Lower Critical Solution Temperature Behavior in Polyelectrolyte Complex Coacervates

Abstract

In light of recent experimental observations of lower critical solution temperature (LCST) in polyelectrolyte complex coacervates (Ali, S. et al. ACS Macro Lett. 2019, 8, 289–293), we explore its possible mechanisms on the basis of a slight modification of our theory (Adhikari, S. et al. J. Chem. Phys. 2018, 149, 163308). We explore the consequences of the temperature dependence of the solvent dielectric constant (ε) and the solvent−polymer interaction parameter (χ) on the complex coacervates’ phase behavior. The results show that the temperature dependence of the solvent dielectric constant and solvent−polymer interaction parameter can result in a complex phase behavior involving two disjoint unstable regions on the temperature (T)−polyelectrolyte concentration (ϕ_p) plane. Comparison of phase diagrams constructed for different possible temperature dependencies of ε and χ shows that the experimentally observed LCST behavior is obtained only if the solvent dielectric constant decreases and the solvent−polymer interaction parameter increases with increasing temperature. Preferential partitioning of salt into the polyelectrolyte poor phase is predicted for all possible combinations of temperature dependencies of χ and ε considered in this work.

Graphical Abstract

INTRODUCTION

Complex coacervation, the liquid−liquid phase separation of a solution of oppositely charged polyelectrolytes into a polyelectrolyte rich complex coacervate phase and a dilute aqueous phase, has been studied for several decades.^1–56 In general, the thermodynamic behavior of a complex coacervate depends on several variables such as the temperature, salt concentration, polyelectrolyte backbone charge density, polyelectrolyte molecular weight, and polycation−polyanion composition asymmetry. Most of the theoretical models of complex coacervation^{38,40–42,46–48,51–53} including the classic Voorn−Overbeek model^2–5 generally focus on the effect of salt concentration and the polyelectrolyte backbone charge density on the phase behavior and predict an upper critical salt concentration. Explicit temperature dependence of complex coacervation is theoretically less well-known mainly because of the difficulty in capturing the variation of the solvent dielectric constant with temperature. Under the assumption of a constant solvent dielectric constant, our previous work⁵³ predicts an upper critical solution temperature (UCST) behavior.

Measurements of polyelectrolyte complex coacervate phase diagrams also have primarily focused on the total salt concentration versus total polyelectrolyte concentration.^17,28,33,34 These phase diagrams are determined by separately analyzing the coexisting liquid phases for polyelectrolyte and salt concentrations as a function of initial concentrations within the two-phase region. This allows an estimate of the binodal curve with corresponding tie lines. Two general features are observed: there is an upper critical salt concentration behavior such that an increasing monovalent salt concentration leads to the one-phase region, and second, the tie-line slopes are not horizontal but negative and dependent upon the quench depth from the apparent critical salt concentration. These observations are general for 1:1 by monomer-charge stoichiometric polyelectrolyte mixtures with added monovalent salt at fixed temperature. Recent measurements³² show that the underlying aqueous phase behavior for potassium poly(styrene sulfonate) (KPSS) and poly(diallyl dimethylammonium) bromide (PDADMAB) with added KBr exhibits phase separation upon heating or lower critical solution temperature behavior, complementing the C_s−C_p (salt−polyelectrolyte concentration) study by Wang and Schlenoff.²⁸ These new observations point out a need to understand the interplay of electrostatics and the van der Waals interactions to understand both sets (UCST and LCST) of phase diagrams.

Whether the thermodynamics of a polyelectrolyte solution exhibits an upper critical solution temperature (UCST), lower critical solution temperature (LCST), or both mainly depends upon how various quantities such as the interaction energies among polymers and solvent molecules, the solvent dielectric constant, and the degree of ionization of the polymer backbones are affected by the temperature. In this paper, we explore complex coacervate phase behavior considering different cases arising from the temperature dependencies of the solvent dielectric constant (ε) and solvent−polymer interaction parameter (χ). We compare the constructed phase diagrams with recent experimental observations³² and suggest possible mechanisms for the LCST phase behavior observed in the experiments.

THEORY

We consider a symmetric solution of polycations, polyanions, salt, and solvent molecules where the polycations and polyanions are equal in number and identical except that they have opposite charges. For simplicity, we consider that there are no counterions from polymers (but salt is present), which is relevant to experiments in which counterions are washed out of the system before adding salt.⁴⁷ The free energy density f of this system, assuming the existence of polycation−polyanion pair complexation in the homogeneous solution,⁵² can be written as

$$f=\frac{{\varphi }_{\text{p}}}{2N}\text{ln}\left({\varphi }_{\text{p}}\right)+{\varphi }_{\text{s}}\text{ln}\left(\frac{{\varphi }_{\text{s}}}{2}\right)+{\varphi }_0\text{ln}\left({\varphi }_0\right)+\frac{1}{2}{v}_{\text{dd}}{\left(\frac{{\varphi }_{\text{p}}}{2}\right)}^2+\chi {\varphi }_{\text{p}}{\varphi }_0-\frac{1}{4\pi }\left(\text{ln}(1+\kappa l)-\kappa l+\frac{1}{2}{\kappa }^2{l}^2\right)$$ (1)

where ϕ_p, ϕ_s, and ϕ₀ are the volume fractions of total polyelectrolytes, salt ions, and solvent molecules, respectively. The first three terms are for mixing entropy, the fourth term accounts for the dipole−dipole interaction energy, the fifth term is for the solvent−polymer interaction, and the last term accounts for the electrostatic energy of correlations of ions. N is the number of Kuhn segments in each chain, and v_dd parameterizes the electrostatic dipole−dipole interaction energy between two dipoles and is given by

$$v_{\text{dd}}=-\frac{\pi }{9}\frac{l_{\text{B}}^2{p}^4}{l^6}{\text{e}}^{-2\kappa l}\left[4+8\kappa l+4{(\kappa l)}^2+{(\kappa l)}^3\right]$$ (2)

where l_B is the Bjerrum length, p is the dipole length, l is the Kuhn length, and 1/κ is the Debye screening length. There are three quantities that depend on temperature (T): l_B, κ, and χ. The temperature dependence of l_B is

$$l_{\text{B}}(T)=\frac{e^2}{4\pi {\epsilon }_0\epsilon (T)k_{\text{B}}T}$$ (3)

where e is the electronic charge, ε₀ is the permittivity of vacuum, ε is the dielectric constant of the medium, and k_B is the Boltzmann constant. For water, the temperature dependence of its dielectric constant can be explained by an empirical relation⁵⁷

$$\epsilon (T)=87.740-0.40008(T-273)+9.398\times {10}^{-4}{(T-273)}^2-1.410\times {10}^{-6}{(T-273)}^3$$ (4)

where ε(T) is a monotonically decreasing function of the absolute temperature (T). The temperature dependence of κ appears through l_B(T) as

$${\kappa }^2(T)=4\pi \frac{l_{\text{B}}(T)}{l^3}{\varphi }_{\text{s}}$$ (5)

We consider two different temperature dependencies of χ given by

$$\chi =\frac{\theta }{2T}$$ (6)

and

$$\chi =a+bT$$ (7)

where θ, a, and b are constant parameters.

By using the free energy given by eq 1, the binodal for two-phase coexistence is computed through the following algorithm. First, some valid composition for a homogeneous solution (meaning the volume fractions of all the components) is specified. With pair complexation, it is a ternary system where ϕ_p, ϕ_s, and ϕ₀ fully specify the system. Second, the system is partitioned into two phases (phases A and B) for which there are three independent variables (the ratio of volumes of two phases, ϕ_p in phase A, and ϕ_s in phase A, for example). From these independent variables, the compositions of both phases can be determined by conditions of incompressibility, mass conservation, and charge neutrality. Third, the free energy of the phase-separated system is minimized using the downhill simplex algorithm by adjusting the three independent variables. This minimization results in either (1) two identical phases in which case the system is stable as a homogeneous mixture or (2) two distinct phases, corresponding to a pair of coexisting points at equilibrium. Collecting the set of these coexisting points gives a binodal curve.

RESULTS AND DISCUSSION

We construct phase diagrams for the following four combinations arising from the possible temperature dependence of ε and χ:

1. ε = ε(T) (eq 4) and $\chi (T)=\frac{\theta }{2T}$

2. ε = ε(T_room) and χ = a + bT where T_room = 298 K is the room temperature.

3. ε = ε(T) and χ = a + bT

4. ε = ε(T_room) and $\chi =\frac{\theta }{2T}$

A. ε = ε(T) and χ(T) = θ/2T.

The first case we explore is when ε(T) is given by eq 4 and χ(T) = θ/2T. In this case, both ε and χ monotonically decrease with increasing T. Figure 1 shows temperature (T) versus polyelectrolyte concentration (ϕ_p) coexistence curves for three different salt concentrations (ϕ_st) of the homogeneous solution (starting solution). For all the three salt concentrations of the starting solution, the polyelectrolyte concentration (volume fraction) is taken to be 0.1, which is marked by a vertical dashed line in the phase diagram. The phase diagrams for all three salt concentrations (indexed by different colors) show that phase separation occurs only above a certain threshold temperature, and the window of phase instability widens on increasing the temperature, exhibiting an LCST behavior.

Figure 1.

Temperature (T) vs polyelectrolyte concentration (ϕ_p) coexistence curves for given salt (ϕ_st) and polyelectrolyte concentrations (ϕ_pt) of starting solutions. Different salt concentrations (ϕ_st), shown in the legend, are considered while the polyelectrolyte concentration (ϕ_pt) is equal to 0.1 (marked by the dashed vertical line) for all the cases. The set of parameters includes θ = 140 K, p/l = 1.165, and l_B/l = 1.2 at room temperature and N = 100.

The phase separation in Figure 1 is driven by the combined effect of dipole−dipole attractions and solvent−polymer interactions. On increasing the temperature, the dielectric constant decreases, resulting in an increase in the magnitude of dipole−dipole interaction energy; this renders the phase separation favorable in higher temperatures, which tends to cause an LCST behavior. On the other hand, the magnitude of solvent−polymer interaction energy increases with lowering the temperature and tends to result in the UCST behavior. However, we get only the LCST behavior because the parameters chosen here favor the dipole−dipole interaction over the solvent−polymer interaction. For a comparison of their relative magnitudes, at 330 K, χ = θ/2T = 0.21 and v_dd = − 3.26 at the same temperature and for the set of parameters from Figure 1 with ϕ_st = 0.020. The dashed vertical line in Figure 1 marks the concentration of the fixed total polyelectrolyte solution. Since the total concentration of the polyelectrolyte is fixed at 0.1, it is not possible to have both the coexisting points lying to the right of the dashed line. This is why the binodal lines stop at the dotted line. From the parameter set used in Figure 1, increasing the value of solvent interaction parameter θ results in two different unstable regions: one exhibiting LCST and the other exhibiting UCST behavior, as shown in Figure 2. It is an interesting demonstration of how two independent interactions in a multicomponent system could result in multiple disjoint unstable regions.

Figure 2.

Temperature (T) vs polyelectrolyte concentration (ϕ_p) coexistence curves showing two disjoint unstable regions, one at the top and another at the bottom. Salt and polyelectrolyte concentrations of the starting solution are ϕ_st = 0.02 and ϕ_pt = 0.1, respectively. The dashed line marks the total polyelectrolyte concentration of the starting solution. The parameter set includes θ = 145 K, p/l = 1.165, and l_B/l = 1.2 at room temperature and N = 100.

Figure 3a shows the variation of the threshold temperature (T_th) for the LCST behavior shown in Figure 1 as a function of salt concentration (ϕ_st) for a fixed polyelectrolyte concentration of the starting solution. The threshold temperature (T_th) for a starting homogeneous solution of specified polyelectrolyte and salt concentrations is the temperature above which the solution phase separates. It shows that the threshold temperature for a given polyelectrolyte concentration of the starting solution increases with the increase in salt concentration, which can be rationalized in terms of the screening of dipole−dipole interactions due to salt. Figure 3b shows the variation of the threshold temperature with polyelectrolyte concentration at a fixed salt concentration of the starting solution. This nonmonotonic variation of T_th with ϕ_pt suggests a complex interplay of entropy and the different interactions involved, which requires more exploration for a full understanding.

Figure 3.

(a) Threshold temperature (T_th) vs salt concentration (ϕ_st) at a constant polyelectrolyte concentration given by ϕ_pt = 0.1 and (b) threshold temperature (T_th) vs polyelectrolyte concentration (ϕ_pt) at a constant salt concentration of ϕ_st = 0.02. The parameters are the same as in Figure 1.

B. ε = ε(T_room) and χ = a + bT.

The second case we consider is when ε = ε(T_room) (a constant) and χ = a + bT. Figure 4 shows temperature (T) versus polyelectrolyte concentration (ϕ_p) coexistence curves for different salt concentrations of the starting solution. Coexistence curves for all three starting solutions of given salt (ϕ_st) and polyelectrolyte (ϕ_pt) concentrations show that the phase separation occurs only above a threshold temperature, and the unstable region in the temperature−polyelectrolyte concentration plane grows wider with increasing the temperature. The values of parameters a and b chosen in Figure 4 result in the solvent−polymer interaction parameter (χ) being less than 0.5 within the range of temperature considered, for example, χ = 0.423 at T = 310 K, and it is 0.488 at 360 K. Although the magnitude of χ is not large enough to cause the phase separation alone, the phase separation occurs from the resultant effect of solvent−polymer interactions and electrostatic dipole−dipole attractions that we call “enhancement of effective hydrophobicity” of polyelectrolytes due to the dipolar interactions. The threshold temperature (T_th) for phase separation increases with increasing salt concentration at a fixed polyelectrolyte concentration of the starting solution, Figure 5a, and with increasing polyelectrolyte concentration at a fixed salt concentration of the starting solution, Figure 5b. The increase of T_th with increasing ϕ_st is due to the screening of electrostatic dipole−dipole interaction due to salt.

Figure 4.

Temperature (T) vs polyelectrolyte concentration (ϕ_p) coexistence curves for different salt concentrations (ϕ_st) and a fixed polyelectrolyte concentration (ϕ_pt) (marked by the dashed line) of starting solutions. The set of parameters used includes a = 0.02, b = 0.0013, p/l = 1, and l_B/l = 1.2 at room temperature and N = 100.

Figure 5.

(a) Threshold temperature (T_th) vs salt concentration (ϕ_st) at a constant polyelectrolyte concentration given by ϕ_pt = 0.1 and (b) threshold temperature (T_th) vs polyelectrolyte concentration (ϕ_pt) at a constant salt concentration of ϕ_st = 0.02. The parameters are the same as in Figure 4.

In Figure 4, if the parameters l_B and p are tuned to increase the magnitude of the dipole−dipole interaction energy while keeping the other parameters constant, a phase diagram similar to Figure 2 with both LCST and UCST behavior is obtained. This is because the dipole−dipole interaction energy increases with decreasing the temperature, resulting in the UCST behavior while the solvent−polymer interaction energy increases with increasing the temperature resulting in the LCST behavior.

C. ε = ε(T) and χ = a + bT.

The third case is when ε(T) is given by eq 4 and χ = a + bT. If the dielectric constant decreases while the solvent−polymer interaction increases with increasing the temperature, it results in only LCST behavior as shown in Figure 6. It is because the strength of both dipole−dipole and solvent−polymer interactions increase with increasing the temperature. Phase separation is driven by a combined effect of solvent−polymer interactions and the dipole−dipole interactions. At the lowest temperature considered in Figure 6, which is 278 K, the value of χ is χ = a + bT = 0.38. This magnitude of χ is not big enough for the phase separation to occur if the dipole−dipole interactions and the ion correlations are absent, suggesting that all the three terms are important to drive the phase separation. With dipole−dipole interactions turned off, we found that the solution phase separates only when χ > 0.6. Also, setting the ion correlation term to zero while keeping the other two terms intact does not cause a significant change in the phase behavior. The above observations together show that the phase separation is caused by the solvent−polymer and the dipole−dipole interactions acting together. The threshold temperature for phase separation (T_th) increases with increasing both the salt concentration (at constant polyelectrolyte concentration of the starting solution), shown in Figure 7a, and polyelectrolyte concentration (at constant salt concentration of the starting solution), shown in Figure 7b.

Figure 6.

Temperature (T) vs polyelectrolyte concentration (ϕ_p) coexistence curves for different salt concentrations (ϕ_st) and fixed polyelectrolyte concentration (ϕ_pt) (marked by the dashed line) of starting solutions. The same parameters are used as in Figure 4.

Figure 7.

(a) Threshold temperature (T_th) vs salt concentration (ϕ_st) at a constant polyelectrolyte concentration given by ϕ_pt = 0.1 and (b) threshold temperature (T_th) vs polyelectrolyte concentration (ϕ_pt) at a constant salt concentration of ϕ_st = 0.02. The parameters are the same as in Figure 4.

D. ε = ε(T_room) and $\chi =\frac{\theta }{2T}$.

The last case we consider is when ε(T) is a constant and $\chi =\frac{\theta }{2T}$, which is discussed in detail in our previous paper,⁵³ and we include it here for completeness. Temperature (T) versus polyelectrolyte concentration (ϕ_p) coexistence curves in this case (see Figure 8) show UCST behavior only. We do not find any LCST behavior in this case. It is because both the dipole−dipole interactions and the solvent−polymer interactions become weaker with increasing the temperature. The threshold temperature for phase separation (T_th) in this case decreases with both increasing salt concentration (at constant polyelectrolyte concentration, as shown in Figure 9a) and increasing polyelectrolyte concentration (at constant salt concentration, as shown in Figure 9b.

Figure 8.

Temperature (T) vs polyelectrolyte concentration (ϕ_p) coexistence curves for different salt concentrations (ϕ_st) and fixed polyelectrolyte concentration (ϕ_pt) (marked by the dashed line) of the starting solution. The parameters are θ = 257 K, p/l = 1, and l_B/l = 1.2 at room temperature and N = 100.

Figure 9.

(a) Threshold temperature (T_th) vs salt concentration (ϕ_st) at a constant polyelectrolyte concentration given by ϕ_pt = 0.1 and (b) threshold temperature (T_th) vs polyelectrolyte concentration (ϕ_pt) at a constant salt concentration of ϕ_st = 0.02. The parameters are the same as in Figure 8.

Slope of Tie Lines in the Polyelectrolyte−Salt Concentration Plane.

For all the four cases considered in this work, we found the preferential partitioning of salt into the polyelectrolyte poor phase. We do not discuss in detail the phase diagrams in the ϕ_s−ϕ_p (salt−polyelectrolyte concentration) plane for all the four cases since the focus of this work is on the LCST behavior. For a representative phase diagram in the ϕ_s−ϕ_p plane, we consider again the case when ε = ε(T) and χ = a + bT (case C discussed above). Figure 10 shows a coexistence curve for salt versus total polyelectrolyte concentration at T = 310 K. It shows that phase separation occurs only if the salt concentration is less than the threshold. Also, the negative slope of tie lines shows that the salt preferentially partitions into the polyelectrolyte poor phase. This partitioning of salt arises due to a combined effect of electrostatic screening and excluded volume. Expulsion of salt from the dipole-rich phase strengthens the dipolar interactions and hence reduces the total electrostatic energy of the system. The removal of salt ions from a particular phase may not be entropically favorable, and competition of the two effects sets the tie line slope. This prediction of preferential partitioning of salt to the dilute phase is consistent with recent experiments,^33,47 theories,^38,46,55,56 and simulations.^33,47

Figure 10.

Salt (ϕ_s) vs polyelectrolyte (ϕ_p) concentration coexistence curve at temperature T = 310 K. Black straight lines are tie lines, and parameters used are the same as in Figure 6.

CONCLUSIONS

We explored the consequences of the temperature dependence of the dielectric constant of water and the solvent−polymer interaction parameter on the complex coacervate phase behavior.

Among the four different cases we considered, for case A (Figure 1), while this shows the LCST behavior, the polyelectrolyte concentration in the equilibrium coacervate phase does not change appreciably with increasing salt in disagreement with the measurements. Only the dilute phase shows an increase in polyelectrolyte concentration with increasing salt concentration, in partial agreement with the measurements. Therefore, case A with the temperature dependence in the dielectric constant and traditional scaling of the Flory−Huggins interaction parameter may not be the correct parameterization, even though it leads to LCST phase diagrams. The LCST trend observed in Figures 4 and 6 and the increase in threshold temperature with increasing salt concentration or polyelectrolyte concentration are consistent with experimental cloud point temperature behavior. Based on these trends, one may not distinguish experiment from theory for cases B and C. However, case B with a fixed dielectric constant does not agree with experiment,⁵⁷ implying that case B may not be a physical result. This leads to case C being the most appropriate parameterization of temperature-dependent dielectric constant and linear increasing Flory parameter. Case D was ruled out because of two effects: the UCST phase diagram and the trends in the threshold temperature with salt and polyelectrolyte concentrations (Figures 8 and 9).

Moreover, we showed that only LCST behavior (no UCST) is obtained if the solvent−polymer interaction parameter χ increases and the dielectric constant ε decreases with increasing the temperature as explained in case C. Other combinations of temperature dependencies of ε and χ considered in this work result in either both LCST and UCST (cases A and B) or only UCST (case D) behavior. The preferential partitioning of salt into the polyelectrolyte poor phase is observed in all the four cases considered in this study. We hope that this demonstration of qualitatively different phase behaviors depending on how the solvent dielectric constant and the polymer−solvent interaction parameter change with temperature helps in interpreting experimental data on complex coacervation.