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Dielectric Permittivity Profiles of Confined Polar Fluids THE JOURNAL OF CHEMICAL PHYSICS 122, 114711 ͑2005͒ Dielectric permittivity profiles of confined polar fluids V. Ballenegger and J.-P. Hansen Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, United Kingdom ͑Received 22 September 2004; accepted 10 November 2004; published online 22 March 2005͒ The dielectric response of a simple model of a polar fluid near neutral interfaces is examined by a combination of linear response theory and extensive molecular dynamics simulations. Fluctuation expressions for a local permittivity tensor ⑀͑r͒ are derived for planar and spherical geometries, based on the assumption of a purely local relationship between polarization and electric field. While the longitudinal component of ⑀ exhibits strong oscillations on the molecular scale near interfaces, the transverse component becomes ill defined and unphysical, indicating nonlocality in the dielectric response. Both components go over to the correct bulk permittivity beyond a few molecular diameters. Upon approaching interfaces from the bulk, the permittivity tends to increase, rather than decrease as commonly assumed, and this behavior is confirmed for a simple model of water near a hydrophobic surface. An unexpected finding of the present analysis is the formation of “electrostatic double layers” signaled by a dramatic overscreening of an externally applied field inside the polar fluid close to an interface. The local electric field is of opposite sign to the external field and of significantly larger amplitude within the first layer of polar molecules. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1845431͔ I. INTRODUCTION In this paper we consider the case of polar fluids con- fined by continuous dielectric media characterized by a per- The dielectric permittivity of a medium is a macroscopic mittivity ⑀Ј. We relate the local dielectric permittivity ⑀͑r͒ to concept which is defined by the relationship between the the dipolar fluctuations within the inhomogeneous fluid, 1 polarization P and the electric field E inside the medium. along the lines of the classic Kirkwood–Fröhlich ͑KF͒ linear When the dielectric medium is inhomogeneous over dis- response treatment of the bulk permittivity.6,7 More specifi- tances much larger than molecular scales, a space-dependent cally, we shall consider the cases of a simple polar fluid in an ͑local͒ permittivity ⑀͑r͒ may be defined when dealing with infinite slab confined by two semi-infinite dielectric media mesoscopic electrostatic problems. The question of how far and of a polar fluid confined to a spherical cavity inside a towards molecular scales a local permittivity remains a uniform, macroscopic dielectric continuum. Numerical re- meaningful concept, and how ⑀͑r͒ is related to dipolar fluc- sults based on long Molecular Dynamics ͑MD͒ simulations 2 tuations is a long-standing problem which we have recently will illustrate the limitations of the concept of a local permit- 3 addressed in the case of a polar fluid near a sharp interface. tivity in the two geometries. We showed that a necessary condition for the existence of a All considerations in this paper will be restricted to meaningful, statistical definition of a local permittivity is that sharp interfaces. Like most previous theoretical and numeri- the local electric field inside the medium does not vary ap- cal work in the field, the present coarse-grained treatment preciably on the scale of the molecular correlation length, as suffers from the inconsistency of ignoring the molecular 2 already noted by Nienhuis and Deutch. graininess of the confining media, while using a fully mo- The ability to give a clear-cut statistical definition of a lecular description of the polar fluid. local permittivity is crucial for any coarse-graining strategy, whereby large parts of a complex multicomponent system II. POLARIZATION IN LINEAR RESPONSE are treated as continuous dielectric media, while the remain- ͑ ␤͒ Consider a classical fluid at temperature T=1/ kB , ing parts are described in molecular detail. An important ex- ␮ made up of N polar molecules carrying dipole moments i, ample is provided by implicit solvent models of biomolecu- confined to a cavity of arbitrary shape and volume V, carved lar assemblies, where water is considered as a continuous out of a macroscopic dielectric medium of uniform permit- dielectric medium, characterized by a local permittivity in tivity ⑀Ј. The molecules may be polarizable; their interac- the immediate vicinity of biomolecules or membranes. A spa- tions are arbitrary at short distances, but tend towards the tially varying permittivity then determines the electrostatic 4 dipolar interaction at larger distances. The microscopic po- interactions between charged residues and ions. Conversely larization density is one may wish to describe a polar solvent trapped within a dielectric matrix, as in the case of water confined between N ͑ ͒ ␮ ␦͑ ͒ ͑ ͒ membranes or clay platelets, or within narrow pores. In these m r = ͚ i r − ri , 1 i=1 circumstances it may be advantageous to describe the con- fining matrix as a dielectric continuum, while the confined where ri is the position of the ith molecule inside the cavity. polar liquid is modeled with molecular resolution.5 The corresponding total dipole moment is 0021-9606/2005/122͑11͒/114711/10/$22.50122, 114711-1 © 2005 American Institute of Physics Downloaded 15 Jun 2005 to 194.57.89.1. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 114711-2 V. Ballenegger and J.-P. Hansen J. Chem. Phys. 122, 114711 ͑2005͒ charges at ͉r͉!ϱ͒ EЈ. In an isotropic phase, P ͑r͒=0 for ͑ ͒ ␮ ͑ ͒ 0 M = ͵ m r dr = ͚ i. 2 points in the bulk of the fluid. Close to the confining bound- D cavity i ͑ ͒ aries, P0 r is nonzero in general, but may vanish for sym- metry reasons, as in the case of linear polar molecules con- ͑ ͒ ͗ ͑ ͒͘ Let P0 r = m r be the average local polarization of the fined in a slab or a spherical cavity ͑see Sec. III͒. When a fluid in the absence of an externally applied electric field ͑by uniform external field is applied to the system, it induces a definition, EЈ is the field far away from the cavity, created by polarization density defined by ͓͐ ͑ ͒ ͗ ͑ ͔͒͘ ͓ ␤͑ ͑ … ͒ ͔͒ m r − m r exp − U⑀Ј 1, ,N − M · Ec d1 dN ⌬P͑r͒ = P͑r͒ − P ͑r͒ = ͗m͑r͒͘ − ͗m͑r͒͘ = ¯ , ͑3͒ 0 EЈ ͐ ͓ ␤͑ ͑ … ͒ ͔͒ exp − U⑀Ј 1, ,N − M · Ec d1¯dN where we have used the short-hand notation i for the degrees c ⌬P␣͑r͒ = ␤ ͚ ͓͗m␣͑r͒M␥͘ − ͗m␣͑r͒͗͘M␥͔͘E␥, ͑7͒ of freedom of the ith molecule. For linear nonpolarizable ␥=x,y,z ͑ ␮ ͒ molecules, i= ri , i reduces to the position and orientation of the permanent dipole moment, and integration with phase where ␣,␥=x, y,orz and the statistical averages are under- 3 space element di=d r d⍀␮ is performed over all possible ͑ ͒ i i stood to be taken at zero external and hence cavity field, positions and orientations of the molecule inside the cavity. i.e., with a Bolzmann weight exp͑−␤U⑀Ј͒. As expected for U⑀Ј is the total interaction energy of the N polar molecules of the linear response to a uniform external field, Eq. ͑7͒ in- the fluid within the cavity in the absence of EЈ; it depends volves the average correlation between a fluctuation in the obviously on the permittivity ⑀Ј of the surrounding dielectric. local polarization density m͑r͒ and a fluctuation in the global The instantaneous total dipole moment M couples to the cav- dipole moment M of the system, as has been recognized 8 ity field Ec, i.e., the electric field inside the cavity in the recently by Stern and Feller. Note that ͗M͘ will be zero by absence of polar fluid, when the external ͑applied͒ field in symmetry in all systems we shall consider. the embedding dielectric is EЈ. The two fields are related by Comparison between Eqs. ͑5͒ and ͑7͒ does not provide a the usual boundary conditions of macroscopic electrostatics. fluctuation formula for ␹͑r͒ or ⑀͑r͒, since they involve the ⌬ ͑ ͒ ͑ ͒ ͑ ͒ Let E r =E r −E0 r be the difference between the total and cavity fields, respectively. The relation between mean local electric field inside the cavity, due to the external these two fields depends on the geometry of the cavity, and field and all the dipoles, and the mean electric field when no can be established within macroscopic electrostatics. We ͓ ͑ ͒ ͑ ͒ external field is applied note that E0 r =0 if P0 r =0 ev- consider successively the case of slab and spherical geom- erywhere͔. Then, within the linear regime ͓i.e., for not too etries. strong ⌬E͑r͔͒, the induced polarization density is related to ⌬E͑r͒ via A. Slab geometry 1 ⌬P͑r͒ = ͵ ␹͑r,rЈ͒ · ⌬E͑rЈ͒drЈ, ͑4͒ We consider a cavity in the form of an infinite slab 4␲ D cavity where a fluid of ␳=N/V polar molecules per unit volume is confined in the z direction by two infinite dielectric walls of where ␹ is the dielectric susceptibility tensor. In the slow permittivity ⑀Ј. The distance between the dielectric walls is modulation limit, i.e., for slowly varying ⌬E͑r͒, the integral L. The confined fluid is inhomogeneous in the z direction factorizes approximately, and Eq. ͑4͒ reduces to the local ͑orthogonal to the walls͒ only. By symmetry, the permittivity form tensor reduces to the diagonal form 1 ⑀ʈ͑z͒ 00 ⌬P͑r͒ = ␹͑r͒ · ⌬E͑r͒, ͑5͒ ␲ ⑀͑ ͒ ⑀ ͑ ͒ ͑ ͒ 4 z = ΂ 0 ʈ z 0 ΃, 8 00⑀Ќ͑z͒ where, formally, ␹͑r,rЈ͒=␹͑r͒␦͑r−rЈ͒.
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