A flexible polymer chain in a critical solvent: Coil or globule?

As has been shown by Koga et al. [1], in the vicinity of a liquid-gas critical point of the solvent a dramatic expansion of a dissolved polymer chain takes place. Such anomalous conformational behavior of the polymer chain occurs due to strong solvent density fluctuations which arise in the neighborhood of the liquid-gas critical point. Up to now, only a few theoretical results have been reported on a thermodynamically stable conformation of a single polymer chain in a supercritical solvent in the neighborhood of the critical point. Dua and Cherayil [2] have developed a first-principles statistical theory of the isolated polymer chain in supercritical solvent and showed that on approaching the liquid-vapor critical point, a polymer chain first collapses (when the correlation radius is of the order of a polymer size) and then returns to its initial dimensions (when the correlation radius is much bigger than the size of the polymer chain). However, the authors fully ignored an effect of the solvent density renormalisation near the polymer backbone, although the latter is important in the case of strong polymer-solvent interactions [3–6]. Moreover, the authors did not specify the dependence of correlation length on thermodynamic parameters of the solvent.

Simmons and Sanchez [7] published a scaled particle theory for the coil-globule transition of a chain of attractive hard spheres in an attractive hard-sphere solvent. As was shown by these authors, in the vicinity of the solvent critical point only the collapse of the polymer chain takes place but not its expansion. In the framework of the theory of Simmons and Sanchez the effect of solvent density renormalization near the polymer chain was taken into account but the effect of critical solvent density fluctuations was missing. The same result was obtained by Erukhimovich [8] in the framework of the field-theoretical approach at the level of the Ginzburg-Landau theory within model of a compressible lattice gas. Sumi and co-authors [9] employing a classical density functional theory showed that on approaching the liquid-vapor critical point along the critical isochore, the polymer chain can undergo collapse at a so-called crossover temperature which is slightly higher than the critical temperature. Subsequently, below the crossover temperature near the critical point a very dramatic expansion of the polymer chain takes place. It was concluded that the dramatic expansion of the polymer chain is related to the so-called solvent-mediated interactions arising from strong solvent density fluctuations [10].

Despite the evident success in rationalising the conformational behavior of the polymer chain in the critical solvent, a clear understanding of the different conformational regimes has still not been reached. How does the globular or coiled state conformation of the polymer depend on the microscopic polymer-solvent interaction parameters in a critical solvent? Addressing this question we develop a simple analytical self-consistent field theory of an isolated polymer chain immersed in a low-molecular weight solvent.

The presented theory simultaneously takes into account two effects:

• effect of solvent density renormalisation near the polymer chain due to polymer-solvent interactions;
• indirect solvent-mediated monomer-monomer attractive interaction due to solvent density fluctuations near the critical point (so-called quasi-Casimir forces).

The presented study is based on the formalism, developed in our previous works [5,6]. We investigate the conformational behavior of the polymer chain approaching the critical point of the solvent along the critical isochore. In the case of a weak polymer-solvent attraction at the liquid-gas critical point the polymer chain is in a globular state. When the polymer-solvent attraction exceeds a threshold value the polymer undergoes a dramatic expansion at the critical point.

We consider an isolated polymer chain immersed in a low-molecular weight solvent at a specified number density ρ and temperature T. As in our previous works [5,6] we assume for convenience that the volume of the system consists of two parts: the gyration volume containing predominantly monomers of the polymer chain and the bulk solution. Our aim is to study the conformational behavior of a polymer chain in the vicinity of the solvent liquid-gas critical point as a function of the polymer-solvent interaction strength. We also assume that pair potentials of interactions monomer-monomer, monomer-solvent and solvent-solvent have the following form:

$$\begin{equation} V_{ij}(\mathbf{r})=\begin{cases} -\epsilon_{ij}\left(\frac{\sigma_{ij}}{r}\right)^6, \quad& |\mathbf{r}|> \sigma_{ij},\\ \infty,\quad&|\mathbf{r}|\leq \sigma_{ij}, \end{cases} \end{equation} \tag{ 1 }$$

where $i,j=m,s$; $\sigma_{ij}$ and $\epsilon_{ij}$ are effective diameters and energetic parameters, respectively. We consider the theory at the level of the mean-field approximation, however, taking into account the solvent density fluctuation effect on the monomer-monomer interaction. We describe the thermodynamics of the bulk solution by the van der Waals equation of state and use a Flory-type expression for the ideal part of the polymer free energy [11,12].

We would also like to stress that in contrast to our previous works [5,6] within the present theory we do not introduce the second virial coefficients as parameters of interactions, but we construct the total free energy by using different expressions which are straightforwardly related to repulsive and attractive parts of interaction potentials.

In order to solve of the posed problem, we start from an appropriate thermodynamic potential of a dilute polymer solution which can be expressed in the following form:

$$\begin{equation} \Omega(R_{g},N_{s})=F_{p}(R_{g})+F_{s}(N_{s},R_{g})+PV_{g}-\mu N_{s}, \end{equation} \tag{ 2 }$$

where R_g is the radius of gyration of the polymer chain, N_s is the number of solvent molecules within the gyration volume $V_{g}=4\pi R_{g}^3/3$, μ is the chemical potential of the solvent, and P is the pressure of the bulk solution. The polymer free energy in the framework of the mean-field approximation takes the form

$$\begin{eqnarray} F_{p}(R_{g})&=k_{B}T\left(\frac{9}{4}\left(\alpha^2+\frac{1}{\alpha^2}\right)-\frac{3}{2}\ln{\alpha^2}\right)\nonumber\\ &\quad -\,Nk_{B}T\ln\left(1-\frac{Nv_{m}}{V_{g}}\right)-\frac{a_{p}N^2}{V_{g}}, \end{eqnarray} \tag{ 3 }$$

where $v_{m}=2\pi\sigma_{m}^3/3$ is the van der Waals volume of monomers ($\sigma_{m}=\sigma_{mm}$ denotes the monomer's effective diameter),

$$\begin{equation} a_{p}=\frac{1}{2}\epsilon_{m}\int\limits_{r>\sigma_{m}}\text{d}\mathbf{r}\left(\frac{\sigma_{m}}{r}\right)^6=v_{m}\epsilon_{m} \end{equation} \tag{ 4 }$$

is the van der Waals attraction parameter of monomers, $\epsilon_{m}=\epsilon_{mm}$ is an energetic parameter of the monomer-monomer attraction, $\alpha=R_{g}/R_{0g}$ denotes the expansion factor, $R_{0g}^2=Nb^2/6$ is the mean-square radius of gyration of the ideal polymer chain, N is the degree of polymerization and b is the Kuhn length of the segment. The first term in (3) is the free energy of the ideal Gaussian polymer chain within the Fixman approximation [13,14]. Thus, to take into account the repulsive and attractive interactions between monomers we use a concept of "separate monomers" [15] and introduce the van der Waals-type expression for excess free energy of the polymer chain. The solvent free energy F_s within a mean-field approximation can be expressed in the following form:

$$\begin{eqnarray} F_{s}(R_{g},N_{s})&=N_{s}k_{B}T\left(\ln{\frac{N_{s}\Lambda_{s}^3}{V_{g}-N_{s}v_{s}-Nv_{ms}}}-1\right)\nonumber\\ &\quad -\,\frac{N_{s}^2a_{s}}{V_{g}}-\frac{NN_{s}a_{ps}}{V_{g}}- \frac{a_{ps}^2N_{s}^2N^2\chi_{T}}{2V_{g}^3}, \end{eqnarray} \tag{ 5 }$$

where $v_{s}=2\pi\sigma_{s}^3/3$ is the van der Waals volume of the solvent molecules ($\sigma_{s}=\sigma_{ss}$ is the effective diameter of the solvent molecules), $v_{ms}=4\pi \sigma_{ms}^3/3\ (\sigma_{ms}=(\sigma_{m}+\sigma_{s})/2)$ is the effective volume of monomer and solvent molecule at contact,

$$\begin{equation} a_{s}=\frac{1}{2}\epsilon_{s}\int\limits_{r>\sigma_{m}}\text{d}\mathbf{r}\left(\frac{\sigma_{s}}{r}\right)^6=v_{s}\epsilon_{s} \end{equation} \tag{ 6 }$$

is a solvent-solvent attraction parameter, $\epsilon_{s}=\epsilon_{ss}$ is an energetic parameter of the solvent-solvent attraction

$$\begin{equation} a_{ps}=\epsilon_{ms}\int\limits_{r>\sigma_{ms}}\text{d}\mathbf{r}\left(\frac{\sigma_{ms}}{r}\right)^6=v_{ms}\epsilon_{ms} \end{equation} \tag{ 7 }$$

is a monomer-solvent attraction parameter, $\epsilon_{ms}$ is an energetic parameter of the monomer-solvent attraction. The first term in (5) describes the ideal contribution and contributions of the solvent-solvent and monomer-solvent exluded volume interactions. It should be noted that in (5) we use the simplest possible method to account for the so-called depletion forces [16] which may lead to an additional repulsion of the solvent molecules from the gyration volume due to the presence of monomers. The second and third terms correspond to the solvent-solvent and monomer-solvent attraction, respectively. The forth term is a first correction to the mean-field approximation in the framework of the cumulant expansion [5] and describes the contribution of solvent-mediated monomer-monomer interaction that is due to solvent density fluctuations (quasi-Casimir forces); $\chi_{T}$ is the isothermal compressibility of the solvent in the gyration volume. The fluctuation correction to the mean-field approximation is not related to the third virial contribution, though it has a similar functional form. It should be emphasized, that within our theory we deal with two independent order parameters —the radius of gyration of the polymer chain and number of the solvent molecules that are in gyration volume.

The solvent chemical potential μ and pressure in the bulk solution P in our model are determined by the following mean-field expressions:

$$\begin{eqnarray} &&\frac{\mu(\rho,T)}{k_{B}T}=\ln(\rho\Lambda_{s}^3)+\frac{\rho v_{s}}{1-\rho v_{s}}-\ln\left(1-\rho v_{s}\right)-\frac{2 a_{s}\rho}{k_{B}T}, \end{eqnarray} \tag{ 8 }$$

and

$$\begin{equation} P(\rho,T)=\frac{k_{B}T\rho}{1-\rho v_{s}}-a_{s}\rho^2, \end{equation} \tag{ 9 }$$

where ρ is a number density of the solvent in the bulk solution. The expression (9) is a well-known van der Waals equation of state. We assume that the gyration volume is sufficiently large, so that the surface layer does not contribute into free energy of the solution.

The equilibrium values N_s and R_g are determined from the minimum conditions of the thermodynamic potential Ω, i.e. from the equations

$$\begin{equation} \frac{\partial{\Omega}}{\partial{N}_{s}}=0, \qquad \frac{\partial{\Omega}}{\partial{R_{g}}}=0. \end{equation} \tag{ 10 }$$

Substituting (2) into eqs. (10) and using the expressions (3) and (5) we arrive at the following system of coupled equations:

$$\begin{eqnarray} &\alpha^5-\frac{2}{3}\alpha^3-\alpha =\frac{3\sqrt{6}}{\pi b^3}\sqrt{N}\left(\frac{v_{m}}{1-\frac{9\sqrt{6}v_{m}}{2\pi\sqrt{N}\alpha^3b^3}}-\beta a_{p}-B\right)\nonumber\\ &-\,\frac{2}{3}N\beta a_{ps}\rho_{s}\alpha^3-\frac{2\pi N^{3/2}\beta\alpha^{6}b^3}{81}\left(P-P_{g}\right), \end{eqnarray} \tag{ 11 }$$

and

$$\begin{eqnarray} &&\rho_{s}=\rho e^{\frac{9\sqrt{6} a_{ps}}{2\pi\sqrt{N}\alpha^3b^3 k_{B}T}- \frac{\mu_{ex,g}(\rho_{s},\alpha,T)-\mu_{ex,b}(\rho,T)}{k_{B}T}}, \end{eqnarray} \tag{ 12 }$$

where we have introduced the solvent number density in the gyration volume $\rho_{s}=N_{s}/V_{g}$, the excess chemical potentials of the solvent $\mu_{ex,g}$, $\mu_{ex,b}$ in the gyration volume and in the bulk solution, respectively, and the inverse temperature $\beta =1/k_{B}T$. The solvent pressure in the gyration volume P_g is given by the following expression:

$$\begin{equation} P_{g}=\frac{\rho_{s}k_{B}T}{\gamma-\rho_{s}v_{s}}-a_{s}\rho_{s}^2, \end{equation} \tag{ 13 }$$

where $\gamma=1-\frac{9\sqrt{6}v_{ms}}{2\pi\sqrt{N}\alpha^3b^3}$. The value B determines the influence of the solvent density fluctuations on the value of radius of gyration and has the following form:

$$\begin{eqnarray} B&=\frac{a_{ps}^2\rho_{s}}{\left(k_{B}T\right)^2}\nonumber\\ &\quad \times\,\frac{(\gamma-\rho_{s}v_{s})\left(\gamma(\gamma-2\rho_{s}v_{s})- \frac{\rho_{s}a_{s}}{k_{B}T}(\gamma-\rho_{s}v_{s})^3\right)}{\left(\gamma-\frac{2a_{s}\rho_{s}}{k_{B}T}(\gamma-\rho_{s}v_{s})^2\right)^2}.\nonumber\\ \end{eqnarray} \tag{ 14 }$$

The excess chemical potential of the solvent in the gyration volume can be obtained by the following relation:

$$\begin{eqnarray} &\mu_{ex,g}(\rho_{s},\alpha,T)=k_{B}T\left(\frac{\rho_{s} v_{s}}{\gamma-\rho_{s} v_{s}}- \ln\left(\gamma-\rho_{s} v_{s}\right)\right)\nonumber\\ &-\,2 a_{s}\rho_{s}-\frac{243 a_{ps}^2\gamma}{4\pi^2 N\alpha^6 b^6k_{B}T}\frac{(\gamma-\rho_{s}v_{s})(\gamma-3\rho_{s}v_{s})}{\left(\gamma-\frac{2a_{s}\rho_{s}}{k_{B}T}(\gamma-\rho_{s}v_{s})^2\right)^2}. \end{eqnarray} \tag{ 15 }$$

The first and second terms in the expression (15) determine the mean-field contribution to the excess chemical potential of the exluded volume and the attraction interactions, respectively. The third term determines the contribution of the solvent density fluctuations in the gyration volume.

It should be noted that similar pure mean-field theories have been developed by Erukhimovich [8] for the case of infinitely long polymer chain and by Simmons and Sanchez [7] for the polymer chain of finite degree of polymerization. However, in both works only collapse of the polymer chain in the vicinity of the liquid-vapor critical point has been discussed. In order to describe the coil-globule transition in the critical solvent Erukhimovich used a model of a compressible lattice gas at the level of the mean-field approximation, whereas Simmons and Sanchez used a scaled particle theory. However, it is unclear how the phenomenological parameters of interaction within the lattice gas model are related to the pair potentials of interactions between components of the solution. Moreover, to determine a parameter of the polymer-solvent attraction $\epsilon_{ms}$ in both works a standard Berthelot rule $\epsilon_{ms}=\sqrt{\epsilon_{m}\epsilon_{s}}$ was used, which obviously cannot be justified for the case of polymer solution. In the present study in contrast to work [8] as well as to work [7] we directly introduce the pair potentials interactions between components of the solution (1) using parameters of interactions $\epsilon_{ms}$, $\epsilon_{m}$, and $\epsilon_{s}$ as independent variables. In addition, in contrast to works [7,8] we have taken into account the solvent-mediated fluctuation interactions between the monomers and obtain a new regime of conformational behavior of the polymer, namely its strong expansion when solvent reaches the liquid-vapour critical point (see the next section).

For convenience within the numerical calculations we introduce dimensionless densities $\tilde{\rho}_{s}=\rho_{s}v_{s}$ and $\tilde{\rho}=\rho v_{s}$, and dimensionless interaction parameters $s=\sigma_{m}/\sigma_{s}$, $t=\epsilon_{m}/\epsilon_{s}$, $u=\epsilon_{ms}/\epsilon_{s}$. Using these definitions we rewrite eqs. (11), (12) in the following form:

$$\begin{eqnarray} &\alpha^5-\frac{2}{3}\alpha^3-\alpha=\frac{3\sqrt{6}\sqrt{N}}{\pi \tilde{b}^3}\left(\frac{s^3}{1-\frac{9\sqrt{6}s^3}{2\pi\sqrt{N}\alpha^3\tilde{b}^3}}- \frac{ts^3}{\tilde{T}}-\tilde{B}\right)\nonumber\\ &-\,\frac{\tilde{\rho}_{s}u(1+s)^3N\alpha^3}{6\tilde{b}^3} -\frac{2\pi N^{3/2}\tilde{b}^3\alpha^{6}}{81\tilde{T}}\left(\tilde{P}-\tilde{P}_{g}\right), \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} &\tilde{\rho}_{s}=\tilde{\rho}e^{\frac{9\sqrt{6}u(1+s)^3}{8\pi\sqrt{N}\alpha^3\tilde{b}^3\tilde{T}}+\frac{\tilde{\mu}_{ex,b}(\tilde{\rho},\tilde{T})-\tilde{\mu}_{ex,g}(\tilde{\rho}_{s},\alpha,\tilde{T})}{\tilde{T}}}, \end{eqnarray} \tag{ 17 }$$

where $\tilde{T}=k_{B}T/\epsilon_{s}$ is a dimensionless temperature, $\tilde{\mu}_{ex,b}=\mu_{ex,b}/\epsilon_{s}$ and $\tilde{\mu}_{ex,g}=\mu_{ex,g}/\epsilon_{s}$ are dimensionless excess chemical potentials of the solvent in the bulk solution and in the gyration volume, respectively; $\tilde{B}=B/v_{s}$; $\tilde{b}=b/v_{s}^{1/3}$ is a dimensionless Kuhn length of the segment. Moreover, we have introduced the dimensionless pressures $\tilde{P}=Pv_{s}/\epsilon_{s}$ and $\tilde{P}_{g}=P_{g}v_{s}/\epsilon_{s}$.

Turning to the numerical analysis of the system of equations (16), (17) we fix the index of polymerization $N=10^{3}$, dimensionless Kuhn length of the segment $\tilde{b}=s\left(\frac{3}{2\pi}\right)^{1/3}$ (so that $b=\sigma_{m}$), and t = 1.

We first discuss the conformational behavior of the polymer chain approaching the liquid-gas critical point of the solvent along the critical isochore for different values of the polymer-solvent attraction parameter u. In fig. 1 two different regimes of the behavior of expansion factor α are shown. At small values of u (weak polymer-solvent attraction) the polymer chain collapses at the critical point. However, when the polymer-solvent attraction is strong a qualitatively different behavior of expansion factor takes place. Namely, decreasing the temperature along the critical isochore the polymer chain first collapses at a temperature higher than the critical temperature and then subsequently strongly expands in a small vicinity of the critical point. Such conformational behavior is in agreement with the numerical results of Sumi et al. [9] and with the experimental results of Koga et al. [1]. Figure 2 shows the expansion factor $\alpha_{c}=\alpha(\tilde{\rho}_{c},\tilde{T}_{c})$ at the critical point as a function of u for different s. In the case of a weak polymer-solvent attraction (small u), the expansion factor α in the vicinity of the liquid-gas critical point is effectively constant and does not depend on u. However, when the parameter u exceeds a certain threshold value the expansion factor monotonically increases. Such nontrivial behavior of the expansion factor is related to the fact that at weak polymer-solvent interaction (small u) the polymer is "solvophobic". In this case the local number density of solvent near the polymer chain is much less than the number density in the bulk solution $(\tilde{\rho}_{s}\ll \tilde{\rho})$, so that the polymer collapses due to the pressure difference between the gyration volume and the bulk solution.

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In the case of a sufficiently strong polymer-solvent attractive interaction the fluctuation contibution to the solvent excess chemical potential becomes important. The latter leads to an additional effective attraction between polymer backbone and solvent molecules thus increasing the number density of solvent within the gyration volume and eventually equilibrates the pressures between the gyration volume and the bulk solution, so that the polymer chain becomes "solvophilic". In this case the density of the solvent within the gyration volume becomes comparable to the solvent density in the bulk solution $(\tilde{\rho}_{s}\sim \tilde{\rho})$, while the polymer chain expands to a coiled conformation. Further increasing of the polymer-solvent attraction leads to enhancement of the solvent density in gyration volume and shrinking of the polymer coil. The latter is related to the fact that strong polymer-solvent attraction can compress the coil from within [5,17,18]. It should be noted that solvent-mediated interactions between monomers that are due to the critical solvent density fluctuations only enhance this effect.

It is instructive to regard the conformational behavior of the polymer chain for strong polymer-solvent attraction in a region above the critical point. Particulary relevant is the case when the temperature is increased along an isobar since this is an often realised situation in experiments.

As shown in fig. 3, the expansion factor with increasing temperature T first monotonically decreases, attaining a local minimum, and then monotonically increases. It should be emphasized, that the observed minimum of expansion factor is most pronounced at the critical isobar, where the fluctuations of solvent density are largest. Such behaviour of the expansion factor is in qualitative agreement with the results of Monte Carlo computer simulations reported in ref. [19].

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Employing a previously published [5,6] self-consisted field theory, incorporating a local solvent density renormalisation near the polymer chain and the effect of solvent-mediated fluctuation interaction monomer-monomer we have described the conformational changes of a single polymer chain in low-molecular weight solvent at critical and supercritical parameters of state. Two qualitatively different regimes were found depending on the attraction strength between the polymer and the solvent. When the polymer-solvent attraction is weak progressing along the critical isochore the polymer undergoes a collapse to a globular state at the liquid-gas critical point. In contrast when the polymer-solvent attraction is strong the polymer first collapses, attains a minimum at a temperature higher than the critical one and strongly expands to a coiled conformation at the critical temperature. These effects are caused by the critical fluctuations of the solvent density in the vicinity of the liquid-gas critical point.

Previously reported results on conformational changes of a polymer chain have shown two qualitative different regimes of the polymer chain —the collapse of the polymer has been obtained by theory and simulations [7,8,19], whereas the expansion of the polymer has been obtained theoretically by several authors [2,9]. The presented theory incorporating critical density fluctuations and local solvent density renormalisation shows that both regimes are indeed possible, depending on the monomer-solvent interaction parameter. We would like to stress that similar results were obtained by Vasilevskaya et al. in ref. [20] by means of the hybrid self-consistent MC/RISM method.

We would also like to mention that a more rigorous theory for the description of the coil-globule transitions has been developed in the works of Lifshitz and co-authors [15] based on the idea that the globule can be treated as a fragment of a semi-dilute polymer solution. In contrast to Flory-type theories the behavior of a globule within the Lifshitz theory has been described in terms of the density functional theory, which allows to describe the globule's surface layer with implicit account of the solvent. Thus, the Lifshitz theory introduces an additional length scale which can be interpreted as the thickness of the surface layer of the globule R_s (an effective width of globule's "fringe" [15]) which smaller than R_g. It is clear that within simple Flory-type approaches the globule can undergo the dramatic expansion as whole in the case when $\xi\simeq \text{const} \times\tau^{-\nu}\geq R_{g}\ (\tau=(T-T_{c})/T_{c}$, ν is one of the critical exponents). However within the more sophisticated Lifshitz theory one can investigate the case in which the correlation length of the solution $\xi \simeq R_{s}$. Thus in this case one can expect the new effect when the expansion of globule's surface will take place. In other words, in this case the expansion of the globule will be at much wider temperature range than what predicted within simple mean-field theories, namely at $\tau \leq \text{const}\times R_{s}^{-1/\nu}$. However, such speculations require more detailed investigations within the Lifshitz-type approaches which can be a subject of forthcoming publications.

The described phenomena may be relevant for technologies where supercritical carbon dioxide is used as a solvent to process or synthesize polymers [21–27], produce polymer particles of controled dimensions [28] or to encapsulate bioactive molecules such as drugs, enzymes, proteins into polymer particles [29,30]. Controling thermodynamic parameters of the solvent allows to change the polymer solubility, its conformation and in turn the concentration of solvent within the gyration volume.

The research leading to these results has received funding from the European Union's Seventh Framework Program (FP7/2007-2013) under grant agreement No. 247500 with project acronym "Biosol". The part concerning development of theoretical model has been supported by Russian Scientific Foundation (grant No. 14-33-00017).