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  Primitive model electrolytes. I. Grand canonical Monte Carlo computations John P. Valleau and L. Kenneth Cohen  Citation: The Journal of Chemical Physics 72 , 5935 (1980); doi: 10.1063/1.439092   View online:   View Table of Contents:   Published by the  AIP Publishing   Articles you may be interested in    An efficient iterative grand canonical Monte Carlo algorithm to determine individual ionic chemical potentials inelectrolytes J. Chem. Phys. 132 , 244103 (2010); 10.1063/1.3443558 Boltzmann bias grand canonical Monte Carlo J. Chem. Phys. 128 , 134109 (2008); 10.1063/1.2883683 Phase behavior of the restricted primitive model and square-well fluids from Monte Carlo simulations in the grandcanonical ensemble J. Chem. Phys. 110 , 1581 (1999); 10.1063/1.477798 Monte Carlo study of mixed electrolytes in the primitive model J. Chem. Phys. 96 , 7656 (1992); 10.1063/1.462366 Grand canonical Monte Carlo simulation of liquid argon J. Chem. Phys. 85 , 2169 (1986); 10.1063/1.451110 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Thu, 18 Dec 2014 19:04:03  Primitive model electrolytes I Grand canonical Monte Carlo computations John P Valleau and L Kenneth Cohen Lash Miller Chemical Laboratories University of Toronto Toronto Ontario Conada M S IAI (Received 20 December 1979; accepted 21 February 1980 Monte Carlo calculations in the grand canonical ensemble are described for coulombic systems. and carried out for 1:1. 2:2, 2:1. and 3:1 aqueous electrolytes in the primitive model with equal ion sizes. Energies and activity coefficients are obtained, and the scope and reliability of the method is discussed. I INTRODUCTION Once upon a time this laboratory published 1ã2 Monte Carlo results for the restricted primitive model of 1 : 1 aqueous electrolytes. The results were useful in eval uating various theoretical treatments of the electrolyte problem. At the same time similar Monte Carlo (MC) investigations were carried out for the 3: 1 and 2: 2 cases. 3 These data are of still greater interest, but they were not published: many of them are reported in later papers of this series. We were unwilling to publish those results earlier be cause we were not confident of the accuracy of the os motic coefficients, and hence of the free energies, that were obtained. The experiments were carried out in the canonical ensemble, as is most Monte Carlo work. Like the pressure in uncharged systems, the osmotic coeffi cient requires the values of the pair correlation func tions at contact of the particles, and these are obtained by extrapolation of the pair function data. In the ionic case, however, the pair functions for unlike ions vary extremely rapidly near contact, and the extrapolation is correspondingly dubious even with quite precise pair function data. USing conventional techniques the free energy and activity coefficients would be obtained by in tegration of the osmotic coefficient (with respect to Inc) and so our values of those quantities were also in doubt, although the internal energy and the structural data were both good in themselves. In the case of the 1 : 1 electrolytes 1ã2 the problem is not so severe, and the extrapolations seemed adequate. Furthermore, those free energies were subsequently obtained by a second and independent method· which confirmed the conventional results. Application of that technique (multistage sampling) to the other systems would have been somewhat tedious, but a later develop ment of the idea using non-Boltzmann sampling tech niques 5ã6 -so-caUed umbrella sampling -would be quite efficient. A still better approach is available, however, for the low ionic densities in which we are interested: the problem is a natural one for the grand canonical Monte Carlo methods 8ã 7 (GCMC). In GCMC work one fixes the cbemical potential, along with the temperature and the volume. The Markov chain allows fluctuation of the concentration, and one finds eventually the mean concentration (and energy, etc.) corresponding to the par ticular values chosen for the fixed parameters. The possibility of such calculations has been pointed out often,7 but it is only recently, with the work of Norman and Filinov,8 Adams, 9,10 and Nicholson, Rowley, and Parsonage, 11,12 that attempts have been made to exploit the method. The GCMC method appears not to be useful at high densities e. g., for liquids). This is because steps in the Markov chain which lead to changes of concentration are then exceedingly rare: one is unlike- ly to find room to insert particles at random positions in a dense system. As a consequence the concentration does not fluctuate adequately and convergence is prohibitively slow. There are no such problems at low den sities, however, and this makes GCMC ideal for study ing dilute electrolyte solutions. The present article (I) describes the application of GCMC methods to electrolyte solutions, and reports results for 1: 1, 2: 2, 2: 1, and 3: 1 aqueous electrolytes in the primitive model with equal ion sizes. The following pa per (II) reports some of the earlier canonical Monte Car lo (CMC) data for 2: 2 electrolytes, and uses both sets of data to examine the (by now many) theoretical approximations for the problem and to comment on the structure of such ionic solutions. The third paper (m) does the same thing for the unsymmetrical 2: 1 and 3: 1 cases. A subsequent paper (IV) will report GCMC results for the 1: 1 aqueous primitive model with unequal ion sizes. This article (I) is therefore primarily methodological, and begins (Sec. II) by discussing the theory of GCMC for the electrolytic case and choosing an appropriate transition matrix for the MC Markov chain. After other details of the computations are described (Sec. m , the results are reported (Sec. IV). Most of the scientific discussion of these results is left for later papers; here we are interested in the efficiency and reliability of the GCMC results. II GRAND CANONICAL MONTE CARLO THEORY To specify a configuration in the grand canonical en semble requires the number and the locations of the par ticles of each species present. Suppose 1Ti is the proba bility of a configuration i in the grand canonical en semble. We require a convenient MarkOV chain among such states having a limiting distribution proportional to the distribution hi} The Markov chain is defined by the stochastic matrix I Piil I of the probabilities Pi} of transitions from state i to state j A sufficient (though not a necessary13,14) condition for the irreducible chain to have the correct limiting distribution is J. Chem. Phys. 72(11). 1 June 1980 0021·9606/80/115935·07 01.00 © 1980 American Institute of Physics 5935 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Thu, 18 Dec 2014 19:04:03  5936 J. P. Valleau and L K. Cohen: Primitive model electrolytes. I E.uo: .L P j 1T I 1) and we seek such transition probabilities. As usual we attempt, at each step of the chain, a trial change of state, and this trial move is then accepted or rejected in such a way as to lead to (1). Suppose the probability of a trial step i to j is qlj and the probability of acceptance of that trial step isf lj for i*j. Then Pi};' qjjfiJ> (ii-j) , (2) PH;' 1 LP jj j~i There is of course wide latitude in the choice of I I qjJII , and then also of I I ftJl I, satisfying (1). We examine simple choices of I I qjjll 'and I I fjjll, of the sort pro posed by Adams, 9 but elaborated to deal with the multi component ionic solutions. Monte Carlo experiments can be regarded as describ ing an ensemble either of labeled particles or of un labeled particles. It turns out to be most convenient to regard the particles as unlabeled, and this is done in what follows. (An alternative formulation in terms of labeled particles is of course possible. 15) We now describe a scheme for carrying out a grand canonical Monte Carlo computation. Consider an elec trolyte which on solution dissociates into v.Lo cations of charge Z. and v.Lo anions of charge Z. per mole (where Lo is Avogadro's nwnber). Evidently v.Z. + v.Z. ;, 0, and v=:; V. + v_ ions can form an electrically neutral combination. At each step of the chain either (i) one tries to add or to delete an electrically neutral combi nation of v ions, or (ii) one tries to move one ion to a new location. The steps are done as follows: (i) Additions or deletions are attempted with equal probability P; v. cations and v. anions are added or de leted in a Single step. In a trial addition each of the v ions is inserted anywhere in the box with equal probabil ity. It is convenient (and valid) to regard the box as consisting of V discrete sites (V extremely large). Then if state j is obtained from state i by addition, Nj = Nj + v+ and Nj : N; + v. in an obvious notation, and evi dently P qjJ : V v+ I v.1 (3) In a trial deletion any set of v. cations and v. anions is removed, with equal probability, so (with i obtained from j by deletion) _ PNjlNil qjj - ' + 1''-' , , lVJ lVi·v . v_. 4) It is interesting that the underlying matrix II qui I is not symmetric. The probability of a configuration i in an ensemble of unlabeled particles will be 1 1 1Tt = IT A Nl A:N'i exp[,9(Il.N'i + Il·Nt) - tm.] (5) where Il. and fl. are the chemical potentials of the ions, n is the grand canonical partition function, A. =:; hi (2mn.kT)1/2, and U j is the configurational energy of the state. Combining Eqs. (1)-(5), we obtain (still with N;=~ v., Nj=Nj-+vJ 6) where Il =:; V.(.1.. + v_f 1 _ is the chemical potential of the elec trolyte. Extending a definition due to Adams, 2 we in troduce B' 3«(.1. - Iltdeal) + lnyu+N-u. V (7) which leads to h _ N. Ni [ ( )] f, -N:' ''''' expB- 3U i -U j Ji j.lV j (8) Evidently [cf. Eq. (7)] fixing B fixes fl at fixed temperature and volume. If the acceptance probabilities fii> fji are made to conform to Eq. 8) with some particular value of B, the corresponding deviation from ideality, f 1 - f.1.Sdeal' may be obtained from it once the expectation values of the concentrations Y and N are known. Of course 3( Il = v In)' * , where 'Y z is the mean ionic activity coefficient in the McMillan-Mayer system. A simple way of realizing the result 8) is to set fjj = min{l, Ii ji} for addition, fii =min{1, fu i}} for deletion 9) (10) of particles, where on the right hand sides of the ex pressiOns the ratio f i) fii is given by Eq. 8) after finding the value of {UJ - U j ). (ii) Particle moves are attempted with a probability (1- 2P) at each step, and are carried out exactly as in the canonical case. That is, a particle is chosen at random and moved to a random position in a volume ele ment surrounding its srcinal position. The volume ele ment is chosen so that the part of the transition matrix describing these trial moves is symmetric: qiJ = qJI for moves. It is usual to choose a position within a cube or sphere centered on the srcinal position of the particle; a cube of side 6 was used in the computations described here. The move is accepted with a probability fii given by fiJ = min{1, exp[ - /3(U i - U  ]} ã (11) III. COMPUTATIONAL DETAILS The calculations are for the primitive model with ions of equal size. Thus the potential energy of interaction U b between two ions a, b separated by a distance rd is given by 12) = if r,,&<R , where Zoe, Z6e are the charges on the ions (with - e the electronic charge), E is the dielectric constant chosen to represent the screening effect of the solvent, and R is J. Chem. Phys., Vol. 72, No. 11, 1 June 1980 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Thu, 18 Dec 2014 19:04:03  J. P Valleau and L K. Cohen: Primitive model electrolytes. 5937 TABLE I. Results for 1: 1 electrolyte with R = 4.25 A and e:T= 2.345 x 10 4ã The columns are as follows: L is the length of the cubical periodic box; the chemical potential input parameter [Eq. (7)]; No. Steps refers to the length of the Me run, while No. Disc. shows how much of the run was regarded as aging and ignored in the averaging; 2P is the proportion of steps in which addi-tions or deletions were attempted, and the next column gives the acceptance rate for such steps; the average number of particles N and the resulting molarity M are followed by the data for the activity coefficients and the reduced configurational energies. 1O-3 x 10- 3 x Acc. rates L A) No. Steps No. Disc. 2P Add/Del 165.21 6.722 40 16 0.20 0.76 80.99 6.471 100 10 0.20 0.64 47.36 6.415 100 10 0.20 0.47 37.59 6.679 100 20 0.20 37.59 6.683 312 26 0.20 0.33 29.84 7.468 100 20 0.20 0.14 26.06 8.660 720 120 0.20 0.04 32.84 10.046 720 60 0.20 0.04 23.68 10.300 3024 504 0.20 0.008 29.84 11.686 1680 336 0.20 0.008 22.238 11.366 2400 200 0.67 0.002 28.02 12.752 1560 156 0.67 0.002 the diameter of each ion. The properties of the system depend on R and on the product ET, as well as on the ionic charge types. Rand E T were chosen to match closely those used earlier in the CMC calculations 3: for the 1: 1 case R=4.25 A and ET=2.345xl0· deg (cf. R =4.25 A and ET=2.339xl0· for CMC), and for the 2:2, 2:1, and 3:1 cases R=4.2 A and ET=2.342xl0· (cf. R = 4. 2 A and E T = 2. 336 X 10·). A t room temperature the parameter ET corresponds to the dielectric constant of water. Of course the results can equally well be interpreted as corresponding to a lower E at a higher tem perature, for example, a plasma at about 2.34 x lOS OK. In a similar way the results we describe as those of 2: 2 electrolyte may be interpreted as a 1 : 1 electrolyte at a value of ET reduced by a factor of 4 (e.g., an E of 19.6 at 25°C). Periodic boundary conditions were used, and the re peating box was cubical. The energy calculations were done in the minimum image apprOXimation (MI): that is the potential energy of a particle is evaluated by sum ming the pair interactions with only one image of each of the other particles, the image closest to the particle in question. This corresponds to a cubical cutoff having the dimensions and orientation of the repeating box. t has been shown iã 16 that a spherical cutoff is totally unacceptable for coulombic systems, and this is readily understood. 14 Many calculations on Coulombic systems have approximated the truly periodic boundary conditions by doing Ewald summations, although it seems that this will introduce very nonphysical forces and cor relations. 14 Fortunately at low ionic densities the Ewald and MI approximations are expected to agree with each other and to be physically meaningful. Brush, Sahlin, and Teller 17 compared the two approximations for the one-component plasma and found that they agreed as long as the dimensionless parameter _/4rr1J \1/3 e2 r \ 6Y) EkT (13) was below about 10. (Here N/V is the number density and Ze the ionic charge.) Our own tests 1 ,14 and those of N) M) -(In) ±) -  U)/ N)kT 62.9 ± 0.8 O. 01158± 0.00015 0.087 ± 0.013 0.116 ± O. 004 64.1±0.4 0.1002 ± 0.0006 0.232 ± 0.006 0.277 ± 0.004 63. 4± O. 6 0.495±0.004 0.249 ± 0.009 0.460 ± O. 007 63.4± 1. 0 0.991 ± 0.016 0.117±0.016 0.556 ± O. 014 64.2± O. 4 1. 003 ± O. 006 O. 127 ± O. 006 O. 558± O. 006 63. 8± 1.1 1.994 ± 0.034 -0.271±0.017 0.672 ± O. 021 65.l± O. 4 3.052 ± O. 019 -0.847±0.006 O. 740±0. 008 129. 9±0. 7 3. 045±0. 016 0. 849± O. 005 0.731 ± 0.007 66.1±0.3 4.13± O. 02 1. 652 ± O. 005 0.800 ± 0.006 131.0±0.6 4.09 ± O. 02 1. 661 ± O. 005 0.786 ± O. 006 63.05±0.24 4. 76± 0.02 -2. 232 ± 0.004 0.830 ± 0.005 125.75 ± O. 57 4. 75±0. 02 - 2.235 ± O. 005 0.813 ± 0.008 Hoskins and Smith 16 on two-component plasmas confirm that for low r the methods agree well while for higher r they give totally different results. Up to r 10 the dis crepancy is comparable to the statistical uncertainty. For all of the calculations reported here r is substan tially less than 10 (e. g., for the 2: 2 system at 2 M, r = 4. 9), so the energy approximation should be trust worthy. For the more concentrated solutions this was tested, however, by carrying out parallel runs with sys tems of double the usual size. One expects the results to vary with the system's size if the energy approxima tion is becoming inadequate; the results are discussed below. Two experimental parameters will affect the conver gence rate of the Markov chain. These are the step size 6 allowed when a particle move is attempted, and the ratio 2P/ l 2P) of attempted additions and deletions to attempted particle moves. The former was chosen to make the acceptance rate for moves well behaved; for the denser systems it was adjusted to give an acceptance rate close to 50 . Little is known about the best way to choose P, and we made no systematic study of this. At low densities the acceptance rates for addition and deletion are substantial, and we chose P = 0.1. A high densities the acceptance rate becomes small, however, and it was necessary to increase P in order to get ade quate variation of concentration during the runs. Data on P and the addition/deletion acceptance rates are re ported. Since pair correlation function data for most of these systems had been obtained in the earlier CMC calculations,3 it was not collected in the runs reported here. (For systems with unequal ion Sizes, reported in Paper IV, CMC results were not available and pair cor relations were studied in the GCMC runs. ) IV. RESULTS AND DISCUSSIONS Tables I-IV refer to the 1: 1, 2: 2, 2: 1, and 3: 1 sys tems, in that order. In each table the last three columns report the physical results: activity coefficients y,. and configurational energies U at various concentrations. J. Chem. Phys., Vol. 72, No. 11, 1 June 1980 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Thu, 18 Dec 2014 19:04:03
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