^{1}and Dmitry V. Matyushov

^{1,a)}

### Abstract

We present the results of an analytical theory and numerical simulations of microscopic fields in dipolar liquids. Fields within empty spherical cavities(cavity field) and within cavities with a probe dipole (directing field) and the field induced by a probe dipole in the surrounding liquid (reaction field) are considered. Instead of demanding the field produced by a liquid dielectric in a large-scale cavity to coincide with the field of Maxwell’sdielectric, we continuously increase the cavity size to reach the limit of a mesoscopic dimension and establish the continuum limit from the bottom up. Both simulations and analytical theory suggest that the commonly applied Onsager formula for the reaction field is approached from below, with increasing cavity size, by the microscopic solution. On the contrary, the cavity and directing fields do not converge to the limit of Maxwell’sdielectric. The origin of the disagreement between the standard electrostatics and the results obtained from microscopic models is traced back to the failure of the former to account properly for the transverse correlations between dipoles in molecular liquids. A new continuum equation is derived for the cavity field and supported by numerical simulations. Experimental tests of the theoretical results are suggested.

This research was supported by the NSF (Grant No. CHE-0616646).

I. INTRODUCTION

II. CALCULATION OF FIELDS IN DIELECTRICS

A. Formalism

B. Cavity field

C. Reaction field

D. Microscopic field

III. MONTE CARLO SIMULATIONS

IV. RESULTS

A. Dielectric constant

B. Cavity field

C. Reaction field

D. Microscopic field

V. DISCUSSION

VI. EXPERIMENTAL TESTS

VII. CONCLUSIONS

### Key Topics

- Liquid dielectrics
- 60.0
- Maxwell equations
- 46.0
- Cavitation
- 39.0
- Polarization
- 33.0
- Mean field theory
- 24.0

## Figures

(a) Onsager’s directing field inside a dipolar liquid and the (b) cavity field inside a physical cavity carved in the liquid placed in the uniform external field . The local field [Eq. (3)] is a mean-field approximation for . is the dipole moment of a target molecule.

(a) Onsager’s directing field inside a dipolar liquid and the (b) cavity field inside a physical cavity carved in the liquid placed in the uniform external field . The local field [Eq. (3)] is a mean-field approximation for . is the dipole moment of a target molecule.

Longitudinal (L) and transverse (T) structure factors of the dipolar hard-sphere liquid from numerical simulations at and (solid lines, ) and 1.0 (dashed lines, ).

Longitudinal (L) and transverse (T) structure factors of the dipolar hard-sphere liquid from numerical simulations at and (solid lines, ) and 1.0 (dashed lines, ).

Response functions (dashed lines) and (dashed-dotted line) calculated from Eq. (22) and compared to [solid line, see Eqs. (25) and (26)]. The upper curves are obtained at , , while the lower curves refer to , ; . The and projections required for the integration in Eq. (22) have been obtained from MC simulations. Panels (a) and (b) correspond to and 1.5, respectively.

Response functions (dashed lines) and (dashed-dotted line) calculated from Eq. (22) and compared to [solid line, see Eqs. (25) and (26)]. The upper curves are obtained at , , while the lower curves refer to , ; . The and projections required for the integration in Eq. (22) have been obtained from MC simulations. Panels (a) and (b) correspond to and 1.5, respectively.

Cavity field calculated from Eq. (31) with two cavity sizes indicated by the distance of the closest approach, , in the plot. The points were obtained by numerical integration in Eq. (31) with from MC simulations , while the dashed lines refer to the calculations using the parametrized MSA (Ref. 31). The integral is calculated numerically before the appearance of the singularity on the real axis [Eq. (35)] and by summation over the poles when the singularity falls on the axis. The two methods give identical results when numerical integration is justified. The upper and lower solid lines refer to two continuum limits, Eqs. (34) and (32), respectively. The dashed-dotted line refers to the lattice summation [Eq. (36)] instead of continuous integration in Eq. (31) taken for a cubic cell of , .

Cavity field calculated from Eq. (31) with two cavity sizes indicated by the distance of the closest approach, , in the plot. The points were obtained by numerical integration in Eq. (31) with from MC simulations , while the dashed lines refer to the calculations using the parametrized MSA (Ref. 31). The integral is calculated numerically before the appearance of the singularity on the real axis [Eq. (35)] and by summation over the poles when the singularity falls on the axis. The two methods give identical results when numerical integration is justified. The upper and lower solid lines refer to two continuum limits, Eqs. (34) and (32), respectively. The dashed-dotted line refers to the lattice summation [Eq. (36)] instead of continuous integration in Eq. (31) taken for a cubic cell of , .

The inverse dielectric susceptibility vs . The points represent simulation data. The solid line refers to Eq. (51), and the dotted line refers to Eq. (50). The remaining two lines are the continuum results obtained from the Debye and Onsager equations, Eqs. (6) and (8), respectively. The solid line is obtained by fitting the parameter in Eq. (51) to the simulation data with the best-fit value of .

The inverse dielectric susceptibility vs . The points represent simulation data. The solid line refers to Eq. (51), and the dotted line refers to Eq. (50). The remaining two lines are the continuum results obtained from the Debye and Onsager equations, Eqs. (6) and (8), respectively. The solid line is obtained by fitting the parameter in Eq. (51) to the simulation data with the best-fit value of .

The depolarization coefficient as a function of the dielectric constant for a dipolar fluid at . The solid line is the fit to the simulation data (points) to Eq. (52). The dashed line is the Onsager result in Eq. (7).

The depolarization coefficient as a function of the dielectric constant for a dipolar fluid at . The solid line is the fit to the simulation data (points) to Eq. (52). The dashed line is the Onsager result in Eq. (7).

The cavity field calculated from MC simulations with varying cavity size: (, circles), 1.5 (, squares), 2.0 (, left triangles), 3.0 (, right triangles), and 5.5 (, up triangles). The solid line corresponds to the new continuum expression given in Eq. (34), while the dashed line refers to the standard Maxwell result [Eq. (32)].

The cavity field calculated from MC simulations with varying cavity size: (, circles), 1.5 (, squares), 2.0 (, left triangles), 3.0 (, right triangles), and 5.5 (, up triangles). The solid line corresponds to the new continuum expression given in Eq. (34), while the dashed line refers to the standard Maxwell result [Eq. (32)].

The cavity field calculated directly from simulations [Eq. (48)] as a function of the cavity radius . The insrt is an expanded section at large . The points represent (circles), 1.0 (squares), 2.0 (diamonds), and 3.0 (up triangles). The corresponding values of are 3.54, 8.52, 30.64, and 93.66.

The cavity field calculated directly from simulations [Eq. (48)] as a function of the cavity radius . The insrt is an expanded section at large . The points represent (circles), 1.0 (squares), 2.0 (diamonds), and 3.0 (up triangles). The corresponding values of are 3.54, 8.52, 30.64, and 93.66.

The correlator from Eq. (48) calculated from MC simulations for the dipoles in the first solvation shell surrounding the cavity (circles), from the second solvation shell (squares), and from the entire simulation box (open triangles). The sum of contributions from the first and second solvation shells is indicated by open diamonds. The cavity radius is .

The correlator from Eq. (48) calculated from MC simulations for the dipoles in the first solvation shell surrounding the cavity (circles), from the second solvation shell (squares), and from the entire simulation box (open triangles). The sum of contributions from the first and second solvation shells is indicated by open diamonds. The cavity radius is .

Reaction field calculated from the continuum electrostatics [Eq. (40)] (solid line marked “Onsager,” is used for the cavity radius) and from Eq. (39) (filled points) with the effective radius [Eq. (41)] used in place of : (circles), 1.0 (squares), and 1.5 (triangles). The open points correspond to the MC simulation data for the same values as the closed points. The dotted line applies Eq. (39) at with used for the cavity radius. The dashed lines connect the points and the structure factors from MC simulations were used for integration in Eq. (39).

Reaction field calculated from the continuum electrostatics [Eq. (40)] (solid line marked “Onsager,” is used for the cavity radius) and from Eq. (39) (filled points) with the effective radius [Eq. (41)] used in place of : (circles), 1.0 (squares), and 1.5 (triangles). The open points correspond to the MC simulation data for the same values as the closed points. The dotted line applies Eq. (39) at with used for the cavity radius. The dashed lines connect the points and the structure factors from MC simulations were used for integration in Eq. (39).

Microscopic field at the position of the dipole at the center of a spherical cavity of radius . The results at different were obtained for (circles), 1.0 (squares), and 1.5 (diamonds) using Eq. (49). The solid line refers to the continuum prediction for the homogeneous liquid [Eq. (46)], while dashed and dashed-dotted lines refer to and , respectively, obtained from Eq. (47).

Microscopic field at the position of the dipole at the center of a spherical cavity of radius . The results at different were obtained for (circles), 1.0 (squares), and 1.5 (diamonds) using Eq. (49). The solid line refers to the continuum prediction for the homogeneous liquid [Eq. (46)], while dashed and dashed-dotted lines refer to and , respectively, obtained from Eq. (47).

The local field [Eq. (3)] (circles), the cavity field (squares), and the directing field (closed triangles) vs for . The solid line indicates the new continuum cavity field [Eq. (34)], the dashed line is the Lorentz local field [Eq. (5)], and the dashed-dotted line is the Maxwell cavity field [Eq. (9)]. The lower panel is an expanded section at small .

The local field [Eq. (3)] (circles), the cavity field (squares), and the directing field (closed triangles) vs for . The solid line indicates the new continuum cavity field [Eq. (34)], the dashed line is the Lorentz local field [Eq. (5)], and the dashed-dotted line is the Maxwell cavity field [Eq. (9)]. The lower panel is an expanded section at small .

Directing field vs the target dipole moment . Points are simulations data at , ; the dotted line connects the points.

Directing field vs the target dipole moment . Points are simulations data at , ; the dotted line connects the points.

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