^{1}, Allan D. Friesen

^{1}and Dmitry V. Matyushov

^{1,a)}

### Abstract

Electric field produced inside a solute by a uniformly polarized liquid is strongly affected by dipolar polarization of the liquid at the interface. We show, by numerical simulations, that the electric “cavity” field inside a hydrated non-polar solute does not follow the predictions of standard Maxwell's electrostatics of dielectrics. Instead, the field inside the solute tends, with increasing solute size, to the limit predicted by the Lorentz virtual cavity. The standard paradigm fails because of its reliance on the surface charge density at the dielectricinterface determined by the boundary conditions of the Maxwelldielectric. The interface of a polar liquid instead carries a preferential in-plane orientation of the surface dipoles thus producing virtually no surface charge. The resulting boundary conditions for electrostatic problems differ from the traditional recipes, affecting the microscopic and macroscopic fields based on them. We show that relatively small differences in cavity fields propagate into significant differences in the dielectric constant of an ideal mixture. The slope of the dielectric increment of the mixture versus the solute concentration depends strongly on which polarization scenario at the interface is realized. A much steeper slope found in the case of Lorentz interfacial polarization also implies a higher free energy penalty for polarizing such mixtures.

This research was supported by the National Science Foundation (NSF) (CHE-0910905). CPU time was provided by the National Science Foundation through TeraGrid resources (TG-MCB080116N). We are grateful to Peter Rossky for useful discussions and comments on the manuscript.

I. INTRODUCTION

II. RESULTS

III. DISCUSSION

IV. EXPERIMENTAL OBSERVABLES

V. SUMMARY

### Key Topics

- Maxwell equations
- 44.0
- Polarization
- 29.0
- Dielectrics
- 24.0
- Double layers
- 23.0
- Surface charge
- 23.0

## Figures

Cartoon of the charge distribution σ_{ P } at the surface of a spherical cavity carved from a liquid dielectric. Panel (a) represents the picture of standard Maxwell's electrostatics of dielectrics, when the normal projection of the dipolar polarization along the external field creates the positive and negative lobes of the surface charge distribution. Those result from the mathematical surface cutting through polarized dipoles of the liquid shown by arrows. The overall electric field inside the cavity is then reduced from the external field by the opposing field of the surface charges [Eq. (1)]. Panel (b) shows the scenario suggested by the orientational dipolar order at the surface of a “Rossky cavity” in a polar liquid (Refs. 6–12). In-plane orientations of the surface dipoles, unaltered by a weak external field, do not produce surface charge. The result is zero surface charge density σ_{ P } and the field at the cavity center following the Lorentz equation [Eq. (2)].

Cartoon of the charge distribution σ_{ P } at the surface of a spherical cavity carved from a liquid dielectric. Panel (a) represents the picture of standard Maxwell's electrostatics of dielectrics, when the normal projection of the dipolar polarization along the external field creates the positive and negative lobes of the surface charge distribution. Those result from the mathematical surface cutting through polarized dipoles of the liquid shown by arrows. The overall electric field inside the cavity is then reduced from the external field by the opposing field of the surface charges [Eq. (1)]. Panel (b) shows the scenario suggested by the orientational dipolar order at the surface of a “Rossky cavity” in a polar liquid (Refs. 6–12). In-plane orientations of the surface dipoles, unaltered by a weak external field, do not produce surface charge. The result is zero surface charge density σ_{ P } and the field at the cavity center following the Lorentz equation [Eq. (2)].

A cartoon of the Kihara solute in SPC/E water and the definition of the projection angles describing the orientations of waters in the first hydration layer in respect to the surface normal . θ_{ b } is the angle between O–H bond and , θ_{ m } is the angle between the water dipole and , θ_{⊥} defines the orientation of the plane of H_{2}O (note that ).

A cartoon of the Kihara solute in SPC/E water and the definition of the projection angles describing the orientations of waters in the first hydration layer in respect to the surface normal . θ_{ b } is the angle between O–H bond and , θ_{ m } is the angle between the water dipole and , θ_{⊥} defines the orientation of the plane of H_{2}O (note that ).

The cavity field inside Kihara solutes (filled points, connected by the dashed lines) compared to hard-sphere cavities inside dipolar fluids (open points, connected by the dash-dotted lines) (Ref. 14). Results of MD simulations in the main panel refer to two values of the solute-solvent LJ attraction, ε_{0s } = 0.65 kJ/mol (circles) and 20.0 kJ/mol (squares). The results for the Kihara solutes are plotted against the radius of closest approach *R* _{0s }/σ_{ s } defined by Eq. (4) (σ_{ s } = 2.87 Å). Open points in the main panel and the inset refer to sets of data obtained for varied reduced dipole moment of the dipolar hard-sphere fluids: 0.5 (diamonds), 1.0 (triangles), and 3.0 (squares). The results for hard-sphere cavities in dipolar fluids are plotted against the radius of the closest hard-sphere solute-solvent approach *R* _{0s }/σ_{ s }; σ_{ s } is the diameter of the solvent hard spheres. The dashed horizontal lines in the main panel and in the inset refer to the Lorentz result (L) of Eq. (2). The dash-dotted horizontal line in the inset refers to the Maxwell result (M) of Eq. (1). Both sets of lines have been produced with ε = 72.2 for SPC/E water at 300 K.

The cavity field inside Kihara solutes (filled points, connected by the dashed lines) compared to hard-sphere cavities inside dipolar fluids (open points, connected by the dash-dotted lines) (Ref. 14). Results of MD simulations in the main panel refer to two values of the solute-solvent LJ attraction, ε_{0s } = 0.65 kJ/mol (circles) and 20.0 kJ/mol (squares). The results for the Kihara solutes are plotted against the radius of closest approach *R* _{0s }/σ_{ s } defined by Eq. (4) (σ_{ s } = 2.87 Å). Open points in the main panel and the inset refer to sets of data obtained for varied reduced dipole moment of the dipolar hard-sphere fluids: 0.5 (diamonds), 1.0 (triangles), and 3.0 (squares). The results for hard-sphere cavities in dipolar fluids are plotted against the radius of the closest hard-sphere solute-solvent approach *R* _{0s }/σ_{ s }; σ_{ s } is the diameter of the solvent hard spheres. The dashed horizontal lines in the main panel and in the inset refer to the Lorentz result (L) of Eq. (2). The dash-dotted horizontal line in the inset refers to the Maxwell result (M) of Eq. (1). Both sets of lines have been produced with ε = 72.2 for SPC/E water at 300 K.

The solute-solvent distribution functions [Eq. (6)] vs the distance from the surface of the Kihara solute (*R* _{0s } = 13 Å). Shown are the solute-oxygen (solid lines) and solute-hydrogen (dotted lines) radial distribution functions (ℓ = 0) and orientational distributions for ℓ = 1 (dash-dotted lines) and ℓ = 2 (dashed lines); (a) refers to and (b) refers to .

The solute-solvent distribution functions [Eq. (6)] vs the distance from the surface of the Kihara solute (*R* _{0s } = 13 Å). Shown are the solute-oxygen (solid lines) and solute-hydrogen (dotted lines) radial distribution functions (ℓ = 0) and orientational distributions for ℓ = 1 (dash-dotted lines) and ℓ = 2 (dashed lines); (a) refers to and (b) refers to .

The first (upper panel) and second (lower panel) orientational order parameters of the first-shell SPC/E waters vs . The solid circles (ε_{0s } = 0.65 kJ/mol) and squares (ε_{0s } = 20 kJ/mol) refer to Kihara solutes in water at *T* = 300 K. The corresponding open points refer to *T* = 273 K. The crosses in the lower panel show for HS cavities in the fluid of dipolar hard spheres (DHS) (Ref. 22). with the reduced dipole moment ; *m* is the dipole moment and σ_{ s } is the HS diameter of the solvent. The filled diamond labeled “plane” marks for a planar water interface from Ref. 9.

The first (upper panel) and second (lower panel) orientational order parameters of the first-shell SPC/E waters vs . The solid circles (ε_{0s } = 0.65 kJ/mol) and squares (ε_{0s } = 20 kJ/mol) refer to Kihara solutes in water at *T* = 300 K. The corresponding open points refer to *T* = 273 K. The crosses in the lower panel show for HS cavities in the fluid of dipolar hard spheres (DHS) (Ref. 22). with the reduced dipole moment ; *m* is the dipole moment and σ_{ s } is the HS diameter of the solvent. The filled diamond labeled “plane” marks for a planar water interface from Ref. 9.

Distributions of three angles used to define the water orientations relative to the radial direction (Fig. 2): (a) θ_{ b } for the O–H bond (), (b) θ_{ m } for the water dipole (), and (c) θ_{⊥} for the normal to H_{2}O plane (); *R* _{0s } = 13 Å.

Distributions of three angles used to define the water orientations relative to the radial direction (Fig. 2): (a) θ_{ b } for the O–H bond (), (b) θ_{ m } for the water dipole (), and (c) θ_{⊥} for the normal to H_{2}O plane (); *R* _{0s } = 13 Å.

Surface charge density σ_{ P }(θ) = *P* _{ n }(θ) vs the polar angle θ measured from the direction of the external field (a). The solid line (M) shows the result of Maxwell's electrostatics [Eq. (16), ε = 72.2 for SPC/E water at 300 K]. The dots refer to σ_{ P } calculated as a linear response to an external field from 40 ns of MD trajectories (see the SM (Ref. 19)). The MD σ_{ P } is collected from a water layer *R* _{0s } ⩽ *r* ⩽ *R* _{0s } + 0.1σ_{ s } for the solute with the size *R* _{0s } = 13 Å and ε_{0s } = 0.65 kJ/mol. The dashed line shows the result of using *E* _{ c } from simulations to calculate χ_{1} from Eq. (13). Panel (b) shows the profile of water's dielectric constant (see text).

Surface charge density σ_{ P }(θ) = *P* _{ n }(θ) vs the polar angle θ measured from the direction of the external field (a). The solid line (M) shows the result of Maxwell's electrostatics [Eq. (16), ε = 72.2 for SPC/E water at 300 K]. The dots refer to σ_{ P } calculated as a linear response to an external field from 40 ns of MD trajectories (see the SM (Ref. 19)). The MD σ_{ P } is collected from a water layer *R* _{0s } ⩽ *r* ⩽ *R* _{0s } + 0.1σ_{ s } for the solute with the size *R* _{0s } = 13 Å and ε_{0s } = 0.65 kJ/mol. The dashed line shows the result of using *E* _{ c } from simulations to calculate χ_{1} from Eq. (13). Panel (b) shows the profile of water's dielectric constant (see text).

Relative dielectric constant increment Δε/ε, for several aqueous solutions: lysozyme (Ref. 29) (filled squares), dioxane (Ref. 30) (open circles), glucose (Ref. 31) (open diamonds), and maltotriose (Ref. 31) (open triangles). The solid line (M) shows the result of Eq. (18), while the dashed line (L) is the dielectric increment for a solute with no surface charge density, σ_{1} = 0, and therefore Lorentz result for the cavity field [Eq. (13)]. Experimental and calculation results used to convert the experimentally reported solute concentrations to volume fractions can be found in the SM (Ref. 19).

Relative dielectric constant increment Δε/ε, for several aqueous solutions: lysozyme (Ref. 29) (filled squares), dioxane (Ref. 30) (open circles), glucose (Ref. 31) (open diamonds), and maltotriose (Ref. 31) (open triangles). The solid line (M) shows the result of Eq. (18), while the dashed line (L) is the dielectric increment for a solute with no surface charge density, σ_{1} = 0, and therefore Lorentz result for the cavity field [Eq. (13)]. Experimental and calculation results used to convert the experimentally reported solute concentrations to volume fractions can be found in the SM (Ref. 19).

Article metrics loading...

Full text loading...

Commenting has been disabled for this content