^{1,a)}, Jean-Pierre Hansen

^{2}and Gerhard Kahl

^{3}

### Abstract

We introduce an ultrasoft core model of interpenetrating polycations and polyanions, with continuous Gaussian charge distributions, to investigate polyelectrolyte aggregation in dilute and semi-dilute salt-free solutions. The model is studied by a combination of approximate theories (random phase approximation and hypernetted chain theory) and numerical simulations. The calculated pair structure, thermodynamics, phase diagram, and polyion dynamics of the symmetric version of the model (the “ultrasoft restricted primitive model” or UPRM) differ from the corresponding properties of the widely studied “restricted primitive model” (RPM) where ions have hard cores. At sufficiently low temperatures and densities, oppositely charged polyions form weakly interacting, polarizable neutral pairs. The clustering probabilities, dielectric behavior, and electrical conductivity point to a line of sharp conductor-insulator transitions in the density-temperature plane. At very low temperatures, the conductor-insulator transition line terminates near the top of a first order coexistence curve separating a high-density liquid phase from a low-density vapor phase. The simulation data hint at a tricritical behavior, reminiscent of that observed for the two-dimensional Coulomb gas, which contrasts with the Ising criticality of its three-dimensional counterpart, the RPM.

D.C. and G.K. acknowledge financial support from the Austrian Science Foundation (FWF) under Project No. P19890-N16.

I. INTRODUCTION

II. THE MODEL

III. PAIR STRUCTURE AND THERMODYNAMICS

IV. CLUSTERING AND DIELECTRIC RESPONSE

V. POLYION DYNAMICS

VI. DISCUSSION AND CONCLUSIONS

### Key Topics

- Monte Carlo methods
- 18.0
- Correlation functions
- 14.0
- Polymers
- 14.0
- Dielectrics
- 13.0
- Polyelectrolytes
- 10.0

## Figures

Total correlation functions *h* _{++}(*r*), *h* _{+−}(*r*), and *h* _{ CC }(*r*) from MC simulations (symbols) and RPA (full lines) for two different state points: (a) *n* = 0.0035, *T* = 0.63 and (b) *n* = 0.35, *T* = 0.25.

Total correlation functions *h* _{++}(*r*), *h* _{+−}(*r*), and *h* _{ CC }(*r*) from MC simulations (symbols) and RPA (full lines) for two different state points: (a) *n* = 0.0035, *T* = 0.63 and (b) *n* = 0.35, *T* = 0.25.

Pair distribution functions along the isochore *n* = 0.0035 for three different temperatures (as specified) from MC simulations (symbols) and RPA (full lines): (a) *g* _{+−}(*r*) and (b) *g* _{++}(*r*).

Pair distribution functions along the isochore *n* = 0.0035 for three different temperatures (as specified) from MC simulations (symbols) and RPA (full lines): (a) *g* _{+−}(*r*) and (b) *g* _{++}(*r*).

Pair distribution functions along the isochore *n* = 0.35 for three different temperatures (as specified) from MC simulations (symbols), RPA (full lines), and HNC (dashed lines): (a) *g* _{+−}(*r*) and (b) *g* _{++}(*r*).

Pair distribution functions along the isochore *n* = 0.35 for three different temperatures (as specified) from MC simulations (symbols), RPA (full lines), and HNC (dashed lines): (a) *g* _{+−}(*r*) and (b) *g* _{++}(*r*).

Structure factors along the isochore *n* = 0.0035 for three different temperatures (as specified) from MC simulations (symbols) and RPA (full lines): (a) *S* _{ CC }(*k*) and (b) *S* _{ NN }(*k*).

Structure factors along the isochore *n* = 0.0035 for three different temperatures (as specified) from MC simulations (symbols) and RPA (full lines): (a) *S* _{ CC }(*k*) and (b) *S* _{ NN }(*k*).

Structure factors along the isochore *n* = 0.35 for three different temperatures (as specified) from MC simulations (symbols), RPA (full lines), and HNC (dashed lines): (a) *S* _{ CC }(*k*) and (b) *S* _{ NN }(*k*).

Structure factors along the isochore *n* = 0.35 for three different temperatures (as specified) from MC simulations (symbols), RPA (full lines), and HNC (dashed lines): (a) *S* _{ CC }(*k*) and (b) *S* _{ NN }(*k*).

Excess internal energy per particle *U* ^{ex}/*N* as a function of temperature from MC simulations (symbols) and RPA (full lines) along the isochores *n* = 0.0035 (squares) and *n* = 0.35 (circles).

Excess internal energy per particle *U* ^{ex}/*N* as a function of temperature from MC simulations (symbols) and RPA (full lines) along the isochores *n* = 0.0035 (squares) and *n* = 0.35 (circles).

Dimensionless chemical potential βμ as a function of the density *n* along several isotherms (as labeled) obtained from the Widom insertion method in NVT MC simulations (open circles), grand canonical MC simulations (full circles), and RPA (full lines).

Dimensionless chemical potential βμ as a function of the density *n* along several isotherms (as labeled) obtained from the Widom insertion method in NVT MC simulations (open circles), grand canonical MC simulations (full circles), and RPA (full lines).

Cluster distribution function *P*(*m*) for (a) *T* = 0.125 and (b) *T* = 0.0125 along the isochore *n* = 0.0035.

Cluster distribution function *P*(*m*) for (a) *T* = 0.125 and (b) *T* = 0.0125 along the isochore *n* = 0.0035.

Fractions of selected cluster types *P*(*m*) as functions of temperature along the isochore *n* = 0.0035: *P*(*m* = 1) (squares), *P*(*m* = 2) (circles), *P*(*m* = 3) (triangles), and *P*(*m* = 4) (reversed triangles). The three panels show results obtained using the following values of the cut-off *r* _{ c }: (a) *r* _{ c } = 1.4, (b) *r* _{ c } = 1.0, and (c) *r* _{ c } = 0.7.

Fractions of selected cluster types *P*(*m*) as functions of temperature along the isochore *n* = 0.0035: *P*(*m* = 1) (squares), *P*(*m* = 2) (circles), *P*(*m* = 3) (triangles), and *P*(*m* = 4) (reversed triangles). The three panels show results obtained using the following values of the cut-off *r* _{ c }: (a) *r* _{ c } = 1.4, (b) *r* _{ c } = 1.0, and (c) *r* _{ c } = 0.7.

Cluster lifetime τ_{ m } at *n* = 0.0035 as a function of temperature for selected cluster types: *m* = 1 (squares), *m* = 2 (circles), *m* = 3 (triangles), and *m* = 4 (reversed triangles).

Cluster lifetime τ_{ m } at *n* = 0.0035 as a function of temperature for selected cluster types: *m* = 1 (squares), *m* = 2 (circles), *m* = 3 (triangles), and *m* = 4 (reversed triangles).

Distribution functions *g* _{ m, n }(*r*) between the centers of mass of selected cluster types for *n* = 0.0035, *T* = 0.025: (a) monomer-monomer *g* _{1, 1}(*r*), (b) monomer-dimer *g* _{1, 2}(*r*), and (c) dimer-dimer *g* _{2, 2}(*r*).

Distribution functions *g* _{ m, n }(*r*) between the centers of mass of selected cluster types for *n* = 0.0035, *T* = 0.025: (a) monomer-monomer *g* _{1, 1}(*r*), (b) monomer-dimer *g* _{1, 2}(*r*), and (c) dimer-dimer *g* _{2, 2}(*r*).

Dimensionless, effective dimer-dimer potential β*v* _{2, 2}(*r*) for *n* = 0.0035, *T* = 0.025. Inset: Effective potentials on a double-logarithmic scale. The dashed line indicates the asymptotic behavior expected at large *r* (see Appendix B).

Dimensionless, effective dimer-dimer potential β*v* _{2, 2}(*r*) for *n* = 0.0035, *T* = 0.025. Inset: Effective potentials on a double-logarithmic scale. The dashed line indicates the asymptotic behavior expected at large *r* (see Appendix B).

Dimensionless, effective monomer-dimer potential β*v* _{1, 2}(*r*) for *n* = 0.0035, *T* = 0.025. Inset: Effective potentials on a double-logarithmic scale. The dashed line indicates the asymptotic behavior expected at large *r* (see Appendix B).

Dimensionless, effective monomer-dimer potential β*v* _{1, 2}(*r*) for *n* = 0.0035, *T* = 0.025. Inset: Effective potentials on a double-logarithmic scale. The dashed line indicates the asymptotic behavior expected at large *r* (see Appendix B).

Dielectric order parameter (ε − 1)/ε (filled circles) obtained from Eq. (49) and fraction of free ions *P*(*m* = 1) (crosses) at *n* = 0.0035 as functions of temperature. Inset: enlarged view of the low-temperature behavior of ε.

Dielectric order parameter (ε − 1)/ε (filled circles) obtained from Eq. (49) and fraction of free ions *P*(*m* = 1) (crosses) at *n* = 0.0035 as functions of temperature. Inset: enlarged view of the low-temperature behavior of ε.

Velocity auto-correlation function *Z*(*t*), as defined in Eq. (50), at *n* = 0.0035 for different temperatures (as labeled). Inset: self diffusion constant *D* evaluated via Eq. (52) as a function of temperature.

Velocity auto-correlation function *Z*(*t*), as defined in Eq. (50), at *n* = 0.0035 for different temperatures (as labeled). Inset: self diffusion constant *D* evaluated via Eq. (52) as a function of temperature.

Mean square displacement 〈|**M**(*t*) − **M**(0)|^{2}〉 of the total electric dipole **M**(*t*) as a function of time at *n* = 0.0035 for several temperatures (as labeled). The dashed line indicates the asymptotic behavior expected for a conducting system.

Mean square displacement 〈|**M**(*t*) − **M**(0)|^{2}〉 of the total electric dipole **M**(*t*) as a function of time at *n* = 0.0035 for several temperatures (as labeled). The dashed line indicates the asymptotic behavior expected for a conducting system.

Electrical conductivity σ_{ e }, evaluated via Eq. (53), as a function of temperature along different isochores (as labeled). *T* _{ c } indicates the estimated temperature where σ_{ e } vanishes, according to power-law fits σ_{ e } ∼ (*T* − *T* _{ c })^{ν} with ν = 1.2 (full lines). The estimated uncertainty on ν is ±0.02.

Electrical conductivity σ_{ e }, evaluated via Eq. (53), as a function of temperature along different isochores (as labeled). *T* _{ c } indicates the estimated temperature where σ_{ e } vanishes, according to power-law fits σ_{ e } ∼ (*T* − *T* _{ c })^{ν} with ν = 1.2 (full lines). The estimated uncertainty on ν is ±0.02.

Current estimate of the phase diagram of the URPM in the (*T*, *n*)-plane. Empty circles: coexistence points obtained via GCMC simulations (from Ref. 29). Full line: fit to the coexistence line, assuming a critical exponent β = 1. Rotated empty square: Estimate of the critical point assuming a critical exponent β = 1. Filled circles: Spinodal line estimated from the condition *S* _{ NN }(*k* = 0) ≈ 50. Open triangles: Pairing transition points estimated from the condition *P*(*m* = 1) = 0.3 with *r* _{ c } = 1.4 (as in Ref. 29). Filled triangles: Pairing transition points estimated from the condition *P*(*m* = 1) ≈ *P*(*m* = 2) = 0.5 with *r* _{ c } = 1.0. Stars: CI transition points estimated from the vanishing of the electrical conductivity (see text for definition).

Current estimate of the phase diagram of the URPM in the (*T*, *n*)-plane. Empty circles: coexistence points obtained via GCMC simulations (from Ref. 29). Full line: fit to the coexistence line, assuming a critical exponent β = 1. Rotated empty square: Estimate of the critical point assuming a critical exponent β = 1. Filled circles: Spinodal line estimated from the condition *S* _{ NN }(*k* = 0) ≈ 50. Open triangles: Pairing transition points estimated from the condition *P*(*m* = 1) = 0.3 with *r* _{ c } = 1.4 (as in Ref. 29). Filled triangles: Pairing transition points estimated from the condition *P*(*m* = 1) ≈ *P*(*m* = 2) = 0.5 with *r* _{ c } = 1.0. Stars: CI transition points estimated from the vanishing of the electrical conductivity (see text for definition).

(a) Concentration-concentration structure factors *S* _{ CC }(*k*) and (b) number-number structure factors *S* _{ NN }(*k*) at *n* = 0.0035 and different temperatures (as labeled), obtained using different values of the Ewald sum precision ε: ε = 10^{−4} (full lines), ε = 10^{−3} (dashed lines), and ε = 10^{−2} (dotted lines).

(a) Concentration-concentration structure factors *S* _{ CC }(*k*) and (b) number-number structure factors *S* _{ NN }(*k*) at *n* = 0.0035 and different temperatures (as labeled), obtained using different values of the Ewald sum precision ε: ε = 10^{−4} (full lines), ε = 10^{−3} (dashed lines), and ε = 10^{−2} (dotted lines).

Article metrics loading...

Full text loading...

Commenting has been disabled for this content