Da Meng 1,, Bin Zheng 1,, Guang Lin 1,2, and Maria L. Sushko 1, West Lafayette, IN 47907, USA.

Transcription

1 Commun. Comput. Phys. do: /ccp a Vol. x, No. x, pp xxx 20xx Numercal Soluton of 3D Posson-Nernst-Planck Equatons Coupled wth Classcal Densty Functonal Theory for Modelng Ion and Electron Transport n a Confned Envronment Da Meng 1,, Bn Zheng 1,, Guang Ln 1,2, and Mara L. Sushko 1, 1 Pacfc Northwest Natonal Laboratory, Rchland, WA 99352, USA. 2 Department of Mathematcs, School of Mechancal Engneerng, Purdue Unversty, West Lafayette, IN 47907, USA. Receved xxx; Accepted (n revsed verson) xxx Avalable onlne xxx Abstract. We have developed effcent numercal algorthms for solvng 3D steadystate Posson-Nernst-Planck (PNP) equatons wth excess chemcal potentals descrbed by the classcal densty functonal theory (cdft). The coupled PNP equatons are dscretzed by a fnte dfference scheme and solved teratvely usng the Gummel method wth relaxaton. The Nernst-Planck equatons are transformed nto Laplace equatons through the Slotboom transformaton. Then, the algebrac multgrd method s appled to effcently solve the Posson equaton and the transformed Nernst-Planck equatons. A novel strategy for calculatng excess chemcal potentals through fast Fourer transforms s proposed, whch reduces computatonal complexty fromo(n 2 ) to O(NlogN), where N s the number of grd ponts. Integrals nvolvng the Drac delta functon are evaluated drectly by coordnate transformaton, whch yelds more accurate results compared to applyng numercal quadrature to an approxmated delta functon. Numercal results for on and electron transport n sold electrolyte for lthumon (L-on) batteres are shown to be n good agreement wth the expermental data and the results from prevous studes. AMS subject classfcatons: 92C35, 35J47, 35J60, 35R09, 65M06, 65N55, 65T50 Key words: Posson-Nernst-Planck equatons, classcal densty functonal theory, algebrac multgrd method, fast Fourer transform, L-on battery. The frst two authors contrbute equally to ths work. Correspondng author. Emal addresses: Guangln@purdue.edu (G. Ln), Mara.Sushko@pnnl.gov (M. L. Sushko) 1 c 20xx Global-Scence Press

2 2 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp Introducton Posson-Nernst-Planck (PNP) equatons are wdely used to descrbe the macroscopc propertes of on transport n electrochemcal systems [1 5] (e.g., lthum-on (L-on) batteres, fuel cells) and bologcal membrane channels [6 13]. PNP equatons are also known as the drft-dffuson equatons for the descrpton of currents n semconductor devces [14 17]. In these models, excess chemcal potental of moble ons drves ther dffuson. However, a hghly smplfed descrpton of the nteractons lmted to Coulomb nteractons between all charged speces, s often used. To overcome ths oversmplfcaton n the representaton of collectve nteractons, classcal densty functonal theory (cdft) can be used. cdft s a powerful analytcal tool to descrbe mesoscopc nteractons, such as excluded volume effects and electrostatc correlaton nteractons, and thermodynamc propertes of nhomogeneous systems from frst prncples [18]. The PNPcDFT model s a generalzaton of the PNP model often used to descrbe fluds of charged hard spheres n a confned envronment. It has been appled to study the selectvty and onc flux n bologcal on channels [19 22] and shown to provde computatonal results n good agreement wth expermental data and/or theoretcal analyss. In sold state on and electron dffuson s also affected by the barrers for elementary transport processes: on hoppng between the adjacent equlbrum stes and electron hoppng between the catons n the lattce. Smlarly, n bologcal on channels, shortrange dsperson nteractons between the ons and functonal groups n the channel protens would also affect ther dffuson. These short-range nteractons have a quantum mechancal nature, whch makes t challengng to evaluate them analytcally. To nclude these short-range nteractons n cdft model, quantum mechancal smulatons can be used to evaluate the barrers for the elementary transport processes and represent the nteractons wth a square-well potental, featurng depth equal to the barrer and the wdth comparable to onc dameters [23, 24]. To summarze, n our approach apart from Coulomb nteractons electrostatc correlaton and excluded volume effects are treated usng cdft wth short-range nteractons quantum mechancally evaluated. Ths approach s equally applcable to study on and electron transport n nanostructured materals, on transport through bologcal on channels, and small molecule dffuson n mesoporous materals. In ths work, we use the sold electrolyte, lthum phosphorus oxyntrde (LPON), for L-on batteres as a test system and study temperature dependence of L + conductvty n LPON flms. Ths materal has a complex L + dffuson pathway [25,26], whch requres a full 3D model for on and electron transport. Our prevous work showed the PNP-cDFT model s unque capablty to capture the physcs of nanostructured electrode materals for L-on batteres, provdng nsghts nto the orgn of sze effects of conductvty and temperature dependence. The model can be used to gude synthess of new nanocomposte materals wth sgnfcantly mproved electrochemcal propertes [23, 26 30]. Ths research also revealed lmtatons n modelng realstc nanocompostes wth complex structures, callng for optmzaton of the effcency of the PNP-cDFT solvers.

3 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp The PNP-cDFT model s governed by nonlnear ntegro-dfferental equatons. The mathematcal analyss and numercal smulaton generate nterestng and challengng problems [19 22, 26, 31 33]. Numercal methods for solvng the PNP system of equatons have been studed extensvely, ncludng the fnte dfference [7, 8], fnte volume [12, 34, 35], and fnte element methods [9,11,36,37]. In [38], a second-order convergent numercal method s constructed to handle dscontnuous delectrc constants and sngular sources n the context of bologcal on channel applcatons. In the current study, a standard fnte dfference scheme s suffcent because the dffuson coeffcents for PNP equatons are assumed to be constant, and, for our appled nterests, the computatonal doman s regular. The man computatonal challenges for the PNP-cDFT smulatons nclude the soluton of large sparse lnear systems resultng from the dscretzatons of PNP equatons, and a sgnfcant amount of 3D ntegrals to be calculated for cdft. In partcular, the computaton of chemcal potentals of charged speces requreso(n 2 ) operatons, where N s the number of computatonal grd ponts. Ths makes applcaton of the model to realstc systems computatonally very expensve. Hence, most of the exstng studes are restrcted to the 1D case [19 22, 31 33, 39]. Numercal smulatons of a 3D PNP-cDFT model have been reported n [7, 26], usng coarse grd to reduce computatonal complexty. In ths work, we apply a state-of-the-art fast Posson solver algebrac multgrd (AMG) method that has computatonal complexty of O(NlogN). To speed up numercal ntegratons, we reformulate them as convoluton sums and then employ the fast Fourer transform (FFT) method to reduce the computatonal complexty too(nlogn). Some ntegrals n cdft calculatons nvolve Drac delta functon. A usual approach s to approxmate the Drac delta functon by a smooth functon (e.g., Gaussan) and then apply a standard quadrature rule. The addtonal error ntroduced by the approxmaton of the delta functon s one drawback to ths approach. Here, we use the defnton of Drac delta functon and change of varables to transform these 3D ntegrals nto 2D ntegrals on spheres and remove the sngularty n the ntegrands. Fnally, 3D large scale numercal smulatons of the PNP-cDFT system are made feasble usng the packages BoomerAMG [40, 41] and F3DFFT [42]. The rest of the paper s organzed as follows: the multscale model for nanostructured materal and ts descrpton through PNP equatons and cdft are presented n Secton 2. Secton 3 descrbes the numercal methods used n ths study, ncludng the fnte dfference dscretzaton, Gummel teraton wth relaxaton, AMG solver for sparse lnear systems of equatons, FFT for calculatng excess chemcal potental, and the specal treatment of ntegrals nvolvng delta functon. The valdty, accuracy, and computatonal complexty of the proposed numercal algorthms are demonstrated n Secton 4 va a real applcaton, LPON flm smulaton. Fnally, Secton 5 presents some concludng remarks. 2 PNP-cDFT model In ths secton, we offer a detaled descrpton of the PNP model coupled wth the cdft, or PNP-cDFT, and ts applcaton to L-on batteres smulaton.

4 4 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp PNP equatons provde a mean-feld contnuum model for the flows of charged partcles n terms of the average densty dstrbutons ρ and the electrostatc potental φ. The chemcal potentals of the charged partcles are evaluated by cdft, whch models dscrete on nteractons and accounts for partcle sze effects. In PNP-cDFT theory, the above two models are combned to descrbe the flow of nteractng on speces drven by the excess chemcal potentals µ ex. L ons dffuse n solds ether through hoppng between the ntersttal stes or vacancy mgraton mechansms. There are certan barrers for on and electron dffuson between equlbrum ntersttal stes, whch are calculated usng quantum mechancal approaches for the correspondng bulk materals [25]. The presence of barrers for L + /e dffuson can be represented by the attracton potental between L + ons or electrons and the correspondng equlbrum stes. The smplest form for such potental s a square-well potental wth the well depth equal to the barrer for L + /e hoppng between these stes. In partcular, we focus on the LPON model, one of the most wdely used sold-state electrolytes for thn flm batteres developed at Oak Rdge Natonal Laboratory. Fg. 1 llustrates a 3D model for the descrpton of L + transport n LPON. The I 0 stes are the equlbrum ntersttal stes for L + dffuson (Fg. 1). However, drect L + hoppng between I 0 stes s energetcally unfavorable due to a relatvely hgh energy barrer (0.21 ev) and the large dstance between these stes (0.41 nm), whch decrease the probablty of on hoppng. Accordng to quantum mechancal DFT smulatons, the most energetcally favorable path for L + dffuson s a zgzag path va the II 0 and II stes [25]. Therefore, n our model except for ste I 0 ntermedate stes II 0 and II also are ntroduced (Fg. 1). In our cdft calculatons, these lattce stes I 0,II 0,II are modeled by statonary partcles. Fgure 1: Dffuson channel of L + n LPON. Blue spheres are ntersttal equlbrums (I 0 ). Yellow spheres (II 0 ) and gray spheres (II ) are metastable stes. a, 2b, and 2c are the szes along three crystallographc drectons.

5 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp The 3D steady-state Posson-Nernst-Planck equatons PNP theory s a contnuum electrodffuson model that represents on fluxes n terms of densty dstrbuton of on speces and potental gradents. It has been wdely used n modelng on transport n bologcal on channels or nanocompostes wth broad applcatons n bology and materal scences [11, 12, 23, 26]. Wthn the PNP formalsm, the on flux J (for the ons of type,.e., L + or electrons) n the statonary condton can be calculated n terms of densty (or concentraton) gradent and potental gradent as follows: [ J = D (r) ρ + 1 ( k B T ρ q e φ+ µ d (r)+ µ ex (r)) ], (2.1) J = 0, (2.2) ( (ǫ(r) φ)=4π ρ f (r)+e q ρ ), (2.3) where r=(x,y,z) s the locaton at whch pont the functons are defned, D (r) s the dffuson coeffcent of the -th on speces, ρ (r) s the partcle number densty, k B s the Boltzmann s constant, T s the absolute temperature (k B T s thermal energy), e s the elementary charge, q s the valence of speces (wth sgn), φ(r) s the electrostatc potental, ǫ(r) s the electrc permttvty (or delectrc functon), µ ex (r) s the excess chemcal potental of charged partcles at poston r (determned wthn the cdft framework by (2.13)), µ d (r) s the deal chemcal potental defned n (2.10), and ρ f (r) s the fxed charge densty n the system. Eqs. (2.1), (2.2) may be wrtten n the followng form: [ ( D (r) ρ + 1 ( k B T ρ q e φ+ µ d (r)+ µ ex (r)) )] = 0. (2.4) Eq. (2.4) often s referred to as the drft-dffuson or electrodffuson equaton. By ntroducng the effectve denstes (also referred to as Slotboom varables n semconductor lterature) ρ = ρ e (q eφ+µ d +µ ex )/k B T, (2.5) the set of steady-state Nernst-Planck equatons (one for each speces) can be smplfed as: ( D ρ )=0, (2.6) where D = D e (q eφ+µ d +µ ex )/k B T. The Slotboom transformaton (2.5) removes the convecton term n Eq. (2.4) and results n self-adjont Laplace equatons whch can be solved effcently by multgrd method. However, possble large varaton of the transformed dffuson coeffcent D could result n large condton number of the stffness matrx [37]. Furthermore, the Slotboom

6 6 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp transformaton may cause overflow problem n ts numercal mplementaton due to the exponental term. Fortunately, ths overflow problem does not happen n the applcatons dscussed n ths paper. 2.2 Boundary condtons Usually, the PNP equatons are accompaned by Drchlet- and/or Neumann-type boundary condtons [43]. Let the computatonal doman be gven by Ω=(0,L x ) (0,L y ) (0,L z ). The external electrostatc potental φ s nfluenced by appled potental, whch can be modeled by prescrbng Drchlet boundary condton n y-drecton as: φ(r)=φ 0 (r), r Γ D Ω, (2.7) where Γ D ={(x,y,z) Ω y=0 or y= L y }. For the remanng part of the boundary Ω\Γ D (.e., n x and z drectons), a no-flux boundary condton s appled: ǫ(r) φ(r) n= 0, r Ω\Γ D. The same types of boundary condtons are mposed for varables ρ n the transformed Nernst-Planck equatons,.e., ρ(r)= ρ 0 (r), r Γ D, (2.8) and J (r) n=0, r Ω\Γ D. The exstence and unqueness of the soluton for the nonlnear PNP boundary value problems have been studed n [32, 44, 45] for the 1D case and n [43, 46] for multdmensons. 2.3 Classcal densty functonal theory cdft s an analytcal tool to evaluate the chemcal potentals of charged speces modeled by hard-sphere mxtures [24, 47, 48]. For a gven cdft, an analytcal expresson for the Helmholtz free energy F s formulated as a functonal of the set of partcle densty dstrbutons for all on speces {ρ (r)}. The Helmholtz free energy separates naturally nto two terms, the deal-gas term (F d ) that s obtaned from classcal statstcal mechancs F d [{ρ }]=k B T ρ (r) ( ln(ρ (r)λ 3 ) 1) dr, (2.9) Ω where Λ s the thermal wavelength of component, and the excess free energy (F ex ), whch has contrbutons from the nternal nteractons n the system. Accordngly, the

7 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp chemcal potental can be expressed as the sum of µ d and µ ex. In general, the excess chemcal potental s a non-local functonal of the on denstes. The deal chemcal potental s expressed as [49, 50]: [ µ d (r)= ln γ ρ (r)/ρ bulk ], (2.10) where the actvty coeffcent γ s descrbed by the extended Debye-Hückel theory [51, 52]: I lnγ = Aq 2 1+Ba I. (2.11) In the precedng formula, I= 1 2 ρ q 2 s the onc strength, A= (εt) 3/2 (ε s the delectrc constant), and B=50.3(εT) 1/2. For a system of charged hard spheres, the excess free energy usually ncludes contrbutons from the free energes of Coulomb nteractons FC ex, electrostatc correlatons Fex el, and hard-sphere repulson Fhs ex [53,54]. In [28], an addtonal term correspondng to shortrange attracton nteractons Fsh ex was ncluded, resultng n the followng decomposton: F ex = F ex C +Fex el +Fex hs +Fex sh. (2.12) The excess chemcal potental can be calculated from the functonal dervatve of the excess free energy wth respect to partcle densty: µ ex (r)= δfex [{ρ k }] δρ where ρ=[ρ 1,,ρ s ], and s s the number of on types Hard-sphere component = δfex (ρ(r)), (2.13) δρ (r) The hard-sphere model s often used n statstcal mechancs to represent the short-range repulson between two partcles, known as the excluded-volume effect. The fundamental measure theory (FMT) [47] and modfed fundamental measure theory (MFMT) [55, 56] are among the most accurate formulatons for the descrpton of the structure and thermodynamc propertes of nhomogeneous hard-sphere fluds. In ths model, the excess Helmholtz free energy functonal due to the hard-core repulson Fhs ex can be expressed as an ntegral of the functonal of weghted denstes,.e., Fhs ex [{ρ }]=k B T Φ hs (r)dr, (2.14) Ω where Φ hs s a functon of weghted denstes n α and n β gven by FMT: n 3 2 Φ hs (r)= n 0 ln(1 n 3 )+ n 1n 2 1 n n 3 24π(1 n 3 ) 2 n n 2n 2 n 2 1 n 3 8π(1 n 3 ) 2 (2.15)

8 8 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp or MFMT: Φ hs (r)= n 0 ln(1 n 3 )+ n [ 1n n 3 36πn 2 ln(1 n 3 )+ 3 36πn 3 (1 n 3 ) 2 n [ 1 n n 3 12πn 2 ln(1 n 3 )+ 12πn 3 3 (1 n 3 ) 2 ] n 3 2 ] n 2 (n 2 n 2 ). (2.16) n α and n β are the weghted average of the densty dstrbuton functons ρ (r) and are defned by: n α (r)= n β (r)= Ω Ω ρ (r )ω (α) (r r)dr, α=0,1,2,3, ρ (r )ω (β) (r r)dr, β=1,2, where the weght functons ω (α) and ω (β) characterzng the geometry of partcles (hard sphere wth radus R for on speces ) are gven by: ω (3) (r)=θ( r R ), (2.17) ω (2) (r)= θ( r R ) =δ( r R ), (2.18) ω (2) (r)= θ( r R )= r r δ( r R ), (2.19) ω (0) (r)=ω (2) (r)/(4πr 2 ), (2.20) ω (1) (r)=ω (2) (r)/(4πr ), (2.21) ω (1) (r)=ω (2) (r)/(4πr ). (2.22) In the precedng formulae, θ s the Heavsde step functon wth θ(x)=0 for x>0 and θ(x) = 1 for x 0, and δ denotes the Drac delta functon. From Eq. (2.13) and Eq. (2.14), t follows that the hard-sphere chemcal potental s gven by: µ hs (r)=k B T ( α Ω ) ( Φ hs (r )ω (α) n (r r )dr +k B T α β Ω ) Φ hs (r )ω (β) n (r r )dr. β By takng partal dervatves of Eq. (2.15) wth respect to n 0, n 1, n 2, n 3, n 1,x, n 1,y, n 1,z, n 2,x, n 2,y, or n 2,z, we can get Φ hs n 0 = ln(1 n 3 ), Φ hs n 2 = n 1 1 n 3 + n2 2 n 2 n 2 8π(1 n 3 ) 2, Φ hs n 1 = n 2 1 n 3,

9 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp Φ hs = n 0 + n 1n 2 n 1 n 2 n 3 1 n 3 (1 n 3 ) 2 + n3 2 3n 2n 2 n 2 12π(1 n 3 ) 3, Φ hs = n 2,x n 1,x n 3 1, Φ hs = n 2,y n 1,y n 3 1, Φ hs = n 2,z n 1,z n 3 1, Φ hs Short-range nteractons = n 1,x n 2,x n 3 1 n 2n 2,x 4π(1 n 3 ) 2, Φ hs = n 1,y n 2,y n 3 1 n 2n 2,y 4π(1 n 3 ) 2, Φ hs n 2,z = n 1,z n 3 1 n 2n 2,z 4π(1 n 3 ) 2. In addton to the hard-core repulson, the short-range attractve partcle-partcle nteractons may be modeled by the followng square-well potental:, 0 r<σ αβ, Φ αβ (r)= ǫ αβ, σ αβ r γσ αβ, 0, r>γσ αβ, where r s the dstance between the centers of the sphercal partcles, σ αβ =(σ α +σ β )/2 (σ α s the partcle hard-core dameter, and r<σ αβ characterzes a hard-core repulson), γσ αβ s the square-well wdth,(γ 1)σ αβ ndcates the range of attracton, ǫ αβ s the well depth (postve value represents attractve nteracton, whle negatve value corresponds to repulsve nteracton), and the attractve wdth γ = 1.2 as n Ref. [24]. The correspondng mean-feld approxmaton of the free energy s gven by: Fsh ex = 1 drdr 2 ρ α (r)ρ β (r )Φ αβ ( r r ), (2.23) Ω Ω α,β=+,s where s denotes the statonary ponts correspondng to the lattce stes I 0,II 0,II shown n Fg. 1. It follows from (2.23) that the short-range chemcal potental s gven by: µ sh α (r)= 1 2 ρ β (r )Φ αβ ( r r )dr. Ω β Short-range nteractons between moble speces (L + ons and electrons) are not present n the current study [26, 27]. The densty profles of the statonary ponts ρ s are gven based on the structure of electrode materals. The depth of the potental well s reasonably equal to the barrer heght for L + hoppng between two adjacent statonary ponts and was set accordng to the quantum mechancal and molecular dynamcs data for the barrer heghts [25].

10 10 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp The densty dstrbutons of the statonary ponts are reasonably represented by the sum of normalzed Gaussan ansatz placed at each correspondng lattce ste R k : ( α ) 3/2 ρ s (r)= π k e α r R k Coulomb nteractons The free energy of long-range Coulomb nteractons s gven by: FC ex= k BTl B 2,j Ω Ω q q j ρ (r)ρ j (r ) r r drdr, (2.24) where q, q j are the valences of the charged speces and the Bjerrum length s defned as l B = e 2 /(4πε 0 εk B T). ε 0 s the vacuum permttvty, ε s the relatve delectrc constant of the meda, and the sum s over all on speces,j. From (2.24), we can derve the correspondng Coulomb chemcal potental: Electrostatc correlatons µ C (r)=q q j ρ j (r k B Tl B ) j Ω r r dr. In most cdft methods, the excess Helmholtz energy due to the electrostatc correlatons Fel ex s gven by an analytcal expresson based on the perturbaton of a sutably chosen, poston-dependent reference bulk flud [57, 58]. Often, t s descrbed by a second-order functonal Taylor expanson n terms of powers of the densty fluctuatons ρ (r)=ρ (r) ρ bulk (r) around a reference system wth gven bulk densty profles{ρ bulk (r)} [48]: Fel ex[{ρ }]=Fel ex[{ρbulk (1), el }] k B T C (r) ρ (r)dr k BT 2,j=+, Ω =+, Ω Ω (2), el C j ( r r ) ρ (r) ρ j (r )drdr +O(( ρ(r)) 3 ), (2.25) where the frst- and second-order electrostatc drect correlaton functons are defned as: (1), el C (r)= 1 δfel ex k B T δρ (r) ρ=ρ bulk= 1 k B T µel [{ρ bulk j (r)}] (2.26) and (2), el C j ( r r ) := 1 δ 2 Fel ex k B T δρ (r)δρ j (r ), (2.27) where µ el s the chemcal potental of the moble ons.

11 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp Accordng to the mean sphercal approxmaton (MSA), the second-order drect correlaton functon C (2), el j [59 61]: C (2), el j (r) { [ q q j e 2 2B k B Tε σ ( ) B 2r ] 1 σ r, r σ, 0, r>σ, (2.28) where r= r r s the dstance between two ons, σ s the dameter, σ=(σ +σ j )/2 s the hard-core nteracton dstance between charged partcles and j, and B s gven by: B=[ξ+1 (1+2ξ) 1/2 ]/ξ, (2.29) where ξ 2 = κ 2 σ 2 = [ e 2 ε 0 εk B T ρ bulk q 2 ] σ 2 and κ denotes the nverse Debye screenng length. The electrostatc correlaton component of the chemcal potental s then gven by: µ el (r)=µel [{ρbulk k 3 Numercal methods (r)}] k B T j Ω C (2), el j ( r r )(ρ j (r ) ρ bulk j )dr. (2.30) The coupled PNP equatons are dscretzed by fnte dfference method and solved teratvely usng a decouplng method, Gummel teraton [62]. The AMG method s appled to solve the Posson equaton (2.3) and the transformed Nernst-Planck equaton (2.6) effcently. The excess chemcal potental of charged partcles are determned by cdft calculaton usng FFTs. Fg. 2 depcts the flow chart of our numercal smulaton. We note that the convergence analyss of the numercal method for solvng PNP-cDFT system s very dffcult and there s no theoretcal result to the best of the authors knowledge. In our numercal convergence test, we are only able to observe that the numercal solutons converge when decreasng the mesh sze. 3.1 Fnte dfference dscretzaton The 3D PNP equatons are dscretzed usng the standard 7-pont fnte dfference scheme on a unform cubc lattce grd. For example, consder the transformed Nernst-Planck equaton (2.6) (for smplcty, we drop the speces ndex ). The value of ρ at a gven grd pont (correspondng to local ndex 0) and ts sx neghbors (correspondng to the ndex j=1,,6) satsfy the followng equaton: ( D ρ D 2 ( x) 2 + D 3 + D 4 ( y) 2 + D ) 5 + D 6 ( z) 2 = ρ1 D 1 + ρ 2 D 2 ( x) 2 + ρ3 D 3 + ρ 4 D 4 ( y) 2 + ρ5 D 5 + ρ 6 D 6 ( z) 2.

12 12 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp Fgure 2: PNP-cDFT smulaton flow chart. The dffuson coeffcents D j (j=1,,6) are computed va the harmonc mean of the correspondng values at the grd ponts [12],.e., D j = 2 D 0 D j D 0 + D j, where D j s D evaluated at the grd pont wth local ndex j. 3.2 Gummel teraton wth relaxaton To obtan the self-consstent soluton of the PNP equatons, the coupled equatons are solved usng ether the Gummel method wth relaxatons [7] or Newton method [11, 34]. Gummel teraton s a decouplng method for solvng coupled systems of equatons. Gven an ntal guess of the on concentraton profles ρ and the electrostatc potental φ, a new φ s computed by solvng the Posson equaton. Then, the updated φ s substtuted nto the Nernst-Planck equaton to update the on concentraton. Ths teratve process termnates when the dfference between the results of two subsequent teratons s less than a predefned threshold value (10 6 for potental and 10 5 for on concentratons n our tests).

13 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp The convergence rate of the Gummel teraton s usually slow. To speed up the convergence, successve under- or over-relaxaton s employed for the soluton updates [7, 63]. Namely, n the n-th PNP teraton, the potental φ (n) s mxed wth the prevous potental φ (n 1) through a relaxaton parameter λ 1 before beng substtuted nto the Nernst-Planck equatons, φ (n) λ 1 φ (n) +(1 λ 1 )φ (n 1). Smlarly, for on concentratons, we ntroduce another relaxaton parameter λ 2, ρ (n) λ 2 ρ (n) +(1 λ 2 )ρ (n 1). The relaxaton parameters λ 1 and λ 2 are selected to acheve rapd convergence whle mantanng numercal stablty. Because the Nernst-Planck equaton s more senstve to potental change than the Posson equaton to concentraton changes, we choose λ 1 = 0.2 and λ 2 = 1.0 n our numercal smulatons. To solve the coupled nonlnear PNP system, the Newton method, whch converges quadratcally when a good ntal guess s avalable, has been employed n [11, 34]. It requres the constructon of a Jacoban matrx that, n practce, may be complcated. In contrast, the Gummel method has a fast ntal error reducton, but the convergence rate may be slow. In [34], the PNP system s solved wth the Newton method, and the resultng lnear systems are solved usng the generalzed mnmal resdual (GMRES) method wth multgrd precondtonng. A better approach may be to combne both methods,.e., start the soluton procedure wth a few Gummel teratons to generate a good ntal guess then swtch to the Newton method to accelerate the convergence The choce of ntal guess Even when usng Gummel method, a good ntal guess enhances the stablty of soluton and speeds up convergence. In our approach, we frst solve the statc equlbrum problem and determne the dstrbuton of moble speces n external electrc potental by mnmzng the total free energy wth cdft. These equlbrum densty dstrbutons and the correspondng electrostatc potental are subsequently used as ntal guesses for the Nernst-Planck and Posson equatons, respectvely. Ths way, we avod large changes n denstes of moble speces when solvng the PNP equatons by startng from the system s equlbrated state before mposng constant flow condtons. Ths two-step approach proved to be computatonally effcent wth typcal convergence of the PNP soluton n a few tens of teratons. 3.3 Algebrac multgrd method In several early studes of PNP models [7, 63], classcal Jacob or Gauss-Sedel teratve methods are used to solve the Posson and Nernst-Planck equatons. These methods suffer from slow convergence, makng t dffcult to run large-scale 3D smulatons. Because PNP equatons are all of ellptc type, many exstng fast solvers can be appled to solve

14 14 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp them effcently on massvely parallel computers. In ths work, we use the state-of-the-art fast teratve solver AMG method. The multgrd (MG) method s well know for tso(n) optmalty (N s the number of degrees of freedom) n solvng large sparse lnear systems resultng from dscretzatons of partal dfferental equatons. MG s motvaton may be descrbed by decomposng the soluton error nto the sum of hgh- and low-frequency components. It conssts of a smoothng procedure (damped Jacob, Gauss-Sedel, etc., also called smoothers ), whch reduces hgh-frequency error components, and a coarse-grd correcton operator for lowfrequency errors. Grd transfer operators (restrcton and nterpolaton) are also defned to connect solutons at dfferent grd levels. The success of MG methods reles on the combnaton of soluton procedures at dfferent scales, where dfferent error components are reduced by usng the smoothng property of basc teratve methods. For a thorough dscusson about the MG method, refer to the books [64 66]. Frst ntroduced n the 1980s [67, 68], the AMG method constructs coarse-grd correcton operator and restrcton/nterpolaton operator from a matrx wthout usng grd nformaton. AMG has been qute successful for solvng large sparse lnear systems, especally those correspondng to dscretzed ellptc problems. In our numercal smulaton, we use BoomerAMG, a parallel mplementaton of AMG from the hypre (Hgh Performance Precondtoners) lbrary developed at Lawrence Lvermore Natonal Laboratory [40, 41]. 3.4 Fast Fourer transform The evaluaton of a large number of 3D ntegrals s another computatonally demandng task n cdft computaton (descrbed n Secton 2.3). Drect evaluaton of these ntegrals usng numercal quadrature rules s computatonally ntractable [43]. Fortunately, these ntegrals are descrbed n convoluton form, whch can be calculated effectvely usng FFT and nverse FFT. In our study, we use P3DFFT, a parallel FFT lbrary optmzed for largescale computer smulatons, developed at the Unversty of Calforna, San Dego [42]. More precsely, the 3D ntegrals nvolved n the cdft model can be wrtten n the followng convoluton form: ρ(r )g( r r )dr, (3.1) Ω where g(s)=1/s for the Coulomb chemcal potental, g(s)=φ j (s) for the chemcal potental correspondng to the short-range nteractons, and g(s)=c j (s) for the electrostatc correlaton chemcal potental. In the followng, we llustrate how to calculate (3.1) usng FFT and ts nverse on a 1D example. Gven a unform grd on the nterval [0,T], t = t, =0,1,,N 1, where t = T/N. Consder the convoluton ntegral: (2), el T h(t)= f(τ)g( t τ )dτ (3.2) 0

15 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp approxmated usng the trapezodal rule,.e., N 1 h k = =0 k =0 f g k x f g k + N 1 =k+1 f g k, k=0,1,,n 1, (3.3) where h k = h(t k ), f = f(t ), and g k = g(t k ). Note that for ease of notaton, n (3.3) we have assumed the ntegrand has equal values at the two endponts. To evaluate Eq. (3.3) by applyng FFT, we need to wrte the sums n the convoluton form. To ths end, we ntroduce the followng auxlary vectors: f, f 0 N 1, ˆf ={ 0, f N 2N 1, { g, f 0 N 1, ĝ = 0, f N 2N 1, { gn 2, f 0 N 2, g = 0, f N 1 2N 1, (3.4) (3.5) (3.6) and defne two crcular convoluton ĥ and h (modulo 2N) as By drect calculatons, we get: Hence, ĥ k = h m = k =0 N 1 =k+1 h 2N 2 = 0. f g k, ĥ k :=( ˆf ĝ) 2N 1 k = ˆf ĝ k, =0 h m :=( ˆf 2N 1 g) m = ˆf g m, =0 k=0,1,,n 1, k, m. f g k, k=m (N 1), m= N 1,,2N 3, h k = ĥk+ h k+n 1, k=0,,n 1, where ĥk and h m can be calculated effcently by FFT and nverse FFT [69] wtho(nlogn) computatonal complexty. The generalzaton of the precedng approach for evaluatng 3D convoluton ntegrals (3.1) s straghtforward. As such, the detals are omtted here.

16 16 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp Integrals nvolvng Drac delta functon When calculatng hard-sphere chemcal potentals, t can be seen from Eqs. (2.18), (2.19), (2.20), (2.21) and (2.22) that several 3D ntegrals nvolvng Drac delta functons need to be evaluated. A straghtforward approach s to approxmate the Drac delta functon by the Gaussan functon: δ(r) δ α (r)= 1 α π e r2 /α 2 (3.7) and then apply any numercal quadrature rule. However, a more effcent and accurate approach s to reduce the 3D volume ntegrals nto 2D sphercal ntegrals usng the defnton of Drac delta functon and change of varables. More precsely, we have: I(r)= = ρ(r )δ( r r R)dr Ω 2π π 0 k k 0 =1j=1 ρ(x+rcosθsn ϕ,y+rsnθsnϕ,z+rcos ϕ)r 2 snϕdϕdθ [ρ(x+rcosθ snϕ j,y+rsnθ snϕ j,z+rcos ϕ j )R 2 snϕ j ]w w j, where {(θ,ϕ j )} are the quadrature ponts and {w, w j } are the correspondng weghts. Zero extenson of the ntegrand s used when part of the sphere les out of the computatonal doman Ω. 4 Numercal results We use the L-on and electron conductng sold electrolyte, LPON, to demonstrate our PNP-cDFT solver s performance. 4.1 Computatonal doman and physcal parameters The computatonal doman s gven by 2a Mb 5c, where a=1.053 nm, b=0.612 nm, and c=0.493 nm are the lattce parameters for LPON and M=10,,200. L + ons are represented as sphercal partcles wth charge q + = 1 and dameter σ + = 0.06 nm. Smlar representaton s used for electrons, dffusng along wth L +, wth the parameters q = 1 and dameter σ = nm. The expermental value for the LPON delectrc constant, 16.6, has been used [70]. The parameters used n our PNP-cDFT smulaton for L + /electron transport are lsted n Table Results and dscusson Ion conductvty obtaned from numercal smulaton s on the order of 10 7 S/cm n y drecton at temperature 298 K. Ths s close to the conductvty of S/cm observed

17 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp Table 1: Parameters for computaton. Parameter Symbol Value Unt Dffuson coeffcent (L + ) D cm 2 /s Dffuson coeffcent (electron) D cm 2 /s Valence (L + ) q + +1 Valence (electron) q 1 Delectrc constant of meda ǫ 16.6 Fxed charge densty (at z=0) ρ f 0.1 1/nm 2 Bulk densty of salt ρ bulk /nm 3 Thermodynamc beta (T=223K) β= k 1 B T /eV LPON nanopartcle sze n a-drecton L x nm LPON nanopartcle sze n b-drecton L y nm LPON nanopartcle sze n c-drecton L z nm Sphere dameter (L + ) σ nm Sphere dameter (electron) σ nm Sphere dameter (statonary ponts) σ s 0.2 nm Square-well potental depth (L +, s 1 ) ǫ +,s ev Square-well potental depth (L +, s 2 ) ǫ +,s ev Square-well potental depth (L +, s 3 ) ǫ +,s ev Bjerrum length (T=223K) l B = e2 4πǫ 0 ǫk B T nm expermentally n [71] for L 0.99 PO 2.55 N 0.30 glass. Moreover, based on our smulaton, conductvty ncreases from to when temperature ncreases from 200 K to 320 K (Fg. 3). Ths temperature-dependence trend for conductvty agrees well wth expermental data [72]. It s good to note that reproducng the exact expermental value for conductvty s hghly mprobable as conductvty strongly depends on the flm mcrostructure (shown n [72]). The smulaton doman we use s fnte wth the sze n the man conductvty drecton of nm, whch corresponds to polycrystallne LPON flms, and the smulated conductvtes are well wthn the lmts of varatons n expermental values for polycrystallne samples. Apart from flm mcrostructure, the value for the elementary dffuson coeffcent, a pre-factor to the conductvty, s another source of uncertanly. In our smulatons, we use the value of 10 6 cm 2 /s measured for rutle ttanum doxde, or TO 2, a materal wth an ntersttal L + dffuson mechansm and smlar barrers for elementary dffuson processes [73, 74]. Therefore, we expect the rutle value to be a good estmate for the L + dffuson coeffcent n LPON. In our prevous work, we have shown that the mechansm of coupled on and electron transport changes from strongly coupled onc and electronc to predomnantly onc as the partcle sze becomes larger than the Debye length. These dfferences reflect the

18 18 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp Ionc conductvty(s/cm) 1.48 x Temperature(K) Fgure 3: L + conductvty n LPON nanopartcles along the y drecton wth varyng temperature. The partcle sze n y drecton s 150b (91.8 nm). 1.5 x nm 61.2 nm Free Energy(F sh, ev) Temperature(K) Fgure 4: Temperature dependence of the L + short-range free energy n LPON nanopartcles. The partcle sze n b drecton s 100b (61.2 nm) and 150b (91.8 nm). changes n the mechansm for the compensaton of the external electrc feld, whch s acheved through hghly correlated on and electron flux n small nanopartcles and by the formaton of the space-charge zone at the surface of large nanopartcles. These dfferent mechansms for conductvty also lead to dfferent temperature dependences. For large nanopartcles consdered here, conductvty ncreases wth temperature due to two reasons: ) thermal moton leads to partal destablzaton of the space-charge layer, supplyng more ons and electrons to the flux through the nanopartcle, and ) lowerng of the effectve barrer for elementary on transport between adjacent statonary ponts. The second effect s manfested n the monotonc decrease n the calculated short-range energy F ex sh, whch domnates the total excess free energy Fex (Fg. 4). Ths effect s weaker

19 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp x 10 5 ρz (1/nm3) y x 2.5 Fgure 5: L+ on densty n xy plane (z = nm). 15 x 10 ρy ρ0 (1/nm3) z x Fgure 6: L+ on densty n xz plane (ρ0 = , y = nm). for smaller nanopartcles due to the change n relatve contrbutons of short-range and electrostatc correlaton free energes. The densty dstrbuton n the conducton plane (plane parallel to the y drecton) reveals the formaton of the space-charge layer at the boundares of the nanopartcle (Fg. 5). In contrast, there s almost no varaton n on and electron denstes n the plane normal to the conducton plane wth the varatons n the densty on the order of nm 3 (Fg. 6) Sze dependency of the L+ on conductvty The sze effects on L+ conductvty at temperature T = 300 K s shown n Fg. 7. The observed monotonc ncrease n conductvty s due to a combnaton of several competng effects. On one hand, the ncrease n nanopartcle sze leads to the decrease n the effec-

20 20 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp x Conductvty(S/cm) Partcle sze n y drecton ( b = 0.612nm) Fgure 7: Sze effect of the conductvty at 300 K. tve gradent of electrc potental or local electrc feld n b drecton, resultng n stronger correlaton of on and electron fluxes. On the other hand, t reduces the drvng force actng on ons and electrons, reducng ther dffusvtes through the nanopartcle. Overall, the sze dependence of the conductvty can be expressed as [75]: σ c = 1 Ly J E y L b (y)dy, y 0 where σ c s the conductvty, E y s the local electrc feld, L y s the nanopartcle s sze along the y drecton, and J b s the flux along the y drecton. Competton from the prevously descrbed effects leads to an almost lnear ncrease n conductvty wth nanopartcle szes rangng from 10 to 100 nm, as observed n our smulatons, to very weak dependence of the conductvty on partcle szes rangng from 100 to 1000 nm. Overall, our smulatons demonstrate excellent agreement wth expermental data and analytc theores, valdatng our approach [71, 72] Computatonal complexty We verfy the computatonal complexty of the methods dscussed n Secton 3 va numercal experments. In Fg. 8, we plot the graph of the soluton tme (n seconds), as well as the tme for AMG and FFT components versus the number of grd ponts N. Here, we consder a nanopartcle sze 2a 10b 5c and run smulatons on a sequence of unformly refned meshes usng a sngle processor. The number of PNP teratons s stable wth respect to the mesh sze. Hence, the CPU tme s a good measurement for the computatonal complexty of FFT, whch so(nlogn) (see Fg. 8 where the dashed lne represents O(Nlog N)). However, we do not observe O(N) multgrd complexty. Ths nonunform convergence of the multgrd s possbly due to the PNP-cDFT ntegral-dfferental system s hgh nonlnearty.

21 D. Meng et al. / Commun. Comput. Phys., x (20xx), pp T: CPU Tme N: number of grd ponts 10 log 2 T (s) Total tme MG tme FFT tme log 2 N Fgure 8: Computatonal complexty (log scale). 5 Conclusons As part of ths effort, we have examned numercal methods for smulatng a PNP-cDFT model usng state-of-the-art AMG and FFT packages. To evaluate chemcal potentals effcently, a novel treatment of 3D ntegrals va FFT and ntegrals nvolvng delta functon s proposed. The computatonal complexty of our smulaton s O(Nlog N), whch makes large-scale 3D PNP-cDFT smulaton feasble. Numercal results are valdated through comparson wth expermental data and results from prevous studes. In future work, we wll apply the proposed methods to smulate tme-dependent PNP-cDFT systems. In addton, parallel scalablty of these methods wll be reported n a forthcomng paper. Acknowledgments Work by MLS and DM was supported by the Materals Synthess and Smulaton across Scales (MS3) Intatve (Laboratory Drected Research and Development (LDRD) Program) at Pacfc Northwest Natonal Laboratory (PNNL). Work by GL was supported by the U.S. Department of Energy (DOE) Offce of Scence s Advanced Scentfc Computng Research Appled Mathematcs program and work by BZ by Early Career Award Intatve (LDRD Program) at PNNL. PNNL s operated by Battelle for the DOE under Contract DE-AC05-76RL The research was performed usng PNNL Insttutonal Computng, as well as the Natonal Energy Research Scentfc Computng Center at Lawrence Berkeley Natonal Laboratory.

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