School of Chemistry, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

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1 pubs.acs.org/jpcb Calculation of Partition Functions and Free Energies of a Binary Mixture Using the Energy Partitioning Method: Application to Carbon Dioxide and Methane Haina Do,* Jonathan D. Hirst, and Richard J. Wheatley School of Cheistry, University of Nottingha, University Park, Nottingha, NG7 2RD, United Kingdo ABSTRACT: It is challenging to copute the partition function (Q) for systes with enorous configurational spaces, such as fluids. Recently, we developed a Monte Carlo technique (an energy partitioning ethod) for coputing Q [J. Che. Phys. 2011, 135, ]. In this paper, we use this approach to copute the partition function of a binary fluid ixture (carbon dioxide + ethane); this allows us to obtain the Helholtz free energy (F) via F = k B T ln Q and the Gibbs free energy (G) via G = F + pv. We then utilize G to obtain the coexisting ole fraction curves. The cheical potential of each species is also obtained. At the vapor liquid equilibriu condition, the cheical potential of ethane significantly increases, while that of carbon dioxide slightly decreases, as the pressure increases along an isother. Since Q is obtained fro the density of states, which is independent of the teperature, equilibriu therodynaic properties at any condition can be obtained by varying the total coposition and volue of the syste. Our ethodology can be adapted to explore the free energies of other binary ixtures in general and of those containing CO 2 in particular. Since the ethod gives access to the free energy and cheical potentials, it will be useful in any other applications. 1. INTRODUCTION Knowing the free energy allows any physical and cheical phenoena to be predicted. The calculation of free energy fro theory has been an active field for any decades, yet reains a challenging proble. Methods such as free energy perturbation theory 1 3 and therodynaic integration 3,4 are frequently used to deterine the free energy difference between two states (states I and II for exaple), and rely on calculations along a path fro state I to state II. These states can be characterized by either different potential energy functions or different structures. Systes where states I and II are far apart, with large and coplex changes in structure, can require a coplicated path between states and can be prohibitively expensive in ters of coputing effort. Thus, it is iportant to develop ethods that can provide the free energy for each state I and II independently; in this case, the free energy difference can be calculated even for significantly different states because the integration path is avoided. Methods such as inia ining, 5 7 hypothetical scanning, 8 11 haronic reference state, 12,13 and nondynaic growth have ade proising progress toward this goal. However, the efficiency of these techniques deteriorates rapidly with the size and the coplexity of the syste. Elegant approaches for calculating free energies via the density of states have been developed in the past few decades, including ubrella sapling, 17 the weighted histogra analysis ethod (WHAM), 18,19 transition atrix, 20 uticanonical, 21 and Wang Landau 22 sapling. The idea behind these techniques is either to perfor ultiple siulations, whose saplings overlap in configurational space and connect the together (reweight) or to perfor the siulations in biased ensebles to achieve broader sapling of particular states that are rarely visited. If the Wang Landau technique is eployed for systes with an infinite energy range, such as fluids, one often has to choose a finite range of energy (cutting off the high-energy range) either via trial and error or by calculation WHAM also uses a finite range of energies. Restricting the density of states to a finite energy range eans that the partition function can only be coputed to within a ultiplicative constant. To circuvent these liitations, we have recently developed an energy partitioning Monte Carlo (MC) technique, 27 which can calculate the absolute density of configurational energy states with no high-energy cutoff, fro which the excess partition function, and hence the excess free energy of a olecular fluid, can be obtained. The absolute free energy can be calculated by adding the ideal gas free energy to the excess free energy. Phase equilibriu properties of odel systes and real fluids are of treendous iportance to scientists and engineers. Since the easureent of phase equilibriu properties is tie consuing and expensive, coputer siulation based on olecular odeling is a proising alternative. Free energy can be used to predict the vapor liquid equilibriu properties and Received: Deceber 16, 2011 Revised: March 15, 2012 XXXX Aerican Cheical Society A

2 Figure 1. Sketch (not to scale) explaining the partitioning of the density of states (a) at the start of the siulation, where there is only one energy subdivision; (b) after placing the first energy boundary; (c) after placing the second energy boundary; and (d) after placing the nth energy boundary. w(e) is the weighting function and int Ω(E) is the integrated noralized density of states. any other therodynaic properties, including critical phenoena and pvt data for single phases. Of particular interest to us are ixtures that contain carbon dioxide (CO 2 ) Systes of CO 2 in ulticoponent ixtures with light paraffins are iportant to the natural gas industry, since it has becoe coon practice to process gas streas with oderate to high levels of CO 2. Supercritical CO 2 is a novel solvent that has attracted uch attention. 34 A property of interest is the unusually high solubility of fluorinated hydrocarbons in supercritical CO 2. Mixtures of hydrofluorocarbons and CO 2 are possible replaceents for ozonedepleting refrigerants. Thus, knowledge of the phase equilibriu properties of these ixtures can help optiize their perforance in industrial processes. 35 In this paper, we calculate the partition function and free energy of the binary ixture CO 2 +CH 4. Mixtures of CO 2 with other sall olecules (and other binary ixtures) can be studied analogously. Using the free energy, the vapor liquid equilibriu properties and the critical points of the ixture are calculated. The cheical potentials of each species are extracted fro the free energy. The critical point is observed fro a plot of the Helholtz free energy versus volue. The paper is organized as follows. In section 2, we describe the theoretical background of the energy partitioning approach. In section 3, we outline the siulation procedure. In section 4, we present results for the partition function, free energy, and vapor liquid equilibriu properties of the CO 2 +CH 4 ixture. We obtain the vapor liquid equilibriu properties fro the free energies, and copare our results with experiental data. We also copare our results with a conventional MC (Gibbs enseble) siulation technique and deonstrate the advantages of our ethod copared to this approach. Finally, in section 5, we give concluding rearks and outline future work. 2. METHOD 2.1. Free Energy and Partition Function. The free energy of a syste in the canonical enseble is given by F = k B T ln Q(N,V,T), where Q(N,V,T) is the partition function of the syste, which is the integral of the Boltzann factor exp( βe) over particle positions (r N ) and oenta (p N ). As B

3 the integration over particle oenta can be solved exactly, the canonical partition function can be expressed as 1 Q( N, V, T) =!Λ exp[ βe( r N )] dr N 3 N N (1) where N is the nuber of particles in the syste. The factor 1/ N! coes fro the fact that particles are indistinguishable, Λ is the theral de Broglie wavelength, E is configurational energy, β is 1/k B T, k B is the Boltzann constant, and T is the teperature of the syste. Unlike other equilibriu therodynaic properties, such as the internal energy, that can be estiated as a tie or enseble average, the Helholtz free energy F is related directly to the volue of the configurational space (as shown in eq 1), and calculating F still reains a challenge in olecular siulation. Equation 1 can be rewritten in the for of reduced coordinates (s = r/l) as N V 1 Q N V T =!Λ β N N (,, ) exp[ E( s )]ds 3N N 0 where L is the length of the cubic siulation box, which contains N olecules, and V is the volue of the box. The ter V N /(N!Λ 3N ) in eq 2 is the translational partition function of an ideal gas and can be calculated analytically. Thus, we seek to copute the excess part of the partition function (Q ex ) 1 N N Q ex = exp[ βe( s )] ds 0 The probability of finding an energy E is proportional to the density of states, Ω(E), where Ω(E) = 0 1 δ(e E(s N )) ds N. Thus, Q ex can be expressed ore conveniently as Q ex = exp( βe) Ω( E) de Ω( E)dE where the integral is over all possible energies of the syste. Equation 4 shows that if the density of states is known, the excess partition function can be calculated fro it. Our goal is to calculate the density of states and hence Q ex Calculation of the Density of States. The ai of the energy partitioning ethod is to divide the energy range recursively into subdivisions (indexed ), such that the E integrated noralized density of states 1 E Ω(E) de/ Ω(E) de is 1/2 1 for the first energy subdivision ( =1, E 1 E E 0, E 0 = ) (Figure 1d), 1/2 2 for the second subdivision ( =2,E 2 E E 1 ), and so on down to 1/2 n for the two lowest-energy subdivisions ( = n, E n E E n 1 and = n +1, E E n ). At the start of the siulation, there is only one energy subdivision (Figure 1a). All MC oves are accepted at this stage (rando sapling), and the sapled energies are saved. After a predeterined nuber of MC oves (usually several thousand), the energy range E is divided: the first energy boundary E 1 is set equal to the edian configurational energy. The choice of the nuber of MC steps used in each division of the energy is explained in section 3. After the first division of the energy (Figure 1b), there are two energy subdivisions with the sae integrated density of states (1/2 and 1/2). We then throw away all the sapled energies and ove on to the second division of the energy. (2) (3) (4) During the second division of the energy (Figure 1b), MC oves are accepted based on a biased weighting function w(e) = 4, where is the energy subdivision into which the configurational energy E falls so the weighting functions for the energy subdivisions 1 (E > E 1 ) and 2 (E E 1 ) are 4 1 and 4 2, respectively. These biased weighting functions are necessary to push the syste toward the low-energy region; the energy subdivision 2 is visited four ties as often on average as the energy subdivision 1. After the sae predeterined nuber of MC oves as above, the second energy boundary E 2 is set equal to the edian configurational energy found in the lowestenergy subdivision E E 1. This produces three energy subdivisions with the integrated density of states equal to 1/2, 1/4, and 1/4 for the highest energy, iddle, and lowest energy, respectively (Figure 1c). Then the sapled energies are discarded once again and the third division of the energy starts. During the third division of the energy (Figure 1c), the weighting for each energy subdivision is 4 1,4 2, and 4 3 for the highest-energy, iddle, and lowest-energy subdivision respectively, which eans that the lowest-energy subdivision (subdivision 3) is visited four and eight ties as often on average as the subdivisions 2 and 1 respectively. At the end of the MC procedure, the third energy boundary E 3 is set equal to the edian configurational energy found in the lowest-energy subdivision E E 2. There are now four energy subdivisions with the integrated density of states equal to 1/2, 1/4, 1/8, and 1/8 for the highest-energy to the lowest-energy subdivisions respectively. The procedure continues iteratively. In general, for the nth division of the density of states (to produce energy boundary E n, n 2), MC sapling is perfored with a weighting function w(e) =4 for the previously calculated subdivisions, 1 n (subdivision 1 being the highest energy). The weighting function is essential to speed up the siulations; it ensures that about 2/3 or ore of the configurations of the syste fall into the current lowest-energy subdivision. 27 The probability of accepting a ove fro an old state with configurational energy E old that falls into subdivision old,toa new state with configurational energy E new that falls into subdivision new, is given by we ( new) P(old new) = in 1, = in 1, we ( old) 4 new 4 old (5) At the end of the predeterined nuber of MC oves, the energy boundary E n is set equal to the edian configurational energy found in the lowest-energy subdivision E E n 1 and all sapled energies are discarded. The siulation is repeated for the next division of the energy until it is terinated. The density of states Ω(E) strongly increases with energy (except at very high energy), exp( βe) strongly decreases with energy, and the function Ω(E) exp( βe), which is integrated in eq 4 to get Q, has a axiu at an average energy. Since this average energy is not known, the nuber of energy boundaries is not fixed in advance. It is found during the siulation using the criterion that when the integrated function [ [Ω(E) exp( βe) de] ] for the current lowest-energy (th) subdivision is uch saller than its axiu value over all energy subdivisions, [ [Ω(E) exp( βe) de] ax ], the siulation can be terinated, as nothing would be gained by further subdividing the density of states. A stopping criterion of C

4 [ [Ω(E) exp( βe) de] ]=10 9 [ [Ω(E) exp( βe) de] ax ] is used. This is teperature dependent, and we use the lowest teperature of interest, in order to cover all the iportant parts of the energy range. Once the siulation is finished, the excess partition function is obtained fro the noralized integrated density of states as (cf. eq 4) n+ E = 1 1 ( 1 Ω( E)d E)exp( β E ) E Q ex = n+ E = 1 1 ( 1 Ω( E)d E) E = n+ = exp( β E ) + = 1 n 1 2 (6) where E =(E + E 1 )/2. For the first energy subdivision ( = 1), E is set equal to E 1, and for the last energy subdivision, E is set equal to E n. Also, in the last energy subdivision, 2 is replaced by 2 n, as the noralized integrated density of states of the two lowest-energy subdivisions are the sae and equal to 2 n. Errors produced in this application by introducing E are negligible. However, ore care would be needed if very few energy subdivisions were found Interolecular Potentials. The quality of the results of a coputer siulation depends on the potential odels describing the interactions between the olecules of the studied substances. Much effort has been devoted to the developent of an accurate potential for CO 2. In our siulations, the rigid fixed-point charge eleentary physics odel (EPM) is eployed, due to its widespread use. 36 In general, one would not expect force fields to predict properties accurately at high pressure. However, the EPM odel perfors reasonably well under oderately high pressure (<200 bar). 37,38 To enable cobination with the CO 2 odel, potentials for CH 4 with the sae functional for are required. The transferable potentials for phase equilibriu with explicit hydrogen atos (TraPPE-EH) odel 39 is used for CH 4 : the CH 4 olecule has five Lennard-Jones interaction sites, which are located at the carbon ato and the centers of the CH bonds. For Lennard-Jones interactions between unlike atos, the Lorentz Berthelot cobining rules are used. 3. SIMULATION DETAILS A series of siulations are perfored at different total ole fractions and densities to calculate the excess partition functions of the ixture under these conditions. The results are a set of Q ex at different copositions ranging fro the vapor phase to the liquid phase (passing through the two-phase region). The Gibbs free energy is obtained directly fro Q ex (G = k B T ln Q ex + pv) and utilized to obtain the coexisting ole fraction curves at constant pressure. More details about this are given in section 4. At the start of each siulation, 300 olecules with the desired total ole fraction are inserted randoly into a cubic periodic box of fixed volue MC steps are eployed for each partition of the energy. Each MC step involves either a translational ove or a rotational ove of a single olecule with the sae probability. At the beginning of the siulation, the axiu displaceent of a translational ove is set to L/4 and that of a rotational ove is set to 180. The acceptance rates of both oves are 100% during the first division of the energy (as iplied by the weighting function). They then decrease as the nuber of energy subdivisions,, increases. Once becoes large enough for the acceptance rates to drop to 30%, the axiu displaceents are adjusted autoatically after each energy division to keep acceptance rates of about 30% throughout the rest of the siulation. A spherical cutoff of half of the length of the siulation box is used to truncate the potential energy. Thus, a long-range correction for the dispersion r 6 ter (tail correction) 3 is used. The r 12 ter decays rapidly with distance, so a correction for this ter is unnecessary. The CO 2 and CH 4 olecules are nonpolar; the quadrupole quadrupole interactions are sufficiently short range that a long-range correction for the electrostatic energy is not needed. This has been exained and confired in our previous study on this syste. 31 The efficiency and accuracy of the siulations depend on the choice of the nuber of MC steps used in each partition of the energy. Fewer MC steps give faster calculations but lower accuracy. Figure 2 shows the effect on the accuracy of the Figure 2. Effect on the partitioning ethod of the nuber of MC steps used in each partitioning of the energy (50% CO 2 and 50% CH 4 at nuber density = Å 3 ). (a) Errors in selecting the energy boundary in the high-energy region. (b) Errors in selecting the energy boundary in the low-energy region. (c) Errors in the excess partition function (at K). The total nuber of energy subdivisions under these conditions is about The standard deviations are deterined by perforing a set of 15 different siulations. D

5 ethod of the nuber of MC steps used to deterine each energy boundary. In the early state of the partitioning process (the first few hundred partitions), or fewer MC steps could be used (Figure 2a), but as the nuber of energy subdivisions grows, at least MC steps are needed, as the siulations take longer to equilibrate (Figure 2b). If fewer than MC steps are eployed in the low-energy region, a systeatic error occurs. The excess partition function is ost accurately calculated with at least MC steps (Figure 2c). In the rest of this work, MC steps are used. To speed up the calculations, fewer MC steps could be used in the highenergy region than in the low-energy region, but we have not investigated this possibility further. In the first iteration of the siulation, all MC steps are in the lowest-energy subdivision, which spans the entire energy range (Figure 1a). In the second iteration, the weighting between the energy subdivision 1 ( = 1) and the energy subdivision 2 ( = 2) is 1:4 and, therefore, we expect 1/5 of the total MC steps ( 9000) to be in the high-energy subdivision and 4/5 of the total MC steps ( ) in the low-energy subdivision. The energy boundary E 2 is drawn fro the MC steps in the low-energy subdivision. In the third iteration, the ratio between the weighting of the energy subdivisions 1, 2, and 3 (lowest-energy) is 1:4:16 and the integrated density of states is 2:1:1. Thus, we expect to collect 2/22 ( 4091), 4/22 ( 8182), and 16/22 ( ) of the total MC steps in the energy subdivisions 1, 2, and 3, respectively. The energy boundary E 3 is the edian configurational energy found in the MC steps in the lowestenergy subdivision. As the nuber of energy subdivisions increases, about 2/3 ( ) MC steps are found in the current lowest-energy subdivision, as dictated by the weighting function, and the energy boundary E n is set equal to the edian configurational energy found in the lowest-energy subdivision. 4. RESULTS AND DISCUSSION The calculated density of states is used to obtain the partition function, free energy, vapor liquid equilibriu properties, and cheical potential of the binary ixture CO 2 +CH 4. After the partitioning process, the entire energy range (for a fixed coposition and volue) has been discretized into n + 1 energy subdivisions (see Figure 1). The excess partition function is obtained using eq 6, and the excess Helholtz free energy (F ex ) is calculated using F ex = k B T ln Q ex. The density dependence of the ideal gas free energy (F id ) is calculated using F id = k B T(x CO2 ln ρ CO2 + x CH4 ln ρ CH4 ), where x is the ole fraction and ρ is the nuber density. The densitydependent part of the absolute free energy is the su of the excess part and the ideal gas part (F = F ex + F id ). The Gibbs free energy is G = F + pv, where p is the pressure and V is the volue of the syste. Figure 3 shows the Gibbs free energy per particle versus volue per particle for CO 2 +CH 4 ixtures at K, bar and K, bar. This shows the preference for each phase of the syste under different conditions. For exaple, at K and bar (Figure 3a), the liquid phase is favored (a lower iniu in the free energy) when there is no CH 4 present. When the ole fraction of CH 4 is 0.2, two inia are observed, but the liquid phase is still favored. As the total ole fraction of CH 4 increases, the vapor phase becoes favored. The liquid phase is less favored and disappears when the ole fraction of CH 4 reaches about 0.6; i.e., the critical coposition is reached for this teperature. Figure 3. Gibbs free energy per particle versus volue per particle of CO 2 +CH 4 at different copositions: (a) K and bar and (b) K and bar. The standard errors are deterined by perforing a set of 15 different siulations for each volue. In ost cases, the errors in G/N are between 0.4% and 0.7%. For the sake of clarity, the plots at the copositions of CH 4 equal to 0.0, 0.2, 0.4, 0.6, and 0.8 are offset by 5, 3.5, 2.5, 1.5, and 0.5 kj/ol, respectively. Siilar observations can also be ade at other conditions, for exaple, at K and bar (Figure 3b). Using the inia in Figure 3, a plot of the Gibbs free energy versus the coposition for both phases is constructed, fro which the vapor liquid equilibriu copositions are deterined (Figure 4). The coexisting copositions are deterined by constructing a double coon tangent connecting both phases. The cheical potential for each species at any coposition can be calculated by taking G and dg/dx fro Figure 4 and solving the siultaneous equations μ + μ = x1 1 x2 2 G =μ μ d G /dx1 1 2 (7) for μ 1 and μ 2, where μ is the cheical potential and can be either vapor or liquid phase. The phase diagra (pressure versus coposition) of the binary ixture CO 2 +CH 4 is extracted fro Figure 4 and plotted in Figure 5, with results fro experient 40 and fro the Gibbs enseble technique. 31 The Gibbs enseble technique is teperature-dependent. Its efficiency depends on the nuber of state points that are required to be calculated. E

6 Figure 4. Gibbs free energy per particle versus CH 4 coposition for CO 2 +CH 4 : (a) K and bar and (b) K and bar. The standard deviations are deterined and shown as error bars by perforing a set of 15 different siulations for each volue. The energy partitioning ethod, on the other hand, is teperature-independent and does not suffer fro this shortcoing. The vapor liquid equilibriu properties coputed using both siulation techniques agree well to within the statistical uncertainties of the siulations apart fro a couple of points at low pressure. This discrepancy is, perhaps, due to the errors in fitting the curves and placing the double tangent lines. Reasonable agreeent between experiental data and siulations is achieved for the liquid phase at both teperatures. However, the siulations slightly overestiate the solubility of CH 4 in the vapor phase, which is due to the liitation of the force fields. This has been discussed in our previous work. 31 Due to the finite size effect, the Gibbs enseble technique cannot be used near the critical point. Using the energy partitioning ethod, we observe the occurrence of the critical point by constructing a plot of the Helholtz free energy versus the volue of the syste. Figure 6 shows such plots for ixtures of CO 2 +CH 4 at and K. The critical point occurs when the second and third derivatives d n F/dV n are both zero, and the critical pressure is df/dv at the sae point. The occurrence of the critical point is observed visually. The critical coposition at K is x CH and the critical pressure is approxiately 70 bar. The critical coposition at K is x CH4 0.55, and the critical pressure is approxiately 81 bar. These results were obtained using ole fractions of CH 4 separated by 0.02 (not shown in Figure 6 for clarity), and the uncertainties in the critical coposition and pressure are estiated to be 0.02 and 5 bar, respectively. The critical pressure can also be extrapolated fro Figure 5. Phase diagra of the CO 2 +CH 4 syste at (a) K and (b) K. Energy partitioning ethod (squares with error bars) versus experiental data (pluses) and Gibbs enseble technique (crosses with error bars). Stars indicate the critical points calculated fro the energy partitioning ethod. the siulated data by plotting p versus x vap x liq. At the critical point, x vap x liq is equal to zero. These results agree with those given above. The apparent critical points of a finite syste depend on the syste size, V. These paraeters obey a scaling law behavior with V To investigate the sensitivity of the calculated critical points with the syste size used, we have calculated the critical points (critical pressures and copositions) for systes with double and half of the size of the studied syste. We found that the apparent critical pressures and CH 4 ole fractions of sall systes are slightly higher than those of bigger systes. However, these differences are still within the uncertainties of the estiation of the critical points. Figure 7 shows the cheical potentials of each species in the ixture CO 2 +CH 4 at the vapor liquid equilibriu condition, calculated using eq 7. At both teperatures, the cheical potential of CH 4 increases steeply while that of CO 2 decreases gradually as the pressure increases. The gradient of the cheical potential of CH 4 decreases as the pressure increases. This indicates that at the vapor liquid equilibriu condition the cheical potential of CH 4 is ore sensitive to pressure than that of CO 2 and that the solubility of CH 4 in CO 2 increases with the pressure. The rate of change of the cheical potentials with respect to the pressure of a given species (species 1, for exaple) at twophase equilibriu can be expressed in ters of the ole F

7 Figure 6. Helholtz free energy per particle versus volue per particle of the ixture CO 2 +CH 4 at different copositions at (a) K and (b) K. The standard errors are deterined by perforing a set of 15 different siulations for each volue. In ost cases, the errors are between 0.4% and 0.7%. For the sake of clarity, the plots in (a) at the copositions of CH 4 equal to 0.5, 0.54, 0.58, 0.62, and 0.64 are offset by 1.0, 0.8, 0.6, 0.4, and 0.2 kj/ol, respectively. The plots in (b) at the copositions of CH 4 equal to 0.3, 0.4, 0.5, and 0.55 are offset by 0.8, 0.6, 0.4, and 0.2 kj/ol, respectively. fraction of the other species in both phases and the olar volue (v) of each phase as β dμ β 1 x = 2v x2 v dp β x2 x2 (8) Thus, since the ole fractions and the volues of each phase are known (Figures 3 and 4), dμ 1 /dp can be calculated. For exaple, dμ CH4 /dp at K and bar is about 870 c 3 / ol and that calculated using eq 8 gives 890 c 3 /ol. The difference between these two nubers coes fro the uncertainties in placing the tangent line. Equation 8 can also be used to interpret the behavior of the cheical potentials shown in Figure 7. The rate of change of the cheical potential of CH 4 with respect to the pressure is (x liq CO2 v vap x vap CO2 v liq )/ liq (x CO2 x vap CO2 ). Since the olar volue of vapor is larger than that of liquid and the ole fraction of CO 2 in the liquid is always greater than in the vapor, this quantity is always positive. At low pressure, the olar volue of the vapor phase is uch larger than that of the liquid phase, and this results in a steep slope. As the pressure increases, the volue of the vapor phase decreases and also the ole fractions of CO 2 in both phases get closer to each other, which decreases the agnitude of dμ CH4 / dp. This is observed fro Figure 7. A siilar analysis can be done for dμ CH4 /dp, which equals (x liq CH4 v vap x vap CH4 v liq liq )/(x CH4 vap ). Although the volue of the vapor phase is uch greater x CH4 Figure 7. Cheical potentials of each species in the ixture CO 2 + CH 4 at the vapor liquid equilibriu condition versus pressure at (a) K and (b) K. than that of the liquid phase, the ole fraction of CH 4 in the liquid phase is saller than that in the vapor phase. Therefore, x liq CH4 v vap x vap CH4 v liq liq is positive, while x CH4 x vap CH4 is negative. Thus, dμ CH2 /dp is negative, which is observed in Figure 7b. 5. CONCLUSION In this paper, we have calculated the density of states, partition function, free energy, and cheical potential of the binary ixture CO 2 + CH 4 using an energy partitioning MC technique. The Gibbs free energy of the binary ixture is coputed as a function of the volue and the coposition. Knowing the Gibbs free energy allows us to predict the vapor liquid equilibriu properties, which agree quite well with experiental data and siulations using the Gibbs enseble technique. One of the advantages of the energy partitioning ethod is that it allows us to observe the occurrence of the critical coposition by constructing plots of the Helholtz free energy versus the volue. We also obtain the cheical potentials of each species in the ixture directly. These quantities are useful, but difficult to copute using other ethods. At the vapor liquid equilibriu condition, we find that the rate of change of the cheical potentials with respect to pressure of CH 4 increases significantly, whereas that of CO 2 slightly decreases. The cheical potential of CH 4 varies significantly with pressure, while that of CO 2 does not vary uch. This is understandable in ters of the coposition and the volue of both phases. The energy partitioning ethod can be applied to other binary ixtures and can be easily extended to study ternary ixtures and the free energy of transfer between two solvent phases. Standard ethods for easuring free energy differences, G

8 such as therodynaic integration and free energy perturbation, are not directly applicable to calculations of the free energy of transfer. 46 A cobination of the Gibbs enseble ethod, Wido s test particle 47 and the configurational biased MC technique 48,49 have been used to tackle this proble for alkanes. 46,50 Using the energy partitioning ethod, the free energy of transfer could be obtained fro the cheical potential. Since the ethod gives access to the free energy, it will be useful in any other applications. Also, the ethod can be extended to study systes with discrete energy levels, for which there is also a large nuber of applications. AUTHOR INFORMATION Corresponding Author *E-ail: haina.do@nottingha.ac.uk. Notes The authors declare no copeting financial interest. ACKNOWLEDGMENTS We thank the University of Nottingha High Perforance Coputing facility for providing coputer recourses and the Engineering and Physical Sciences Research Council (EPSRC) for funding (Grant No. EP/E06082X). We thank Prof. Martyn Poliakoff and Dr. Jie Ke for useful discussions. H.D. is grateful to the EPSRC for a Ph.D. Plus Fellowship. REFERENCES (1) Henderson, D.; Barker, J. A. Phys. Rev. A 1970, 1, (2) Zwanzig, R. W. J. Che. Phys. 1954, 22, (3) Frenkel, D.; Sit, B. Understanding Molecular Siulation: fro Algoriths to Applications, 2nd ed.; Acadeic Press: San Diego, CA, (4) Hansen, J.; Verlet, L. Phys. Rev. 1969, 184, (5) Head, M. S.; Given, J. A.; Gilson, M. K. J. Phys. Che. A 1997, 101, (6) Krisvov, S. V.; Karplus, M. J. J. Che. Phys. 2002, 117, (7) Evans, D. A.; Wales, D. J. J. Che. Phys. 2003, 119, (8) White, R. P.; Meirovitch, H. J. Che. Phys. 2003, 119, (9) White, R. P.; Meirovitch, H. Proc. Natl. Acad. Sci. U.S.A 2004, 101, (10) Meirovitch, H. Phys. Rev. A 1985, 32, (11) Meirovitch, H. J. Che. Phys. 1999, 111, (12) Tyka, M. D.; Clarke, A. R.; Sessions, R. B. J. Phys. Che. B 2006, 110, (13) Tyka, M. D.; Sessions, R. B.; Clarke, A. R. J. Phys. Che. B 2007, 111, (14) Ytreberg, F. M.; Zuckeran, D. M. J. Che. Phys. 2006, 124, (15) Zhang, X.; Maonov, A. B.; Zuckeran, D. M. J. Coput. Che. 2009, 30, (16) Bhatt, D.; Zuckeran, D. M. J. Phys. Che. 2009, 131, (17) Torrie, G. M.; Valleau, J. P. J. Coput. Phys. 1977, 23, (18) Ferrenberg, A. M.; Swendsen, R. H. Phys. Rev. Lett. 1988, 61, (19) Ferrenberg, A. M.; Swendsen, R. H. Phys. Rev. Lett. 1989, 63, (20) Wang, J. S.; Tay, T. K.; Swendsen, R. H. Phys. Rev. Lett. 1999, 82, (21) Berg, B.; Neuhaus, T. Phys. Lett. B 1991, 267, (22) Wang, F. G.; Landau, D. P. Phys. Rev. Lett. 2001, 86, (23) Ganzenuller, G.; Cap, P. J. J. Che. Phys. 2007, 127, H (24) Shell, M. S.; Debenedetti, P. G.; Panagiotopoulos, A. Z. Phys. Rev. E 2002, 66, (25) Shell, M. S.; Debenedetti, P. G.; Panagiotopoulos, A. Z. J. Che. Phys. 2003, 119, (26) Yan, Q.; Faller, R.; de Pablo, J. J. J. Che. Phys. 2002, 116, (27) Do, H.; Hirst, J. D.; Wheatley, R. J. J. Che. Phys. 2011, 135, (28) Oakley, M. T.; Wheatley, R. J. J. Che. Phys. 2009, 130, (29) Oakley, M. T.; Do, H.; Wheatley, R. J. Fluid Phase Equilib. 2009, 290, (30) Oakley, M. T.; Do, H.; Hirst, J. D.; Wheatley, R. J. J. Che. Phys. 2011, 134, (31) Do, H.; Wheatley, R. J.; Hirst, J. D. J. Phys. Che. B 2010, 114, (32) Do, H.; Wheatley, R. J.; Hirst, J. D. Phys. Che. Che. Phys. 2010, 12, (33) Do, H.; Wheatley, R. J.; Hirst, J. D. Phys. Che. Che. Phys. 2011, 13, (34) Poliakoff, M.; King, P. Nature 2001, 412, (35) Skaroutsos, I.; Hunt, P. A. J. Phys. Che. B 2010, 114, (36) Harris, J. G.; Yung, K. H. J. Phys. Che. 1995, 99, (37) Vorholz, J.; Hariseadis, V. I.; Rupf, B.; Panagiotopoulos, A. Z.; Maurer, G. Fluid Phase Equilib. 2000, 170, (38) Liu, Y.; Panagiotopoulos, A. Z.; Debenedetti, P. G. J. Phys. Che. B 2011, 115, (39) Chen, B.; Siepann, J. I. J. Phys. Che. B 1999, 103, (40) Wei, M. S. W.; Brown, T. S.; Kidnay, A. J.; Sloan, E. D. J. Che. Eng. Data 1995, 40, (41) Chen, J. H.; Fisher, M. E.; Nickel, B. G. Phys. Rev. Lett. 1982, 48, (42) Ferrenberg, A. M.; Landau, D. P. Phys. Rev. B 1991, 44, (43) Wilding, N. B. Phys. Rev. E 1995, 52, (44) Wilding, N. B. Phys. Rev. E 1997, 55, (45) Potoff, J. J.; Panagiotopoulos, A. Z. J. Che. Phys. 1998, 109, (46) Martin, M. G.; Siepann, J. I. J. A. Che. Soc. 1997, 119, (47) Wido, B. J. Che. Phys. 1963, 39, (48) Siepann, J. I.; Frenkel, D. Mol. Phys. 1992, 68, (49) Frenkel, D.; Mooji, G. C. A. M.; Sit, B. J. Phys.: Condens. Matter 1992, 4, (50) Martin, M. G.; Siepann, J. I. Theor. Che. Acc. 1998, 99,