^{1}, Rajeev Kumar

^{2}and Glenn H. Fredrickson

^{3,a)}

### Abstract

Complex coacervation, a liquid-liquid phase separation that occurs when two oppositely charged polyelectrolytes are mixed in a solution, has the potential to be exploited for many emerging applications including wet adhesives and drug delivery vehicles. The ultra-low interfacial tension of coacervate systems against water is critical for such applications, and it would be advantageous if molecular models could be used to characterize how various systemproperties (e.g., salt concentration) affect the interfacial tension. In this article we use field-theoretic simulations to characterize the interfacial tension between a complex coacervate and its supernatant. After demonstrating that our model is free of ultraviolet divergences (calculated properties converge as the collocation grid is refined), we develop two methods for calculating the interfacial tension from field-theoretic simulations. One method relies on the mechanical interpretation of the interfacial tension as the interfacial pressure, and the second method estimates the change in free energy as the area between the two phases is changed. These are the first calculations of the interfacial tension from full field-theoretic simulation of which we are aware, and both the magnitude and scaling behaviors of our calculated interfacial tension agree with recent experiments.

This work was partially supported by the MRSEC Program of the National Science Foundation under Award No. DMR05-20415 and the Institute for Collaborative Biotechnologies through Grant No. W911NF-09-D-0001 from the U.S. Army Research Office. The content of the information does not necessarily reflect the position or the policy of the Government, and no official endorsement should be inferred.

I. INTRODUCTION

II. THEORY AND METHODS

A. Polyelectrolytemodel

B. Pressure operator

C. Calculation of the interfacial tension

D. Gaussian fluctuations

E. Numerical methods

III. RESULTS

A. UV convergence

B. Density Profiles

C. Interfacial tension and phase behavior

IV. DISCUSSION

A. Interfacial tension calculations

B. Scaling relationships

C. Comparison with RPA

V. SUMMARY

### Key Topics

- Polymers
- 42.0
- Field theory models
- 26.0
- Electrostatics
- 16.0
- Polyelectrolytes
- 13.0
- Tensor methods
- 13.0

## Figures

UV convergence of the dimensionless pressure (a) and the excess chemical potential of the polycations (b) at *C* _{±} = 0.5, *B* = 1.0, and *C* _{ s ±} = 0; at these conditions, the system exists as a homogeneous phase. The *x*-axis in both cases is the grid spacing divided by the parameter *a*. Systems both with electrostatics (*E* = 1000 in red) and without electrostatic interactions (*E* = 0 in black) are shown. The solid horizontal lines show the RPA predictions for the model while the dashed lines show the results from the CL simulations that numerically sample the fully fluctuating fields. Each property was calculated in a cubic simulation box with periodic boundary conditions and *L* _{α} = 3.0*R* _{ g }.

UV convergence of the dimensionless pressure (a) and the excess chemical potential of the polycations (b) at *C* _{±} = 0.5, *B* = 1.0, and *C* _{ s ±} = 0; at these conditions, the system exists as a homogeneous phase. The *x*-axis in both cases is the grid spacing divided by the parameter *a*. Systems both with electrostatics (*E* = 1000 in red) and without electrostatic interactions (*E* = 0 in black) are shown. The solid horizontal lines show the RPA predictions for the model while the dashed lines show the results from the CL simulations that numerically sample the fully fluctuating fields. Each property was calculated in a cubic simulation box with periodic boundary conditions and *L* _{α} = 3.0*R* _{ g }.

Density profiles of the polymer (left axis) and the small ions (right axis) for a system with *E* = 20000, *B* = 0.05, and *C* _{±} = 1.5. a) The density profile for a case with symmetric salt (*Z* _{ s ±} = ±1) and *C* _{ s ±} = 25. The curves for the polycations and polyanions lie directly on top of each other; similarly, the profiles for the small cations and anions overlap. b) The density profiles for a system with an asymmetric salt with *Z* _{ s +} = 2, *Z* _{ s −} = −1, *C* _{ s +} = 10, and *C* _{ s −} = 20. The density profile for the small cations has been scaled so that it lies on the same y-axis as the anions.

Density profiles of the polymer (left axis) and the small ions (right axis) for a system with *E* = 20000, *B* = 0.05, and *C* _{±} = 1.5. a) The density profile for a case with symmetric salt (*Z* _{ s ±} = ±1) and *C* _{ s ±} = 25. The curves for the polycations and polyanions lie directly on top of each other; similarly, the profiles for the small cations and anions overlap. b) The density profiles for a system with an asymmetric salt with *Z* _{ s +} = 2, *Z* _{ s −} = −1, *C* _{ s +} = 10, and *C* _{ s −} = 20. The density profile for the small cations has been scaled so that it lies on the same y-axis as the anions.

Polymer chain concentration profiles of the polycations (ρ_{+}) and the polyanions (ρ_{−}) at *E* = 20000 and *B* = 0.05 for two cases where the charge arising from the two polyelectrolytes is asymmetric. (a) *C* _{+} = *C* _{−} = 1.5, β_{+} = 1.0, and β_{−} = 0.9. (b) β_{+} = β_{−} = 1.0, *C* _{+} = 1.5, and *C* _{−} = 1.35. In both cases, overall charge neutrality is maintained by adding *C* _{ s −} = 15, and *Z* is fixed at ±1 for both species.

Polymer chain concentration profiles of the polycations (ρ_{+}) and the polyanions (ρ_{−}) at *E* = 20000 and *B* = 0.05 for two cases where the charge arising from the two polyelectrolytes is asymmetric. (a) *C* _{+} = *C* _{−} = 1.5, β_{+} = 1.0, and β_{−} = 0.9. (b) β_{+} = β_{−} = 1.0, *C* _{+} = 1.5, and *C* _{−} = 1.35. In both cases, overall charge neutrality is maintained by adding *C* _{ s −} = 15, and *Z* is fixed at ±1 for both species.

(a) Polycation concentration in the coacervate phase and (b) interfacial tension as a function of the dimensionless Bjerrum length *E*. In (a), the circular points are the results of our CL simulations, and the solid and dashed lines are the binodal and spinodal predictions of the RPA, respectively. In (b), red circles are calculated using the pressure tensor method and the blue diamonds are calculated using Bennett's method; the solid line represents the γ ∝ (*E* − *E* _{ cr })^{0.52}, with *E* _{ cr } ≈ 1490. All calculations represented in these figures were with *B* = 0.05, β_{+} = β_{−} = 1, *Z* = ±1, *C* _{ s ±} = 0, and the total polymer concentration was adjusted so that the coacervate phase occupied approximately half of the simulation box.

(a) Polycation concentration in the coacervate phase and (b) interfacial tension as a function of the dimensionless Bjerrum length *E*. In (a), the circular points are the results of our CL simulations, and the solid and dashed lines are the binodal and spinodal predictions of the RPA, respectively. In (b), red circles are calculated using the pressure tensor method and the blue diamonds are calculated using Bennett's method; the solid line represents the γ ∝ (*E* − *E* _{ cr })^{0.52}, with *E* _{ cr } ≈ 1490. All calculations represented in these figures were with *B* = 0.05, β_{+} = β_{−} = 1, *Z* = ±1, *C* _{ s ±} = 0, and the total polymer concentration was adjusted so that the coacervate phase occupied approximately half of the simulation box.

Log-log plot of the polycation concentration in the coacervate phase (a) and the interfacial tension (b) as the excluded volume parameter *B* is increased. In (a), the solid and dashed lines represent the binodal and spinodal coexistence curves calculated using the RPA. The estimates of γ in (b) were obtained using Bennett's method. All calculations in these figures were performed with *E* = 20000, β_{+} = β_{−} = 1, *Z* = ±1, *C* _{ s ±} = 0, and the total polymer concentration was adjusted so that the coacervate phase occupied approximately half of the simulation box.

Log-log plot of the polycation concentration in the coacervate phase (a) and the interfacial tension (b) as the excluded volume parameter *B* is increased. In (a), the solid and dashed lines represent the binodal and spinodal coexistence curves calculated using the RPA. The estimates of γ in (b) were obtained using Bennett's method. All calculations in these figures were performed with *E* = 20000, β_{+} = β_{−} = 1, *Z* = ±1, *C* _{ s ±} = 0, and the total polymer concentration was adjusted so that the coacervate phase occupied approximately half of the simulation box.

(a) Changes in the polycation concentration in the coacervate phase and (b) the interfacial tension as the salt concentration is increased. In (a), the solid line is simply a guide to the eye, while in (b), the solid line represents the scaling γ ∝ (*C* _{ s, crit } − *C* _{ s })^{3/2} with *C* _{ s, crit } ≈ 110. All calculations in these figures were performed with *E* = 20000, *B* = 0.05, *C* = 1.5, β_{±} = 1 and *Z* = ±1.

(a) Changes in the polycation concentration in the coacervate phase and (b) the interfacial tension as the salt concentration is increased. In (a), the solid line is simply a guide to the eye, while in (b), the solid line represents the scaling γ ∝ (*C* _{ s, crit } − *C* _{ s })^{3/2} with *C* _{ s, crit } ≈ 110. All calculations in these figures were performed with *E* = 20000, *B* = 0.05, *C* = 1.5, β_{±} = 1 and *Z* = ±1.

Scaling of the interfacial tension in the absence of salt with the product *CE*, where *C* is the concentration of the polyelectrolytes in the coacervate phase. The solid line represents γ ∝ (*CE*)^{1/2}. These data are taken from the simulations that are summarized in Fig. 4 where *B* was held constant and *E* was systematically varied.

Scaling of the interfacial tension in the absence of salt with the product *CE*, where *C* is the concentration of the polyelectrolytes in the coacervate phase. The solid line represents γ ∝ (*CE*)^{1/2}. These data are taken from the simulations that are summarized in Fig. 4 where *B* was held constant and *E* was systematically varied.

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