Dielectric Permittivity Profiles of Confined Polar Fluids

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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 confined 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 confining 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

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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 symmetry 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. The local permittivity tensor is deﬁned by where ⑀ʈ and ⑀Ќ denote the components parallel and orthogonal to the walls. Equations ͑5͒ and ͑6͒ then combine into two ␹͑r͒ = ⑀͑r͒ − I. ͑6͒ independent relations,

͑ ͒ ⑀ʈ͑z͒ −1 Linearization of Eq. 3 with respect of Ec leads to the fol- Pʈ͑z͒ = Eʈ͑z͒, ͑9a͒ ⌬ ͑ ͒ ͑ ͒ ␲ lowing relation between the components of P r and Ec r : 4

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⑀Ќ͑z͒ −1 The key finding is that a local expression of the permit- ⌬PЌ͑z͒ = ⌬EЌ͑z͒. ͑9b͒ 4␲ tivity involves correlations of the local and total polarization of the form ͗m␣͑z͒M␥͘, and not of the local polarization We dropped the symbol ⌬ in Eq. ͑9a͒ because isotropy in the alone, as has sometimes been wrongly assumed in the litera- ͑ ͒ ͑ ͒͑ 8 x, y -plane implies that P0 r the average polarization in ture. This was already recognized by Stern and Feller, but the absence of external field͒ has no parallel components. their expression for the permittivity tensor ⑀͑z͒ differs from Using the standard boundary conditions on the normal and the one derived here, because they did not consider a single tangential components of the electric field, one finds the fol- slab, but a system which is periodically replicated in space to lowing relations between the components of the uniform ex- form an infinite spherical array of the original slab. ternal field EЈ and the cavity field Ec, We stress that formulas ͑13͒ and ͑17͒ were derived for a c c uniform external field under the local assumption ͑5͒.Ifthe Eʈ = EЈʈ , EЌ = ⑀ЈEЌЈ , ͑10͒ local assumption is not valid, definitions ͑9a͒ and ͑9b͒ be- c Ј ͑ ͒ where Eʈ and Eʈ are two-dimensional vectors in the x,y come purely formal, and the permittivity ⑀͑z͒ may take val- plane; the orthogonal components are along the z direction. ues that are unphysical and specific to the case of a uniform .١ϫE͑z͒=0 implies external field permeating a planar interface Maxwell’s equation E ͑z͒ץ E ͑z͒ץ x = y =0, ͑11͒ z B. Spherical geometryץ zץ ͑ ͒ ͑ ͒ Ј We now consider a system of N polar molecules con- so that Eʈ = Ex ,Ey is independent of z, i.e., Eʈ z =Eʈ every- where in space. In other words, Eq. ͑9a͒ leads to fined to a spherical cavity of radius R, surrounded by a dielectric medium of permittivity ⑀Ј. We first recall the fluc- ⑀ ͑ ͒ ⑀ ͑ ͒ ʈ z −1 ʈ z −1 c tuation formula for the bulk dielectric constant of this Pʈ͑z͒ = EЈʈ = Eʈ . ͑12͒ 4␲ 4␲ system. The cavity field is now Ec =3⑀ЈEЈ/͑2⑀Ј+1͒. For a macroscopic spherical sample of uniform permittivity ⑀, the Comparison of Eqs. ͑7͒ and ͑12͒, together with isotropy in local field, far from the boundaries, is uniform and equal to the ͑x,y͒ plane then leads to the desired fluctuation formula E=3⑀ЈEЈ/͑2⑀Ј+⑀͒. Substituting Ec and E into Eq. ͑7͒ and for ⑀ʈ͑z͒: into the definition ͑5͒ of the polarization, one arrives, upon ⑀ʈ͑z͒ =1+2␲␤͓͗mʈ͑z͒ · Mʈ͘ − ͗mʈ͑z͒͘ · ͗Mʈ͔͘. ͑13͒ identification and use of the isotropy of the system, at The orthogonal component may be determined from Max- ͑⑀ −1͒͑2⑀Ј +1͒ 4␲⑀ = ͓͗m͑r͒ · M͘ − ͗m͑r͒͘ · ͗M͔͘. ͑18͒ D͑z͒=0, where D=E+4␲P is the dis- ͑2⑀Ј + ⑀͒ 3·١ well’s equation placement vector, leading to In this formula r can be any point in the bulk of the sample, d so that m͑r͒ may be replaced by M/V if boundary effects are ͓EЌ͑z͒ +4␲PЌ͑z͔͒ =0. ͑14͒ dz negligible. This leads back to the well known KF formula for the bulk dielectric constant in terms of fluctuations ͑͗M2͘ Integration of Eq. ͑14͒ from −ϱ to z yield −͗M͘2͒/V of the total dipole moment of the system.7 Since EЌ͑z͒ − EЌЈ =−4␲PЌ͑z͒ +4␲PЌ͑z =−ϱ͒ boundary conditions in computer simulations are designed to minimize finite size effects, the KF formula can be used, as =− 4␲PЌ͑z͒ + ͑⑀Ј −1͒EЌЈ . ͑15͒ expected, to compute the dielectric constant in a simulation performed with a reaction field or periodic boundary condi- Substracting from Eq. ͑15͒ the same equation without exter- c tions ͑if the Ewald sums are performed using the spherical nal field and using Eq. ͑9b͒ gives ⑀Ќ͑z͒⌬EЌ͑z͒=⑀ЈEЈ =E , Ќ Ќ convention for the order of summation͒.9 For a confined where the second equality follows from Eq. ͑10͒. Equation spherical system which is not periodically repeated, the di- ͑9b͒ may hence be rewritten as electric constant should be computed from Eq. ͑18͒͑see Sec. ⑀ ͑ ͒ ͒ 1 Ќ z −1 c III B . PЌ͑z͒ = EЌ. ͑16͒ 4␲ ⑀Ќ͑z͒ Attempts have been made to generalize the KF relation, valid for a macroscopic spherical system, to mesoscopic ͑ ͒ ͑␣ Ќ͒ Comparison of Eq. 16 with the transverse = version samples, where P͑r͒, E͑r͒, and the resulting ⑀͑r͒ are nonuni- ͑ ͒ ⑀ ͑ ͒ 10,11 of Eq. 7 leads to the desired fluctuation formula for Ќ z , form near the confining surface. Strictly speaking, ⑀͑r͒

⑀Ќ͑z͒ −1 is, however, no longer a scalar near the sample boundary, but =4␲␤͓͗mЌ͑z͒MЌ͘ − ͗mЌ͑z͒͗͘MЌ͔͘. ͑17͒ a tensor with radial and tangential components. ⑀Ќ͑z͒ We consider here the case where the external field is Equations ͑13͒ and ͑17͒ are the appropriate fluctuation for- radial, preserving thus the spherical symmetry of the prob- mulas to compute the permittivity tensor ⑀͑z͒ of a system lem. Such a radial field can arise from an external charge q inhomogeneous in one direction. Note that these formulas placed at the center of the spherical cavity filled with polar depend only implicitly, via the statistical averages weighted molecules, or polar residues of a globular macromolecule by the Bolzmann factor exp͑−␤U⑀Ј͒, on the permittivity ⑀Ј of ͑e.g., a protein͒; in that case the nonuniform external field is the confining medium. This is to be contrasted with the re- EЈ͑r͒=͑q/r2͒rˆ where rˆ =r/r. The dipoles of the confined sults for spherical samples to be discussed below. fluid or macromolecule couple to this field with energy,

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tended dipole ␮ made up of two opposite charges ±q placed ␮ Ј͑ ͒ ͵ ͑ ͒ Ј͑ ͒ ͑ ͒ Uext =−͚ i · E r =− m r E r dr, 19 at ±d/2 from the center, such that ␮=qd; the elongation was D i cavity chosen to be d/␴=1/3, where ␴ is the molecular diameter. where m͑r͒=m͑r͒·rˆ is the radial component of the micro- The bulk dielectric behavior of fluids with such extended scopic polarization density ͑1͒, and EЈ͑r͒=q/r2. Substituting dipoles has been shown to be very similar to that of fluids 12 ͑19͒ into the Boltzmann factor in the definition ͑3͒ of the with point dipoles, as long as d/␴Ͻ1/2. The short-range polarization, and linearizing ͑which is valid in the limit of repulsion between the spherical molecules is chosen to be of small q͒, one arrives at the following relation between the the “soft sphere” form radial component of P͑r͒ and the radial component of the ␴ 12 external field: u ͑r͒ =4uͩ ͪ ͑25͒ Sr r P͑r͒ = P͑r͒ · rˆ = ͗m͑r͒͘ + ␤͵ drЈ͓͗m͑r͒m͑rЈ͒͘ with ␴=0.366 nm and u=1.85 kJ/mole ͓the charges ±q D cavity carry a mass m=5 amu; a molecule has hence a total mass of * ␴2 − ͗m͑r͒͗͘m͑rЈ͔͒͘EЈ͑rЈ͒, ͑20͒ 2m and a reduced moment of inertia I =I/2m =1/9. The simulations were performed with the simulation package 13 where the statistical averages are once more taken at zero GROMACS. The equations of motion were integrated with a external field. We consider the case of molecules carrying time step dt=2 fs ͑reduced time step dt* =dt/͑m␴2 /u͒1/2 linear dipoles, so that ͗m͑r͒͘=0 by symmetry. We now as- =0.0024͔͒. The walls exert a force on the center of mass of sume a local relationship between P͑r͒ and the radial local the molecules that derives from the potential, field E͑r͒ in the general form defined by Eqs. ͑5͒ and ͑6͒, 4␲u ␴9 ␴9 and involving a local dielectric permittivity ⑀͑r͒, U ͑z͒ = ͫ + ͬ. ͑26͒ walls 45 z9 ͑L − z͒9 ⑀͑r͒ −1 P͑r͒ = E͑r͒. ͑21͒ 4␲ This potential follows from integrating the soft-sphere repulsion potential ͑25͒ over the regions zϽ0 and zϾL, for a wall ␲ ␦͑ ͒ ␳ ␴3 ͒ ͑ ١ This relationship, together with ·D r =4 q r , implies of density wall =1. The long range Coulomb interactions that the fields E͑r͒ and EЈ͑r͒ are related by between the charges ±q and the infinite array of periodic EЈ͑r͒ images are computed by a slab-adapted version of the usual E͑r͒ = . ͑22͒ 3D Ewald summation, as explained in the Appendix. ⑀͑r͒ The structure of the fluid inside the slab is best charac- Combination of Eqs. ͑20͒, ͑21͒, and ͑22͒ then leads to the terized by the density-orientation profile ␳͑z,␪͒, where ␪ is following relation for ⑀͑r͒: the angle between a molecular dipole and the z axis. This may be expanded in Legendre polynomials according to 1 ⑀͑r͒ −1 EЈ͑r͒ = ␤͵ drЈ͗m͑r͒m͑rЈ͒͘EЈ͑rЈ͒͑23͒ ϱ 4␲ ⑀͑r͒ D cavity ␳͑z,␪͒ = ͚ ␳ᐉ͑z͒Pᐉ͑cos ␪͒, ͑27͒ ᐉ or, substituting EЈ͑r͒=q/r2, =0 +1 where ␳ᐉ͑z͒=1/2͑2ᐉ +1͒͐ ␳͑z,␪͒Pᐉ͑cos ␪͒d͑cos ␪͒.Inthe ⑀͑r͒ −1 r 2 −1 =4␲␤͵ drЈ͗m͑r͒m͑rЈ͒ͩ͘ ͪ . ͑24͒ case of uncharged walls, only even coefficients appear in the ⑀͑r͒ D rЈ cavity series ͑27͒ because of the symmetry ␳͑z,␪͒=␳͑z,␲−␪͒. The ᐉ ␳ ͑ ͒ ␳͑ ͒ Note that, contrary to Eq. ͑18͒, this relation does not depend =0 coefficient 0 z = z /2 is one half of the density pro- ␳͑ ͒ ͐␲␳͑ ␪͒ ␪ ␪ explicitly on the permittivity ⑀Ј of the confining medium. file z = 0 z, sin d . The ratio The present space-dependent dielectric constant ⑀͑r͒ de- ␳ᐉ ͑z͒ ␣͑z͒ = =2 ͑28͒ scribes the screening by the polar fluid of the external field ␳͑z͒ created by the point charge q, as is obvious from Eq. ͑22͒.It reduces to the bulk dielectric constant when r is sufficiently provides a measure of the mean alignment of the dipoles. ͑ ͒ ͑ 2 ͒ ␣͑ ͒ large, but still small compared to the radius R of the spheri- Since P2 x = 3x −1 /2, z is negative if the dipoles are cal cavity. predominantly aligned parallel to the interface ͓␣͑z͒=−5/4 for complete alignment͔, while ␣͑z͒ is positive for predomi- III. MOLECULAR DYNAMICS RESULTS nantly orthogonal alignment ͓␣͑z͒=5/2 for full alignment orthogonal to the interface͔. A. Slab geometry We performed the simulations at a constant temperature We have carried out a number of long MD simulations T=300 K ͑reduced temperature T* =kT/u=1.35͒, and for

͑spanning tens of nanoseconds͒ to obtain estimates of ⑀ʈ͑z͒ two values of the dipole moment: ␮=1.47 and 2.45 D, cor- * ͱ 2 3 and ⑀Ќ͑z͒ from the fluctuation formulas ͑13͒ and ͑17͒.Ina responding to a reduced dipole ␮ = ␮ /␴ u=1.2 and 2, re- slab of width L, 3500 dipolar soft spheres ͑DSS͒ were spectively. The width of the slab was adjusted to L placed. The confining walls at z=0 and z=L are assumed to =16.62␴, so that the reduced density of the fluid far from the ͑⑀Ј ͒ ␳* ␳ ␴3 be nonpolarizable =1 . The simulation cell is a cube with walls is bulk= bulk =0.8. The bulk dielectric constant of edges of length L, and periodic boundary conditions are im- this polar fluid is 98±2 at ␮* =2,12,14 and about 10 at ␮* posed in the ͑x,y͒ directions. Each molecule carries an ex- =1.2.

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FIG. 1. Density and orientational profile of a DSS fluid in a slab ͑T* ␳* ͒ =1.35, bulk=0.8 for two values of the reduced dipole moment. FIG. 2. Parallel component of the permittivity tensor ͑same system as in Fig. 1͒, from fluctuation formula ͑13͒ and from the response to an external The resulting density and orientation profiles ␳͑z͒ and field EЈ=0.1 V/nm along x axis. ␣͑z͒ in zero applied field are plotted in Fig. 1. Layering along the walls occurs in an interfacial region of five to six wall with ⑀Ј=1. This is contrary to the prediction of a gen- molecular diameters. The first layer of molecules is seen to eralization of the familiar Onsager cavity model to the case align its dipoles parallel to the interface, but orientational of a dipolar fluid near a dielectric wall.17 ordering is rapidly lost further away from the dielectric Turning to the perpendicular permittivity ⑀Ќ͑z͒, we con- walls. The ordering of molecular dipoles parallel to the wall sider first the MD results obtained in the absence of external in the first layer may be understood qualitatively in terms of ͑ ͒ 15 field. The structure of the fluctuation formula 17 is very electrostatic interactions of these dipoles with their images. unfavorable for the estimation of large values of the permit- Strictly speaking, there are no images on the molecular scale tivity since the ratio of the left-hand side is then close to 1, since the walls are not polarizable, but image charge interac- so that statistical uncertainties on the fluctuation expression tions arise on the mesoscopic scale because of the dielectric on the right-hand side, which we shall henceforth denote by discontinuity between the polar fluid ͑⑀Ͼ1͒ and the walls f͑z͒, will be dramatically amplified on the quantity of interest ͑⑀Ј ͒ =1 . The behavior of the density-orientation profile far ⑀Ќ͑z͒=1/͓1−f͑z͔͒. This shortcoming is illustrated in Fig. 3 * from a single dielectric wall has been studied by Badiali, for the case ␮ =1.2 ͑we did not succeed in extracting ⑀Ќ͑z͒ who showed that ␳͑z͒ is given asymptotically by its bulk from dipolar fluctuations in the case ␮* =2͒. The signal f͑z͒ value plus an A/z3 tail arising from the dielectric discontinu- calculated by dividing the slab width L into 300 “bins” is 16 ity between ⑀ and ⑀Ј. seen to be rather noisy, despite the length of the simulation Parallel and perpendicular permittivity profiles ⑀ʈ͑z͒ and ͑28 ns͒. The fluctuations are strongly enhanced near the ⑀Ќ͑z͒ were estimated from the simulations in zero external field using the fluctuation formulas ͑13͒ and ͑17͒, as well as from simulations in the presence of an external field, by evaluation of the ratio P͑z͒/E͑z͒ of the induced local polarization and electric field ͓cf. Eqs. ͑9͔͒.

Results for the parallel permittivity ⑀ʈ͑z͒ obtained by both routes are shown in Fig. 2 ͑the simulation with EЈ=0 lasted 28 ns, while that with EЈ=0.1 V/nm along the x axis was 3.5 ns long͒. The agreement between the two independent estimates is seen to be perfect. The pronounced oscillations of ⑀ʈ͑z͒ near the walls closely mirror the oscillations in the density profile apparent in Fig. 1. In fact the ratio ⑀ ͑ ͒␳ ␳͑ ͒ ʈ z bulk/ z , also shown in Fig. 2, shows much less structure. Towards the middle of the slab, ⑀͑z͒ is seen to be constant and to take a value ⑀Ӎ10 ͑for ␮* =1.2͒ and ⑀Ӎ96 ͑for ␮* =2͒, in agreement with the bulk value derived from MD simulations of a periodic nonconfined fluid at the same state point. Note that on average, ⑀ʈ͑z͒ tends to increase above its FIG. 3. Dipolar fluctuations f͑z͒͑thin line͒ and smoothed curve f͑z͒͑thick bulk value close to the confining walls; in other words, par- line͓͒see text͔. Inset: orthogonal component of the permittivity tensor ͑␮* * ␳* ͒ allel dipolar fluctuations tend to be enhanced near a dielectric =1.2, T =1.35, bulk=0.8 .

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FIG. 4. Continuous line: polarization charge density c͑z͓͒e/nm3͔ induced ͑␮* ␳* * ͒ inside a slab of polar ﬂuid =1.2, bulk=0.8, T =1.35 by an external electric ﬁeld EЈ=1 V/nm along the z direction. Dashed line: ten times the ͐z ͑ Ј͒ Ј integral 0c z dz .

walls, but these values lead to nonphysical, negative values FIG. 5. Electric field E͑z͒͑thick line͒ and polarization density P͑z͒ com- of ⑀Ќ͑z͒, pointing to the inadequacy of the local assumption puted using Eq. ͑30͒͑thin line͒ and from the statistical average ͗m͑z͒͘ ͑dot- ͒ ⑀ ͑ ͒ ͑ ͒ ͑9͒. A smoothed curve f͑z͒ is obtained by averaging the sig- ted line , for the same system as in Fig. 4. Inset: Ќ z from Eq. 31 and ͑ ͒͑ ͒ nal f͑z͒/␳͑z͒ over intervals of width 3␴, and multiplying the from fluctuation formula 17 dashed line . result by the local density ␳͑z͒. The resulting estimate of ⑀Ќ͑z͒ in the central region of the slab is shown in the inset of ͑Ӎ−2 V/nm͒, i.e., the external field EЈ is overscreened by a Fig. 3. factor of 2! We refer to this remarkable effect as the forma- The statistical uncertainties are still large, but the aver- tion of an “electrostatic double layer” ͑EDL͒. Beyond the

age value is compatible with a bulk permittivity ⑀Ќ =⑀ʈ first layer, the oscillations in E͑z͒ are gradually damped, but Ӎ10. Outside this central “bulk” interval, ⑀Ќ͑z͒ becomes are still visible in the central part of the slab, where the unphysical whenever f͑z͒ exceeds 1. oscillations are around a mean value of about 0.1 V/nm. The We have investigated the transverse dielectric response polarization P͑z͒ oscillates out of phase with E͑z͒ as ex- by applying an external field EЈ=1 V/nm along the z axis of pected, and the two estimates, based on Eq. ͑30͒ and on ͑␮* ͒ ͗ ͑ ͒͘ the moderately polar fluid =1.2, simulation time: 12 ns . m z EЈ, are in excellent agreement. So are the estimates of This large value was chosen such that the local ͑screened͒ ⑀Ќ͑z͒ based on Eqs. ͑17͒ and ͑31͒, despite the large statistical field near the center of the slab, E͑z͒=EЈ/⑀Ќ͑z͒, is still suf- uncertainties. As pointed out above, the relation between ficiently strong to induce a sizeable polarization. The local PЌ͑z͒ and EЌ͑z͒ is nonlocal ͓i.e., of the more general form charge density c͑z͒ induced by the applied field is plotted in ͑4͔͒ near the walls, where ⑀͑z͒=1+4␲P͑z͒/E͑z͒ can take Fig. 4. As expected it is antisymmetric with respect to mid- negative values, and diverges whenever E͑z͒=0. Hence plane ͑z=L/2͒, and integration of c͑z͒ over the left-hand and ⑀Ќ͑z͒ is not a useful quantity near the walls. A more relevant right-hand halves of the slab leads to induced surface charge quantity is the normalized EDL profile E͑z͒/EЈ. For weak Ӎ 2 ͑ ͒ densities cS ±0.05 e/nm . The local electric field E z is fields, it is given by linear response theory, viz., calculated from

z EЌ͑z͒ =1−4␲␤͓͗mЌ͑z͒MЌ͘ − ͗mЌ͑z͒͗͘MЌ͔͘ E͑z͒ = EЈ +4␲͵ c͑zЈ͒dzЈ ͑29͒ EЈ 0 =1−f͑z͒. ͑32͒ and the polarization density may then be deduced from Figure 6 illustrates the good agreement between the EDL EЈ − E͑z͒ z P͑z͒ = =−͵ c͑zЈ͒dzЈ ͑30͒ profiles computed from the fluctuations f͑z͒͑properly ␲ 4 0 smoothed as in Fig. 3͒ and directly from the ratio EЌ͑z͒/EЈ. The small discrepancies close to the walls are probably due while the permittivity profile follows from to the fact that the extended dipoles are treated as point di- EЈ poles in the evaluation of f͑z͒. In order to investigate the ⑀Ќ͑z͒ = . ͑31͒ ␴ E͑z͒ influence of the dipole extension d/ on the striking overscreening effect observed in the EDL profile near the walls, Results obtained in this way for E͑z͒, P͑z͒, and ⑀Ќ͑z͒ are we have also carried out simulations for an extension d/␴ plotted in Fig. 5. The polarization profile P͑z͒ can also be =0.1 ͑compared to 0.3 in the previous simulations͒, keeping determined from the statistical average ͗m͑z͒͘ of the micro- the dipole moment fixed at ␮* =1.2. An oscillatory profile of scopic polarization density alone ͑dotted line in Fig. 5͒. The E͑z͒ similar to that in Fig. 5 is once more observed, but the local electric field E͑z͒ exhibits large oscillations close to the amplitude of the oscillation closest to the two walls increases walls. Inside the first layer, E͑z͒ is strongly negative by Ϸ50% ͑from Ϫ2 to about Ϫ3 V/nm͒, thus pointing to an

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FIG. 8. Geometry for the Berendsen formula: a droplet of radius R is surrounded by a continuous medium of dielectric constant ⑀Ј. Dipolar fluctuations in the fluid are measured inside a concentric subsphere of radius r, while the remaining fluid in the outer shell is assumed to behave as a dielectric continuum of permittivity ˜⑀.

FIG. 6. EDL ratio EЌ͑z͒/EЈ from Eq. ͑29͒͑continuous line͒ and from the rounding medium of dielectric constant unity, the local dipo- fluctuation formula ͑32͒͑dashed line͓͒same system as in Fig. 3͔. lar fluctuations in the vicinity of the interface remain almost bulklike, except in the very first layer. This behavior is in enhancement of the overscreening effect as the point dipole marked contrast to that observed for linear extended dipoles, limit is approached. illustrating the dominant influence of hydrogen bonding in We also carried out one long ͑30 ns͒ MD run of 2076 the case of water. SPC water molecules,18 confined between the same hydrophobic walls ͑26͒. Figure 7 shows the resulting profiles for B. Spherical geometry the Oxygen density, molecular orientation, and parallel permittivity ͑the orthogonal permittivity could not be obtained The bulk dielectric constant of a spherical droplet of a ͑ from the fluctuations͒. The density profiles are not unlike polar fluid possibly surrounded by a continuous medium of ⑀ ͒ those observed in the simulations of Lee, McCammon, and permittivity Ј is given in terms of dipolar fluctuations by ͑ ͒ Rossky,19 who used a different water model, different wall- Eq. 18 . Previous workers have used another approach, water interaction, and a narrower slit. The symmetry which is approximate: they get the dielectric constant of a ͗ 2 ͑ ͒͘ ␳͑z,␪͒=␳͑z,␲−␪͒ no longer holds, and there is indeed a droplet from the mean square dipole mB r of an inner nonzero average polarization P ͑z͒=␳͑z͒␮͗cos ␪͘ close to the spherical region of radius r, assuming that the remaining z ͑ ͒ walls, even with no applied external field. The permittivity surrounding fluid a shell of thickness R−r can be treated as ⑀ ͑ ͒ a dielectric continuum of permittivity ˜⑀ ͑see Fig. 8͒. The profile Ќ z shows that the bulk permittivity of SPC water 21–23 ͑⑀Ӎ65͒,20 is approximately reached after just one molecular latter approach yields the Berendsen formula, layer inside the fluid. It is a striking result that, despite the ͑2⑀Ј + ˜⑀͒͑2˜⑀ +1͒ −2͑r/R͒3͑⑀Ј − ˜⑀͒͑1−˜⑀͒ large dielectric discontinuity between the fluid and the sur- ͑⑀ −1͒ ͑2⑀Ј + ˜⑀͒͑2˜⑀ + ⑀͒ −2͑r/R͒3͑⑀Ј − ˜⑀͒͑⑀ − ˜⑀͒ 4␲␤͗m2 ͑r͒͘ = B , ͑33͒ 3Vr ␲ 3 ͑ ͒ when Vr =4 r /3. Equation 33 reduces to the KF formula if ˜⑀=⑀Ј or if r=R. When ˜⑀=⑀, it interpolates between the KF formula for a sphere surrounded by a medium of permittivity ⑀ ͑for rӶR͒, and the KF formula for a sphere surrounded by a medium ⑀Ј ͑for r=R͒. We compared the predictions of Eq. ͑18͒ with those of Eq. ͑33͓͒with ˜⑀=⑀͔, for a dipolar soft sphere fluid with pa- *␳ * ء␮ rameters =2, T =1.35, and bulk=0.2. The fluid was confined to a spherical region by external forces deriving from the potential, 1 1 9R − r 9R + r 4 V ͑r͒ =4u␴9␲ͩ ͫ − ͬ − ͪ, R 360 r ͑R − r͒9 ͑R + r͒9 9R9 ͑34͒ which arise from integrating the soft-phere repulsion potential ͑25͒ over the region rϾR, assuming a confining medium ␳ ␴3 of density wall =1. The results of a 30 ns long MD simulation of a droplet of N=1000 molecules confined in a sphere of radius R=11.4␴ surrounded by vacuum ͑⑀Ј=1͒ are shown FIG. 7. Density, orientation, and permittivity profiles of 2076 SPC water molecules confined in a slab of width 4.65 nm by hydrophobic walls in Fig. 9. Interactions were computed without the introduc- ͑␳ −3 ͒ ⑀ ͑ ͒ bulk=33.6 nm , T=300 K . tion of any cutoff. The estimate for based on Eq. 18

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FIG. 10. Radial electric field, polarization density, molecular density, and ͓ ␳͑ ͒␴3͔ FIG. 9. Density profile thick curve: ten times r and estimate of the permittivity profiles for a spherical droplet of polar fluid ͑␮* =2, T* =1.35, ͑ ͒ ͑ ͒ bulk dielectric constant of a droplet of a polar fluid from Eq. 18 with m r ␳* =0.2͒ when an ion of unit electronic charge ͑reduced charge q* =28.7͒ is ͑ ͒ bulk the total dipole moment of a concentric subsphere of radius r thin curve , present at the origin. The dotted line indicates the bulk dielectric constant ͑ ͒͑ ͒ and from the Berendsen formula 33 dashed line . The expected bulk value ⑀=6.3 ͑divided by 10͒. is indicated by the dotted line.

The electric field oscillates strongly close to the central agrees very well with the bulk dielectric constant ⑀ ion. Overscreening of the ion’s charge occurs in the first =6.3±0.2 obtained from a simulation performed under peri- molecular layers, similarly to the behavior observed near a odic boundary conditions using Ewald sums. The Berendsen planar interface ͑Fig. 5͒. At distances 3␴ϽrϽR, the electric approximation is seen to underestimate the dielectric con- field and polarization density decay as 1/r2, as required by stant by about 8%. It yields the most accurate estimate when macroscopic electrostatics. The radial permittivity profile, r is large enough to encompass the structural angular corre- obtained from the ratio P͑r͒/E͑r͒, shows qualitatively the lations, but small compared to R so that the outer shell of the same behavior than the orthogonal permittivity close to a fluid can screen effectively the dipolar fluctuations. This is in planar interface: it reduces to the bulk dielectric constant far ͑ ͒ ͑ ⑀ Ӎ ͒ contrast with Eq. 18 , which provides in principle an esti- from the interfaces here bulk 6.3 , while it is ill defined mate for ⑀ that is independent of the chosen radius r of the close to the central ion and close to the surface of the droplet inner subsphere, as long as it is small enough for boundary since it diverges whenever E͑r͒ changes sign. effects to be negligible. The variations of the estimate observed in Fig. 9, especially at small r, are to be attributed to ⑀Ј ͑ ͒ statistical uncertainties. Notice that when =1, Eq. 18 in- IV. CONCLUSION volves the unfavorable ratio ͑⑀−1͒/͑⑀+2͒, so that uncertainties are greatly enhanced in this case. Better estimates for the We have combined linear response theory with extensive dielectric constant would be obtained by surrounding the molecular dynamics simulations of a simple model for dipo- droplet with a medium of permittivity ⑀Ј=⑀, or even ⑀Ј=ϱ lar molecules, in an attempt to validate the concept of a local ͑metallic boundary conditions͒, similarly to the case of peri- dielectric permittivity of a confined ͑inhomogeneous͒ polar odically repeated systems ͑see discussion in Ref. 12͒. fluid. The geometries which were specifically investigated We turn now to the radial dielectric permittivity profile correspond to a polar fluid confined between two parallel ⑀͑r͒ of the spherical droplet, which is defined formally by walls ͑slit geometry͒, and in a spherical cavity surrounded by Eq. ͑22͒. The profiles for the average electric field, polariza- a dielectric continuum. Rather large samples ͑thousands of tion density, molecular density, and radial permittivity are polar molecules͒ were considered, with confinement lengths shown in Fig. 10. These results were obtained from a 35 ns on the scale of a few nanometers, while statistics were gath- long MD simulation of the previous droplet, when an ion of ered over tens of nanoseconds. The key findings of our work reduced charge q* =q␮/␴2u=28.7 is present at the origin. are the following. The total charge enclosed in a sphere a radius r around the ͑a͒ There is excellent agreement between the results for central ion is q͑r͒=q−4␲r2P͑r͒, since the induced charge the permittivity tensor ⑀͑r͒ obtained from the zero field fluc- P͑r͒. The polarization, and hence also the tuation formulas derived in Sec. II, and from the explicit· ١− density is electric field profile E͑r͒=q͑r͒/r2, can be measured directly response of the system to an external field. Simulation results in the simulation from the average radial polarization density based on the latter method converge generally faster if the P͑r͒=͗m͑r͒·rˆ͘. A better method, however, is to compute the amplitude of the applied field is large enough. The efficiency profiles from the measured charge q͑r͒ enclosed in a sphere of both methods is comparable when we take into account of radius r, as this yields smoother profiles thanks to the the computational cost of ensuring that the applied field re- integration over the sphere. sults are not affected by nonlinearity.24

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͑b͒ The purely local assumption is found, perhaps not summations, leaving empty space in the simulation box out- surprisingly, to break down in the vicinity of an interface. In side the slab, in an attempt to decouple the interactions be- the case of the slit geometry, the parallel component ⑀ʈ͑z͒ of tween the original slab and its periodic images in the z the dielectric tensor is found to oscillate strongly, mirroring direction.29 Adopting this approach, Yeh and Berkowitz the oscillatory density profile. Contrary to a wide-spread be- showed that the interactions with the periodic images ͑in all lief, the coarse-grained envelope of ⑀ʈ͑z͒ tends, if anything, three dimensions͒ must not be summed with the usual spheri- to increase rather than to decrease near the dielectric inter- cal convention, but rather in a slabwise manner.30 The effect faces. This trend is observed both for dipolar soft spheres of this change in summation order is simply to replace the ͑ ͒ ␲ 2 ͑ ⑀͒ ͑ ͒ and for the SPC model of water. The tendency is opposite of boundary term Usphere M =2 M / 1+2˜ V by Uslab M 25 ␲ 2 ⑀ that recently observed in water close to mica surfaces, but =2 Mz in the Coulomb energy, where ˜ is the dielectric in those experiments, the mica surface is highly charged, constant at infinity ͑not be be confused with the ˜⑀ introduced leading to local electric fields on the order of 0.2 V/nm, a in Sec. III B͒. This simple result can be seen as a particular strength at which nonlinear effects cannot be neglected. The case of the general formula orthogonal component ⑀Ќ͑z͒ is subject to very large statisti- 2 ͑ ͒ B␣ 2␲M cal fluctuations. It tends to take unphysical negative values ͑ ͒ ␣ ͑ ͒ Uellipsoid M = ͚ A1 close to the interfaces, illustrating the necessity of consider- ␣=x,y,z ˜⑀ + ͑1−˜⑀͒B␣ V ing a nonlocal relation between the local polarization and the local electric field. However both ⑀Ќ͑z͒ and ⑀ʈ͑z͒ go over to for the boundary term when the Coulombic interactions are the bulk values beyond a few molecular diameters from the summed with ellipsoidal summation order. In Eq. ͑A1͒, surfaces. Similar conclusions hold in spherical geometry. which is a generalization to arbitrary values of the dielectric ͑c͒ The fluctuation formula ͑18͒, which is closely related constant ˜⑀ at infinity of a result due to Smith,31 the dimen- to the classic Kirkwood–Fröhlich formula, is an exact result sionless coefficients B␣ are related to the semiaxes ax, ay, az that can be used to compute the bulk dielectric constant of a of the ellipsoid by homogeneous spherical drop of a polar fluid. It yields an ϱ estimate of ⑀ that is in better agreement with the results from axayaz ͵ 1 1 B␣ = dx. 2 ͓͑ 2͒͑ 2͒͑ 2͔͒1/2 simulations performed under periodic boundary conditions, 2 0 x + a␣ x + ax x + ay x + az than the traditional route based on the approximate Ber- ͑A2͒ endsen formula. For highly polar fluids, metallic boundary conditions should be used to minimize the propagation of These numbers satisfy Bx +By +Bz =1. By symmetry, Bx =By statistical errors when solving Eq. ͑18͒ for the dielectric con- =Bz =1/3 in the case of a sphere; Bx =By =1/2, Bz =0 in the stant. case of a cylinder ͑a →ϱ͒; and B =B =0, B =1 in the case ͑ ͒ ͑ ͒ z x y z d If the local electric field E r varies on molecular of a slab ͑a , a →ϱ͒. Expression ͑A1͒ is simply the interac- ⑀͑ ͒ x y scales, a local r may always be formally defined from the tion energy of the charges with the depolarizing field pro- ͑ ͒ ͑ ͒ ⑀͑ ͒ ratio of P r over E r , but the resulting r then generally duced by the uniform polarization density M/V of the infi- depends on the particular interface under consideration. nite ellipsoidal collection of cells, when the latter is ͑ ͒ e In the case of a uniform external field EЈ, the ratio surrounded by a medium of dielectric constant ˜⑀ ͑see Ref. 32 ͑ ͒ E r /EЈ is a more informative quantity, which can be com- for the formula of the depolarizing field in a uniformly po- ͓ ͑ ͔͒ puted from dipolar fluctuations see Eq. 32 . Our simula- larized ellipsoid͒. In the particular case of a slabwise sum- tions point to a dramatic and unexpected “overscreening” of ͑ ͒ ͑ ͒ mation order Bx =By =0, Bz =1 , expression A1 becomes the applied field in the vicinity of the interfaces, which is independent of the permittivity ˜⑀ at infinity, as had been sus- reminiscent of overscreening of surface charges by electric 33 26 pected by J de Joannis, Arnold, and Holm. In the latter double layers in highly correlated ionic systems. reference, explicit formulas to correct for the unwanted in- The main message is that the use of local permittivities terlayer interactions in the 3D Ewald sums have been ob- in “implicit solvent” models of biomolecular aggregates is tained, making the present approach exact in principle, while highly questionable on the nanometer scale. Future work will preserving its O͑N ln N͒ complexity. consider the dielectric response of polar fluids near charged Our simulation cell contained an interlayer gap of width ͑ ͒ rather than hydrophobic surfaces, and will examine the in- 2L, so that it was a parallelipiped of sides LϫLϫ3L.No fluence of the molecular graniness and the polarizability of correction term for substracting interlayer interactions was the interface on dielectric properties of the adjacent polar used, apart from the slabwise summation order. The Ewald fluid. sums were computed using the smooth particle mesh Ewald method ͑Ewald coefficient ␣=3.4705 nm−1, grid size 13 ϫ ϫ ͒ 34 APPENDIX: EWALD SUMS IN THE SLAB GEOMETRY 13 40 cells, interpolation order 6 . The interactions were truncatd beyond 0.9 mm, both in the real space Ewald There exist several exact methods for computing electro- sum and in the soft-sphere interactions. Assuming that the static interactions in systems periodic in two dimensions forces computed using a gap of width 10L are the exact ͑2D͒, but most are only slowly convergent: 2D Ewald sums answers, a gap a width 2L leads to a relative rms error in the and the Lekner method both have an O͑N2͒ scaling, while forces of the order of 10−6 for the system simulated in Sec. O͑ 5/3͒ 27,28 ͑ ␮* ␳* the MMM2D method scales as N . A faster method III A polar fluid with =2, bulk=0.8, confined to a slit of is to use an efficient implementation of the full 3D Ewald width 16.6␴͒.

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