Molecular Dynamics Simulation of the Shear Viscosity of Molten Alkali Halides

Transcription

1 3658 J. Phys. Chem. B 2004, 108, Molecular Dyamcs Smulato of the Shear Vscosty of Molte Alkal Haldes uo Galamba, Carlos A. eto de Castro, ad James F. Ely*, Departameto de Qumca e Boqumca e Cetro de Cecas Moleculares e Materas, Faculdade de Cecas da UVersdade de Lsboa, Lsboa, Portugal, ad Chemcal Egeerg Departmet, Colorado School of Mes, Golde, Colorado ReceVed: July 30, 2003; I Fal Form: December 13, 2003 Equlbrum molecular dyamcs (EMD) Gree-Kubo smulatos of the shear vscosty coeffcet of molte sodum chlorde ad potassum chlorde are reported the mcrocaocal esemble (,V,E). The potetal model used s the Bor-Mayer-Huggs-Tos-Fum teroc potetal, ad several state pots have bee vestgated. The form of the stress tesor elemets for oc fluds whose electrostatc teractos are calculated through the Ewald method s dscussed detal. The results show that the agreemet betwee the smulato ad the avalable expermetal data s satsfactory, ad the vscosty s overpredcted (geerally to wth 10%-15%). * Author to whom correspodece should be addressed. E-mal: jely@ mes.edu. Uversty of Lsbo. Colorado School of Mes. 1. Itroducto Molte salts have bee studed for may years; however, reewed terest them has creased, because of ther use as electrolytes hgh-temperature carboate fuel cells. Uder those hgh-temperature codtos, t s extremely dffcult, f ot mpossble, to accurately measure trasport propertes (such as the vscosty ad thermal coductvty) ad data are very lmted. A premse of ths work s that, because of the expermetal problems ecoutered at hgh temperature, t may be possble to smulate those propertes more accurately tha they ca be measured. Thus, ths vestgato, we have explored the accuracy wth whch the shear vscosty of smple oc systems ca be computed through molecular dyamcs smulatos, wth the ultmate goal to exted these smulatos to more complex ad/or dustrally relevat systems. I ths work, we have chose the alkal haldes as the subject of our vestgato. Molte alkal haldes are amog the smplest oc fluds; they are composed oly of uvalet moatomc os, a fact whch has log motvated the applcato of smulato methods to vestgate ther propertes. We ote that, eve though smple molte salts have bee the topc of computer smulato for may years, very few vestgatos of the trasport propertes of these materals have bee reported the lterature. oteworthy (but ot clusve) examples of these prevous smulatos clude both equlbrum molecular dyamcs (EMD) 1-3 ad Mote Carlo 4,5 studes. Geerally, these studes have oly reported the thermodyamc state, the mcroscopc structure, ad the self-dffuso coeffcet. Of the dfferet teroc potetals that have bee proposed for the smulato of molte alkal haldes, the Bor-Mayer- Huggs model, 6-9 wth parameters calculated by Tos ad Fum 10,11 (BMHTF), has bee preferred most of the studes reported the lterature. Eve though ths model s based o the rgd o approxmato (thus eglectg the o-duced dpole (polarzato) type of teractos) ad uses parameters obtaed from sold-state data oly, t reproduces the thermodyamc ad structural propertes of molte alkal haldes reasoably well. For the case of the self-dffuso coeffcets, the results are satsfactory, but the small set of expermetal self-dffuso data avalable s a lmtato makg coclusos about the accuracy of the model. Despte possble mprovemets of the BMHTF teroc potetal through the cluso of polarzato effects (e.g. shell model teroc potetals), the model s capablty to reproduce the thermal coductvty or the shear vscosty coeffcets s relatvely uexplored. Ths artcle dscusses the computato of the shear vscosty of molte acl ad KCl, usg EMD smulatos cojucto wth the Gree-Kubo method for dfferet temperatures; the volume at each temperature s calculated from expermetal desty data. 15 Smulated molte alkal halde shear vscostes have bee reported prevously, but oly at a sgle state pot. The temperature ad/or desty depedece has ever bee the prmary focus of those studes, or has the mcrocaocal esemble bee used. I partcular, Cccott et al. 16 calculated the shear vscosty of sx molte alkal haldes (LF, acl, ai, KI, RbCl, ad RbI), usg a oequlbrum molecular dyamcs (EMD) method. The reported thermodyamc states correspoded to a sgle temperature ad desty close to the trple pots of each of the salts studed. The corporato of a polarzato cotrbuto to the teroc forces was vestgated by Jacucc et al. 14 for the vscosty smulato of oe molte salt (KI). The polarzato model led to a decrease the shear vscosty ad a crease the dfferece betwee the smulatos ad avalable expermetal data. A EMD-VT study of the vscosty of molte acl ad KCl was performed by Sdzgre ad Glla. 17 However, the prmary objectve of that study was the thermal coductvty of sold acl ad KCl, ad, aga, oly oe pot s reported for the vscosty of molte acl ad KCl. More recetly, EMD smulatos of the shear vscosty of acl were reported by Delhommelle ad Petravc, 18 although the prmary focus of that work was o a comparso of the performace of ketc ad cofguratoal temperature thermostats used EMD smulatos. The state pot reported that artcle has bee smulated ths work ad s gve for comparso purposes. The objectve of ths work s to vestgate /jp036234x CCC: $ Amerca Chemcal Socety Publshed o Web 02/21/2004

2 Shear Vscosty of Molte Alkal Haldes J. Phys. Chem. B, Vol. 108, o. 11, the capablty of the BMHTF potetal model to estmate trasport coeffcets that are more complex tha the selfdffuso coeffcet at state pots for whch comparso wth avalable expermetal data s possble. The paper s orgazed as follows: Frst, a descrpto of the model ad the parameters used the smulatos s gve. The Gree-Kubo method s dscussed, ad the form of the stress-tesor elemets for the case of oc systems for whch the Ewald sum s used for hadlg the log-rage teractos s cosdered detal. The reaso for ths dscusso s that, applyg the Ewald method to the evaluato of the stress tesor, there are two possble terpretatos of how the recprocalspace force should be used the calculato. The correct terpretato s ot dscussed stadard smulato refereces, ad we foud t worthwhle to compare the two terpretatos. The results obtaed ths work are the gve ad compared wth avalable expermetal data ad results of prevous smulatos. Fally, the coclusos of ths study are gve. 2. The Gree-Kubo Vscosty for Ioc Systems The thermal trasport coeffcets ca be calculated from the Gree-Kubo formulas through tegrato of the approprate autocorrelato fuctos (ACFs), whch, tur, ca be computed from EMD smulatos Oe of the problems calculatg the vscosty through ths approach s the fact that, ulke self-dffuso, whch s a sgle-partcle pheomeo, the shear stress volves the etre system. Thus, t s ot possble to mprove the statstcal precso of the calculato by averagg the results over the partcles that compose the system. Ths, addto to the possble system sze depedece of the smulatos, leads to rather log rus. Improvemet of the statstcal precso ca be acheved by averagg over the three dfferet off-dagoal elemets of the stress tesor J2. For potetal models for whch the force has the form F j ) F j r j / r j, as t s for the case of the BMHTF model, J ab ) J ba, whch lmts the average to three elemets oly. 22 Further mprovemet of the statstcs ca be obtaed usg the method of Davs ad Evas, 23 whch takes advatage of full symmetry propertes of the stress-tesor elemets. Ths method allows the cluso of the dagoal elemets of the stress tesor o the calculato of the averaged stress autocorrelato fucto (ACF). However, ths study, we have ot used ths method, ad the stress ACFs were obtaed from the three dstct off-dagoal elemets. Thus, the shear vscosty ca be wrtte as η ) 1 0 J xy (t)j xy (t) dt (1) 3VkT xy where the stress-tesor elemet J xy, wth aalogous expressos for the other compoets, s gve by 24 J xy ) -1 m V x V y + j> r jy F jx (2) For ooc fluds, eqs 1 ad 2 are used drectly to obta the shear vscosty of the flud of terest by calculatg ad tegratg the averaged stress correlato fucto. For a oc flud, these equatos ca strctly oly be used the computato of the short-rage porto of the forces ad for the real compoet of the Ewald-Coulomb forces that are eeded to compute the potetal compoet of the stress-tesor elemets. otce that, for ths porto of the force, the dstace betwee pars of partcles s calculated usg the mmum mage coveto ad radal trucato; thus, for stace, the force F j correspods to the force betwee the partcle ad the partcle j or ts closest mage wth a sphere of rad L/2 cetered at. For the case of the recprocal-space force, the correspodg force betwee a par of os j s gve by 2Z F F jx ) h x Z j ( exp - π 2 h ) ( 2 s 2π L 2 h*0 h 2 R 2 L 2 L j) h r (3) where r j s the dstace betwee the os ad j wth the cetral box. However, the force gve by eq 3 s ot just the force betwee the os ad j; rather, t s the force betwee ad j ad all of ts mages the fte recprocal-space lattce (whch, practce, s trucated usg the codto 0 < h 2 e 2 h max for a cubc box). For a system wth log-rage forces, addtoal cosderato must be gve to teractos betwee partcles more tha oe-half box legth away. Thus eq 2, whch, for a system wth perodc boudary codtos (pbc), s wrtte as J xy ) m V x V y j (r jy - y L)F jx (r - L) (4) ca be trucated for values of ) 0, (1 for the case of shortrage forces, but more values of should be take to cosderato for log-rage forces, whch are computed the recprocal space f the Ewald method s used. I eq 4, the prme symbol dcates the excluso of the teracto of a partcle wth tself, * j for ) 0. Equato 4 shows that, f the drect product F F ja r jb s calculated usg the dstace betwee the os the cetral box, ths wll ot correspod to the dstaces betwee the pars of os cosdered wth the recprocal-space summato for the computato of the forces. Ths problem was frst ecoutered by Vellefosse ad Hase 25 ad Beru ad Vellefosse, 26 who have show that, although the Gree-Kubo relatos for the trasport propertes of ooc fluds could be appled to the case of the oe-compoet plasma model, some care had to be take defg the mcroscopc currets, to avod dvergeces. The problem ca be hadled f the Ewald method s appled to the product F F ja r jb, rather tha applyg the method to just the force ad the computg ts product wth the dstace betwee the os wth the cetral MD box. 27,28 Thus, for the case of the BMHTF model for a molte alkal halde, the stresstesor elemets ca be calculated by J ab ) where -1 m V a V b + j> 1 2πLh*0 r j,a F R j,b + A(h)B ab{[ )1 Z cos( 2π L h r )]2 + ad, for the ab compoet of the tesor B2 (h), 28 Z [ s( 2π )1 L )]2} h r (5) A(h) ) exp[- π2 h 2 /(R 2 L 2 )] h 2 (6)

3 3660 J. Phys. Chem. B, Vol. 108, o. 11, 2004 Galamba et al. B ab ) δ ab - 2 h a h b h 2 - ( 2π L ) 2 h a h b 2R 2 (7) I eq 5, F R cludes the short-rage forces ad the real compoet of the Ewald-Coulomb forces, both computed real space. Fgure 1 compares the dmesoless o-ormalzed stress ACF, calculated usg eq 5 ad the followg (correct) expresso: J xy ) -1 m V x V y + j> r jy F jx R + r y F x F (8) where the Fourer potetal term of the stress-tesor elemet s correctly defed, because the postos cosdered are those oly subject to pbc ad ot to the Ewald summato. otce that the followg trasformato was used (usg ewto s thrd law): Fgure 1. Comparso betwee the dmesoless stress autocorrelato fucto (ACF) calculated usg eq 5 (sold le) ad eq 8 (symbols). The correlato fuctos were calculated from a smulato of acl wth F) g/cm 3 ad T sm ) 1091 K. 1 2 j F j (r j - L)(r j - L) ) F (r j - L)(r - L/2) (9) I eq 8, F F s the Fourer compoet of the force o partcle due to the teracto wth all the partcles the recprocalspace lattce. However, the posto coordate r y s the y-coordate of partcle sde the cetral cell that does ot replcate the mage partcles postos the recprocal-space lattce. The et result s that a completely dfferet form for the stress ACF s obtaed by wrogly usg the lattce sum gve by eq 3 as the force betwee a sgle par of os wth the MD box. Although oe of the pots dscussed here are ew, we fd t worthwhle to demostrate the effect of usg eq 8, terms of the computed stress ACF. Furthermore, the reasos behd the eed to use eq 5 are ot dscussed stadard refereces We also ote that a smlar problem s observed the calculato of the vral pressure of oc systems; 29 however, that problem ca be avoded usg the relato betwee the pressure ad Coulombc eergy Model ad Smulato Parameters The Bor-Mayer-Huggs-Tos-Fum (BMHTF) potetal model has bee used the smulatos dscussed here. Ths model has the followg form: φ j (r) ) Z Z j e 2 r + A j exp[b(σ + σ j - r)] - C j r 6 - D j r 8 (10) where the frst term represets the Coulomb teracto, ad the secod term represets the Bor-Huggs expoetal repulso, wth parameters obtaed by Tos ad Fum. 10,11 The thrd ad fourth terms respectvely correspod to the dpoledpole ad dpole-quadrupole dsperso eerges, wth parameters obtaed by Mayer. 8 The smulatos have bee performed the mcrocaocal esemble for a cubc sample composed of 216 os usg pbc ad the mmum mage coveto. The postos at tme zero were defed as those correspodg to the face-cetered cubc lattce of sold acl, wth the catos occupyg octahedral holes, ad the zero-tme veloctes were defed radomly ad Fgure 2. Itegrated stress ACFs for acl for the dfferet state pots smulated. The smulato average temperatures, from top to bottom, are T sm ) 1091, 1217, 1339, 1413, ad 1495 K. scaled to esure a total lear mometum of zero. The Verlet leapfrog algorthm has bee used to solve ewto s equatos of moto, 19 wth a tme step of s, whch esures coservato of the total eergy of the system. The system was equlbrated for 150 ps at the desred temperature by scalg the veloctes. The producto stage was ru for 5 s for every pot ad the off-dagoal stress-tesor elemets were accumulated at every tme step. From the stress-tesor elemets, the stress ACF was calculated ad the shear vscosty coeffcet obtaed through umercal tegrato for a tme wdow of 5 ps. The choce of the relatvely small tme step of 1 fs, cojucto wth the short-tme spacg of the stress-tesor elemets, geerates eglgble errors assocated wth the umercal tegrato of the stress ACF. For the smulatos reported here, the value of the Ewald covergece parameter R was used (R )5.6/L, where L s the legth of the sde of the molecular dyamcs box), wth a trucato of the real compoet of the Ewald-Coulomb potetal at r c ) L/2 (thus usg the same set of o pars as that for the short-rage compoet of the potetal eergy ad force), ad the Fourer compoet of the force ad potetal eergy was summed up to the vector 2 ) 27, wth the k-space vector gve by k ) 2πh/L. h max 4. Results ad Comparsos Fgure 2 shows the results of the tegrato of the stress ACFs at the dfferet state pots smulated for the case of acl

4 Shear Vscosty of Molte Alkal Haldes J. Phys. Chem. B, Vol. 108, o. 11, TABLE 1: Smulato Results for acl a T equl T sm F P E Vscosty (mpa s) (K) b (K) (g/cm 3 ) (MPa) (kj/mol) η η exp ( ( ( ( ( a The stress autocorrelato fuctos were evaluated from 5-s productos rus ( tme orgs) ad a tme wdow of 5 ps. The expermetal results are from Jaz. 15 b Equlbrato temperature. TABLE 2: Smulato Results for KCl a T equl T sm F P E Vscosty (mpa s) (K) b (K) (g/cm 3 ) (MPa) (kj/mol) η η exp ( ( ( ( ( a The stress autocorrelato fuctos were evaluated from 5-s productos rus ( tme orgs) ad a tme wdow of 5 ps. The expermetal results are from Jaz. 15 b Equlbrato temperature. TABLE 3: Comparso of Results Obtaed Dfferet Smulato Studes of the Shear Vscosty of acl ad KCl salt referece os T (K) F (g/cm 3 ) η (mpa s) acl Cccott et al acl ths work acl Sdzgre ad Glla acl ths work acl Delhommelle ad Petravc acl ths work KCl Sdzgre ad Glla KCl ths work (smlar curves, ot dsplayed here, were obtaed for KCl). The smulato results for the pressure, teral eergy, ad shear vscosty of acl ad KCl are gve Tables 1 ad 2. Because o thermostat was used the smulatos, the average temperatures for some pots are very dfferet from the put temperatures gve the frst colum of the tables. For the pots where the average temperature s hgher (lower) tha the put temperature, a hgher (lower) vscosty value s expected, because the expermetal lqud destes at those pots are lower (hgher) tha the values used the smulatos. The expermetal data for the case of acl are slghtly overpredcted; 15 the error creases as the temperature creases. For the case of KCl, the equato from whch the vscosty has bee calculated 15 s vald oly for the temperature terval of K ad extrapolated data have bee used for the comparsos gve Table 2. Tables 1 ad 2 show that the BHMTF rgd o model overpredcts the shear vscosty for the two molte salts. As metoed prevously, the cluso of a polarzato model ad ts effect o the shear vscosty was vestgated by Jacucc et al., 14 ad the results show a decrease the smulated shear vscosty. That result dcates the possblty of obtag eve better agreemet wth the expermetal data for these systems f polarzato effects are take to cosderato. Table 3 compares the values obtaed prevous smulato studes wth those obtaed here. A addtoal pot (see Table 3) has bee smulated to compare these results wth the recet smulatos of acl through EMD. 18 The BMHTF model has bee used all the smulatos whose results are gve Table 3. The value gve Table 3 from the work of Delhommelle ad Petravc 18 was obtaed usg the EMD method wth the so-ketc Gaussa thermostat. The values obtaed by these authors, through EMD (usg a cofguratoal temperature thermostat) ad through EMD, were ad mpa s, respectvely. Table 3 shows that the results obtaed the dfferet studes are cosstet wth the results reported here. 5. Coclusos Equlbrum molecular dyamcs smulatos of molte acl ad KCl the mcrocaocal esemble have bee performed ths work, to assess the degree of accuracy wth whch the shear vscosty of these systems ca be computed through the rgd o Bor-Mayer-Huggs-Tos-Fum (BMHTF) model. The results obtaed are satsfactory agreemet wth the avalable expermetal data, whch are geerally overpredcted wth 10%-15%. Furthermore, a detaled dscusso has bee gve o the form of the recprocal-space porto of the stress-tesor elemets for oc systems for whch the Ewald sum s used to hadle the log-rage forces. A comparso wth prevous smulatos o the shear vscosty of molte alkal haldes showed that the results obtaed ths work are cosstet wth those of other authors. Although the cluso of polarzato effects s expected to mprove the results for the shear vscosty of molte alkal haldes, further vestgato of ths pot should gve a accout of the degree of mprovemet of the smulato data. Ackowledgmet. The authors would lke to thak Dr. D. M. 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