , where χ31t
(0)(0)
= χ31t − χ31t
and χ31t
is the transition point without additives.
An increase of χ 31t is observed for χ 23 ≈ −1.0, whereas
χ 31t is lowered with an increasing φ 2s for the other additives. In the plotted range of φ 2s , χ 31t has linear dependenceon φ 2s . The transition shifts for φ 2s = 0.001 are plotted withrespect to χ 23 in Fig. 3(b). It indicates a non-monotonic behavior of the transition point.
The difference of the additive effects between χ 23
= −1.0 and χ 23 = −10.0 is notable, since the both additives have tendencies to be adsorbed onto the polymer network. When |χ 23 | is small, the difference between φ 2g andφ 2s is rather small. As explained by Eq. (18), the addition of
the solute with negative χ 23 makes the mixture solvent moregood to the polymer. Thus, the gel is simply swollen with anincreasing φ 2s . If χ 23 (or G) is negatively large enough, on theother hand, the concentration of the solute inside the gel becomes much larger than that outside gel, i.e., φ 2g  φ 2s [seeEq. (19)]. By shrinking its volume, the polymer network tends
to increase the contact points to the additive molecules. Thisnonlinear effect would give rise to the difference between χ 23
= −1.0 and χ 23 = −10.0.
Figure 2 also shows that the gap of the volume transition depends on the additive concentration when the additive
has strong selectivity. As φ 2s is increased, the volume gap
is increased for χ 23 = −10.0 [see Fig. 2(a)], while it is decreased for χ 23 = 10.0 [Fig. 2(b)]. For χ 23 = 10.0, in particular, the gap disappears eventually at a certain concentration
φ 2s = φ 2st , above which the gel does not undergo the VPT[see below]. Figure 4 plots the gap of the volume fraction
φ 3t (= φ 3t+ − φ 3t− ) at the transition point, where φ 3t+ andφ 3t− are the volume fractions just below and above the tran-

sition point. For a large positive value of χ 23 , φ 3t decreases
toward zero with an increasing φ 2s .
It is known that the elasticity parameter B influences thenature of the VPT.3 Figure 5 shows the swelling curves of agel of B = 1.10 with varying χ 31 . In the absence of additive,the volume of this gel changes continuously. By dissolving the
pro-gel additive of χ 23 = −10.0, the swelling curve becomes
non-monotonic with respect to χ 31 , so that the gel undergoes
the VPT. Figures 2(b) and 5 indicate that we can arbitrarily
induce or erase the VPT by adding solutes with the strong
selectivities.
We assume that only a subtle amount of the additive isdissolved in the outer solvent, i.e., φ 2s  1. From Eq. (12),the volume fraction of the additive inside the gel is expressedby
2.
φ2g = (1 − φ3 )e−Gφ3 φ2s + O φ2s

(19)

0.03

0.02

0.01

0.00
0

10.0

7.5

5.0

1.0

-10.0

-7.5

-5.0

-1.0

0.00040.0008

FIG. 4. The gap of the volume fraction φ 3 at the transition is plotted with
φ 2s . We set φ 30 = 1.0 × 10−2 , νv0 = 1.0 × 10−5 , B = 1.15, and χ 12 = 0.0
as in Figs. 1 and 2.

024902-5

Y. Uematsu and T. Araki

J. Chem. Phys. 137, 024902 (2012)

Regarding β as a variable, a tri-critical point of the VPT
is given by∂ 2 χ31 ∂χ31 
= 0,
= 0.
(22)
∂φ3 φ3c ,βc∂φ32 φ3c ,βc

0.06
0.05
0.04

Here, α is treated to be fixed. By solving Eq. (22), we obtain
φ 3c and β c , which give the swelling curve passing throughthe tri-critical point. In the vicinity of the tri-critical point, theswelling curve is approximated by

0.03
0.02

χ31 ≈ χ31t −

0.01
0.522

0.526

0.53

0.534

FIG. 5. The swelling curves of a gel immersed in a mixture solvent. We set
φ 30 = 1.0 × 10−2 , νv0 = 1.0 × 10−5 , B = 1.10, and χ 12 = 0.0. We dissolve
the pro-gel solute of χ 23 = −10.0. The black solid lines are calculated from
Eq. (21). When φ 2s = 0, the gel changes its volume continuously without
VPT.

2+ Gφ3 (1 − φ3 )e−Gφ3 φ2s + O φ2g2= [e−Gφ3 {1 + Gφ3 (1 − φ3 )} − 1]φ2s + O φ2g. (20)(23)

where u is a positive constant. If β < β c , the curve of χ 31changes monotonically with φ 3 , such that the gel volume
changes continuously with changing χ 31 . On the other hand,inflection points appears when β > β c . If so, the gel becomesmechanically unstable, so that√ it exhibits the discontinuousvolume changes as φ3t = 2 (β − βc )/u/φ3c at χ 31 = χ 31t .3
The part of the osmotic pressure ˜ ad shifts the tri-criticalpoint in Eq. (23) as(0)
χ31t = χ31t
+ g(φ3c , Ḡ)φ2s ,(24)2βc = βc(0) + h(φ3c , Ḡ)φ3c
φ2s .

(25)

Substituting Eq. (19) into Eq. (17), we obtain
˜ ad = ln(1 − φ2s ) − ln(1 − φ2s e−Gφ3 )β − βcu(φ3 − φ3c ) + (φ3 − φ3c )3 ,26φ3cHere, βc(0) gives the tri-critical point in the case of φ 2s = 0.
From Eq. (21), the prefactors g and h are given byThen, the swelling curve is approximately given from
Eqs. (15), (16), and (20) byg(φ3 , G) = φ3−2 [e−Gφ3 {1 + Gφ3 (1 − φ3 )} − 1],

−5/3
χ31 ∼
+ (β − 1)φ3−1= −φ3−2 ln(1 − φ3 ) − αφ3h(φ3 , G) = φ3−3 2 − e−Gφ3 G2 φ32 (1 − φ3 ) + 2Gφ3 + 2 .

+ [e−Gφ3 {1 + Gφ3 (1 − φ3 )} − 1]φ3−2 φ2s ,

(21)

1/3
where α = νv0 /φ30 and β = νv0 B/φ30 . Since the right handside of Eq. (21) includes χ 31 via G, we cannot obtain ananalytical solution of χ 31 . Then, we replace G to Ḡ = χ23(0)− χ31t− χ21 and regard Ḡ as a fixed parameter. The approximated curves of Eq. (21) are drawn in Figs. 1, 2, and 5. They
are well in agreement with the numerical solutions.
1(27)
g and h change their signs depending on φ 3 and G. Their dependences are shown in Fig. 6. If g(φ 3 , G) is positive, the transition point shifts to higher χ 31 with an increasing φ 2s and viceversa. Eq. (26) for φ 2s = 0.001 is drawn in the broken curvein Fig. 3(b). Here, we set φ 3c = 0.0267 and χ 31t = 0.5327.
This approximated curve is in agreement with the numerical
1

(a)

(b)

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

(26)

-4

-2

0

2

4

0

-4

-2

0

2

4

FIG. 6. The contours of the functions g [Eq. (26)] and h [Eq. (27)] are drawn in G-φ 3 planes (a) and (b), respectively. The transition point is increased linearlywith φ 2s in the region of for positive g. In the region of negative h, the volume gap of the transition point is increased.

024902-6

Y. Uematsu and T. Araki

J. Chem. Phys. 137, 024902 (2012)0.8

3
Eq.(21)

0.7
0.6

2nd VPT

2

0.5
0.4
0.3

1

0.2
0.1
0.0
0.5

0

1st VPT

0.54

0.58

0.62

0.66

0.7

FIG. 7. The swelling curves of a gel immersed in a mixture solvent in a
wider range of χ 31 . We set φ 30 = 1.0 × 10−2 , νv0 = 1.0 × 10−5 , B = 1.15,
and χ 12 = 0.0. We dissolve the strongly good solute of χ 23 = −10.0 and φ 2s
= 1.0 × 10−4 . The black solid lines are calculated by Eq. (21) . The second
volume phase transition is induced.

solution. When φ 3c  1, g ≈ −G(G + 2)/2 and it changes its
sign at G = 0 and G = −2. A large negative h enhances thediscontinuity of the volume change as observed in Fig. 2(b). If
φ 3c  1, h is expressed as h ≈ G2 (G + 3)/3 and it changes its
sign at G = −3. Thus, we can possibly induce the VPT evenin a gel, which originally shows a continuous volume change,
by adding soluteof −G  3. The gap of the VPT disappears√−2. Thisas φ3t ∝ φ2st − φ2s , where φ2st = {β − βc(0) }h−1 φ3cdisappearing behavior is observed in Fig. 4.
In this paper, only the numerical solutions for χ 12
= 0.0 are presented. However, we confirmed that the essentially same features are observed for any set of χ 12 and χ 31 if
the resultant G is the same.
B. The second volume phase transition

In Fig. 7, we plot the swelling curves in a wider range ofχ 31 . We set φ 30 = 1.0 × 10−2 , νv0 = 1.0 × 10−5 , B = 1.15,
χ 12 = 0.0, and χ 23 = −10.0. It is indicated that another discontinuous volume phase transition can occur at χ 31 larger
than χ 31t corresponding to the first transition. We confirmedthat this second phase transition is observed even if ν = 0 (or,α = β = 0), while the first one disappears. This fact indicatesthat this second instability is independent of the network elasticity and has a physical mechanism different from those forwell-studied volume phase transitions.3
Scott reported that various types of phase diagrams are
realized for ternary mixtures (polymer solutions in binary
mixtures).34 They can have several critical points, below
which three phases coexist. Analogous to the ternary mixtures, we consider that the second volume transition observedin Fig. 7 is attributed to the bulk instability of the mixing freeenergy. Since there are some differences between ternary mixtures and gels in mixture solvents, the Scott’s argument cannot
be simply applied to the gel systems. The most important differences are that a gel has the elasticity and never exhibits a
one-phase homogeneous state.

0

0.2

0.4

0.6

0.8

(0)

FIG. 8. The theoretical curves of ∂χ31 /∂φ3 and −h(φ 3 , G)φ 2s are shown.
(0)
When ∂χ31 /∂φ3 < −h(φ3 , G)φ2s , the gel becomes mechanically unstable,
showing the second volume phase transition.

We have not obtained a simple explanation on the second transition, since nonlinearities, which are hard to treat
analytically, would play important roles in the bulk instability. In Fig. 7, we also draw the swelling curve obtained by
Eq. (21), which neglects the nonlinearity of φ 2s . The approximated curve also exhibits the second transition, although it
does not coincide quantitatively with the numerical solutions
for large φ 3 . However, this fact implies that the nonlinearityof φ 2 has a minor contribution to the essential mechanism of
the second transition.
As discussed above, the gel becomes mechanically un(0)/∂φ3 andstable when ∂χ 31 /∂φ 3 < 0. In Fig. 8, we plot ∂χ31
(0)−h(φ 3 , G)φ 2s , where χ31(φ3 ) is the swelling curve of the gelin the solvent of the first component. In the range satisfying(0)/∂φ3 < −h(φ3 , G)φ2s , the gel would exhibit a discontin∂χ31
uous volume change. It is indicated that strong nonlinearity ofh(φ 3 , G) with respect to φ 3 is a possible origin of the second
transition.

IV. SUMMARY AND REMARKS

We studied the volume phase transition of gels immersed
in binary mixtures, based on the three-component FloryRehner model. Assuming that the volume fraction of the second solvent component is small, we reformulated the FloryRehner model into a simple model with a new parameterG(= χ 23 − χ 31 − χ 12 ). From numerical solutions and thesimplified theory, we found the following behaviors of thevolume phase transition of the gel.
(i)

When the selectivity of the second component to thepolymer network is small, the composition of the mixture solvent in the gel is close to that out of the gel.
Here, the single liquid approximation works well. The
renormalized interaction parameter between the solvent
and the network depends linearly on the compositionof the binary mixture. Thus, we can swell the gel by024902-7

Y. Uematsu and T. Araki

dissolving the additive good to the network, whereas,
the gel is shrunken when the poor additive is dissolved.
On the other hand, the difference of the composition becomes large when the additive selectivity isstrong. Owing to the nonlinearity of the compositiondifference, the gel tends to be shrunken as the additive concentration is increased, regardless of whether
it is good or poor. From the simplified theory, the dependence of the transition point on the additive concentration is found to be proportional to ∂χ31t /∂φ2s≈ g(φ3c , Ḡ), which is approximated as g ≈ −Ḡ(Ḡ+ 2)/2 for a hyper swollen gel.
(ii) By dissolving the strongly poor additive, we can extinguish the discontinuous volume phase transition. Furthermore, we can induce the volume phase transition in
the gel, which does not exhibit the discontinuous gap in
the absence of the additive, by dissolving the strongly
good additive. In the vicinity of the (first) tri-criticalpoint, this behavior is well described by the simplified
theory as shown in Fig. 6(b).
(iii) Far from the transition point, we found that another volume phase transition can occur. This second instability
is observed even if the elasticity is negligible. The transition is caused by the mixing instability. Our theory
indicates that the nonlinearity of φ 3 , which is well described by Eq. (27), does have a major contribution to
the transition.
Here, we make some remarks about our results.
(1) If only van der Waals interaction is taken into account,the interaction parameters should be positive, i.e., χ ij ≥0. However, a variety of molecular forces, such as iondipole interaction, hydrogen bonding and hydrophobic
interaction, would also influence the volume phase transition of the gel. When studying their contributions explicitly, we have to consider the microscopic degrees ofthe freedom in a more specific manner.21, 23, 24 In thiswork, we assume that the macroscopic χ parameter canhave a large negative value, by renormalizing these microscopic degrees into them. In a future work, we would
like to investigate the connections of such microscopic
interactions and our phenomenological theory. Effectsof the selective salts31 on polyelectrolyte gels are alsointeresting.
(2) Effects of additives on thermo-sensitive NIPA and PAAgels have been intensively studied.11–21 However, there
remain some problems which are difficult to explain by
conventional theories. Although our model is very simple, it can predict counterintuitive behaviors. It would be
helpful to elucidate some of unresolved problems.
For instance, Ishidao et al. studied the volume transition
of NIPA gels in aqueous solutions.25 They observed thatthe concentration of 1-propanol inside the gel is larger
than that outside the gel. This indicates that this additive has more good affinity to the polymer network thanwater, so that one can imagine that addition of the 1propanol makes the mixture solvent more good. However, the NIPA gel is shrunken by dissolving a small
amount of this additive. In this paper, we show consid-

J. Chem. Phys. 137, 024902 (2012)

ering the nonlinearities of G and φ 3 enables us to understand such a counterintuitive behavior. Here, we notethat the mixing of two solvents decreases the effective
interaction parameter in usual cosolvents with χ 12 > 0[see Eq. (18)].
Furthermore, Annaka et al. reported that the volume change of NIPA becomes continuous by dissolving
NaI.14 Ordinary theories cannot describe any additiveinduced disappearance of discontinuous transition. As
commented in Ref. 14, the disappearance of the VPT
is considered to be attributed to the interaction between
the salt and the polymer network. We consider that resultant inhomogeneities of the ion concentrations also
play a key role in this continuous volume change, as discussed in this paper. Since our model considers only one
species of solute and neglects the electrostatic interactions, however, it cannot explain this experimental result
quantitatively.
Since only a few experiments have reported the difference of the additive concentrations between in and outof the gels,25, 36 however, we cannot verify the quantitative validity of our model yet. Further experimental studies are highly desired.
(3) So far, thermo-sensitivities of gel volumes have beenfound only in hydrophilic gels such as polyelectrolytes
and NIPA. In them, the electrostatic interactions (including hydrogen bondings) would play important roles
more and less. Our model is based on the Flory-Huggins
free energy, so that it can be applied ubiquitously tomany types of polymer networks including hydrophobic (lipophilic) ones. We hope our model will help one
to find volume phase transitions in gels other than polyelectrolytes, NIPA, PAA, and their derivatives.

ACKNOWLEDGMENTS

We acknowledge valuable discussions with Akira Onuki.
This work was supported by the Japan Society for the Promotion of Science Core-to-Core Program “International research network for non-equilibrium dynamics of soft matter”
and Grants-in-Aid for Scientific Research. The computational
work was carried out using the facilities at the Supercomputer
Center, Institute for Solid State Physics, University of Tokyo.
1 P. J. Flory and J. Rehner, J. Chem. Phys. 11, 512 (1943); 11, 521 (1943).
2 K. Dušek and D. Patterson, J. Polym. Sci., Part A-2. 6, 1209 (1968).
3 T. Tanaka, Phys. Rev. Lett. 40, 820 (1978).
4 T. Tanaka, D. Fillmore, S.-T. Sun, I. Nishio, G. Swislow, and A. Shah, Phys.

Rev. Lett. 45, 1636 (1980).
5 T. Tanaka, Sci. Am. 244, 124 (1981).
6 Y. Hirokawa and T. Tanaka, J. Chem. Phys. 81, 6379 (1984).
7 I. Ohmine and T. Tanaka, J. Chem. Phys. 77, 5725 (1982).
8 J. Rička and T. Tanaka, Macromolecules 17, 2916 (1984).
9 A. Fernández-Nieves, A. Fernández-Barbero, and F. J. de las Nieves, J.

Chem. Phys. 115, 7644 (2001).
10 T. Amiya, Y. Hirokawa, Y. Hirose, Y. Li, and T. Tanaka, J. Chem. Phys. 86,

2375 (1987).
11 S. Hirotsu, Y. Hirokawa, and T. Tanaka, J. Chem. Phys. 87, 1392 (1987).
12 T. G. Park and A. S. Hoffman, Macromolecules 26, 5045 (1993).
13 H. Inomata, S. Goto, K. Otake, and S. Saito, Langmuir 8, 687 (1992).
14 M. Annaka, K. Motokawa, S. Sasaki, T. Nakahira, H. Kawasaki, H. Maeda,

Y. Amo, and Y. Tominaga, J. Chem. Phys. 113, 5980 (2000).

024902-8

Y. Uematsu and T. Araki

15 S. Hirotsu, J. Phys. Soc. Jpn. 56, 233 (1987).
16 E. Kokufuta, Y.-Q. Zhang, T. Tanaka, and A. Mamada, Macromolecules

26, 1053 (1993).
17 H. Kawasaki, S. Sasaki, H. Maeda, S. Mihara, M. Tokita, and T. Komai, J.
Phys. Chem. 100, 16282 (1996).
18 S. Sasaki, H. Kawasaki, and H. Maeda, Macromolecules 30, 1847 (1997).
19 D. Dhara and P. R. Chatterji, Langmuir 15, 930 (1999).
20 S. Koga, S. Sasaki, and H. Maeda, J. Phys. Chem. B 105, 4105 (2001).
21 F. Tanaka, T. Koga, and F. M. Winnik, Phys. Rev. Lett. 101, 028302(2008).
22 S. Sasaki and H. Maeda, Phys. Rev. E 54, 2761 (1996).
23 H. Kojima and F. Tanaka, Macromolecules 43, 5103 (2010).
24 H. Kojima and F. Tanaka, Soft Matter 8, 3010 (2012).
25 T. Ishidao, Y. Hishimoto, Y. Iwai, and Y. Araki, Colloid Polym. Sci. 272,
1313 (1994).

J. Chem. Phys. 137, 024902 (2012)
26 V. V. Vasilevskaya, V. A. Ryabina, S. G. Starodubtsev, and A. R. Khokhlov,

Polym. Sci. U.S.S.R. 31, 784 (1989).
27 T. Iwatsubo, K. Ogasawara, A. Yamasaki, T. Masuoka, and K. Mizoguchi,

Macromolecules 28, 6579 (1995).
28 W. R. Krigbaum and D. K. Carpenter, J. Polym. Sci. 14, 241 (1954).
29 G. M. Bristow, Trans. Faraday Soc. 55, 1246 (1959).
30 O. Okeowa and J. R. Dorgan, Macromolecules 39, 8193 (2006).
31 R. Okamoto and A. Onuki, Phys. Rev. E 82, 051501 (2010).
32 A. Onuki and R. Okamoto, Curr. Opin. Colloid Interface Sci. 16, 525(2011).
33 A. Onuki, R. Okamoto, and T. Araki, Bull. Chem. Soc. Jpn. 84, 569 (2011).
34 R. L. Scott, J. Chem. Phys. 17, 268 (1949).
35 P. J. Flory, Principles of Polymer Chemistry (Cornell University, Ithaca,

New York, 1953).
36 T. Iwatsubo and T. Shinbo, J. Macromol. Sci., Phys. B40, 1017 (2001).