On the calculation of single ion activity coefficients in homogeneous ionic systems by application of the grand canonical ensemble

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1 On the calculation of ingle ion activity coefficient in homogeneou ionic ytem by application of the grand canonical enemble Peter Sloth Jjik-Kemik Intitut, Technical Univerity of Denmark, Bldg. 206, DK 2800 Lyngby, Denmark (Received 11 June 1992; accepted 12 January 1993) The grand canonical enemble ha been ued to tudy the evaluation of ingle ion activity coefficient in homogeneou ionic fluid. In thi work, the Coulombic interaction are truncated according to the minimum image approximation, and the ion are aumed to be placed in a tructurele, homogeneou dielectric continuum. Grand canonical enemble Monte Carlo calculation reult for two primitive model electrolyte olution are preented. Alo, a formula involving the econd moment of the total correlation function i derived from fluctuation theory, which applie for the derivative of the individual ionic activity coefficient repect to the total ionic concentration. Thi formula ha previouly been propoed on the bai of omewhat different conideration. 1. INTRODUCTION Single ion activitie are important in the theory of variou electrochemical phenomena. A an example of ignificant technical interet, we here jut mention the ph concept, which i intimately connected to the individual activity of the hydrogen ion. Unfortunately, it appear that no rigorou experimental procedure ha yet been devied by which reliable ingle ion activity coefficient could be meaured. Thi fact trongly motivate a more thoroughly theoretical invetigation. The preent work focue on two apect aociated the application of the grand canonical enemble (GCE) to homogeneou ionic fluid. Firt, the feaibility of etimating ingle ion activity coefficient from GCE Monte Carlo (GCEMC) calculation i dicued, and numerical reult are preented for two primitive model electrolyte. Second, we conider the problem of calculating ingle ion propertie for infinite homogeneou ytem in the framework of the Kirkwood-Buff theory. In paing, it i noted that thee topic have very recently been conidered for confined, pherical ytem.2 For the ake of implicity, the dicuion in thi paper i retricted to pherical ymmetrical ion in. a dielectric continuum. Since the pioneering work of Valleau and Cohen3 it ha become common to conider only electroneutral configuration in GCEMC calculation of homogeneou ionic ytem. By uch a procedure, it i not poible, however, to obtain information regarding individual ionic activitie. In the preent paper, it i hown that it i poible to etimate ingle-ion activity coefficient by GCEMC calculation on homogeneou ytem tandard periodic boundary condition if the Coulombic interaction are truncated according to the minimum image (MI) approximation. The ytem conidered in thi work conit of ion in a central cubic box of volume Y ide length L = V1 3. The ytem i urrounded by, periodically repeated image of itelf containing mirror particle moving like the original one. In the MI approximation, every ion in the cen- tral box i aumed to interact only other ion (including mirror ion ) in a cube of ide length L the elected ion at the center. By formally conidering uch a MI ytem in the limit V+ CO, an expreion i derived for a ytem containing two ionic pecie, which applie for the ingle-ion activity coefficient. Thi expreion i the ame a the one previouly propoed on the bai of omewhat different conideration.4 However, the preent approach (although o far retricted to a dielectric continuum) might appear to be more appealing from a phyical point of view. It could be mentioned that the correponding Kirkwood-Buff expreion for the mean ionic activity coefficient wa conidered quite ome year ago by Raaiah and Friedman. II. THEORY In the grand canonical enemble, the electrochemical potential of the individual pecie are held fixed. Let j&, &, and a, be the electrochemical potential, the tandard chemical potential, and the activity of pecie a, repectively, and let Y be the electric potential of the phae. Then,iii, i given by pa=,uz+kgt In a,+q,y. (1) In Eq. ( 1 ), qa i the charge of a particle of pecie a, T i the abolute temperature, and kb i the Boltzmann contant. By Eq. ( 1 ), it i aumed that the activity a, i a dimenionle quantity. The electric potential Y of the phae i not known a priori. Thu, in order to determine the individual activitie, it i neceary to evaluate Y a poteriori, by ome mean. Thi problem i greatly implified if the ion are placed in a tructurele dielectric continuum (e.g., if we conider the primitive model of electrolyte olution). The value of Y in Eq. ( 1) i identified the average electrotatic potential acting on a hypothetical tet ion, of vanihing charge and ize, ituated in the MC ytem. In our ytem, the average electrotatic potential (Y(r) ) at any poition r i generated by ion in the central Downloaded J. Chem. 26 Oct Phy to (Q), May 1993 Reditribution /93/ $06.00 ubject to AIP licene or 1993 ee American Intitute of Phyic 7319

2 7320 Peter Sloth: Ion activity coefficient box a well a by mirror ion. Since all electrotatic interaction are truncated according to the MI approximation, only ion in a cubic region fi of ize L3, centered at r, contribute to (Y(r) ). Let p( r ) be the average electric charge denity at r and let E be the (uniform) dielectric permittivity. The average electrotatic potential (Y(r) ) temming from the charge inide the cloed region fi i then given = Ja 4Tzyr@r,, dr = JZ2 J-z2 JZ2 $$fg x dx dy dz. (2) Since the average ditribution of ion (including the mirror ion) i homogeneou, by contruction, p,(r ) =p&y,z) i a contant the value p,=(q)/l3, where (Q) i the net average charge in the central box. Equation (2) then give Y=(Y(r))=(Q)K/16mL, where K i a contant given by jr= I, L L (x2+3+2) - dx dy dz =2[6 hr(2+ 6) -r-j (4) Equation (3) provide the deired expreion for the effective (uniform) electric potential in our ytem. Having olved thi problem, we are now equipped to proceed ome formal conideration. Let ( *** > denote an average in the GCE and let N, be the number of particle of pecie a in a given configuration. For given value of Z V, and the electrochemical potential of the individual pecie, it i then poible, from a GCEMC calculation, to obtain value of (iv,) and (ivjvd), etc. The particle number fluctuation (N$VD) -(NJ (ND) are related to the derivative of the individual electrochemical potential repect to the average compoition of the ytem. In the preent work, we explicitly conider ytem two different kind of particle only, i.e., one kind of cation (denoted + > and one kind of anion (denoted - ). In thi cae, it i convenient to expre the average compoition of a ytem in term of (Q) = (q+n+ +q-iv_) and p= (iv)/v, where N=N+ +N-. For ytem which are not too mall, the following relation can be obtained:2*8 (3) M+-= G,-=h I w21 - w)2=p+p-g+lim Vdrn (q+-qj2 v I ;9, m [g+-(r),i]? dr. (10) 0 In Eq. (lo), g&r) i the radial ditribution function between pecie a and pecie fl. It i poible to obtain a imple, analytical expreion for the quantity C, defined by Eq. ( 8 ), in the limit V+ 00. From Eq. ( 3 >, it i een that the charge denity vanihe a V+ CO for any given finite value of $. Thu, it can be expected that the calculated activitie become independent of $ a the ytem become large. From Eq. ( 1 ), (3)) and (6)) we then obtain www (N) = wam2) jcn) wwd+) = 16mLkBTqa/CK+q, j C-P 16mkBTL/K (for V-B 00 ). (11) It i een later on that Eq. ( 11) i conitent the econd moment electroneutrality condition of Stillinger and Lovett,g and alo our numerical Monte Carlo reult. From Eq. (l), (3), (5), and (8), we Anally obtain I qa I 1 - )(Q)=4+ M+-. (12) Equation ( 12) provide an expreion for (a In a,/+) tel in term of &/C and M+-. Here it i noted that any influence of the artificial MI ytem i expected to be removed when the ytem become infinitely large. Equation ( 12) might be ueful in connection GCEMC calculation, but i not directly applicable in connection, e.g., integral equation approximation calculation. In the following, we hall try to remedy thi hortcoming by rewriting the term &/C. To do thi, we firt hnd an alternative way to expre the net charge Q, of a given configuration, in the central box. Let q(r) be the electric potential at r for a given configuration of the ytem. Integration of $(r) over the central box yield (the prime indicate that the ummation i to be taken over the N ion-including mirror ion-in a cube of edge length L r at the center) I Y(r)dr= I Fqk/(4mlr--rkl )dr (4a--4p) (a&)(q) =(Q/4m) I l/lrldr =kbtq+q-&m+-c (a#p), (5) = (~)=k~tq,/c L=WaQ>-W,>(Q), c=q& 9 (6) (7) (8) 3 lim Q/C= ( l/vkbt) Y(r)dr, v-r.20 (13) where the lat identity i obtained by application of Eq. (11). J. Chem. Phy., Vol. 98, No. 9, 1 May 1993 Downloaded 26 Oct 2009 to Reditribution ubject to AIP licene or copyright; ee

3 Peter Sloth: Ion activity coefficient 7321 Here, we hall conider only ytem which are overall electroneutral, i.e., we hall et <Q) =O. Thi relation i fulfilled for a real electrolyte bulk ytem. In paing, it i noted that thi amount to put \I, =0, in the preent context, according to Eq. (3). Then, the following identity i obtained from Eq. (7) and ( 13) in the limit Vd CO : &z/c= (N,Q)/C= ( l/k,tv) ( N,~P(r)dr). (14) Let U,,, be the total potential energy of a given configuration and let 2 be the grand canonical partition function. Then (N,JY(r)dr) can be given by ( NaJy(r)dr) =Nzo Nzo $l$rz...ssy(r)drexp(--uni/k.t)dr,...dr~ tcr,(rbr.) = q+ =oq =o ay+an-n - a j--j-y(r>exp( N+!N-! NCa!-Na! %,=oq - 43 a+ N+IN-! - UN/kBT)drZ*..drN J-*-J exp(-un/k,t)dr2***drn * (16) In Eq. ( 15)) pa(rl) i the local average denity of pecie a at rl. By examination of Eq. (16), it i een that $,Jrr,r) i the average electrotatic potential at r, uppoing that an ion of pecie a i fixed at rl. In a homogeneou ytem, p,(q> i a contant the value p,=(n,)/v, and &(ri,r) i a function of the ditance r= Jr-r, ) only. In thi cae, Eq. ( 15) yield (NaJY(r)dr) =UV,) J&Jr)dr. (17) By combination of Eq. (14) and ( 17) (the following equation, in thi ection, apply for V+ oo ), i obtained. From Eq. (18) and the definition obtain (18) of C, we Aay= - (4~/6) m 0 [g,,(r) - l]r4 dr. (21) Inertion of Eq. (9) and (20) into Eq. ( 12) yield dlna a= lclal 1 - (pcxq&k,t&qyp&zr (22) dp P+P-(q+--4-)G+- ~. _~ where G, _ i given by Eq. ( 10). Equation (22) applie for an infinite electroneutral bulk ytem. Introducing the activity coefficient yn=aa/pa, Eq. (22) can be rewritten a dlny, dp= 1 -f ag+- - (%cpa~ekbt~z~yp~q~ PPaG+- (23) The above equation i equivalent to the expreion propoed in Ref. 4. &gap, &bwr=kbt. (19) In the cae of a dielectric continuum, the above equation ha been hown to be equivalent to the econd moment condition. Following the procedure outlined by Outh- Waite in Ref. 10, we may how where pa hhwr= (p,/e>zyqypyacry=kbt~~/c, (20) Ill. MONTE CARLO RESULTS In thi ection, we invetigate the GCEMC method for the primitive model (PM) of electrolyte olution. The PM i defined by the pair potential function between a particle of pecie a at rl and a particle of pecie fl at r2 (r12=r2-rd9 u,&2) = ~0, hl < &+dpv2 I 4d7dW r12 1, I r12 12 (da +$)/2. (24) Downloaded 26 Oct 2009 to Reditribution J. Chem. Phy., ubject Vol. 98, to AIP No. licene 9, 1 May or 1993 copyright; ee

4 7322 Peter Sloth: Ion activity coefficient TABLE I. GCEMC reult baed on computed value of (N,) and (N-) for a l:l-electrolyte B=1.681, d+/d-=3, and exp(ap+/ kj) =exp(apjk,t) = A total number of 1 million configuration wa generated for L/a= 5 and L/a= 10. In the cae of the larger ytem, a total number of 1.5 million configuration wa generated in each run. The tirt configuration were neglected in all of the run. L/a * PS hy+ lny- -e,,y/k,t Here, d, i the hard core diameter of a particle of pecie a. GCEMC calculation have been performed for a 1: 1 electrolyte d,/d- = 3 and for a 2: 1 electrolyte d+/d- = 1. The GCEMC procedure applied here i imilar to the one decribed in Ref. 11. It i convenient to introduce a length cale in term of the ditance of contact between a cation and an anion a= (d, +d-)/2. Then we have the dimenionle denitie pe=pg3 and the dimenionle activitie a, = &vyar. From Eq. ( 1) and (3)) we obtain the relation or Apa+ia-pz=kBT In p3a+q,(q)k/16ml exp(a~~/kbt>=p~y,expeq,(q)bk/iq+q-14(l/a)l, (25) where B i the Bjerrum parameter defined by BE [ q+q- I/ 4mkBTa. For given value of A&, B, and L/a, it i poible to obtain pa * = ( N,)a3/ V and (Q) from a GCEMC calculation, and eventually the individual ionic activity coefficient y, by application of Eq. (25). Here it i noted that for the relatively mall ytem ued in Monte Carlo calculation, the calculated value of y, might depend on (Q) to ome extend. We hall return to thi apect later on. The 1:l electrolyte, conidered in thi work, i pecified by B=1.681 and a mean ionic activity a, = G = Similarly, the 2:l electrolyte i pecified by B=3.4CKI and a, = qz Thee parameter correpond to hypothetical aqueou electrolyte olution at room temperature molar concentration around 1 mol/dm3, uppoing that a=425 and 4.2 A, repectively. Monte Carlo reult for the logarithm of the ingle-ion TABLE II. GCEMC reult baed on computed value of (N,) and (N-) for a 2:1-electrolyte B=3.400, f+/d-= 1, exp(ap+/kt) = , and exp(ajijk,t) = A total number of 1 million configuration wa generated in each run. The firt 20 Ooo configuration were neglected in all of the run. L/a P,* hl Y, hy- eoy/kbt ; b TABLE III. GCEMC reult baed on fluctuation theory for the ytem defined in the heading to Table I. L/a C/L& L/all activity coefficient, the electric potential, and the dimenionle alt denitie p: = (p*, + p _My+ + Y-) (v,being the toichiometric coefficient of pecie a) are preented in Table I and II for different value of L/a. The reult in Table I and II were obtained by (quite arbitrarily) etting exp(ap+/kt) =exp(apjk,t) = in the cae of the 1:l electrolyte, and by etting exp(ap+/k,t) = and exp( AjiJk,T) = in the cae of the 2:l electrolyte. It hould be emphaized that the reult of a MC run are to be conidered a experimental data and a uch are ubject to ome uncertainty. In general, the GCEMC reult in Table I and II are in atifactory agreement reult obtained by the hypemetted chain (HNC) approximation.i2 Alo, the reult in Table I are in good agreement the correponding GCEMC reult for confined pherical ytem obtained in Ref. 2. In Table III and IV, we how the reult for the (a ln.h/ap*) (Q) coefficient a well a the value obtained for C/[&L/a)] (ee being the charge of a proton). The value of (8 Iny,/ap*) te, were etimated by (p* = p*+ + p*_>, =- I 4a I 1-4ar~cf~C M#+ 4a -(pa-q&$ M# _ I W2) - w2 +- (q+-q-l2 (V/a) * (a#b (26) (27) At thi point, we jut note that the value of (a my,/ dp*) (Q) are ubject to rather large uncertaintie. From Table III and IV, it i een that the value of C/[ei( L/a)], for the larger ytem, are very cloe to the limiting value TABLE IV. GCEMC reult baed on fluctuation theory for the ytem defined in the heading to Table II. L/a C/l&L/a) J.Chem. Phy.,Vol. 98,No.9,1 May1993 Downloaded 26 Oct 2009 to Reditribution ubject to AIP licene or copyright; ee

5 Peter Sloth: Ion activity coefficient 7323 TABLE V. Variou GCEMC reult for a l:l-electrolyte B= 1.681, d.+/d- =3, exp(ap+/kbt) = , and exp(ai-/kt) = A total number of 1.5 million configuration wa generated in each run except for L/a=24 in which cae, a total number of 2 million configuration wa generated. The firt 20 Ooo configuration were neglected in all of the run. L/a p,* lny+ hy.- (8 lny+/ap*)(o) (aln y-/ap*)(~j given by Eq. ( 11 ), which are and 0.247, repectively. Thi indicate that the reult for In y, become independent of (Q). A real bulk electrolyte i electroneutral. In order to examine to what extend the (artificial) MC ytem diturb the reult in cae where (Q>#O additional run were carried out for the 1: 1 electrolyte exp( A,?i+ ) ~ and exp( A@-) = Thee value were choen on the bai of the reult in Table I to correpond to (Q) : 0. Reult for thee run are reported in Table V. The reult for p,* are remarkably independent of the ize of the ytem in contrat to the correponding reult in Table I. Thi indicate (a one could expect) that the reult of a MC run on a finite ytem i cloer to the thermodynamic value if (Q) -0. The reult for the a In ya/ap* coefficient in Table V are in good agreement the correponding reult in Table III. The reult for (d In y+/ap*> (Q) are fairly independent of the ize of the ytem, wherea the value of (a ln y-/dp*) (Q) eem to increae L/a. Thu, in order to etimate (a 1nyJ dp*) in the thermodynamic limit, computation for larger ytem hould probably be carried out. However, the preent GCEMC method might not be practical/efficient for the calculation of more precie value of the (a In JJ,/ ap) coefficient. A reaon for thi i that the & value are obtained a mall difference between two large quantitie and are thu expected to be ubject to quite ome uncertainty. The value of C eem here to be determined rather accurately, though. J. G. Kirkwood and F. P. Buff, J. Chem. Phy. 19, 774 (1951). *P. Sloth, Mol. Phy. 77, 667 ( 1992). 3J. P. Valleau and L. K. Cohen, J. Chem. Phy. 72, 5935 (1980). 4P. Sloth, Chem. Phy. Lett. 164, 491 (1989). J. C. Raaiah and H. L. Friedman, J. Chem. Phy. 48,2742 (1968); 50, 3965 (1969). H. Margenau and G. M. Murphy, The Mathematic of Phyic and Chemitry, 2nd ed. (Van Notrand, Princeton, 1956), pp P. Sloth and T. S. Sorenen, Chem. Phy. Lett. 173, 51 (1990). *Although the dicuion in Ref. 2 i epecially concerned confined ytem, thee equation are alo valid in the preent context. F. H. Stillinger and R. Lovett, J. Chem. Phy. 49, 1991 (1968). C. W. Outhwaite, Chem. Phy. Lett. 24, 73 (1974). P. Sloth and T. S. Sorenen, J. Chem. Phy. 96, 548 (1992). P. Sloth and T. S. Sorenen, J. Phy. Chem. 94, 2116 (1990). Downloaded 26 Oct 2009 to Reditribution J. Chem. Phy., ubject Vol. to 98, AIP No. licene 9, 1 May or copyright; 1993 ee