^{1,a)}and Karl F. Freed

^{1}

### Abstract

The physical content of and, in particular, the nonlinear contributions from the Langevin-Debye model are illustrated using two applications. First, we provide an improvement in the Langevin-Debye model currently used in some implicit solvent models for computer simulations of solvation free energies of small organic molecules, as well as of biomolecular folding and binding. The analysis is based on the implementation of a charge-dependent Langevin-Debye (qLD) model that is modified by subsequent corrections due to Onsager and Kirkwood. Second, the physical content of the model is elucidated by discussing the general treatment within the LD model of the self-energy of a charge submerged in a dielectric medium for three different limiting conditions and by considering the nonlinear response of the medium. The modified qLD model is used to refine an implicit solvent model (previously applied to protein dynamics). The predictions of the modified implicit solvent model are compared with those from explicit solvent molecular dynamics simulations for the equilibrium conformational populations of 1,2-dimethoxyethane (DME), which is the shortest ether molecule to reproduce the local conformational properties of polyethylene oxide, a polymer with tremendous technological importance and a wide variety of applications. Because the conformational population preferences of DME change dramatically upon solvation, DME is a good test case to validate our modified qLD model. The present analysis of the modified qLD model provides the motivation and tools for studying a wide variety of other interesting systems with heterogeneous dielectric properties and spatial anisotropy.

We thank Dr. Dmitry Bedrov for providing the data from the simulation of aqueous DME. We also thank Professor Caroline Taylor and Dr. Bhimlapuram Prabhakar for critical reading of the manuscript. This work is supported by NSF Grant No. CHE-0312226 and NIH Grant No. GM081642. A.K.J. acknowledges the support of Burroughs Wellcome Fund Interfaces No. 1001774.

I. INTRODUCTION

II. LANGEVIN-DEBYE THEORY OF IONIC SATURATION

III. CORRECTION DUE TO ONSAGER AND KIRKWOOD

IV. CONTINUUM MODELS AND NONLINEAR RESPONSE

V. CHARGE-DEPENDENT LANGEVIN-DEBYE MODEL

VI. GENERAL EXPRESSION FOR SELF-ENERGY OF A CHARGE

VII. APPLICATION TO CONFORMATIONAL POPULATIONS OF 1,2: DIMETHOXYETHANE

VIII. DISCUSSION

IX. CONCLUSION

### Key Topics

- Solvents
- 100.0
- Dielectric constant
- 47.0
- Biodiesel
- 27.0
- Dielectrics
- 26.0
- Electrostatics
- 17.0

## Figures

Two dimensional cartoon representation of the fictive mathematical device called a “Lorentz sphere.” The Langevin-Debye model assumes the Lorentz sphere to be homogeneous and isotropic. This assumption implicitly implies that the electric field from the dipoles vanishes inside the Lorentz sphere. The volume of the sphere, represented in the figure by the circle, is large compared to the solvent molecule size. The ovals in the circle represent the polar solvent molecules that can be polarized in two ways: (a) due to the distortion of the charge distribution of the molecules in the presence of an external field and (b) from the alignment by the applied field of the randomly oriented permanent dipole moment of the molecules. These effects are represented by elongation and alignment, respectively.

Two dimensional cartoon representation of the fictive mathematical device called a “Lorentz sphere.” The Langevin-Debye model assumes the Lorentz sphere to be homogeneous and isotropic. This assumption implicitly implies that the electric field from the dipoles vanishes inside the Lorentz sphere. The volume of the sphere, represented in the figure by the circle, is large compared to the solvent molecule size. The ovals in the circle represent the polar solvent molecules that can be polarized in two ways: (a) due to the distortion of the charge distribution of the molecules in the presence of an external field and (b) from the alignment by the applied field of the randomly oriented permanent dipole moment of the molecules. These effects are represented by elongation and alignment, respectively.

Permittivity profile. (a) Curves are derived from the Langevin-Debye formalism with the input values for the number density of molecules per (for a molecular weight of and a mass density of for water), optical dielectric constant , static dielectric constant, , and . The black data points correspond to the general solution, whereas the red and green points are obtained for the limiting cases of zero dipole and linear response regime of the solvent, respectively. (b) The radial permittivity following corrections in the LD model due to Onsager (red) and Kirkwood (green). The parameters and are used to give the water dipole moments of and in the gas phase and liquid phase. The inset is a magnified view of region from . (c) The different functional forms used to model the distance dependence of the dielectric constant. The parameters used for each functional form are described in the text. (d) The modified Langevin-Debye model calculated for all the four charges for the atoms of DME. We use the fits from Eq. (14) (long dash – –) and Eq. (15) (short dash - - -) and the parameters in Table I. The curve fits for are displayed. The inset is a magnified view of regions from .

Permittivity profile. (a) Curves are derived from the Langevin-Debye formalism with the input values for the number density of molecules per (for a molecular weight of and a mass density of for water), optical dielectric constant , static dielectric constant, , and . The black data points correspond to the general solution, whereas the red and green points are obtained for the limiting cases of zero dipole and linear response regime of the solvent, respectively. (b) The radial permittivity following corrections in the LD model due to Onsager (red) and Kirkwood (green). The parameters and are used to give the water dipole moments of and in the gas phase and liquid phase. The inset is a magnified view of region from . (c) The different functional forms used to model the distance dependence of the dielectric constant. The parameters used for each functional form are described in the text. (d) The modified Langevin-Debye model calculated for all the four charges for the atoms of DME. We use the fits from Eq. (14) (long dash – –) and Eq. (15) (short dash - - -) and the parameters in Table I. The curve fits for are displayed. The inset is a magnified view of regions from .

Relative magnitude of nonlinear contribution to the self-energy. The contribution of the nonlinear term to the self-energy of a charge placed in aqueous medium [Eq. (22)]. The grey stars correspond to the three atom types in DME.

Relative magnitude of nonlinear contribution to the self-energy. The contribution of the nonlinear term to the self-energy of a charge placed in aqueous medium [Eq. (22)]. The grey stars correspond to the three atom types in DME.

## Tables

Parameters for the functional forms of the distant-dependent dielectric “constant” obtained by fitting to the Langevin-Debye model with corrections due to Onsager-Kirkwood. The magnitudes of the partial charges are those for the atoms in DME. The parameters and are fixed to the static and optical dielectric constants of water, respectively.

Parameters for the functional forms of the distant-dependent dielectric “constant” obtained by fitting to the Langevin-Debye model with corrections due to Onsager-Kirkwood. The magnitudes of the partial charges are those for the atoms in DME. The parameters and are fixed to the static and optical dielectric constants of water, respectively.

Comparison of populations (%) from implicit and explicit solvent simulations for DME. The charge-dependent dielectric model used corresponds to the parameters shown in Table I, while the implicit model uses .

Comparison of populations (%) from implicit and explicit solvent simulations for DME. The charge-dependent dielectric model used corresponds to the parameters shown in Table I, while the implicit model uses .

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