Semidilute Principle for Gels Naoyuki Sakumichi1Takashi Yasuda1and Takamasa Sakai1y 1Graduate School of Engineering The University of Tokyo 7-3-1 Hongo Bunkyo-ku Tokyo Japan.

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Semidilute Principle for Gels

Naoyuki Sakumichi,1, ∗Takashi Yasuda,1, ∗and Takamasa Sakai1, †

1Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan.

(Dated: December 12, 2022)

Polymer gels such as jellies and soft contact lenses are soft solids consisting of three-dimensional

polymer networks swollen with a large amount of solvent. For approximately 80 years, the swelling of

polymer gels has been described using the Flory–Huggins mean-ﬁeld theory. However, this theory

is problematic when applied to polymer gels with large solvent contents owing to the signiﬁcant

ﬂuctuations in polymer concentration. In this study, we experimentally demonstrate the superiority

of the semidilute scaling law over the mean-ﬁeld theory for predicting the swelling of polymer gels.

Using the semidilute scaling law, we experimentally determine the universal critical exponent νof

the self-avoiding walk via swelling experiments on polymer gels. The experimentally obtained value

ν'0.589 is consistent with the previously reported value of ν'0.588, which was obtained by precise

numerical calculations. Furthermore, we theoretically derive and experimentally demonstrate a

scaling law that governs the equilibrium concentrations. This scaling law contradicts the predictions

made by de Gennes’ c∗theorem. A major deﬁciency of the c∗theorem is that the network elasticity,

which depends on the as-prepared state, is neglected. These ﬁndings reveal that the semidilute

scaling law is a fundamental principle for accurately predicting and controlling the equilibrium

swelling of polymer gels.

I. INTRODUCTION

The swelling of polymer networks by absorption of

surrounding solvent is a ubiquitous phenomenon [1,2];

for example, it is used in water-absorbent materials in

disposable diapers and portable toilets. Here, polymer

networks swollen with a large amount of solvent are

called polymer gels, such as jellies and soft contact

lenses. As-prepared polymer gels after fabrication are

generally not in an equilibrium swollen state and swell

when immersed in a solvent. Revealing the governing

law for the swelling of polymer gels has long been an

important issue [3,4], and considerable studies have been

conducted to predict the equilibrium swollen state (e.g.,

the liquid content in swollen polymer gels) [4–21]. This

is because it is essential for a fundamental understanding

of polymer gels and their applications. For example, in

practical biomedical applications such as drug delivery

[22], adhesion barriers [23], and artiﬁcial vitreous humor

[24], understanding the swelling behavior of polymer

gels is essential because the liquid content signiﬁcantly

inﬂuences the softness and mass diﬀusivity of polymer

gels in the external environment.

According to Flory and Rehner [4], the swelling equi-

librium of (electrically neutral) polymer gels in good sol-

vents is determined by the balance between the elas-

tic (Πel) and mixing (Πmix) contributions in the total

swelling pressure of polymer gels [Fig. 1(a)]:

Πtot = Πmix + Πel.(1)

∗These authors contributed equally: N. Sakumichi, T. Yasuda

†Correspondence should be addressed to N. Sakumichi or T. Sakai:

sakumichi@gel.t.u-tokyo.ac.jp; sakai@gel.t.u-tokyo.ac.jp

The elastic contribution satisﬁes Πel =−G0(c/c0)1/3,

[21,25] because uniform swelling reduces the density of

the polymer network. Here, G0is the shear modulus in

the as-prepared state, and c0and care the polymer mass

concentrations in the as-prepared and equilibrium states,

respectively. Therefore, we consider the expression for

Πmix.

A conventional approach to describe Πmix in Eq. (1)

is the Flory–Huggins (FH) mean-ﬁeld theory, which

was originally developed for polymer solutions [26–28].

The FH mean-ﬁeld theory is applicable to concentrated

polymer systems such as non-dilute polymer solutions

and polymer blends with small concentration ﬂuctua-

tions [29], because it uses the mean-ﬁeld approximation.

Thus, its application to dilute systems, including poly-

mer gels with low polymer volume fractions (typically

less than 0.1), is problematic, as pointed out by Flory

himself1. However, in numerous subsequent studies,

the FH theory has been inappropriately applied to

polymer gels containing a large amount of solvent [5–8],

resulting in signiﬁcant inconsistencies in the reported

polymer–solvent interaction parameters [30–33].

Another well-known claim, de Gennes’ c∗theorem [2],

asserts that the polymer concentration ceq of a polymer

gel in the equilibrium swollen state is proportional to the

overlap concentration c∗of a group of polymer chains

in a good solvent. Notably, this so-called c∗“theorem”

is not a mathematical theorem but only a physical

conjecture that requires experimental validation. This

1Regarding the limitations of the FH mean-ﬁeld theory, Flory

stated, “Of importance here is the realization that the theory as

developed so far is inappropriate, generally speaking, for dilute

polymer solutions” on p. 505 in Ref. [1].

arXiv:2210.15275v2 [cond-mat.soft] 9 Dec 2022

conjecture follows from the assumption that subchains

(chains between adjacent crosslinks) in an equilibrium

swollen network of a polymer gel disinterpenetrate, i.e.,

subchains do not interpenetrate each other and remain

in contact at the overlap threshold. Here, polymer gels

are considered as a set of closely packed polymer chains

sealed together by crosslinks, similar to polymer solu-

tions at the overlap threshold c∗. Several experimental

observations have supported this conjecture, revealing

similarities in the scaling relations between polymer gels

and semidilute solutions [9–11]. However, subsequent

systematic measurements of the elastic modulus of gels

contradict the disinterpenetration of the swollen network

chains assumed in this conjecture [8,12,13]. Hence,

the c∗theorem, although seemingly plausible, remains

controversial.

In this study, we measure Πmix of chemically

crosslinked polymer gels with precisely controlled

homogeneous network structures [34] throughout the

quasistatic swelling process from the as-prepared to equi-

librium state [Fig. 1(b)], using the osmotic deswelling

method proposed in 1940s [35] and developed in the

1980s [36]. Consequently, we successfully demonstrate

that the semidilute scaling law of polymer solutions

[2,37] governs Πmix of polymer gels in Eq. (1). Fur-

thermore, by assuming the semidilute scaling law as the

fundamental principle for polymer gels, we theoretically

derive and experimentally validate the following three

arguments. First, the semidilute scaling law is superior

to the FH mean-ﬁeld theory in accurately predicting the

equilibrium state of polymer gels throughout the qua-

sistatic swelling process. Second, swelling experiments of

polymer gels provide a novel method for experimentally

determining the excluded volume exponent ν; we obtain

ν'0.589, which is consistent with those reported

previously (see Table Ibelow). Here, νis also known as

the universal critical exponent of the self-avoiding walks

(SAW), which corresponds to n→0 in O(n)-symmetric

university classes in three dimensions [38–42]. Third, we

derive a novel scaling law for the equilibrium concentra-

tions ceq (i.e., the polymer mass concentrations in the

equilibrium swollen states) that depends explicitly on

the polymer mass concentration c0and shear modulus

G0in the as-prepared state. By contrast, ceq predicted

by the c∗theorem does not depend on c0and G0. Our

experimental results support the derived scaling law

and contradict the prediction made by the c∗theorem [2].

The remainder of this paper is organized as follows.

In Sec. II, we formulate the FH mean-ﬁeld theory and

the semidilute scaling law of Πmix. In Sec. III, we ex-

plain the materials and methods. In Sec. IV, we analyze

our results on swelling experiments by the FH theory

and semidilute scaling law. In Sec. V, we compare the

prediction of the quasistatic swelling process using the

FH theory and semidilute scaling law. In Sec. VI, we

demonstrate the scaling law governing equilibrium con-

Quasistatic swelling process by tuning

Polymers (PVP)

Swollen stateAs-prepared state Swelling state

Swell

~ Few days

Volume :

Gel

Equilibrium swollen state

Volume :

As-prepared state

Gel

Veq

Πtot >0Πtot = 0

Πmix

Πel

Πmix

Πel

Πext

Q= 1 Q=V/V0Qeq =Veq/V0

(a)

(b)

FIG. 1. Swelling behavior of polymer gels. (a) Volume

change of a polymer gel from the as-prepared to equilib-

rium swollen states in a pure solvent. A gel after fabri-

cation is called the as-prepared state, where the volume is

V0. Upon immersing into a pure solvent, the as-prepared gel

swells because its osmotic pressure (Πtot = Πmix + Πel >0)

is larger than the osmotic pressure outside (Πext = 0). Af-

ter a few days, the gel reaches the equilibrium swollen state

at which Πtot = 0, changing its volume (V0→Veq ) and

polymer mass concentration (c0→ceq). (b) Quasistatic

swelling process of a polymer gel by tuning the external os-

motic pressure Πext. We tuned Πext using controlled aqueous

poly(vinylpyrrolidone) (PVP) solutions, whose polymer mass

concentration dependence of osmotic pressure was previously

measured by Vink [43]. At each equilibrium state, the osmotic

pressure inside (Πtot) and outside (Πext) a gel is equivalent:

Πtot = Πmix + Πel = Πext. The volume change of a gel is

quantiﬁed by the volume swelling ratio Q≡V/V0, where V

is measured from a gel taken out from the solvent.

centration, which contradicts the predictions made by de

Gennes’ c∗theorem. In Sec. VII, we summarize the main

results of this study. Several details are described in the

Appendices to avoid digressing from the main subject.

II. FLORY–HUGGINS MEAN-FIELD THEORY

AND SEMIDILUTE SCALING LAW OF Πmix

To determine the law governing Πmix of polymer gels,

we experimentally evaluated Πmix throughout the qua-

sistatic swelling process, from the as-prepared to equilib-

rium swollen states, via osmotic deswelling [35,36,44].

As shown in Fig. 1(b), we immerse a gel sample in an ex-

ternal polymer solution with a certain external osmotic

pressure Πext. The equilibrium condition is Πtot = Πext,

at which the osmotic pressure inside and outside the gel

is equivalent. Especially, Πext = 0 for a pure solvent,

as shown in Fig. 1(a). Hence, by gradually tuning Πext

with the controlled aqueous polymer solutions [43], we

can evaluate Πmix as

Πmix = Πext +G0c

c01/3

,(2)

where G0and ccan be experimentally measured. We

assume Eq. (2) throughout this study. In this study, we

consider only the equilibrium states of the polymer gels

in external polymer solutions, such as the as-prepared

(Q= 1 and Πext >0), swelling (Q > 1 and Πext >0),

and equilibrium swollen (Q > 1 and Πext = 0) states.

Here, Q≡V/V0is the volume swelling ratio [see

Fig. 1(b)].

In a solution with suﬃciently long polymer chains, the

Flory–Huggins (FH) mean-ﬁeld theory [26–28] predicts

that

Πmix =−kBT

vφ+ ln (1 −φ) + χφ2,(3)

where kB,T,v, and φare the Boltzmann constant, abso-

lute temperature, volume of solvent molecules, and poly-

mer volume fraction, respectively. The FH interaction

parameter χquantiﬁes the energies of the interactions

between the polymer units and the solvent molecules per

lattice site as follows: χ≡z(pp +ss −2ps)/(2kBT),

where zis the coordination number of the lattice, and

pp,ss, and ps are the polymer–polymer, solvent–

solvent, and polymer–solvent interaction energies,

respectively. Although the FH theory is applicable only

to concentrated polymer solutions [29], it has been in-

appropriately applied to polymer gels containing a large

amount of solvent, by suitably adjusting χ[5–8]. Conse-

quently, signiﬁcant inconsistencies in χwere reported in

previous studies, although χshould be constant for the

same polymer–solvent system and temperature [45,46].

For example, χ= 0.426 for the poly(ethylene glycol)

(PEG) hydrogel [30] and χ= 0.366, 0.441, and 0.465

for the aqueous PEG solution with the molar masses

M= 4.6 [31], 10 [32], and 33 kg/mol [33], respectively,

were reported at T= 298 K, using v= 18×10−6m3/mol.

By contrast, we propose that the semidilute scaling law

of polymer solutions [2,37] governs Πmix in polymer gels,

based on a recent observation [44]. In Ref. [44], it was ob-

served that a universal osmotic equation of state of poly-

mer solutions [2,37,47–49] describes Πmix throughout

the gelation process. For polymer solutions, the semidi-

lute scaling law is expressed as ΠmixM/(cNAkBT) =

K(c/c∗)1/(3ν−1), where K'1.1 is the universal constant

for the semidilute polymer solutions, NAis the Avogadro

constant, c∗is the overlap concentration, and ν'0.588

is the universal critical exponent of the SAW [38–42].

However, for polymer gels, c∗→0 and c/c∗→ ∞ [44],

because the molar mass of a polymer network is inﬁnite.

Thus, we introduce the regulator Mseg to propose the

following semidilute scaling law:

Πmix

nkBT=Kn

n∗

seg 

3ν−1

,(4)

as a fundamental principle for polymer networks that

contain a large amount of solvent. Here, both sides of

Eq. (4) are dimensionless; n≡cNA/Mseg is the number

density of a “segment” of polymers, where Mseg is

the molar mass of the segment. In this study, we use

Mseg = 44 ×10−3kg/mol, which is the molar mass of an

ethylene glycol unit as a segment. In Eq. (4), n∗

seg is a

parameter that depends on Tand the type of polymer

network and solvent considered. We determined n∗

seg by

adjusting to the experimental data. This procedure is

similar to the determination of χin the FH mean-ﬁeld

theory from the experimental data.

Notably, a few pioneering studies [11,16–20] attempted

to describe the equilibrium swelling of polymer gels using

the semidilute scaling law such as Πmix ∼φ9/4. However,

they did not focus on demonstrating the superiority of

the semidilute scaling law over the FH mean-ﬁeld theory,

owing to the limited accuracy of the gel experiments and

the lack of theoretical comprehension of polymer gels. By

contrast, in this study, we experimentally demonstrate

the superiority of Eq. (4) over the FH mean-ﬁeld the-

ory [Eq. (3)] in terms of predicting the equilibrium state

of polymer gels throughout the quasistatic swelling pro-

cess. We verify the superiority of the semidilute scaling

law using precisely controlled homogeneous networks [34]

and a diﬀerent deﬁnition of the polymer mass concentra-

tion [44] that enables us to extend the universality of the

osmotic equation of state of polymer solutions.

III. MATERIALS AND METHODS

A. Fabrication of model gels

As a model system to examine the quasistatic swelling

process of chemically crosslinked polymer gels, we used

a tetra-branched PEG hydrogel [34], synthesized via the

AB-type cross-end coupling of two prepolymer (tetra-

arm PEG) units of equal size. Each end of the tetra-arm

PEG was modiﬁed with a mutually reactive maleimide

(tetra-PEG MA) and thiol (tetra-PEG SH). We dissolved

tetra-PEG MA and tetra-PEG SH (NOF Co., Japan &

XIAMEN SINOPEG BIOTECH Co., Ltd., China) in

phosphate citrate buﬀer with an ionic strength of 100 mM

and a pH of 3.8. For gelation, we mixed these two so-

lutions with equal molar masses Mfor equal prepoly-

mer mass concentrations c0at various mixing fractions

s. Here, sis the molar fraction of the minor prepolymers

(tetra-PEG SH) to all prepolymers (0 ≤s≤1/2). By

tuning s, the desired connectivity pcan be obtained in

accordance with p= 2s[50,51], where pis deﬁned as

the fraction of reacted terminal functional groups to all

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SemidilutePrincipleforGelsNaoyukiSakumichi,1,TakashiYasuda,1,andTakamasaSakai1,y1GraduateSchoolofEngineering,TheUniversityofTokyo,7-3-1Hongo,Bunkyo-ku,Tokyo,Japan.(Dated:December12,2022)Polymergelssuchasjelliesandsoftcontactlensesaresoftsolidsconsistingofthree-dimensionalpolymernetworksswollenwith...

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Semidilute Principle for Gels Naoyuki Sakumichi1Takashi Yasuda1and Takamasa Sakai1y 1Graduate School of Engineering The University of Tokyo 7-3-1 Hongo Bunkyo-ku Tokyo Japan.

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