---
abstract: 'The equation of state (EOS) of the osmotic pressure for linear-polymer solutions in good solvents is universally described by a scaling function. We experimentally measured the osmotic pressure of the gelation process via osmotic deswelling. It is found that the same scaling function for linear-polymer solutions also describes the EOS of the osmotic pressure throughout the gelation process, involving both sol and gel states. Furthermore, we reveal that the osmotic pressure of polymer gels is universally governed by the semidilute scaling law of linear-polymer solutions.'
author:
- Takashi Yasuda
- Naoyuki Sakumichi
- 'Ung-il Chung'
- Takamasa Sakai
title: Universal Equation of State of Osmotic Pressure in Gelation Process
---

[^1]

The statistical mechanics of groups of chains is the basis of polymer physics [@flory1953principles; @de1979scaling; @lifshitz1978some; @panyukov1996statistical]. A remarkable example of this basis is the universality of linear-polymer solutions in good solvents [@de1979scaling; @oono1985statistical]. Their macroscopic collective properties are independent of the microscopic details of the system, because of the great length of polymer chains. This example of the universality of critical phenomena in the $O(n)$-symmetric universality classes ($n=1,2,3$ corresponding to the Ising, XY, and Heisenberg classes, respectively) is found in many systems, ranging from the fields of soft and hard condensed-matter physics to high-energy physics [@pelissetto2002critical]. The above polymer solutions correspond to the limit of $n\to 0$ (self-avoiding walks) in three dimensions [@de1979scaling; @pelissetto2002critical], for which the critical exponent (the excluded volume parameter) $\nu \simeq 0.588$ can be computed by Monte Carlo simulations [@clisby2010accurate; @clisby2016high], the $\epsilon$ expansion method [@kompaniets2017minimally], and the conformal bootstrap method [@shimada2016fractal; @hikami2018conformal]. Furthermore, not only the critical exponents but also the asymptotic scaling functions themselves can be experimentally measured, such as the osmotic pressure [@noda1981thermodynamic; @higo1983osmotic] and the correlation lengths of the density fluctuations [@wiltzius1983universality].

Here, we focus on the equation of state (EOS) of osmotic pressure for linear-polymer solutions in good solvents, which is universally described by the scaling function [@noda1981thermodynamic; @higo1983osmotic; @des1975lagrangian; @des1982osmotic; @ohta1982conformation; @ohta1983theory]: $$\hat{\Pi} = f\left({\hat{c}}\right),
\label{eq:EOS}$$ where $\hat{\Pi} \equiv \Pi M/(cRT)$ is the reduced osmotic pressure, and $\hat{c}\equiv c/c^*$ is the reduced polymer concentration normalized by the overlap concentration $c^{*}\equiv1/(A_2 M)$. Here, $M$, $R$, $T$, and $A_2$ are the molar mass, gas constant, absolute temperature, and the second virial coefficient, respectively. The above definition of $c^{*}$ is proportional [@burchard1999solution] to the conventional definition of the overlap concentration $c^*_g \equiv 3M/(4\pi N_A R_g^3)$, at which the polymer chains begin to overlap each other to fill the space. Here, $N_A$ and $R_g$ are the Avogadro constant and the gyration radius of the polymer chain, respectively.

![ Universal EOS of polymer solutions and gelation process in a good solvent. Main image shows the $\hat{c}$-dependence of $\hat{\Pi}$ in a log–log plot, and the inset shows the $\hat{c}^{-1}$-dependence of $\hat{\Pi}/\hat{c}^{1.31}$. The triangles represent two kinds of linear polymer (poly(styrene) of $M=51$–$1900$ kg$/$mol [@higo1983osmotic] and poly($\alpha$-methylstyrene) of $M= 70.8$–$1820$ kg$/$mol [@noda1981thermodynamic]) in toluene solutions. These converge to the universal EOS (\[eq:EOS\]) (black solid curve), which is asymptotic to the van ’t Hoff law ($\hat{\Pi}=1$) as $\hat{c}\to 0$ and to the scaling law in Eq. (\[eq:scaling\]) as $\hat{c}\to\infty$ (black dotted lines). The black circles represent four-branched polymer (poly(ethylene glycol)) solutions of $M=10$ and $40$ kg$/$mol. The orange filled circles represent the gelation process in sol states with various degrees of connectivity ($p=0, 0.1, \dots, 0.5$) at a constant concentration ($c=20$ g$/$L). The red star in the inset corresponds to the universal EOS for polymer gels. []{data-label="fig:EOS"}](Fig1.pdf){width="\linewidth"}

In the case of branched polymer solutions, it was reported that each EOS of regular star polymers with up to 18 arms exhibits only minor differences from the universal EOS (\[eq:EOS\]) of linear polymers [@higo1983osmotic; @adam1991concentration; @merkle1993osmotic; @burchard1999solution]. Here, $\hat{c}\equiv c/c^*$ is the only universal scaling parameter (up to multiplication by a constant) [@burchard1999solution]. In other words, $c/c^*_{g}$ is not a universal scaling parameter because $c_{g}^{*}/c^*=3\sqrt{\pi}\Psi^{*}$ includes the interpenetration factor $\Psi^{*}$, which is nonuniversal for a number of arms (e.g., $\Psi^{*}\simeq 0.24$ and $0.44$ for linear and four-branched polymer solutions, respectively [@rubio1996monte; @okumoto1998excluded]). Figure \[fig:EOS\] demonstrates that the two kinds of linear polymer solution and four-branched polymer solutions converge to the single universal EOS (\[eq:EOS\]). In the dilute regime ($c<c^{*}$), each molecular chain is sufficiently isolated such that the universal EOS (\[eq:EOS\]) is well described by the virial expansion [@flory1953principles]: $$\hat{\Pi} = f\left({\hat{c}}\right) = 1 + \hat{c} + \gamma\,\hat{c}^{2}+ \dots 
\quad(\mathrm{for}\,\,\, 0<\hat{c}<1),
\label{eq:virial}$$ where $\gamma \simeq 0.25$ [@flory1953principles; @noda1981thermodynamic] is the dimensionless virial ratio. In the semidilute regime ($c^{*}<c$), molecular chains become interpenetrated and the universal EOS (\[eq:EOS\]) is asymptotic to the scaling law [@des1975lagrangian; @de1979scaling]: $$\hat{\Pi} = f\left({\hat{c}}\right) \simeq K\hat{c}^{\frac{1}{3\nu -1}}
\qquad(\mathrm{for}\,\,\, \hat{c}\gg 1),
\label{eq:scaling}$$ where $K\simeq 1.1$ is the numerical constant and $1/(3\nu -1)\simeq 1.31$ if $\nu=0.588$.

In the present study, we experimentally investigate the EOS of the osmotic pressure of polymer gels, including the whole gelation process. We measured the osmotic pressure in both the sol and gel states via osmotic deswelling in external polymer solutions [@bastide1981osmotic; @horkay1986studies; @horkay2000osmotic]. Our findings are summarized in Fig. \[fig:EOS\]; the universality of EOS (\[eq:EOS\]) holds for both the sol (orange filled circles) and gel (red star) states with only minor variations, although these systems are comprised of highly branched three-dimensional polymer networks. When gelation proceeds at a constant concentration $c$, the average molar mass $M$ increases, and $c^{*}$ decreases. Thus, both $\hat{\Pi}$ and $\hat{c}$ continuously increase along the universal EOS (\[eq:EOS\]) in the sol states. After the gelation (i.e., sol–gel transition), because polymer gels correspond to $M\to\infty$ and $c^{*}\to 0$, both $\hat{\Pi}$ and $\hat{c}$ diverge to infinity in the gel states. According to the semidilute scaling law given by Eq. (\[eq:scaling\]), $\hat{\Pi}/\hat{c}^{1.31}$ is always constant in gel states (red star in the inset of Fig. \[fig:EOS\]).

To statically reproduce the gelation process, we non-stoichiometrically tuned the mixing fractions $s$ ($0\leq s\leq 1/2$) of two kinds of precursor solution in an AB-type polymerization system (schematics in Fig. \[fig:gelation\]). Here, $s$ is the molar fraction of the minor precursors to all precursors. We define the connectivity $p$ ($0\leq p\leq1$) as the fraction of the reacted terminal functional groups, assuming reaction completion. By tuning $s$ in accordance with $p = 2s$ [@sakai2016sol; @yoshikawa2019connectivity], we can obtain a desired $p$. Before gelation ($0\leq p <p_\mathrm{gel}$), polymer chains crosslink to form a polydisperse mixture of highly branched polymers with increases in the average molar mass $M$. After gelation ($p_\mathrm{gel}\leq p\leq 1$), polymer networks crosslink to complete the reaction as the elasticity increases.

Based on our findings shown in Fig. \[fig:EOS\], the “non-reduced” osmotic pressure $\Pi$ during the gelation process is illustrated in Fig. \[fig:gelation\]. Unlike the sol states, the gel states have elastic contributions to the swelling pressure. According to Flory and Rehner [@flory1943jr], the total swelling pressure in the gel states ($\Pi_{\mathrm{tot}}$) consists of two separate contributions as $\Pi_{\mathrm{tot}}=\Pi_{\mathrm{mix}}+\Pi_{\mathrm{el}}$, where $\Pi_{\mathrm{mix}}$ and $\Pi_{\mathrm{el}}$ are the mixing and elastic contributions, respectively. We regard $\Pi_{\mathrm{mix}}$ as being the osmotic pressure in the gel states because $\Pi_{\mathrm{mix}}$ corresponds to the osmotic pressure in the sol state $\Pi$. As the connectivity $p$ increases at a constant concentration $c$, the osmotic pressure $\Pi$ in the sol states decreases, because the chemical reaction decreases the number density of the molecules. When the samples enter the gel states, the osmotic pressure $\Pi_{\mathrm{mix}}$ reaches a constant; polymer gels are always in a semidilute regime with an infinite molar mass.\

![ Osmotic pressure during the gelation process at a constant polymer concentration ($c=60$ g$/$L) and a constant molar mass of precursors ($M=10$ kg$/$mol). By measuring $\Pi_\mathrm{tot}$ and $G$, we obtain $\Pi_\mathrm{mix}=\Pi_\mathrm{tot}+G$ in gel states. As the connectivity $p$ increases, $\Pi$ and $\Pi_\mathrm{tot}$ decrease, but $\Pi_{\mathrm{mix}}$ remains constant (blue curves). After gelation ($p_{\mathrm{gel}}\leq p\leq1$), the elasticity (red curve) increases. Here, $p$ ($0\leq p\leq1$) is controlled by mixing two kinds of precursors non-stoichiometrically in an AB-type polymerization system. Gel samples with a low connectivity ($p_{\mathrm{gel}} \leq p < 0.7$) were difficult to characterize, because of the outflow of small polymer clusters. []{data-label="fig:gelation"}](Fig2.pdf){width="\linewidth"}

*Materials and Methods*. — For a model system of AB-type polymerization in gelation, we used a tetra-PEG gel, synthesized by the AB-type cross-end coupling of two tetra-arm poly(ethylene glycol) (tetra-PEG) units of the same size [@sakai2008design]. Each end of the tetra-PEG is modified with mutually reactive maleimine (tetra-PEG MA) and thiol (tetra-PEG SH). We dissolved tetra-PEG MA and tetra-PEG SH (Nippon Oil & Fat Corporation) in a phosphate-citrate buffer with an ionic strength and pH of $200$ mM and $3.8$, respectively. For gelation, we mixed these solutions with equal molar masses $M$ and equal concentrations $c$ in various mixing fractions $s$. We held each sample in an enclosed space to maintain humid conditions at room temperature ($T\simeq 298\ K$) to allow the reaction to complete.

![ Osmotic deswelling in external polymer solutions to measure $\Pi$ and $\Pi_{\mathrm{mix}}$ in (a) sol and (b) gel samples, respectively. For each plot, precursors are $M=10$ kg$/$mol. Each line is the least squares fit to the data for each $p$. (a) Square root plots of $\Pi$ of sol samples on $c_{0}=20$ g$/$L for the degrees of connectivity $p=0, 0.1, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5$. We immersed samples with a micro-dialyzer in external polymer (PVP) solutions. We can determine $\Pi$, because $\Pi = \Pi_{\mathrm{ext}}$ at equilibrium. (b) Swelling ratio $Q$ of gel samples on $c_{0}=60$ g$/$L for $p=0.7,0.8,0.9,1$ in equilibrium state in external polymer (PVP) solutions. We directly immersed samples in the external solutions of various concentrations $c_\mathrm{ext}$. We can determine $\Pi_{\mathrm{mix}}$, because $\Pi_{\mathrm{mix}}=\Pi_{\mathrm{ext}}+G$ at equilibrium. []{data-label="fig:measurement"}](Fig3.pdf){width="\linewidth"}

We prepared the four-branched polymer (precursor) solutions ($p=0$) by dissolving tetra-PEG MA with molar masses of $M=10$ and $40$ kg$/$mol and initial concentrations $c_{0} = 20$–$120$ g$/$L. Herein, we define the polymer concentration ($c_{0}$ and $c$) as the precursor weight divided by the solvent volume, rather than the solution volume, to extend the universality of the EOS (\[eq:EOS\]) to higher concentrations (see Supplemental Material, Sec. S1). We prepared the sol and gel samples in the gelation process, dissolving precursors with $M=10$ kg$/$mol. For $c_{0} = 20$ g$/$L, we set $p=2s=0.1, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5$ (sol samples). For $c_{0}=40, 60, 80, 120$ g$/$L, we set $p=2s=0.1, 0.2, 0.3$ (sol samples) and $0.7, 0.8, 0.9, 1$ (gel samples). Sec. S2 of Supplemental Material describes the determination of these measurement ranges.

We measured the osmotic pressures of the sol states $\Pi$, using controlled aqueous poly(vinylpyrrolidone) (PVP, K90, Sigma Aldrich) solutions whose concentration dependence of osmotic pressure $\Pi_{\mathrm{ext}}$ was measured by Vink [@vink1971precision] (Supplemental Material, Sec. S3). As shown in the schematic in Fig. \[fig:measurement\](a), each solution sample was placed in a micro-dialyzer (MD300, Scienova), which had a semipermeable membrane with a mesh size of $3.5$ kDa. We immersed each dialyzer in aqueous polymer (PVP) solutions at a certain concentration $c_\mathrm{ext}$ with stirring. This system achieved equilibrium at $\Pi = \Pi_{\mathrm{ext}}$. (Achievement of swelling equilibrium was assured as described in Sec. S4 of Supplemental Material.) At that time, each solution sample was changed in weight and concentration from its initial to equilibrium states as $w_0 \to w$ and $c_0 \to c$, respectively. Assuming a constant weight density and small deformation for the sample, we obtained the concentration at equilibrium as $c=c_{0}/Q$, where $Q=w/w_{0}$ is the swelling ratio. In examining the gelation process (orange filled circles in Fig. \[fig:EOS\] and Fig. \[fig:gelation\]), we compared $\Pi$ of the “as-prepared” sol samples at equal concentrations $c=c_0$ with various values of $p$. We determined $\Pi$ of each as-prepared sol sample by measuring the swelling ratio $Q$ as a linear function of $c_\mathrm{ext}$. (The method used to determine $\Pi$ of the as-prepared samples is the same as that used for gels, as detailed below.)

To evaluate the parameters $M$ and $c^{*}$ from $\Pi = \Pi (c)$ measured at each $p$, we used the square root plots [@flory1953principles], as shown in Fig. \[fig:measurement\](a). From the virial expansion (\[eq:virial\]), we have $\hat{\Pi}
 = \left[1+\hat{c}/2
 + \left(\gamma-1/4 \right)\hat{c}^2/2
 \right]^2
 + O\left(\hat{c}^3\right)
$. Together with $\gamma \simeq 1/4$ (Supplemental Material, Sec. S5) for certain solutions of few-branched polymers, we have $\sqrt{\Pi/c}
\simeq \sqrt{RT/M}
\left[1+c/(2c^{*})\right]$ for small $c/c^{*}$. Thus, the intercepts and slopes of each fitting line in Fig. \[fig:measurement\](a) give $M$ and $c^{*}$, respectively, for each $p$. The obtained $M$ and $c^{*}$ values are consistent with the scaling prediction of $c^{*} \sim M^{1/(3\nu-1)}$ with $\nu=0.588$ (Supplemental Material, Sec. S6).

We measured the osmotic pressure in the as-prepared gel states $\Pi_\mathrm{mix}$ via the osmotic deswelling. As shown in the schematic in Fig. \[fig:measurement\](b), we immersed each gel sample directly in the external aqueous polymer (PVP) solutions with various concentrations $c_\mathrm{ext}$, because the surfaces of the gels function as semipermeable membranes. Then, each gel sample swells or deswells from the as-prepared state to the equilibrium state as $\Pi_{\mathrm{mix}}+\Pi_{\mathrm{el}}=\Pi_{\mathrm{ext}}$, changing its weight and concentration from the initial to equilibrium states as $w_0 \to w$ and $c_0 \to c$, respectively. The swelling ratio $Q=w/w_{0}$ was measured and interpolated as a linear function of $c_\mathrm{ext}$ for each gel sample (e.g., the samples of various connectivity $p$ at $c_0=60$ g$/$L, as given in Fig. \[fig:measurement\](b)). By using the $c_\mathrm{ext}$-dependence of $Q$, we evaluated $c_\mathrm{ext}$ and $\Pi_\mathrm{ext}$ such that each gel sample maintained its weight ($Q = 1$) and concentration ($c = c_{0}$). We assumed that $\Pi_{\mathrm{el}} = -G$ [@james1949simple] and evaluated $\Pi_{\mathrm{mix}} = \Pi_{\mathrm{ext}} + G$ of each as-prepared gel sample, where $G$ is the shear modulus as measured by rheometry (Supplemental Material, Sec. S7).\

![ Osmotic pressure during the gelation process. The molar mass of precursors is $M=10$ kg$/$mol, corresponding to the overlap concentration $c^*\simeq58$ g$/$L at $p=0$. (a) Osmotic pressure in the unreacted four-branched polymer solutions (black circles) and in the reaction-completed polymer gels (red filled circles). The former and latter are in very good agreement with the universal EOS (\[eq:EOS\]) (black curve) and with the semidilute scaling law $\Pi\propto c^{2.31}$ (red line), respectively. Here, $3\nu/(3\nu -1)\simeq 2.31$ for $\nu \simeq 0.588$. The black dotted curve is the virial expansion (\[eq:virial\]) up to the third-order terms. As $p$ increases (green triangles), $\Pi$ decreases in sol states ($0\leq p<p_\mathrm{gel}$) and becomes constant in gel states ($p_\mathrm{gel}<p\leq1$). The inset shows the osmotic pressure during the gelation process at a constant polymer concentration $c=c_{0}=40, 60, 80, 120$ g$/$L. The green triangles ($c=40$ g$/$L) are the same as those in the main panel. The blue circles ($c=60$ g$/$L) are used in Fig. \[fig:gelation\]. (b) Connectivity ($p$) dependence of $\hat{\Pi}/{\hat{c}}^{1.31}$. The symbols and data are the same as those in the inset of (a). In gel states, $\hat{\Pi}/{\hat{c}}^{1.31}$ converge to the universal value $1.1$, which is independent of $p$ and $c$. []{data-label="fig:result"}](Fig4.pdf){width="\linewidth"}

*Results and Analysis*. — The main panel in Fig. \[fig:result\](a) shows the $c$-dependence of the osmotic pressure in the unreacted four-branched polymer solutions ($p = 0$) and in the reaction-completed polymer gels ($p = 1$). In the wide concentration range $c$, the experimental results of the former and latter are in remarkably good agreement with the universal EOS (\[eq:EOS\]) for *linear* polymer solutions and with the semidilute scaling law $\Pi\propto c^{3\nu/(3\nu -1)}$, respectively. With the increase in $c$, $\Pi$ in polymer solutions (black curve) is asymptotic to $\Pi_{\mathrm{mix}}$ in polymer gels (red line). This asymptotic relationship suggests that the $\Pi_{\mathrm{mix}}$ of polymer gels is governed by the semidilute scaling law in Eq. (\[eq:scaling\]) with $K\simeq 1.1$ for polymer solutions.

The inset in Fig. \[fig:result\](a) shows the $p$-dependence of $\Pi$ and $\Pi_\mathrm{mix}$ throughout the gelation process ($0\leq p \leq1$). In the sol states ($0\leq p<p_\mathrm{gel}$), $\Pi$ decreases as $p$ increases, because the average molar mass $M$ increases. As $c$ increases, the extent of the decrease in the osmotic pressure itself decreases. In particular, for $c=120$ g$/$L, $\Pi$ and $\Pi_\mathrm{mix}$ are constant throughout the gelation process ($0\leq p \leq1$), because the precursor solution is in the semidilute regime even at $p=0$. In the gel states ($p_\mathrm{gel}<p\leq1$), $\Pi_\mathrm{mix}$ is constant even if $p$ increases. In general, the osmotic pressure is dependent and independent of the average molar mass in the dilute and semidilute regimes, respectively [@de1979scaling]. Thus, the constant $\Pi_\mathrm{mix}$ in the gel states ($p_\mathrm{gel}<p\leq1$) indicates that polymer gels are always in the semidilute regime, because of the infinite molar mass of the polymer networks.

We can interpret $\Pi$ during the gelation process in the sol states ($0\leq p<p_\mathrm{gel}$) in terms of the universal EOS (\[eq:EOS\]). By using $M$ and $c^{*}$ evaluated in Fig. \[fig:measurement\](a) at each $p$, we changed the state variables (from $c$ and $\Pi$ to $\hat{c}$ and $\hat{\Pi}$), yielding the orange filled circles in Fig. \[fig:EOS\]. Remarkably, the gelation process in sol states ($p=0, 0.1, \dots, 0.5$) obeys the universal EOS (\[eq:EOS\]), although these systems continue to form multi-branched polymer clusters. Considering this in tandem with the semidilute scaling law observed in the gel states ($\Pi_{\mathrm{mix}}\propto c^{2.31}$), it is expected that $\hat{\Pi}_\mathrm{mix}$ in the gel states ($p_\mathrm{gel}<p\leq1$) will conform to the semidilute scaling law given by Eq. (\[eq:scaling\]) of *linear*-polymer solutions (red line in Fig. \[fig:result\](a)) with $K\simeq 1.1$.

Based on the above expectation, we propose a universal EOS of osmotic pressure $\Pi_\mathrm{mix}$ for polymer gels as $$K=\frac{\hat{\Pi}_\mathrm{mix}}{\hat{c}^{1/(3\nu-1)}}
\equiv \frac{M{c^{*}}^{1/(3\nu-1)} \Pi_\mathrm{mix}}{RTc^{\,3\nu/(3\nu-1)}},
\label{eq:scaling-gel}$$ where $K\simeq 1.1$. We note that $\hat{\Pi}_\mathrm{mix}/\hat{c}^{1/(3\nu-1)}$ is finite, although both $\hat{c}\equiv c/c^{*}$ and $\hat{\Pi}_\mathrm{mix}\equiv \Pi_\mathrm{mix} M/(cRT)$ diverge to infinity, because gels correspond to infinite molar mass $M\to\infty$ and $c^{*}\to 0$. In Fig. \[fig:result\](b), we demonstrate that $\hat{\Pi}_\mathrm{mix}/\hat{c}^{1/(3\nu-1)}$ converge to the universal value $K\simeq1.1$, which is independent of $p$ and $c$, after the gelation ($p_\mathrm{gel}\leq p\leq 1$). Thus, in the inset of Fig. \[fig:EOS\], the gel states are positioned at $(1/\hat{c},\hat{\Pi}_\mathrm{mix}/{\hat{c}}^{1.31}) \simeq (0, 1.1)$ (red star). We obtained Fig. \[fig:result\](b) by setting a constant value for $M{c^{*}}^{1/(3\nu-1)}$ and substituting $\Pi$ and $\Pi_\mathrm{mix}$ (shown in the inset of Fig. \[fig:result\](a)) into Eqs. (\[eq:scaling\]) and (\[eq:scaling-gel\]), respectively. (Further details are given in Sec. S6 of Supplemental Material.) This procedure demonstrates that we can determine $\Pi_\mathrm{mix}$ for any polymer gel by measuring a non-universal parameter $M{c^{*}}^{1/3\nu-1}$.\
*Concluding remarks*. — We have experimentally measured the osmotic pressure of polymer gels throughout the gelation process. We find that the universal EOS (\[eq:EOS\]) of the osmotic pressure for *linear*-polymer solutions describes the osmotic pressure throughout the gelation process involving both the sol and gel states (Fig. \[fig:EOS\]). In the sol states, both $\hat{\Pi}$ and $\hat{c}$ continuously increase according to the universal EOS (\[eq:EOS\]) with an increase in the average molar mass (orange filled circles in Fig. \[fig:EOS\]). In the gel states, the osmotic pressure of polymer gels is universally governed by the semidilute scaling law (\[eq:scaling-gel\]) (red star in Fig. \[fig:EOS\] and Fig. 4(b)). Here, both $\hat{\Pi}$ and $\hat{c}$ diverge to infinity, because the gel states correspond to the average molar mass $M\to\infty$ and the overlap concentration $c^{*}\to 0$. In addition, we have demonstrated that Eq. (\[eq:scaling-gel\]) enables the determination of $\Pi_\mathrm{mix}$ for any polymer gel by measuring a non-universal parameter $M{c^{*}}^{1/3\nu-1}$.

Because polymer gels are open systems that can swell and deswell by exchanging solvents with the human body, an understanding of the osmotic pressure is essential for controlling the swelling of polymer gels. Our findings are not only conceptually important to polymer physics, but also practically useful, encouraging biomedical applications of polymer gels such as soft contact lenses, adhesion barriers, sealants, and artificial vitreous humor.

We thank Masao Doi, Yuichi Masubuchi, Takashi Uneyama, and Xiang Li for their useful comments. This work was supported by the Japan Society for the Promotion of Science (JSPS) through the Grants-in-Aid for Early Career Scientists grant number 19K14672 to N.S., Scientific Research (B) grant number 18H02027 to T.S., and Scientific Research (S) grant number 16H06312 to U.C. This work was also supported by the Japan Science and Technology Agency (JST) CREST grant number JPMJCR1992 to T.S. and COI grant number JPMJCE1304 to U.C.

[32]{} url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}

, ** (, , ). , ** (, , ). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, ().

[^1]: These authors contributed equally: T. Yasuda, N. Sakumichi
