APoisson P 3 MForce Field Scheme for Particle-Based Simulations of Ionic Liquids

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1 Journal of Computatonal Electroncs 3: , 2004 c 2004 Kluwer Academc Publshers. Manufactured n The Netherlands. APosson P 3 MForce Feld Scheme for Partcle-Based Smulatons of Ionc Lquds S. ABOUD Molecular Bophyscs Department, Rush Unversty, Chcago, IL, USA D. MARREIRO AND M. SARANITI Electrcal and Computer Engneerng Department, Illnos Insttute of Technology, Chcago, IL, USA sarant@t.edu R. EISENBERG Molecular Bophyscs Department, Rush Unversty, Chcago, IL, USA Abstract. In ths work we propose a force-feld scheme for the self-consstent partcle-based smulaton of electrolytc solutons. Wthn ths approach, the electrostatc nteractons are modeled wth a partcle-partcle-partclemesh (P 3 M) algorthm, where the long-range components of the force are resolved n real space wth an teratve mult-grd Posson solver. Smulatons are performed where the solute ons are treated as Brownan partcles governed by the full Langevn equaton, whle the effects of the solvent are accounted for wth the mplct solvent model. The man motvaton of ths work s to effcently extend the modelng capablty of the standard partclebased approaches to lqud systems characterzed by a spatally nhomogeneous charge dstrbuton and realstc, non-perodc boundary condtons. Examples of such systems are large polymer chans, bologcal membranes, and on channels. Keywords: onc solutons, Brownan dynamcs, molecular dynamcs, force feld, on channels 1. Introducton The work presented n ths document was motvated by the need of an mproved force feld scheme for the partcle-based smulaton of on channel systems. Ion channels are protens embedded n the lpd membrane of bologcal cells, they nteract n a complex way wth ther envronment and are responsble for fnely regulatng the flux of onc charge across the membrane. For nstance, the generaton and transmsson of potentals n nerves and muscles, as well as the hormone release from endocrne cells, are beleved to be mechansms governed by the transport of onc charge through these proten gates [1]. To whom correspondence should be addressed. Snce the frst demonstraton n 1976 [2] of a relable expermental methodology for the detecton of currents flowng through ndvdual on channels, several refnements of the expermental setup have been successfully appled to a varety of membrane and cell confguratons, both n vvo and n vtro. The extraordnary progress of the expermental technques trggered an ncreasng theoretcal effort amed at the understandng of the role of on channels n the physology of complex bologcal systems, and, more generally, ther nfluence on the electrcal equlbrum between the cells and ther envronment. Besdes the purely theoretcal aspect, the enormous pharmacologcal advances expected from the knowledge of the operaton mechansms n on channels has been one of the strongest arguments n favor of ther research [3]. Furthermore,

2 118 Aboud from an engneerng vewpont, on channels are beng envsoned as a key component of a new generaton of bo-sensors that ntegrate the selectvty and the extreme senstvty of on channels wth the processng capabltes of modern mcroelectroncs [4]. A qute pecular aspect of the research on on channels s that t frequently nvolves researchers workng n tradtonally dfferent dscplnes. The sold-state electroncs communty, for example, s well aware of the fact that tradtonal scalng.e. the reducton of the feature-sze of transstors n order to ncrease the performance of ntegrated crcuts [5] wll soon be nadequate to satsfy the requrements of emergng technologes [6]. A natural soluton s to ncrease the complexty rather than the speed of the basc components, and much can be learned from on channels, whch are extremely specalzed and mnaturzed low power devces. Transstors are defntely faster than on channels, but the advantage due to ther operatonal speed s compensated by the complexty of the operatons performed by on channels. The full understandng of on channels propertes wll allow for ether the modfcaton of ther desgn for novel applcatons, or for manufacturng analogous structures capable of emulatng ther functonalty. It s safe to say that the capablty of explanng the functons of on channels n relaton to ther mcroscopc structure would make possble the realzaton of manmade devces, such as nanotubes, that are based on the same prncples that regulate ther natural counterparts, and perform operatons wth the same level of complexty. A herarchy of smulatve approaches has been appled to the study of on channels durng the last two decades. Contnuum models, such as the Posson- Boltzmann [7] and the Posson-Nernst-Plank [8], have been used to defne the electrostatc landscape of onchannel systems and to analyze on electrodffuson n terms of contnuous fluxes. Indvdual onc trajectores have been studed usng partcle-based approaches subjected to both Brownan [9] and Newtonan dynamcs [10]. Ths document focuses on the approach used to compute the self-consstent feld of forces used by partcle-based smulaton algorthms. In ths context, the adjectve self-consstent refers to the fact that the forces due to the electrostatc nteractons wthn the components of the system strctly depend on the spatal confguraton of the components themselves, and must be contnuously updated as the dynamcs of the system evolves. The most straghtforward approach to compute the force on a gven charged partcle s based on Coulomb s law, and conssts of drectly summng the parwse contrbutons from all the other partcles n the system. The performance of ths Partcle-Partcle (PP) approach scales quadratcally wth the partcle populaton, makng ts mplementaton mpractcal for the smulaton of large systems. Based on a screenng hypothess [11], a fnte cut-off radus can be used to mprove the overall performance of the PP method by reducng the number of partcles nvolved n the calculaton. Ths technque neglects long-range coulombc forces that can be determnant [12,13], partcularly when the charge s nhomogeneously dstrbuted wthn the computatonal doman, or when the effects of the force feld are nvestgated on large molecular structures. Furthermore, accountng for nhomogeneous delectrc boundares and external bas s extremely dffcult wthn the framework of Coulomb s equaton. Indeed, a complcated dynamc dstrbuton of mage charges [14] needs to be added to the system n order to represent the effects of realstc boundary condtons. Alternatvely, Partcle-Mesh (PM) methods [15] are used to compute the electrostatc forces due to the charge dstrbuton of the whole system, and naturally allow for the ncluson of boundary condtons by settng the potental or ts dervatve on the surfaces of the computatonal doman. Wthn the PM framework, the charge dstrbuton s mapped onto a dscrete set of ponts (the mesh or grd ponts), and the correspondng electrostatc potental s computed on the same ponts wth a standard numercal technque. Forces are then nterpolated from the grd to the arbtrary postons of the charged partcles n the system. An evdent advantage of the PM approach s that t scales lnearly wth the number of charged partcles and, n some cases [16], wth the number of grd ponts. The PM method has been successfully used for the smulaton of semconductor devces [15], and has the advantage of effcently modelng systems wth delectrc nterfaces, hghly complex geometres, and arbtrary boundary condtons. However, wthn the PM approach the resoluton of the charge dstrbuton s lmted to the mesh sze, therefore short-range effects are not accounted for n the force feld. Ths does not consttute a problem for the smulaton of electron devces n the framework of the ndependent electron approxmaton [17], or when short-range coulombc nteractons are treated statstcally wth perturbaton theory [18]. It s worthwhle to note that works have

3 Posson P 3 MForce Feld Scheme for Partcle-Based Smulatons of Ionc Lquds 119 been publshed recently [19,20] where the short-range nteracton between charge carrers n semconductor devces s computed wth the PP method and coupled wth the PM approach. Neglectng the short-range structure of the force feld s unrealstc n the case of molecular lqud systems because of the fnte sze of the varous components. In other words, the exponental van der Waals forces arsng from the overlappng electronc orbtals of adjacent molecules (or atoms) [21] play a crucal role n the mcroscopc dynamcs of onc solutons. Ths s also true for charge transport across on channels, whch use atomc scale nteractons to control the macroscopc flow of ons drven by long-range forces. For ths reason, a realstc force feld scheme must have a spatal resoluton adequate to model the van der Waals nteracton, and therefore couple effectvely and accurately both the short- and long-range electrostatcs of the system [15]. Two mathematcally smlar [22] methods that combne the PM and PP approaches n an unfed feld solver are the the Ewald summaton method [23] and the partcle-partcle-partcle-mesh (P 3 M) approach [15]. In spte of ther theoretcal affnty, the mplementaton of the two approaches shows substantal dfferences that motvate our preference of the latter for the smulaton of nhomogeneous onc solutons. Wthn both models, the electrostatc force feld s separated nto two smoothly varyng functons representng the shortrange and long-range forces. The PP approach s used to model the short-range component wthn a relatvely small cutoff regon, whle dfferent methods are used to compute the long-range nteractons. In partcular, the Ewald method computes the short-range forces n real space, whle long-range nteractons are rapdly accounted for n the recprocal space. A crucal effect of ths approach s the rather strngent requrement of spatal perodcty for the charge dstrbuton functon [24]. Ths fact does not consttute a problem n the smulaton of bulk systems, provded that the perodcty does not ntroduce artfacts (e.g., spurous numercal heatng of the system), and suffcent spatal resoluton s mantaned to descrbe large scale order relevant for atomc scale behavor, but lmts ts applcablty n systems wth complex boundares. On the other hand, wthn the P 3 M method, the long-range nteractons are computed n real space wth a PM approach [15], makng t the optmal choce for modelng nhomogeneous systems wth external bas values. In both the Ewald and the P 3 M method a geometrc overlap exsts between the long-range and short-range computatonal domans,.e. the short-range doman s fully contaned by the long-range one. Consequently, both schemes nclude a correcton that accounts for the fact that charges wthn the short-range doman are also used for the long-range calculaton. The approach we propose n ths work extends and valdates the orgnal P 3 M method of Hockney [15] by ncludng a hghly effcent real-space 3D Posson solver based on the teratve mult-grd method [16,25]. The use of a Posson solver n determnng the long range electrostatc forces for a molecular lqud system s not a new approach, but ts use has been lmted n the past due a clam that obtanng ts numercal soluton s computatonally prohbtve [26 28]. We assert that the numercal soluton of the Posson equaton can be mplemented n a computatonally effcent manner for systems of arbtrary geometry and boundary condtons. To valdate the proposed force feld scheme, the Posson-based P 3 M algorthm s coupled wth a Brownan dynamcs (BD) [29,30] kernel. The BD approach s an effcent partcle-based method for smulatng large onc lqud systems for relatvely long tmes. The BD method has been appled to a wde range of many-body problems, such as the smulaton of bulk onc solutons [29 31], the permeaton of ons through membrane channels [9,28,32 37], and the steady-state propertes of electron transport n semconductor devces [38]. Wthn the BD framework, the trajectores of the ndvdual ons are tracked through phase-space, whle the surroundng solvent medum s modeled as a contnuum delectrc accordng to the mplct solvent model. The ons and the solvent are then coupled through electrostatc, hydrodynamc and stochastc forces. Observable propertes of the system are calculated by averagng over a suffcently large number of on trajectores. The computatonal burden s greatly reduced by ncludng the effects of solvent molecules mplctly, so allowng for long smulaton tmes. Ths fact motvated our choce of a BD kernel for the valdaton of the proposed force-feld scheme. However, the proposed approach s fully applcable to molecular dynamcs (MD) [39] smulatons. It s worthwhle to note that many varants of the BD smulaton kernel are possble, ncludng the ones based on the Smoluchowsk equatons of moton [40] and the Langevn equaton [11]. Memory effects can be ncluded n the model [41], as well as hydrodynamc nteractons [42 44].

4 120 Aboud Ths paper s organzed as follows. In the proceedng secton the dynamcs smulaton kernel used to valdate the force-feld scheme s brefly presented. The approach used to resolve the electrostatc nteracton s then dscussed n the followng secton. Detals about the mplementaton of the proposed force-feld scheme wll be suppled. Fnally, the results of the smulaton of dfferent electrolytc solutons are dscussed, and comparsons wth other models and wth expermental data are offered. 2. Brownan Dynamcs Wthn the chosen BD framework, the solvent (water) s treated as a contnuum delectrc, whle each on s explctly modeled as a Brownan partcle wth ts dynamcs beng tracked through the 6-dmensonal phase space. The feld dstrbuton s perodcally updated on the 3D computatonal doman, where the on-solvent nteractons are accounted for by ncludng macroscopc water propertes n the model. In other words, the dynamcs of the ons s explctly determned by the electrostatcs of the system through Newtonan mechancs, whle ther nteracton wth water molecules s mplctly modeled wth the Langevn equaton. The observable propertes of the system are then extracted from the mcroscopc representaton of the ons by averagng over the onc ensemble and over tme after the steady state s reached [45]. It should be noted that the further ncluson of complex molecules such as lpd conglomerates or protens can be acheved both n a macroscopc fashon by addng delectrc dscontnutes to the computatonal doman, and mcroscopcally, by buldng collectons of van der Waals partcles subjected to an approprate constraned dynamcs. The latter approach allows for the study of effects related to the tme evoluton of the structural propertes of the molecules n the system. The next sectons wll be devoted to the bref dscusson of the dynamcs equatons used by the mplemented BD kernel and the methodology used to ntegrate them Langevn Equaton The dynamcs smulaton engne used to valdate the proposed force feld scheme models on trajectores wthn the Langevn formalsm [29,30]. In partcular, the strct or full Langevn equaton s used, whch assumes Markovan random forces and neglects correlatons (both spatally and temporally) of the onc moton: m d v (t) dt = m γ v (t) + F ( r (t)) + B (t), (1) where m s the reduced mass of the th on, v (t)sts velocty at tme t, and γ s the frcton coeffcent (.e., the nverse of the velocty relaxaton tme). The defnton of the frcton coeffcent n the Langevn equaton vares n lterature [11,45,46], the notaton used here s the same as n the work of Gunsteren and Berendsen [46]. The force on partcle due to all other partcles n the system and boundary condtons (ncludng nternal delectrc dscontnutes) s F, whle B s a fluctuatng force that mmcs the molecular bombardment of water on the ons, and s modeled wth a Markovan random varable. The Langevn equaton s dscretzed temporally by a set of equally spaced tme ntervals, and spatally onto a tensor-product Cartesan grd. At predetermned tmes the on dynamcs s frozen, and the spatal dstrbuton of the force s calculated from the vector sum of all ts components, ncludng both the long- and short-range contrbutons. The components of the force are then kept constant whle the dynamcs resumes under the effect of the updated feld dstrbuton. Self-consstency between the force feld and the onc moton n the phase-space s obtaned by teratng ths procedure for a desred amount of smulaton tme. The choce of the spatal and temporal dscretzaton schemes plays a crucal role n terms of computatonal performance and model accuracy. The ntegraton scheme for Eq. (1) s chosen based on two requrements: ensurng energy stablty and allowng for long tme-steps [47]. The latter requrement s related to the need for nvestgatng the propertes of the system over the typcally long bologcal and/or chemcal tme scales, whch can extend up to mllseconds. The use of long tme-steps reduces the number of operatons for each unt of smulated tme, consequently ncreasng the performance of the smulaton code. On the other hand, the tme-step must be small compared wth the mean tme between partcle collsons. An excessvely coarse tme dscretzaton would not account for rapd varatons of the short-range force, and fals to correctly account for the coulombc sngularty. Ths typcally results n a spurous heatng of the partcle ensemble that becomes energetcally unstable [15,48].

5 Posson P 3 MForce Feld Scheme for Partcle-Based Smulatons of Ionc Lquds 121 Two ntegraton schemes have been mplemented: the standard frst-order Euler scheme and the Verletlke method by Gunsteren and Berendsen [46], whch s a thrd order model that reduces to the Verlet algorthm [49] when the frcton coeffcent n the Langevn equaton s zero. Ths approach allows for a larger tmestep as compared to the Euler method. Both schemes are dscussed n the followng sectons, and comparsons are made Euler Integraton The frst order Euler ntegraton scheme reduces the Langevn equaton to the followng expresson: [ F v (t + t) = v (t) t γ v (t) 2γ k B T m t m ] N(0, 1), (2) where t s the ntegraton tme-step and N(0, 1) s a three dmensonal Gaussan random varable wth zero mean and varance 1. The spatal trajectores are calculated wth Newtonan mechancs. A crucal aspect of the Euler scheme s that n order to represent the fluctuatng force as a statonary Markovan Gaussan process, the tme-step t duraton must be much smaller than the recprocal of the frcton coeffcent γ n the Langevn equaton (Eq. (1)) [46]. Ths results n a fne (and computatonally expensve) tme dscretzaton when onc solutons are smulated Verlet-Lke Integraton The short tme-steps lmtaton of the Euler ntegraton scheme s addressed wthn the Gunsteren and Berendsen approach [46] by accountng for the evoluton of the fluctuatng force durng the ntegraton tme-step. In ths method, the force on the th partcle at tme t n+1 s frst expanded n a power seres about the prevous tme t n : F (t n+1 ) F (t n ) + Ḟ (t n )(t n+1 t n ), (3) where Ḟ denotes the tme dervatve. The power seres expanson s substtuted nto Eq. (1), and the resultng soluton of the Langevn equaton s gven by v (t n+1 ) = v (t n )e γ t + (m γ ) 1 F (t n )(1 e γ t ) + (m γ 2 ) 1 Ḟ (t n )(γ t (1 e γ t )) t + (m ) 1 e γ t e γ (t t n ) B (t )dt, (4) t n where t = t n+1 t n s the ntegraton tme-step. Note that the fluctuatng force B (t)sleft nsde the ntegral. The onc poston s calculated wth the followng expresson: x (t n+1 ) = 2x (t n ) x (t n 1 )e γ t + tn + t t n v (t )dt + e γ t tn t n t v (t )dt, (5) and, fnally, the updated partcle poston s wrtten as x (t n+1 ) = x (t n )[1 + e γ t ] x (t n 1 )e γ t where + (m γ ) 1 F (t n )( t)[1 e γ t ] + (m γ 2 ) 1 Ḟ (t n )( t)[0.5γ t(1 + e γ t )] [1 e γ t ] + X n (0, t) + e γ t ]X n (0, t). (6) tn X n (0, t) = (m γ ) 1 + t [ 1 e γ (t n + t t ) ] t n B (t )dt (7) s also a Markovan stochastc process wth zero mean and varance t. The functon X n (0, t)scorrelated wth X n 1 (0, t) through a bvarate Gaussan dstrbuton. In the lmt that the frcton coeffcent goes to zero ths set of equatons corresponds to the trajectores obtaned wth the Verlet MD algorthm [46,50]. The set of trajectores resultng from the Verlet-lke ntegraton scheme are not lmted by the velocty relaxaton tme, and a longer tme-step can be used as compared to the Euler scheme. Fgure 1 shows a plot of the steady-state average onc energy versus the tme-step length for a 150 mm KCl soluton smulated for 1 ns n the absence of an external electrc feld. The Euler and Verlet-lke algorthms gve smlar results for tmesteps below approxmately 10 fs, but larger tme-steps result n a hgh energy drft for the Euler ntegraton.

6 122 Aboud where F C s the coulombc force due to all the partcles wthn a predefned short-range doman (see Secton 3.2), F W represents the effects of the van der Waals forces, and, fnally, R s a reference force [15] that corrects the double countng of charges due to the overlap between the short-range doman and the entre computatonal regon over whch Posson s equaton s solved. A detaled descrpton of all the components of the force F ( r )sgven n the followng sectons. Fgure 1. Steady-state energy of an ensemble of anons and catons n a 150 mm soluton of KCl as a functon of the tme-step length, for both the Euler and Verlet-lke ntegraton schemes. 3. Posson P 3 MForce Feld Wthn the proposed approach, the force on an ndvdual on s dvded nto a smoothly varyng long-range component and a short-range part. Both components are perodcally updated n a fashon consstent wth the onc dynamcs, as explaned n Secton 2.1. The long-range nteracton accounts for the force due to the collectve onc populaton and the boundary condtons, whle the short-range nter-partcle nteractons result from the coulombc and van der Waals forces between close ons. The total force actng on a partcle s then wrtten as follows: F = F pm + F pp. (8) The long range partcle-mesh force F pm s obtaned by assgnng the charge densty to the grd ponts, solvng Posson s equaton [14], and dfferentatng the potental: F pm ( r p ) = q φ( r p ) (9) where F pm and φ( r p ) are the force and the electrostatc potental, respectvely, at the grd pont p located at r p. Ths component of the force also accounts for external boundary condtons, delectrc dscontnutes, and statc charges. The force F pm at the specfc poston r of the on s then computed by an approprate nterpolaton scheme (see Secton 3.1). The partcle-partcle force s decomposed n three parts: F pp = F C + F W + R, (10) 3.1. Long Range Interacton, Posson s Equaton In order to solve Posson s equaton on a mesh, a charge assgnment scheme must be devsed to buld a charge dstrbuton from the onc coordnates. Furthermore, once the electrostatc feld has been computed on the grd from the soluton of Posson s equaton, the force must be nterpolated to each on locaton n a way that s consstent wth the orgnal charge assgnment scheme. In other words, a geometrc shape s assgned to each on charge though a space-dependent weghtng functon W ( r) [15], and the relaton between the charge shape and the dscretzaton grd s accounted for n all the transformatons used to transfer quanttes (.e. charge and force) to and from the dscrete mesh centered at r p. The generalzed algorthm follows the treatment of Hockney [15]: 1. Assgn charge: ρ( r p ) = 1 V p N p 2. Solve Posson s equaton: 3. Calculate electrc feld: 4. Interpolate force: q W ( r r p ); (11) ɛ r φ( r p ) = ρ( r p) ɛ 0 ; (12) F pm ( r ) = E( r p ) = φ( r p ); (13) N p p q W ( r r p ) E( r p ); (14)

7 Posson P 3 MForce Feld Scheme for Partcle-Based Smulatons of Ionc Lquds 123 where V p and N p are the volume of the grd cell and the number of partcles n t, respectvely, and F pm s the long-range component of the force actng on the partcle located at r (see Eq. (8)). It should be noted that the same functon W ( r) must be used both for the charge assgnment and for the force nterpolaton, because the use of a mxed scheme would result n an unphyscal self-force of the partcle upon tself. The choce of the weghtng functon depends on the propertes of the system. The three models mplemented and tested n ths work treat the partcle as a pont charge, an unformly charged sphere, and a sphere wth a lnearly decreasng densty; the correspondng assgnment schemes are called the nearest-grd pont (NGP), the cloud-n-cell (CIC) and the trangular-shaped cloud (TSC) schemes, respectvely [15]. Once a charge shape has been chosen, the correspondng weghtng functon s determned by the followng ntegral, W ( r r p ) = S( r r) d r, (15) V p where the functon S( r) represents the shape of the charge cloud assocated wth the partcle. In one dmenson, the weghtng functons computed from Eq. (15) are gven by the followng relatons for the three charge shapes mentoned above: { 1 W NGP (x) = x 1 H 2, (16) 0 otherwse { 1 x x 1 W CIC (x) = H H 0 otherwse, (17) 3 4 x 2 x 1 ( H H 2 W TSC (x) = x ) 2 1 H 2 x 3, (18) H 2 0 otherwse where H s the mesh sze. In three dmensons the weghtng functon s obtaned as follows, W ( r) = W (x)w (y)w (z). (19) In agreement wth the consderatons of Hockney [15], the TSC weghtng functon s the optmal compromse between accuracy and computatonal performance for the systems studed n ths work. The use of a Posson solver for the soluton of the long-range nteracton results n two man advantages: (1) the possblty of mposng boundary condtons through externally appled potentals and (2) the ablty to smulate systems wth arbtrary onc concentratons at the boundares. Whle the smulaton of bulk homogeneous systems do not explot these capabltes and can (should) be performed wth perodc boundary condtons, the use of a Posson solver allows for a hgher degree of realsm n reproducng the electrostatc confguraton of nhomogeneous systems Mult-Grd Posson Solver. The need for selfconsstency between the spatal charge dstrbuton and the feld of forces drectly mples a frequent soluton of Posson s equaton. For ths reason, the mplementaton of an effcent and robust Posson solver plays a crucal part n the proposed P 3 M scheme. Indeed, for three dmensonal systems the tme spent for the repeated soluton of Posson s equaton becomes a sgnfcant part of the total CPU tme, and the effcency of the solver becomes an ssue. Because of the frequent soluton of Posson s equaton wthn the self-consstent scheme, teratve solvers [51,52] are the most natural opton, due to the avalablty of the prevously computed soluton as the ntal guess for the solver [53]. Therefore, the method of choce n ths work s a fnte-dfference Posson solver based on the teratve verson of the mult-grd algorthm [16,25]. Wthn the mult-grd approach, the matrx equaton resultng from the dscretzaton of Posson s equaton [54,55] s solved smultaneously on a set of grds wth varyng coarseness. The herarchy of grds allows for the smultaneous reducton of dfferent Fourer components of the error assocated wth each teraton of the solver [56]. Ths results n a much faster convergence rate as compared to other standard teratve methods such as the successve-over-relaxaton (SOR) method [15,57]. A comparson of the total CPU tme spent solvng Posson s equaton for one selfconsstent teraton as a functon of the convergence error s shown n Fg. 2 for the mult-grd method and an optmzed SOR algorthm [16]. The computatonal doman conssts of a 100 mm KCl soluton represented ona homogeneous grd wth a mesh sze of 0.5 nm n all three dmensons. The slope of the error ndcates the performance behavor of the solver. As can be seen, the convergence rate of the mult-grd method s much better than the SOR, partcularly at

8 124 Aboud Fgure 2. Comparson of the CPU tme requred to solve Posson s equaton by the mult-grd and SOR methods. The computatonal doman conssts of a homogeneous mesh. small values of the relatve error. Convergence thresholds are typcally chosen wthn the range [ ], and result n CPU tmes of 8 10 seconds for the soluton usng the mult-grd approach, and seconds for the SOR. Algorthmc detals on the mplementaton of the mult-grd method can be found n the excellent works of Hackbusch [16] and Brandt [25,58]. It should be noted that the mult-grd approach can be easly appled to adaptve non-tensor-product grds [25,58], allowng for a varable resoluton n regons of the computatonal doman where the charge concentraton s hgh. Beng focused on bulk propertes, the smulatons of ths work are performed usng a relatvely smple tensor-product dscretzaton grd. A dscretzaton scheme based on adaptve grds can result n a further ncrease of the performance when smulatng hghly nhomogeneous systems such as bologcal membranes or complex protens. The choce of usng the SOR solver n the P 3 M proposed by Beckers [59] s not advocated here because of ts slower convergence compared to the multgrd approach, and because of ts neffcency for large problems. It s recognzed, however, that the extreme smplcty of the SOR algorthm makes t an attractve choce. A typcal SOR solver can be mplemented wth a few tens of lnes of code, whle our 3D mult-grd solver s several thousands lnes long. Furthermore, the mult-grd choce for the Posson solver s certanly not unque, other effcent algorthmc choces are avalable, such as the strongly mplct schemes [60] and the many flavors of the conjugate gradent technque [61]. Mxed schemes based ether on heterogeneous precondtonng or smoothng are also possble [62]. In ths work, we have chosen to use the mult-grd approach because of ts effcency, robustness, and scalablty. As a fnal remark, we lke to brefly dscuss the ssue of computatonal performance. Clearly, the force feld calculaton s an mportant component n the budget of CPU resources used n partcle-based smulaton of lquds. The computatonal burden due to the algorthms used for long- and short-range nteractons depends on the nature of the system and ts sze. Wthn the framework of on channel smulaton, one normally expects a relatvely large n-homogeneous Posson grd wth at least 10 4 cells, and a larger number of partcles (ncludng the proten fxed charges). In ths case, even though most of the burden s due to the computaton of the short-range nteracton, the tme spent for the longrange calculaton would be a sgnfcant porton of the tme devoted to the full force-feld algorthm. For ths reason, both performance and robustness are crucal requrements when modelng the long-range nteracton for on channel systems Electrc Feld. The electrc feld used to calculate the long-range PM force s computed at each grd pont as the gradent of the potental obtaned from the Posson solver. The dfferentaton of the potental takes nto account delectrc dscontnutes, possble nterfacal charges and boundary condtons. The force F pm at the specfc partcle locaton s then nterpolated wth the same weghtng functon used for charge assgnment (see Eq. (14)) Short-Range Interacton As stated by Eq. (10), the partcle-partcle force s comprsed of three parts: the Coulomb force F C, the van der Waals force F W, and the reference force R : F C = F W = j j j q q j 4πɛ r ɛ 0 r r j rˆ j, (20) 2 [ ( ) 24ɛ 12 ( ) 6 ] j σj σj 2 rˆ j r r j r r j r r j β j q q j 4πɛ r r j (p + 1) Lennard-Jones, ( ) s + s p j rˆ j r r j nverse power, (21)

9 Posson P 3 MForce Feld Scheme for Partcle-Based Smulatons of Ionc Lquds 125 q q j R = 4πɛ r ɛ 0 j S( r 1 )S( r 2 r j ) ( r 1 r 2 ) r 1 r 2 3 d r 1d r 2, (22) where s the doman of the short-range nteracton (see below), ɛ r s the relatve delectrc constant, ɛ 0 s the permttvty of vacuum, q s the charge, and r j s the dstance between the ons. The van der Waals force F W s often modeled wth the Lennard-Jones functon or by an nverse power relaton [63]. The former has been used n ths work, and s based on the two fttng parameters σ j and ɛ j, representng respectvely the maxmum attracton dstance and the strength of the nteracton [64]. For ons of dfferent speces, the Lennard-Jones parameters are calculated by combnng the values of the ndvdual speces [64], σ j = 1 2 (σ + σ j ), and, ɛ j = ɛ ɛ j. (23) In the expresson of the nverse power law, β j s an adjustable parameter, s s the radus of the th partcle, and p s a hardness parameter that also represents the strength of the nteracton. A comparson of the nter-onc potental profle s shown n Fg. 3 for the two dfferent par potental schemes n an aqueous KCl soluton. The parameters used for the short range potentals are taken from [28] for the Lennard- Jones functon and from [15] for the nverse power relaton. The fnal component R of the partcle-partcle force s the reference force, whch depends on the shape S of the onc charge. As prevously stated, the partclepartcle porton of the force s calculated for ons wthn the relatvely small sphercal regon. The role of the reference force s to correct for the overlap between and the entre system over whch the mesh force F pm s calculated. In other words, the sources of electrostatc force actng on a gven charged partcle are classfed as far sources (ncludng boundary condtons) that are accounted for effcently by the Posson solver, and close sources generatng forces that are not resolved by the Posson solver and must be computed by the expensve O(N 2 ) partcle-partcle scheme. The doman defnes the fne resoluton regon around a gven on. For obvous reasons, the soluton of Posson s equaton can not be obtaned by subtractng the charges wthn ths would ndeed requre a full soluton for each partcle at each teraton so the effect of those sources s subtracted from the potental dstrbuton after the soluton has been obtaned. Ths correcton s accomplshed by the reference force. Clearly, the sze of the regon should be chosen as small as possble based on performance consderatons. The key aspect that lmts the mnmum sze of s the sze of the onc charge used for the charge assgnment scheme (see Secton 3.1). As stated above, the charge dstrbuton s computed by assgnng a cloud of charge to each ndvdual on. The cloud has a specfc geometrc shape and a predefned charge densty. When calculatng the total force on a gven on, all the charged partcles j whose charge cloud s overlappng wth the one of must be consdered close sources of the electrostatc force, and must be ncluded n the doman. As stated n Secton 3.1, the onc electrostatc shape S chosen for ths work s a sphere wth an unformly decreasng charge densty, correspondng to the TSC weghtng scheme [15]: 3 (r S(r) = c r) r c r πr c, (24) 0 otherwse Fgure 3. Comparson of short range Lennard-Jones and nverse power potental schemes for K+ and Cl n an aqueous soluton. where r c s the radus of the sphercal charge cloud. Therefore, the natural choce for the mnmum cutoff radus that defnes the short-range regon s twce r c. The reference force s then found analytcally by substtutng the shape functon S(r) nto

10 126 Aboud Fgure 4. Components of the force between two ons of opposte charge n a 500 mm soluton of KCl wth no external bas. Eq. (22): R(r) = q q j 4πɛ r ɛ 0 4 (224ζ 224ζ ζ ζ 5 21ζ 6 ) 35rc 2 0 ζ 1 4 (12/ζ ζ 840ζ ζ 3 35rc ζ 4 48ζ 5 7ζ 6 ) 1 ζ 2 1 otherwse r 2 (25) where ζ = r/r c.toreduce the computatonal burden, the reference force s precomputed and tabulated as a functon of the dstance between on pars as suggested by Hockney [15] and, successvely, by Wordelman [31]. The components of the force between an anon and a caton nsde the short-range doman (2r c =2 nm) are shown n Fg. 4 as a functon of the nter-onc separaton. The two ons are placed n a 500 mm KCl soluton, wth no external bas. As expected, the reference force and mesh force have the same ampltude, and therefore wll cancel wthn the short-range doman Tme Dscretzaton Scheme The ssue of tme dscretzaton s partcularly relevant for the performance of the smulaton code. As dscussed n Secton 2.1, the duraton of the tme-step must be carefully chosen to ensure self-consstency whle mnmzng the use of computatonal resources. It should be notced that, even f self-consstency requres a perodc update of the force felds, dfferent components of the force evolve on dfferent tme scales, therefore allowng for dfferent tme-steps for the calculaton of the mesh force, F pm, and the short-range partcle-partcle force, F pp. Such dfferentaton s used to optmze the use of computatonal resources. The tme-step used for updatng the PP force s dctated by the tme ntegraton scheme of the onc dynamcs, as dscussed n Secton 2.1. In ths case, the use of the Verlet-lke scheme allows for relatvely longer tme-steps (20 fs) as compared wth the standard Euler ntegraton (5 fs). Concernng the PM part of the force feld, one observes that the evoluton of the charge densty over the grd can be correctly modeled wth a less frequent update of the potental dstrbuton [47,50,65]. Therefore, the Posson tme-step can be longer than the update tme of the short-range force. As an estmator of the mnmum characterstc tme requred to resolve the fluctuatons n the long-range force we use the nverse of the plasma frequency [15], c q ω = 2 ɛ r ɛ 0 m, (26) where c s the onc charge concentraton, m s the on mass and q s the magntude of the on charge. The plasma frequency represents the electrostatc response of the system to a perturbaton n the charge densty. Therefore, the Posson solver cannot properly accommodate electrostatc changes n the system f the PM tme-step s larger than the nverse of ths quantty. For an aqueous soluton of KCl at 100 mm the nverse of the plasma frequency s approxmately 5 ps whch s several orders of magntude longer than the tme-step requred to update the PP force. Although n ths work we use Eq. (26) to estmate the maxmum tolerable duraton of the Posson tmestep, t should be noted that the over-damped systems studed here are not rgorously descrbed by plasma theory [15,66], and that an energetcally stable onc populaton can be modeled wth less strngent lmts for the Posson tme-step. 4. Computatonal Doman A small cubc test volume representng a porton of a larger aqueous electrolytc soluton s smulated

11 Posson P 3 MForce Feld Scheme for Partcle-Based Smulatons of Ionc Lquds 127 to valdate the smulaton tool. Drchlet boundary condtons are set on opposte planes of the 3D smulaton volume, whle Neumann condtons [15] are mposed on the other four boundares. In ths way, an external electrostatc potental can be appled across the computatonal doman, and s ncluded n the soluton of Posson s equaton. Ions are allowed to traverse the Drchlet contact cells, and are specularly reflected by the Neumann boundares. Snce perodc boundary condtons are not mposed, an njecton mechansm must be devsed at the Drchlet boundares to mantan a gven on concentraton n the computatonal doman wthout perturbng ts energetc stablty. In ths work, snce the smulated volume s assumed to represent only a porton of a larger electrolytc bath, the Drchlet boundares are kept at a constant concentraton to mmc the effect of two far electrodes. At each Posson tmestep an approprate on flux s mposed at the electrodes, where the velocty of the njected ons s calculated wth a Maxwellan dstrbuton n the drectons parallel to the contact cells and a half-maxwellan n the normal drecton. Wthn ths njecton scheme the average velocty of the njected partcles does not correspond to the macroscopc flux, and the partcles veloctes (and energes) must relax to the steady-state values. Ths process s generally fast, and steady-state behavor s obtaned wthn 2-3 grd cells from the Drchlet boundary. To avod artfacts ntroduced by the njectng electrodes, several cells neghborng the contacts are excluded n the calculatons of the bulk propertes. Such an njecton mechansm ensures a constant onc concentraton over long smulaton tmes. 5. Smulaton Results In order to valdate the proposed force-feld scheme, the thermodynamc propertes of an electrolytc soluton are calculated under equlbrum condtons as a functon of concentraton, and are compared to values obtaned wth analytc approxmatons and expermental results, where avalable. Addtonal smulatons are also performed by ntroducng a2nmdelectrc slab n the center of the computatonal doman, to mmc the presence of a lpd membrane wth an externally appled transmembrane potental. An mportant pont of mert s that all the smulaton results presented n ths document have been obtaned wthout any external dsspatve mechansms that enforce energy stablty. For ths reason the analyss of the stablty of the onc populaton plays a crucal role n the algorthmc choce Thermodynamc Propertes As an ntal test, the equlbrum thermodynamc propertes of an onc soluton are determned. Ths nvolves the calculaton of the radal dstrbuton functon (RDF) [11], whch relates the probablty of fndng a par of ons at a specfc separaton to the probablty n a homogeneous dstrbuton at the same densty [39]. The RDF as a functon of the nter-partcle dstance s g(r) = 1 ρ N j δ( r r j ), (27) where r j s the partcle separaton, ρ s the on densty, and N s the tme-averaged number of ons. The RDF provdes mportant structural nformaton about the system, and can be used to calculate the ensemble average of any par functon, ncludng the free energy, pressure, and chemcal potental [39]. The knowledge of these three functons allows for the calculaton of all other observable thermodynamc quanttes, and the comparson wth the experment. The RDF for KCl and NaCl electrolytc solutons are shown n Fg. 5 and are compared wth the numercal soluton of the Ornsten-Zernke equaton [45] solved usng the hypernetted chan approxmaton (HNC) as a closure relaton [11,67]. As can be seen, the RDFs obtaned wth the smulaton approach proposed n ths work show excellent agreement wth the HNC results over a varety of concentratons and onc speces. The results of the HNC solver have been valdated through comparsons wth prevously reported HNC smulatons [28]. Detals of the HNC method used to calculate the RDF are gven n App. A. Further comparsons have also been made of the osmotc coeffcent [11,45] as computed wth the proposed method, wth the HNC, and wth expermental values. The osmotc coeffcent for a mult-on system s calculated wth the expresson [45]: ψ = 1 1 6ρk B T r max ρ ρ j, j u j(r k ) r k r g j (r k )4πr 3 k r k, (28)

12 128 Aboud Fgure 5. Radal dstrbuton functon as a functon of dstance for dfferent concentratons of KCl and NaCl. Comparson wth results from the HNC, shown as sold lnes, show excellent agreement for dfferent concentratons. where ρ s the densty of the th speces, u j s the partcle-partcle nteracton potental, and r max s the dstance at whch the contrbuton from spatal dervatve of the potental u j s neglgble. The RDF s wrtten here as g j to dstngush between the dfferent speces and j. Aplot of the osmotc coeffcent as a functon of concentraton s shown n Fg. 6 for KCl and NaCl, and ncludes the expermental values as well as the results of the HNC calculaton. Although the agreement between the HNC and the Brownan dynamcs s very good, there s a devaton from the expermental values, partcularly for NaCl. Several factors can explan the dscrepancy. Frst, the equlbrum pressure s a crucal component of the osmotc coeffcent, and the ambent pressure of the computatonal representaton dffers wth respect to the expermental system because of the mplct water model [45]. Also, the RDF s sgnfcantly dependent on the parameters used n the short-range nteracton, and these vary consderably n lterature, partcularly for NaCl [68]. The computatonal doman used for these smulatons s a cube wth 20 nm sdes and s dscretzed wth a homogeneous tensor-product grd. The PM force s updated every 2 ps whle the PP force s calculated every 20 fs. The ntegraton scheme used for the smulatons s the Verlet-lke algorthm. The total smulated tme s 2 ns and the system propertes are determned by takng the ensemble average over the fnal 1.5 ns. Fgure 6. Osmotc coeffcent versus concentraton for (a) KCl and (b) NaCl. Results are compared wth expermental values and wth results from the HNC. The osmotc coeffcent n ths work s n very good agreement wth the analytc model Transmembrane Potental Another mportant valdaton of the force-feld scheme comes from the calculaton of the transmembrane potental. An mpermeable delectrc slab representng a lpd membrane s embedded n the center of an electrolytc soluton, and a smulaton s run to determne the charge dstrbuton at the water/lpd nterface.

13 Posson P 3 MForce Feld Scheme for Partcle-Based Smulatons of Ionc Lquds 129 Fgure 7. Average concentraton of anons and catons n a 150 mm soluton of KCl wth a2nmdelectrc membrane n the center. The relatve delectrc constant of the membrane and the surroundng bath s 2 and 80, respectvely. External bas s appled at the electrodes. Due to the large dfference n delectrc constants, most of the potental drop s across the membrane. The standard contnuty condton of the electrc dsplacement across the delectrc boundary [14] gves rse to an accumulaton or depleton of ons at the edge of the membrane. The sgn of the onc charge and the external bas determnes whether the ons are attracted to or repelled by the membrane surface. A plot of the average concentraton of anons and catons for dfferent bas voltages s shown n Fg. 7. The on concentraton s plotted along the drecton normal to the membrane nterface, and a spatal average s taken n the other two drectons. The membrane s 2 nm wde and s contaned n an electrolytc bath of 150 mm KCl. At low voltages, the dstrbuton of ons s approxmately homogeneous on both sdes of the membrane (see Fg. 7(a)), but, wth ncreasng bas, the K + ons accumulate on one sde and deplete on the other, whle the Cl ons assume the opposte confguraton, as a consequence of Gauss law. The concentraton on the two sdes of the doman s mantaned durng the smulaton by the njecton mechansm. As opposed to other force feld schemes, the proposed approach allows for smulatons wth asymmetrc onc concentratons. Fgure 8 shows a plot of the average onc dstrbuton and the potental profle n a Fgure 8. Average onc concentraton and potental dstrbuton n a KCl bath separated by a 2 nm delectrc membrane. The concentraton on the left sde s 300 mm, whle the one on the rght s 150 mm. system wth 300 mm KCl on the left sde of the membrane and 150 mm KCl on the rght. Accordng to the behavor of the Boltzmann dstrbuton [11], the net accumulaton or depleton of onc charge s equal on the two sdes of the membrane. The transmembrane potental smulatons are performed on the same geometrc confguraton used for the study of the thermodynamc propertes, as descrbed n Secton 5.1. The Euler ntegraton scheme s used, and the PP force s updated every 5 fs, whle the Posson tme-step s 2 ps. Smulatons are run for 5nswth averages taken over the last 4 ns.

14 130 Aboud 6. Conclusons In ths work, a force-feld scheme based on a Posson P 3 M algorthm s proposed as an accurate and effcent tool to smulate onc charge transport n electrcally nhomogeneous systems. The force-feld scheme s selfconsstently coupled wth a dynamcs smulaton kernel based on the Langevn equaton. Calculatons of thermodynamc propertes of dfferent electrolytc solutons are performed, and show excellent agreement wth other models. In addton, the spatal charge dstrbuton profle n the presence of a double delectrc nterface s obtaned. Ths work represents an ntal step toward the development of an effcent and robust smulaton tool for the partcle-based modelng of complex nhomogeneous bologcal systems, such as lpd membranes and on channels. The proposed approach wll be appled to MD smulatons to further valdate t and nvestgate ts advantages and lmtatons. Appendx A: Hypernetted-Chan Method The hypernetted-chan approxmaton (HNC) supples a closure relaton for the Ornsten-Zerncke (OZ) equaton, whch s a non-lnear ntegral equaton for the radal dstrbuton functon n terms of the ntermolecular potental [45]. The OZ equaton for a mxture of several speces s gven by [11], h ss (r j ) = c ss (r j ) + ρ l h sl (r k )c ls (r jk )dr 3, l (29) = g ss (r j ) 1, (30) where c ss (r j ) and g ss (r j ) are the drect correlaton functon and par correlaton functon (radal dstrbuton functon), respectvely between partcle of speces s and partcle j of speces s. The ntegral term n Eq. (29) s an ndrect component of the correlaton functon, and represents the correlaton of partcle wth partcle j ntally propagated through a thrd partcle (ether drectly or ndrectly). A closure relaton for Eq. (29) s obtaned by determnng an expresson for c ss (r j )nterms of g ss (r j ), and substtutng ths new expresson nto Eq. (29). Wthn the HNC formulaton, the drect correlaton functon s expressed as the dfference between the total correlaton functon and the ndrect correlaton functon, or, c ss (r j ) = g ss (r j ) g nd ss (r j), (31) where g nd ss (r j)sthe radal dstrbuton functon correspondng to a system wthout the drect (.e., parwse) nteracton, also called the ndrect radal dstrbuton functon: g ndrect ss (r j) = e [w ss (r j) u ss (r j )]/K B T, (32) where w s the total nteracton potental (potental of mean force) and u s the parwse partcle-partcle nteracton. By Taylor-expandng the ndrect radal dstrbuton functon, g ndrect ss (r j) = 1 k B T [w ss (r j) u ss (r j )] 1, (33) and substtutng t nto Eq. (31), one fnally obtans the HNC equaton: where c ss (r j ) = g ss (r j ) 1 n ss (r j ), (34) n ss (r j ) = 1 k B T [w ss (r j) u ss (r j )]. (35) The radal dstrbuton functon s then calculated teratvely from the followng set of coupled matrx equatons [69]: c(r) = g(r) 1 n(r) (36) ˆn(q) = ĉ(q)/(1 ρĉ(q)) ĉ(q) (37) g(r) = exp[(n(r) u(r))/k B T ], (38) where ĉ(q) denotes the three dmensonal Fourer transform. The bold face notaton s used to represent matrx quanttes. Ths set of equatons s dscretzed on a real-space grd wth 0.1 Å unform spacng. The maxmum partcle separaton s set to 2 nm. The soluton s obtaned when the followng convergence crteron s satsfed [69]: [ ɛ th c k (r) c k+1 (r) 1/2 dr] 2, (39) 0 where ɛ th s the convergence threshold and k represents the teraton step. Accordng to publshed values [69], the convergence threshold n ths work s set to

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