Standard thermodynamic properties of solutes in supercritical solvents: simulation and theory

Chemical Physics Letters 381 (2003) 771 776 www.elsevier.com/locate/cplett Standard thermodynamic properties of solutes in supercritical solvents: simulation and theory J.L. Alvarez a,c, R. Fernandez-Prini a,b, *, E. Marceca b a Unidad Actividad Quımica, Comision Nacional Energıa Atomica, Av. Libertador 8250, uenos Aires, Argentina b INQUIMAE, Facultad Ciencias Exactas y Naturales, Universidad de uenos Aires, Ciudad Universitaria, Pabellon II, uenos Aires, Argentina c Dto. Ingenierıa Quımica, FRA, Universidad Tecnologica Nacional, Medrano 951, uenos Aires, Argentina Received 11 September 2003; in final form 15 October 2003 Published online: 4 November 2003 Abstract We report a modified simulation procedure to calculate partial molar quantities of dilute solutions in supercritical fluids through the Krichevskii function. This procedure circumvents the difficulties posed by the very large solvent compressibility and expansivity in the near-critical region of the solutions where other simulation procedures cannot be used. Ó 2003 Elsevier.V. All rights reserved. 1. Introduction In 1987 Ciccotti, Frenkel and coworkers [1] proposed a procedure to calculate partial molar quantities of binary fluid mixtures by simulation using a modified Widom insertion method [2]. eing partial molar quantities derivatives of extensive properties, the procedure used up to then to calculate their values required two simulation experiments carried out in the NpT Gibbs ensemble for two different compositions of the binary mixture. That procedure duplicated the error and the number of runs, because of the need to use two * Corresponding author. Fax: +541167727121. E-mail address: rfprini@cnea.gov.ar (R. Fernandez-Prini). simulation experiments. The procedure proposed by Ciccotti, Frenkel and coworkers involves the (virtual) exchange of a particle of component A by one molecule of compound in the Gibbs ensemble yielding the difference between the partial molar quantities of the two components by means of a single computational experiment. The equations for the partial molar volumes and enthalpies were derived [3] and these expressions were used to calculate the partial molar properties of Lennard Jones Ar Kr mixtures with the exchange procedure (EXP). The calculations were successful, but the equations derived for the EXP have an intrinsic limitation, the authors considered that the volume and the enthalpy of the mixtures were independent of pressure and of temperature, respectively. Since 0009-2614/$ - see front matter Ó 2003 Elsevier.V. All rights reserved. doi:10.1016/j.cplett.2003.10.050

772 J.L. Alvarez et al. / Chemical Physics Letters 381 (2003) 771 776 there is a special interest in the determination of standard (infinite dilution) partial molar quantities which are the corner-stone properties to describe the behaviour of solutes, we shall illustrate the limitations of the EXP equations with the simpler expressions for the partial molar volume of a solute at infinite dilution. The difference between the partial molar volumes V A and V of the two components, is given by [1,3] ex V expð bdua! V A V ¼ Þ N A ;N expð bdua! ex Þ hvi NA ;N ; NA ;N ð1þ where DUA! ex is the change in the energy of the system of N A molecules A and N molecules when one A is (virtually) exchanged for one molecule; h i denotes an average taken over the ensemble of ðn A þ N Þ particles. If the change of volume with pressure is taken into account, ðv A V Þ becomes V A V Dh i E V þ p ov expð bdu ex op A! Þ T N ¼ A ;N expð bdua! ex Þ N A ;N V þ p ov : ð2þ op T N A ;N When the dilute solutions are close to the solventõs critical point, ðov =opþ T is very large becoming infinity at the critical point for solutions at infinite dilution. Taking into account the difficulties of the experimental determination of the properties of solutes dissolved in supercritical fluids, which have become of great importance in chemical processes and in studying the physical chemistry of solvation phenomena, it would be very valuable to use simulation methods to determine, or at least estimate, the values of those quantities. The method devised by Ciccotti, Frenkel and coworkers will intrinsically fail for supercritical solutions which are characterized by large compressibilities. For attractive solute A (that interacts more strongly with the solvent than two solvent molecules do) VA 1 is large and negative for wide ranges of temperature and density. A similar situation occurs for partial molar enthalpies at infinite dilution since the heat capacities also become very large in the near-critical region, so the dependence of the enthalpy on temperature cannot be neglected. On the other hand, it has been established that partial molar quantities are not convenient to base the description of the near-critical behaviour because these quantities diverge at the solventõs critical point [4]. Using the volume as the archetypical property, the soluteõs partial molar VA 1 is given by " # 1 V 1 A ¼ V 1 op þ j T ¼ V ox ð1 þ j T JÞ; ð3þ A V ;T where x A is the mole fraction of A, J is called the Krichevskii function, a well behaved quantity at the solventõs critical point and j T is the isothermal compressibility of solvent, a strongly diverging quantity at the critical point of. This is the reason why the favoured description of the thermodynamic properties of near-critical and supercritical solutes is based upon J [5,6]. From the definition of J it is natural to use a modified EXP for the calculation of J by simulation experiments in the NVT ensemble. The main goal of this preliminary communication is to test the feasibility of this approach and evaluate its performance for simple attractive solutes and different reduced temperature T red ¼ ðt =T C Þ, where T C is the solventõs critical temperature. Consequently we have restricted this study to binary Lennard Jones systems, because it is possible to compare the results of the simulation procedure hereby proposed for J with an equation of state that fits many simulation data for Lennard Jones fluids [7] and also with the results of an inhomogeneous integral equation [8], the latter equation has recently been proved successful to describe the behaviour of near-critical systems [8,9]. 2. asis of the calculation procedure The application of the EXP to a function f ðr i Þ in the NVT ensemble leads to the following general expression:

J.L. Alvarez et al. / Chemical Physics Letters 381 (2003) 771 776 773 hf ðr i Þi NA þ1;n 1 ¼ f ðr iþ expð bdua! ex Table 1 N A ;N A lres ÞŠ : Lennard Jones parameters used r (nm) (e=k) (K) Ref. ð4þ Ne 0.282 36.86 [14] The average value of the function for the system Kr 0.3684 174.68 [15] consisting of N A þ 1; N 1 particles f ðr i Þ in the NVT ensemble, may be calculated by averaging the function f ðr i Þ expð bdua! ex Þ for the ensemble of N A ; N particles and dividing that quantity by Xe 0.4067 224.83 [15] A lres Š, where lres I is the residual chemical potential of component I. Eq. (4) was derived by a straightforward application of the published equations [1,3]. In our case f ðr i Þ is the pressure of the system when one A (solute) is added replacing a (solvent) molecule. For our purpose we need to calculate the difference in pressure p of the system having ðn A þ 1; N 1Þ particles p NA þ1;n 1 N A þ1;n and p 1 NA ;N N A ;N. According to Eq. (4) this difference is obtained averaging the two quantities in the N A ; N system. Using Eq. (4) we get J ¼ p N A þ1;n 1 expð bdua! ex Þ, A lres ÞŠ p NA ;N x A : N A ;N ð5þ The pressure was calculated in each Monte Carlo simulation step with the virial equation [10]. In every step a solvent molecule was chosen at random and exchanged by a solute particle and DU ex A! and the pressure were calculated for the virtual N Aþ1 ; N 1 system.the quantity inside the angular brackets of Eq. (5) was calculated in every step of the simulation and then averaged. With the same program the value of the difference of residual chemical potentials was calculated with the expression [1,3] A lres ÞŠ ¼ expð bdu ex A! Þ N A ;N : ð6þ The systems studied in this work consisted of 125 Lennard Jones solvent molecules having the intermolecular parameters of neon. One of the solvent molecules was exchanged in every Monte Carlo step by a Lennard Jones Kr or Xe atom. Runs were made for fluid densities within the range of interest for applications of supercritical solvents and at two values of T red > 1 for each ÔsoluteÕ atom. The size of the simulation box was adjusted in every run to accommodate the 125 particles in the box at the chosen density, and the interactions were truncated at a distance equal to half of the box length. The number of steps of each run was between 0.2 and 1 10 6 depending on T red and strength of solute solvent interaction. The contribution to J of particles beyond the cutoff distance was calculated using for the tail contribution to pressure the equation given by Frenkel and Smit [11]. Applying this expression for the exchange of one molecule by that of the A solute molecule, we obtain J tail Dptail A! x A ¼ 32pq 2 e r 3 " e A e r A r 3 T A T #; ð7þ where T IJ gives the contribution of the integral of the distance dependent function between the cutoff reduced distance z cut IJ ¼ r=r IJ and infinity T IJ ¼ 1 2 : ð8þ 3z 3 IJ 9z 9 IJ The parameters of the Lennard Jones potential for the three atoms are given in Table 1. The parameters for the interaction of solvent molecules with the solute where obtained from those of the pure components using the simple Lorentz erthelot combining rule. 3. Equation of state and integral equation used for comparison There are two other tools that we have used to check the results obtained with the EXP simulation experiments. The first one is the JZG equation

774 J.L. Alvarez et al. / Chemical Physics Letters 381 (2003) 771 776 Table 2 Values of J (MPa) calculated with the Monte Carlo exchange procedure Solute T red qr 3 0.2 0.25 0.33 0.40 Xe 1.30 )45.1, )51.5 )84.1 )128.2, )126.2 Xe 1.10 )44.6 )46.5 )103.3 Kr 1.10 )26.7, )32.7 )71.6 Kr 1.03 )30.5 )72.2, )80.7 qr 3 0.45 0.50 0.55 0.60 Xe 1.30 )172.6 Xe 1.10 167.2 )187.6 )186.6 Kr 1.10 )128.3 )138.4 Kr 1.03 )86.4, )92.6 )109.4 )146.7 of state developed by Johnson et al. [7] from results of many MD and MC simulations for Lennard Jones pure fluids. The equation of state fitted to the simulation results was based upon a modified enedict Webb Rubin formulation, thus it is able to describe also the critical region of the fluid. Johnson et al. [7] remark that their equation of state may be also applied to conformal binary mixtures, hence it should be applicable to our Lennard Jones atomic binary mixtures. For this purpose we used the JZG equation employing the van der Waals one-fluid theory, as recommended by the authors, the pressure was then differentiated with respect to composition (cf. Eq. (3)) and finally the limit to infinite dilution was taken. The second tool used in the present work to compare with simulation results was the integral equation for inhomogeneous fluids, known as hydrostatic hypernetted chain (HHNC) equation [12]. We have shown recently [13] that HHNC can be successfully employed to account for many observations of equilibrium and structural properties of near-critical solutions. In this case J was calculated using the solvent solvent and the solvent solute direct correlation functions according to the equation bj ¼ q 2 ½^c ð0þ ^c A ð0þš; ð9þ where q is the density of the solvent and ^c IJ ð0þis the Fourier transform of the direct correlation function for the wave vector equal to zero. The fact noted previously [13] that to get a more reliable result for J with HHNC it was convenient to truncate the integral ^c A ð0þ after three molecular diameters from the solute, introduced some uncertainty about the correctness of the values Fig. 1. Values of J for Xe dissolved in Ne. (}) MC results; solid curve, JZG equation of state; dashed curve, HHNC equation. (a) T red ¼ 1.30, (b) T red ¼ 1.10.

J.L. Alvarez et al. / Chemical Physics Letters 381 (2003) 771 776 775 obtained for J. The results of the present work show conclusively that the truncation procedure does not introduce significant changes. 4. Results and discussion The values of J obtained for the two binary mixtures are reported on Table 2, the thermodynamic state points studied correspond to densities typical of those used in studies involving supercritical solvents. The reproducibility of the calculated values is 10% in the worst case and more frequenty close to 5%. It should be stressed that the reproducibility decreases substantially when T red approaches unity and also when the solute solvent interaction is larger [8,16], this is why we did not calculate J for Xe in Ne at T red < 1:10 with the EXP. Hence, it would be misleading to give an overall value for the error, we preferred to make duplicate runs for five state points, shown in Table 2, which are representative of data reproducibility. For Kr in Ne even at T red ¼ 1:03 and reduced density 0.25, which is very close to the critical density of the solvent, the value of J is very reasonable. Figs. 1 and 2 are plots of the calculated J with EXP and with the JZG equation [7] and with HHNC; the results calculated with the Monte Carlo EXP are in very statisfactory agreement with the others. The two other procedures yield values of J that are very close indeed with the exception of the high density region (cf. Figs. 1b and 2a,b), since that density region is outside of our main interest to cover conditions typical of supercritical solutions, we shall not discuss the possible causes for this discrepancy. 5. Conclusion Our work shows that the EXP is a valuable tool to calculate J and that this route also enables the calculation of VA 1 using the value of the Krichevskii function, the compressibility of the pure solvent and Eq. (3). The precision of the results of EXP is limited by the closeness to the solventõs critical point and by the strength of the solute solvent interaction. However the use of a larger simulation box, that will imply the use of more particles, will allow approaching closer to the solventõs critical point, and/or enable to calculate properties for systems with larger solute solvent interactions [13]. This will be our next goal which will be used to calculate values of the Kritchevskii function at the solventõs critical point for dilute aqueous solutions. Acknowledgements Fig. 2. Values of J for Kr dissolved in Ne. (}) Monte Carlo results; solid curve JZG equation of state; dashed curve HHNC equation. (a) T red ¼ 1.10, (b) T red ¼ 1.03. The authors are grateful to ANPCyT for partial economic support and to Dr. D.H. Larıa for making available to us the basic Monte Carlo program.

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