Downloaded by guest on September 29, 2021 ilcrcresponse dielectric interaction structure. solvation developing liquid of understanding those descriptions local to for on relevant lend based approach potentials are only an also not as but results behavior, LMFT These for. to accounted short- support apparent readily its and are counts, of comings number a on long- remarkably performance a of good gives model treatment short-ranged This mean-field electrostatics. well-controlled ranged is a This (LMFT), provides theory investigated. molecular-field which local of are framework electrostatic the short-ranged within intermolecular done entirely whose are model dielectric the water interactions article, a this of In com- simulations. properties complicated conditions molecular partic- in boundary is is used periodic This description monly the interactions. its under dipolar pronounced but of ularly nature solvents, long-ranged useful 2020) the 28, as March abil- by review their act for of (received much to 2020 underpins 14, liquids ity July polar approved of and nature MD, dielectric Park, The College Maryland, of University Weeks, D. John by Edited a Cox J. electrostatics Stephen short-ranged with response Dielectric www.pnas.org/cgi/doi/10.1073/pnas.2005847117 exist- insight the the liquids, polar within mean- demonstrating for understood framework a mechanical from be statistical ing can in Aside performance is interactions fashion. LMFT’s 31) how electrostatic well-controlled, (30, yet LR (LMFT) field, recast theory to molecular-field entirely exploited local framework are the by Specifically, provided interactions investigated. are (SR) electrostatic short-ranged intermolecular whose to espe- interpretations hard not physical is are simple behavior. they masking it liquid-state hand, of risk other hand, and enjoyed the intuitive have one On cially approaches tool. the Ewald computational On that a as success (29). the standard against considered argue widely facto under now de interactions only are the the techniques electrostatic not summation with certainly Ewald deal prop- PBC, While to dielectric (19–28). available history for method long has a this has electrostatic implications erties, of the nature ratios. LR and the surface-to-volume high interactions, for account and spuriously appropriately curvature mitigate to How to interfacial employed of often degrees con- are samples boundary (PBC) of periodic that result, ditions below a far As the sizes experimentally. the system investigated with to at often application associated however, their liquids provide, limits cost simulations of that computational behavior resolution microscopic The the level. investigate molecular to approach taken the is It disciplines. across article. liquids, this ubiquity polar of its biological, focus all to owing Of the out sciences. stands across water materials implications and physical, self-assembly, broad chemical, solvation, are Con- as there (15–18). nanopores such sequently, and membranes phenomena through play in transport they and role as overstated, crucial be dielec- cannot a fluid the liquids fascinating polar understanding polar of both of properties its a study tric importance on their The define depends makes complicated. that system This and polar interactions (LR). a long-ranged dipolar of are the energy (14), from free deduced the shape immediately theoreti- that be and fact can (1–3) the As experimental investigations. both (4–13) for cal motivation the be to U fluids confined eateto hmsr,Uiest fCmrde abig B E,Uie Kingdom United 1EW, CB2 Cambridge Cambridge, of University Chemistry, of Department nti ril,tedeeti rpriso iudwater liquid of properties dielectric the article, this In widely a is simulation computer experiments, to addition In ftepicplam flqi-tt hoyadcontinues one and theory is liquid-state fluids of polar aims of principal the nature of dielectric the nderstanding | oa oeua edtheory field molecular local a,1 | iudwater liquid | lcrsai interactions electrostatic | doi:10.1073/pnas.2005847117/-/DCSupplemental at online information supporting contains article This 1 the under Published Submission.y Direct PNAS a is article This interest.y competing no declares author The uhrcnrbtos ...dsge eerh efre eerh nlzddt,and data, analyzed research, paper. performed the wrote research, designed S.J.C. contributions: Author a on relies LMFT of success Ewald the in contrast, taken In procedure efficiency. same tational where the 28), ref. as (e.g., many approaches to familiar be will where contributions, LR and SR into exactly given features. separated salient been most its have to derivations limited is consid- detailed discussion is (31), as elsewhere interactions generally Moreover, electrostatic more here. to applied ered application be its can only LMFT (40), the Although system construction, captured. full the by are of correlations potential; comprises portion higher-order one-body and a SR structure system chosen average the exists mimic suitably from be there a This solely and that can system. arising and subset) interactions “mimic” portions intermolecular a a LR (or to and mapping interactions SR LMFT intermolecular into of partitioned premise the The that (31). is forces corre- intermolecular molecular with relate lations that Yvon– equations the on of based hierarchy framework Born–Green mechanical statistical a is LMFT Constant Dielectric the and LMFT charged of 39). (38, solvation species the a understanding as The LMFT for to follow approaches. approach support theoretical further that theoretical provide such results also here of the presented results development that dif- the hoped for help is 38 and will it 37 and refs. suggestions), (see theory fering functional ideas LMFT density between classic exist and connections strong Moreover, that to modern likely observables. seems us simulated it with on not, help consequences does and case severe also does, have electrostatics will LR the of neglect It often when inter- understand (32–36). SR is of approaches development which machine-learning the aid potentials, will molecular study this from obtained mi:[email protected]. y Email: clbss hswr illkl aiiaetedevelopment the our confinement. and under facilitate potentials fluids of likely interaction understanding short-ranged will efficient theoret- work both a of This on basis. rationalized is ical short- this behavior only and model’s with investigated, short-ranged system are model interactions intermolecular a ranged of properties complicates Here dielectric which simulations. the its ranges, molecular and from long descriptions stems very theoretical both solvent over act a polar can as between molecules act forces intermolecular to The ability properties. liquid’s dielectric a of Much Significance e sbgnb oigta h olm oeta a be can potential Coulomb the that noting by begin us Let κ −1 enstelnt cl vrwhich over scale length the defines r 1 y = NSlicense.y PNAS erfc(κr r ) + κr erf( r κ ) scoe ootmz compu- optimize to chosen is . y ≡ https://www.pnas.org/lookup/suppl/ v 0 NSLts Articles Latest PNAS (r + ) v 1 v (r 0 ), eas This decays. | f7 of 1 [1]

CHEMISTRY choice of κ such that the mimic system accurately captures the Dielectric Response of Bulk Liquid Water one-body density and correlations of the full system. In what fol- An infinite system is not a realizable object, even in a computer lows κ−1 = 4.5 A,˚ which has been previously demonstrated to be simulation that employs PBC. Thus while  can be “determined” a reasonable choice (31). Instead of computing LR electrostatic within the SCA framework from arguments based on an infinite interactions explicitly, the effects of v1 are accounted for by a sample, it still needs to be established how measurable quantities static restructuring potential, like the polarization response, or fluctuations at zero field, are affected. In this section, such properties are investigated for bulk Z 0 0 0 liquid water under PBC. This is probably the closest realizable VR(r)= V(r) + dr nR(r )v1(|r − r |), [2] system to the infinite geometry considered above. Nevertheless, it is important to bear in mind that there is now an implicit where nR is the average charge density in the mimic system, “boundary at infinity” (19, 20, 22, 46). V is an external electrostatic potential that would be applied Fig. 1 A and B shows how the polarization P responds to either to the full system, and the integral is understood to be taken a uniform electric field E or electric displacement field D applied over all space. As VR is to be chosen such that nR = n, where along z, respectively (e.g., refs. 48–50). Results are shown for n is the average charge density of the full system, Eq. 2 defines the cases that LR electrostatics are both calculated explicitly a self-consistent relationship between VR and nR. In a “pure” (“Ewald”) and neglected entirely (“SCA”). In fact, using SCA LMFT approach, Eq. 2 can be solved either by brute force or by instead of a full LMFT treatment can perhaps be justified here: exploiting linear-response theory (41). As the focus of this arti- VR = V on account of the fact that uniform fields induce charge cle is on understanding dielectric properties within the LMFT density only at physical boundaries, which, at least explicitly, are framework, here a more pragmatic approach is instead taken: absent in the current geometry. (−∂z V = E or D accordingly.) At The self-consistent cycle is “short-circuited” by using n obtained constant E a degree of nonlinear response is observed at larger from a simulation of the full system as the initial, and only, guess fields, while the response to constant D is linear to an excellent (42). While Eq. 2 has a simple mean-field form, it is important approximation. In either case, the Ewald and SCA approaches to stress that it is not derived from a mean-field ansatz. It repre- are virtually indistinguishable over the range of field strengths sents a controlled approximation provided that the mimic system studied. This observation is corroborated by the probability dis- is chosen carefully. It should also be noted that Eq. 2 has been tributions at zero field, pE and pD , of the z component of the derived with nonuniform systems in mind and that, for uniform total dipole moment of the simulation cell M = ΩP, shown in systems, more sophisticated LMFT approaches exist (42–44). Fig. 1 C and D. (Ω is the volume of the simulation cell.) These When considering uniform systems in this study, however, the have been obtained with histogram reweighting, and in the case strong-coupling approximation (SCA) will be used, in which the of pD , the wings of the distribution extend far beyond values of integral in Eq. 2 is simply ignored; i.e., VR = V. P suggested by Fig. 1B. The agreement between the SCA and The central quantity describing the dielectric behavior of Ewald results is remarkable. materials is the static dielectric constant . A natural question The observation that hM 2i obtained from SCA agrees well thus arises: Can we expect SCA to accurately capture  of the with Ewald echoes previous studies using Wolf-based electro- full system? Following Madden and Kivelson (45), it is taken statics (51) to compute  (52–56). These studies appealed to the as an empirical fact that  is an intensive material property and therefore does not depend on the shape of the sample under con- sideration. This provides the freedom to choose any geometry for which it is convenient to calculate , including an infinite sys- tem in which boundaries are not present. In this case, it is well established that Z ∞ AB (2 + 1)( − 1) 4πρ 2 = 1 + drr h∆(r), [3] 9y 3 0

with y = 4πβρµ2/9, where ρ is the number density, µ is the magnitude of the permanent molecular dipole moment, and β = 1/kBT .(T is the temperature, and kB is Boltzmann’s con- 2 stant.) h∆ = 3hh(1, 2)µ1 · µ2/µ iΩ1Ω2 is the projection of the total correlation function h(1, 2) on to the rotational invariant 2 µ1 · µ2/µ , where µi is the dipole vector of molecule i, and h· · · iΩ1Ω2 denotes an unweighted average over the orientations CD of molecules 1 and 2. Crucial to the current study is that, for an infinite system, h∆ ∼ 0 beyond some microscopic distance ` (46). Hence,  is determined by SR correlations between dipoles, in line with Kirkwood’s original arguments (47). By construction, SCA accurately describes such SR correlations, from which one can infer that  is indeed the same as for the full system. The above argument skirts around a subtle issue that, as is dis- cussed in more detail below, manifests itself as an inconsistency in the response of the SCA system to uniform fields vs. the k → 0 limit of its external susceptibility, where k is a reciprocal space wavevector. One might therefore already anticipate problems Fig. 1. Dielectric response in homogeneous bulk water. A and B show, where inhomogeneous systems are concerned. Fortunately, the respectively, the polarization response P to imposed E and D fields (along framework provided by LMFT accounts for such inconsistencies z). C and D show, respectively, the probability distribution of M at E = 0 and on average. Moreover, it will also be shown that the fluctuations D = 0. Agreement between SCA and Ewald is excellent. Dashed lines indi- deviate from those of the full system in a predictable manner cate the expected response (A and B) or variance (C and D) from DCT using consistent with dielectric continuum theory (DCT).  = 71.7 (Fig. 2 and Eq. 6).

2 of 7 | www.pnas.org/cgi/doi/10.1073/pnas.2005847117 Cox Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 e eut rmrf 6aerqie.Tefis eutrelates result first The required. are 46 to ref. from results key volume sadpl tteoii,and origin, the at where dipole system, the a in is dipoles between correlations entational induce not does i.e., infinity perturbations; at structural boundary the Moreover, identical. are odto tifiiy .. the i.e., boundary infinity; the at alter not condition does compensating SCA Gaussian Crucially, ). Appendix own (SI its charge in embedded each charges point meddi t w ooeeu opnaigcag,while charge, compensating ψ homogeneous own its in embedded system, short-range the the in affecting correlations without ignored be can space reciprocal SCA, In constant tric hsfiieaypoi au sceryafiiesz fet vanish- effect, finite-size limit a the in clearly ing is value asymptotic finite This between relation relating mula dimension volume The is This SCA. conditions, by case boundary tinfoil modified under been seen has easily most distribution charge the that Eq. As where fluids polar the conditions. of boundary prescribe functions different which correlation under pair results, naturally the (46) of more Caillol’s forms of itself asymptotic terms system—lends the in full captures analysis of the accurately to premise of that underlying correlations system the mimic SR here, a strategy devise similar LMFT—to a Instead following approaches. Wolf-based of using relating when formula unaffected fluctuation largely the Steinhauser is that and Neumann arguing and 25), (21) (24, Neumann of works seminal Cox of behavior finite, asymptotic now is the system, infinite gives the unlike result third The with replaced are potential, Hamiltonian Ewald the the enter that interactions static † fetvl ape h nnt geometry. infinite the samples effectively h smttcbhvo of behavior asymptotic The SCA (0) h Kirkwood The = (Ω cell simulation cubic a For  and yG ψ sisedteGensfnto o e fproi unit periodic of set a for function Green’s the instead is n (0) K 4 ψ v  savco fitgr,and integers, of vector a is (r oudrtn C’ efrac ne B,three PBC, under performance SCA’s understand To . κ 0 ( steGensfnto o h ulsse,i sclear is it system, full the for function Green’s the is , λ) ecie eidcsto ntpitcags each charges, point unit of set periodic a describes ψ −1 r = ) Setting . SCA = (r) ( v λ)  h|M| h 0 scoe ufiinl ag uhta h u in sum the that such large sufficiently chosen is G a eete peeo radius of sphere a either be can L (2 ∆ ecie h urudn eim“tinfinity.” “at medium surrounding the describes = (r) K (r ∞. → G G X + 2 (r 1)( + K n i ) factor, L v + 1 = ) ∼ and 1 and 9 X h 3 erfc(κ| = ∆ † n 3y k X  si a led enage htboth that argued been already has it As 1 L 0 = 6 a lob aet aihb setting by vanish to made be also can  − |r h ρ erfc(κ|r 3 o given a for ∆ G + ie h prpit utainfor- fluctuation appropriate the gives ( 1) 4π k ρ 3 , K M r n cnann em nEqs. in terms λ-containing |r 2 nL| − + Z R (r + v e  + 0 i 1) 0 = nL|) = ) r stettldpl oeto a of moment dipole total the is k·r ( nL| dr + 2 e L − . 9 (2 −k λ hµ nL|) 0 2( 3  1) dr ne B,teelectro- the PBC, under ) 3 = dv 0 2 1 h eodrsl sthe is result second The . /4κ 0 2 0 · 0 − + M /(2 h − 2( 2 ∆ 2 ) ) v 2πλ − (r 0 0 i/µ 3 0 + − L 0 1 h dielec- The 1). + 2πλ ). 3  ) 3 2 |r . r L ecie ori- describes , v 3 2 λ racb of cube a or |r L L (r . 0 = 3 h 3 2 hM  ) ∆ 0 , . = nthis In . 4 which, 2 which , and i to G [6] [7] [4] [8] [5] µ K 5 1  the susceptibility external anisotropic the factor The work: immediately Eqs. not is that it fact, obvious In (19–28). proper electrostatics with LR of associated account history the given polar remarkable nonetheless of princi- the properties with consistent dielectric that entirely ple is the this While describing water. like in liquids SCA of mance the indicate 1 Fig. using in DCT SPC/E from lines for 1 dashed (Fig. values response The literature expected 57–59). existing (48–50, with agreement water good in is constant: tric Averaging Kirkwood composite the 49, of G value ref. asymptotic Following the from constant result. dielectric Ewald distance-dependent the the to compared tinguishable 49). (48, respectively both where for correla- 2A, subvolumes for Fig. SR spherical support by provided a different empirical is Further considered notion SCA. a be such to still amenable function can tion it behavior, asymptotic thus  vrgn eut for results Averaging composite the from Eq. by determined at obtained 2. Fig. B A and Kc h eut rsne ofrsgetana als perfor- flawless near a suggest far so presented results The despite that, suggest arguments above The k (2G = h|M| →  Roinainlcreain determine correlations orientational SR 0  0 r nhne ySA tdrcl olw that follows directly it SCA, by unchanged are 2 E sdtrie yS rettoa orltos ti still is it correlations, orientational SR by determined is i ii 4) hl o ytmcmrsn exclusively comprising system a for while (45), limit  K,E=0 = E K r lounaffected. also are =  0 for 0 71. = and itnedpnetdeeti constant dielectric Distance-dependent (B) 6. ( +  G r (2 0 r 71.7 = 3 > atr(see factor G = ` > D 7 and K, 1)( + and ∞) 6 G o ohteSAadEadsses This systems. Ewald and SCA the both for  = A D=0 ˚ = K orsodn to corresponding 0, = 6 6 A bandfo C svrulyindis- virtually is SCA from obtained hsrsl speetdi i.2B. Fig. in presented is result This )/3.  gives A ` ˚ hudhl ihnteSAframe- SCA the within hold should −  and ilcrcRsos fBl iudWater Liquid Bulk of Response Dielectric D ie h arsoi ttcdielec- static macroscopic the gives 1)/9 = v 0 rvrac Fg 1 (Fig. variance or B)  4 = ( = 0 rgntsfo h rc of trace the from originates o ohEadadSCA. and Ewald both for 71.7 = π χ r )hv naypoi form asymptotic an have 0) (0) 3 NSLts Articles Latest PNAS eut r shown are Results /3. Kirkwood (A) . (k)  G K  h h ∝ K (r 0 ∆ = speetdfor presented is snonvanishing ’s ) m(k) ˜ ∞ a efound be can and  m ˜ C G K G ∗ G obtained and (k)i | factor,  K factors 0 f7 of 3 0 = and D) in ). ,

CHEMISTRY SR interactions, one would expect χ(0) to be isotropic at long In this section, the dielectric properties of water confined wavelengths. (m˜ is the Fourier transform of the molecular dipole between structureless, repulsive walls are investigated. This is density, using the water oxygen atom as the molecular center.) a prototypical model for understanding nanoconfined water in ∗ Such behavior is indeed hinted at by Fig. 3, where hm˜ α(k)m ˜ α(k)i hydrophobic environments. In such a geometry, the interface is is shown, with α = x, y, z and k = kˆz. While for the best part approximately planar, and Eq. 2 can be recast as good agreement between the SCA and Ewald approaches is seen, discrepancies are observed at long wavelengths in the longitudi- 1 X 4π 2 2 VR(z) = V(z) + n˜R(k) exp(ikz) exp(−k /4κ ), [9] α = z L k 2 nal ( ) fluctuations, with the SCA results sharply increasing k=6 0 as k → 0. It is interesting that these deviations appear at length scales far larger than the range separation prescribed by SCA where n˜R denotes a Fourier component of nR, and L is now the (Fig. 3, Inset). total length of the simulation cell in the direction perpendicular (0) In a full treatment of electrostatics, setting E = 0 ensures χxx to the interface (taken to be z). The system is still understood (0) (0) (0) and χyy are continuous at k = 0; e.g., χxx (k → 0) = χxx (0). On to be replicated in all three dimensions. A schematic is shown in (0) (0) (0) Fig. 4A . the other hand, χzz is discontinuous; i.e., χzz (k → 0) 6= χzz (0) Even in the absence of an external field, liquid water has (23). The situation is reversed for D = 0. In contrast, Fig. 3 (0) (0) a nonvanishing charge density close to the interface. Conse- suggests that with SCA, χαα(k → 0) = χαα(0) at E = 0 and quently, VR is finite, along with a corresponding restructuring (0) (0) χαα(k → 0) 6= χαα(0) at D = 0, irrespective of whether α = x, y, field ER = −∂z VR. Neglecting ER has severe consequences for or z. The fact that k = 0 response to both E and D in SCA well the orientational statistics of water in a confined geometry. This describes that of the full system therefore suggests an inconsis- was already discussed by Rodgers and Weeks (30) and their tency within the SCA framework. We will see the consequences results are recapitulated in a slightly different form in Fig. 4B, of this when considering an inhomogeneous system below. There where the average molecular dipole density along z is shown. it will also be shown that the k → 0 longitudinal and transverse While the average polarization obtained with LMFT agrees well external susceptibilities are indeed equal in SCA; i.e., the for- with the Ewald system, the SCA system on its own (ER = 0) mer is too large by a factor  (SI Appendix). Fortunately, LMFT yields a qualitatively incorrect picture. Crucial to what follows readily provides a route to account for this inconsistency. is that LMFT also gives the correct average polarization in the presence of a uniform field, which is also shown in Fig. 4B for Dielectric Response with Extended Interfaces D = −∂z V = 0.15 V/A.˚ Placing systems under PBC is a useful construction for investi- It is clear that LMFT provides a means to correct for the gating bulk properties of materials such as , especially when effects of neglecting LR electrostatics on the average dielectric computational resources are limited. Real systems, however, response in inhomogeneous systems. Results from previous stud- have boundaries that will induce structural inhomogeneities. ies (60–64) suggest the fluctuations will also be affected, and These structural inhomogeneities may also be accompanied by establishing how they are affected is likely to provide useful phys- regions of nonvanishing charge density of the polar liquid. This ical insight. To set about tackling this issue, let us consider a could arise from the boundary itself preferentially orienting the continuum model in which a uniform dielectric slab with thick- molecules of the liquid (e.g., due to functional groups at a solid ness w is centered at z = 0 such that its boundaries occur at surface) or from an asymmetric charge distribution in the liquid’s z± = ±w/2. A vacuum region exists either side of the slab. If the constituent molecules (e.g., water). In any event, it is simply not slab has a uniform polarization P, this leads to a charge density enough to evaluate the performance of SR interaction potentials at the boundaries, n(z) = P [δ(z − w/2) − δ(z + w/2)]. Recall- on their ability to reproduce properties of homogeneous systems. ing that nR = n at self-consistency, taking the Fourier transform Rather, it is imperative to assess and understand their behavior of n, substituting into Eq. 9, and differentiating to find ER gives in the presence of extended interfaces. 8πP X cos(kz) sin(kw/2) 2 2 E (z) = E(z) − exp(−k /4κ ). R L k k=6 0 [10] In the limit L → ∞, this can be solved analytically,

lim ER(z) = E(z) − 2πP {erf [(w/2 − z)κ]+ erf [(w/2 + z)κ]}. L→∞ [11]

In this case it is instructive to consider the limiting values of κ,

E(z) (as κ → 0), E (z) = [12] R E(z) − 4πP (as κ → ∞).

The result for κ → 0 simply states that all electrostatic inter- actions have been accounted for explicitly in the SCA system. In the case κ → ∞, the result can be interpreted as follows: Due to the neglect of LR electrostatics, the SCA system omits the depolarizing field established by the induced surface charge density at the boundaries, which is then accounted for by the * Fig. 3. Dipole density correlations in reciprocal space hm˜ α(k)m˜ α(k)i deter- term (−4πP) in Eq. 12. For finite κ, it is found empirically that mined for both Ewald and SCA electrostatics (α = x, y, z and k = kˆz). On the ER = E − 4πP is an excellent approximation in the slab’s inte- whole, good agreement between SCA and Ewald is seen. ( ) At low , Inset k rior, provided w  κ−1. In the general case of finite L, Eq. 10 SCA’s longitudinal correlations (α = z) deviate from Ewald, tending toward the transverse correlations (α = x, y). These deviations occur on a length can be solved numerically in a straightforward manner. This is scale greater than the range separation prescribed by SCA, as indicated by shown in Fig. 4C for L = 75, 150, and 300 A,˚ along with the ana- the vertical dotted line at k = 2πκ. lytic result (Eq. 11) for L → ∞, using  = 71.7 obtained above.

4 of 7 | www.pnas.org/cgi/doi/10.1073/pnas.2005847117 Cox Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 h pnaeu otiuinsbrce.Tedte iesosthe shows line (Eq. dotted result The subtracted. contribution spontaneous the finite for w rnedte ie ( line. dotted orange codn oislniuia xenlsusceptibility, external longitudinal its to according as increases variance to insensitive the relatively predicts is DCT this LMFT, Ewald, With moment. dipole slab’s indicates the cross black The fit. linear Eq. evaluating by obtained Neglecting LMFT. ierywith linearly is however, to, responds it same field the gives LMFT construction, By accuracy. remarkable with result simulation this molecular captures from above simple presented model The continuum Appendix). dielectric (SI subtracted contribution spontaneous is shown Also lines. V/ profiles 0.15 polarization along Average cell (B) simulation walls. the confining of the length in total vacuum the with interfaces two forms in schematically shown 4. Fig. Cox Eq. Eq. substituting Eq. evaluating boundaries the By to ignored. close are variations as approximate is relationship where = hnsbetdt netra edalong field external an to subjected When 33.2 13 sitdvrial y3 by vertically (shifted A ˚ η n eragn yields rearranging and ilcrcrsos fwtrcnndbtenhdohbcwalls, hydrophobic between confined water of response Dielectric D B A L L .Sldlnshv enotie ynmrclyeautn Eq. evaluating numerically by obtained been have lines Solid A. ˚ .Tesae einindicates region shaded The 11). seky.Tedse iei iuainrsl for result simulation a is line dashed The key). (see ≤ 1 sse nFg 4 Fig. in seen as 1/L, san is E E R 4π R 12 for 4π C at P ) dpnetsaa relating scalar L-dependent E gives D P 4 = L ae lbo thickness of slab water A A. R L = rdce yDTwith DCT by predicted lim 4 = 75 = →∞ eut npo gemn,a niae ythe by indicated as agreement, poor in results 0 10 πχ 4π πχ at R,zz (0) 4πχ × bandfo iuain ihthe with simulation, from obtained A ˚ P z zz (0) 10 η = 4 = ∞ E 3 R,zz (0) E E oe ice in circles (open 0 10 R e/ R = LMFT = ≈ A πχ C E ˚ 1/ C ahrthan rather at 2 =  xy 4πχ o lrt)aewl ecie by described well are clarity) for R,zz (0) and −w   z (E . ln at plane η z − − 0 = P  h otdlnsdpc the depict lines dotted The . L −1 R,zz (0) rbblt itiuin of distributions Probability ) /2 − [E 1. 1 E ntelimit the In D. stefl ytm The system. full the as sidctdb h solid the by indicated as , P ,  ≤ = z η η z E z tboth at 5V/ 0.15 L L 4π h lbresponds slab the , w . z ≤ E C E = sfudt scale to found is etrdat centered , E .Tesldln sa is line solid The ). , w , P R ±w Ewald .(D) /2. ] Substituting . A, to ˚ D L /2.  = = E = n the and , and 0 75 L η ,and 71.7, L L denotes ,with A, With L. vs. ∞, → L ˚ z ∞ → [13] [14] [15] = D L −1 10 = 0, ieqatt htde o eeduo apesae hl this While shape. sample upon depend not does that quantity sive electrostatics. SR full with with where system a DCT system of by that been a well from unchanged has described of it are properties correct interactions Instead, to electrostatic made. dielectric attempt been that such has with no demonstrated LMFT interfaces article, in of captur- this fluctuations In the effects by (64). fluctuations dynamical symmetry the and high recover equilibrium to (60–63) both means LMFT) ing of a extension provides of (an likely fluctuations theory symmetry- polarization particular, mean-field have In the could preserving studies. which previous fields that from affected, external anticipated clear be been of also also will absence is systems the inhomogeneous It in LMFT even 66). early water, (30, from be liquid obvious simply on already cannot studies was electrostatics This LR entirely. of neglected effects clear, abundantly the is that It however, interactions. highly electrostatic be LR explicit without to could liquids reference describe potentials to interaction seeking those SR to advantageous with captured be can on depend that pressure, and also energy can uniform  internal of same the properties The thermodynamic e.g., solvent. to systems, corrections SR for underlying said the dielec- be static show of the they constant with that consistent tric is is above interaction presented renormalized results this (∼ the of limit aspect shown asymptotic pleasing was an which interaction, introduc- have from by ion–ion to results corrected direct was renormalized on discrepancy a Building this ing PMFs. (38), al. the et “v on attributed Remsing this screening was dielectric of This of treatment. inability electrostatic the full to a correlations interactions with electrostatic ion agreed all probe treating to SR that used the found with was was LMFT It which water. in in (39), al. approach et suitable a development as its facilitate LMFT forward. should going to and solvation weight presented investigating results for further the solvation, solutes, lend of dielectric charged understanding of solvent’s above our case the solva- in the success As in role ionic scales. particularly central recent and length a much hydrophobic of plays range both constant enjoyed a includes has across This tion LMFT 65). solvation where 39, of always (38, descriptions area not theoretical one treatments is our mean-field is it conventional Improving when that work. priori, cases will a in simulation least applied molecular at be clear, standard framework can with mechanical and compatible approaches statistical readily elegant is that an provides LMFT Discussion electrostatics. that full so with notion system system the the LMFT for of the that support of further behavior is the sim- well this describes that model fact DCT The Appendix). ple (SI sums Ewald using obtained data: simulation the and to fits not are continuum these  that dielectric stressed this the is It by where model. predicted 4E, variances Fig. with tributions in confirmed different with is simulations This of Ewald. distributions with probability is it that Specifically, formalism. that LMFT follows the immediately it in fluctuations the on mation is susceptibility external factor longitudinal by the large that too case the indeed is Eqs. with Comparing 71.7 = (44). ale nti ril,i a ae sgvnthat given as taken was it article, this in Earlier liquids polar of properties dielectric certain that fact The Gao of study recent the is work this to relevance particular Of infor- elicits directly model continuum dielectric simple This w 33.2 = a endtrie rmtesmltoso ukwater, bulk of simulations the from determined been has v 0 a endtrie rmtevrac of variance the from determined been has A ˚ e optnilo enfre PF)ta dis- that (PMFs) forces mean of potential to led  as 13 k and → L 0 hM 0 ol”apoc ocpueeffects capture to approach -only” M r lte ln ihGusa dis- Gaussian with along plotted are ti la that clear is it 14, IAppendix). (SI 2 o h slab the for i 1/r nLF safactor a is LMFT in ossetwt C.A DCT. with consistent ) NSLts Articles Latest PNAS  p sucagdfrom unchanged is slab η ∞ bandfrom obtained 1 =  sa inten- an is 1/η hsit Thus /. L | larger p f7 of 5 slab  is

CHEMISTRY is supported by rigorous theoretical calculations (e.g., ref. 67), in Fig. 3 were obtained from 45- to 50-ns simulations of 6,912 molecules with the fact that dielectric properties can be understood within the L = 59.1912 A˚ and D = 0. For simulations of water between hydrophobic LMFT framework can be viewed as a demonstration of this walls (Fig. 4), 400 water molecules were confined between Lennard-Jones result and is perhaps more open to intuitive physical interpre- 9-3 walls,

tation. This may prove useful as we continue to develop our "  9  3# 2 σw σw understanding of dielectrics under confinement (1, 4, 9, 10). uwall(z) = εw − 15 ∆loz ∆loz "  9  3# Materials and Methods 2 σw σw + εw − , [16] All simulations used the SPC/E water model (68), whose geometry was con- 15 ∆hiz ∆hiz strained using the RATTLE algorithm (69). Dynamics were propagated using

the velocity Verlet algorithm with a time step of 2 fs. The temperature where ∆loz = |z − zlo| and ∆hiz = |z − zhi| with zlo ≤ z ≤ zhi located within was maintained at 298 K with a Nose–Hoover` chain (70, 71), with a damp- the primary simulation cell. All simulations used zhi = −zlo = 17.25 A,˚ εw = ing constant 0.2 ps. Where applicable, the particle–particle particle–mesh 0.6 kcal/mol, and σw = 2.5 A.˚ The potential was truncated and shifted Ewald method was used to account for long-ranged interactions (72), with for ∆lo(hi)z ≥ 2.1459 A.˚ Simulations were between 25 and ∼90 ns. Hybrid parameters chosen such that the root-mean-square error in the forces was boundary conditions were used (48); i.e., D = Dˆz and Ex = Ey = 0. For D = 0, a factor 105 smaller than the force between two unit charges separated this is formally equivalent to the Yeh–Berkowitz correction for the slab by a distance of 1.0 A˚ (73). A cutoff of 10 A˚ was used for nonelectro- geometry (76). static interactions: For simulations using LMFT/SCA, this cutoff was used for all interactions. The LAMMPS simulation package was used through- Data Availability. Source code beyond the standard LAMMPS distribution out (74). For simulations with an imposed electric displacement field, the used to perform simulations described here can be accessed at https:// implementation given in ref. 50 was used. Simulations using LMFT/SCA github.com/uccasco/LMFT. Input files to generate the trajectories are required further modification of the LAMMPS source code, which has been openly available at the University of Cambridge Data Repository, https:// made freely available. For results presented in Fig. 1, the system comprised doi.org/10.17863/CAM.52565. Data have been deposited in Apollo (Cam- 256 molecules in a cubic simulation cell of dimension L = 19.7304 A.˚ For bridge research repository) and Github (https://doi.org/10.17863/CAM. Fig. 1 B and D, an electric displacement field was imposed in all three 52565, https://github.com/uccasco/LMFT). dimensions; i.e., D = Dxˆx + Dy ˆy + Dˆz with Dx = Dy = 0. See ref. 48 for fur- ther discussion of this point. Simulations at constant E were run for 150 ns ACKNOWLEDGMENTS. Michiel Sprik and Rob Jack are thanked for insightful postequilibration, while those at constant D were run between 2.5 and discussions. Simon´ Ram´ırez-Hinestrosa is thanked for technical discussions 5.0 ns. The probability distributions p and p were obtained using the mul- regarding the LAMMPS implementation. Computational support from the E D UK Materials and Molecular Modeling Hub, which is partially funded by the tistate Bennett acceptance ratio method (75), with simulations performed Engineering and Physical Sciences Research Council (EPSRC) (EP/P020194), ˚ ˚ at E = 0, ±0.005, ... , ±0.020 V/A and D = 0, ±0.05, ... , ±1.45 V/A, respec- for which access was obtained via the United Kingdom Car-Parrinello (UKCP) tively. The Kirkwood G factors (Fig. 2) were obtained from the same set of Consortium and funded by EPSRC Grant EP/P022561/1, is gratefully acknowl- simulations, although those at E = 0 were performed for a further 150 ns edged. I am supported by a Royal Commission for the Exhibition of 1851 with configurations stored more frequently (every 30 ps). Results presented Research Fellowship.

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6 of 7 | www.pnas.org/cgi/doi/10.1073/pnas.2005847117 Cox Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 9 .Go .C esn,J .Wes hr ovn oe o o orltosand correlations ion for model solvent Short Weeks, free D. J. solvation Remsing, to C. R. contributions Gao, Long-ranged A. Weeks, 39. D. J. Liu, S. Remsing, C. R. 38. 0 .D ek,Cnetn oa tutr oitraefrain oeua cl van scale molecular A formation: interface to structure local Connecting Weeks, D. J. 40. 8 .Zag .Srk optn h ilcrccntn flqi ae tconstant at water liquid of constant dielectric the Computing Sprik, M. Zhang, C. 48. the short-ranged determine to accurate theory and field mean efficient Using the Weeks, D. On J. Katsov, Weeks, K. Vollmayr-Lee, D. J. K. Hu, 43. for Z. equations Rodgers, field M. mean J. self-consistent 42. of solutions Efficient Weeks, D. J. Hu, Z. 41. 9 .Zag .Hte,M pi,CmuigteKrwo -atrb obnn con- combining by G-factor Kirkwood the Computing Sprik, M. Hutter, J. 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CHEMISTRY