- Email: [email protected]
Monte Carlo Simulation of Methanol Diffusion in Critical Media1
Chinese J. Chem. Eng., 14(3) 413-418
(2006)
RESEARCH NOTES
Monte Carlo Simulation of Methanol Diffusion in Critical Media* JIA Yuxiang( 3 ;$)a*b and GUO Xiangyun(@ I$ Z)%** 'State Key Laboratory of Coal Conversion, Institute of Coal Chemistry, CAS, Taiyuan 030001,China bSchool of Chemical and Materials Engineering, Southern Yangtze University, Wuxi 214122, China
Abstract The diffusion behavior of methanol in different critical media (n-pentane, n-hexane, n-heptane and acetone) was investigated by the Monte Carlo (MC) method. From the simulation results, the diffusion constant of methanol molecule in the critical n-hexane is much larger than those in n-pentane, n-heptane and acetone. By analyzing the microscopic configurations of the critical mixtures, it is found that the diffusion constant of methanol is related to the local solvent clustering around methanol, but it does not exhibit strong dependence on the size of solvent cluster around methanol. Moreover, the survival time of the solvent cluster plays an important role in determining the diffusion constant. Keywords critical fluids, Monte Carlo simulation, methanol, diffusion
1 INTRODUCTION Supercritical fluids (SCFs) have already been known for more than 150 years. However, only in the latest decades, they attracted considerable attention and obtained important applications"]. The impetus for the applications of SCFs originates from their unique solvent properties. Low viscosity, high diffusivity and adjustable solvency have enabled SCFs to be used as media for chemical reactions, adsorptionheparation and extraction Many studies have shown that the use of SCFs as reaction media leads to significant increase in reaction rate and obvious variation in product distrib~tiod~]. These features have attracted many interests in employing SCFs as a possible alternative instead of traditional liquid solvents. When using conventional liquid solvents, one has to employ multifarious solvents in order to obtain required physicochemical properties (e.g. solvency, density, and dielectric ~onstant)'~]. However, SCFs are substantially flexible, and thus provide a means to actually tune physicochemical properties by adjusting the fluid temperature and/or pressure. As a result, SCFs are generally considered as continuously property-tunable solvents. In addition, compared with classical organic solvents, SCFs have less health and environment riskst6]. In view of the unique properties of SCFs, Zhong et aLL7' studied the methanol synthesis process in near-critical and supercritical n-pentane, n-hexane,
n-heptane and acetone, and found that the CO conversion for one way was greatly increased up to 90% by employing supercritical n-hexane as reaction medium and the reaction achieved a maximum of CO conversion when operating the process near the critical point of n-hexane. Theoretical calculations about the methanol synthesis showed that the addition of suitable solvents such as n-hexane and n-heptane could greatly improve the CO conversion under supercritical conditions by using the Soave-Redlich-Kwong equation of state[*].Zhang et uZ.'~' studied the microscopic structures of the binary mixture of methanol-hexane under different conditions by the Monte Carlo (MC) method. The work suggested that there existed significant clustering of n-hexane among methanol molecules under the supercritical condition. They also simulated the supercritical methanol synthesis in copper catalyst with ZGB model and investigated the influence of methanol aggregation number (MAN) and the mole fraction of n-hexane on the catalyst activity"']. Previously, we studied the effect of n-hexane on methanol desorption from copper catalyst surface and found that there existed evident clustering between solvent and solute molecules under the critical condition by the MC method'"]. In this work, the diffusion behavior of methanol in different critical fluids is investigated, and the reasons resulting in the different diffusion behavior of methanol molecule are discussed.
Received 2005-08-02, accepted 2006-01-08.
* Supported by the One-Hundred-Talent Program of the Chinese Academy of Sciences.
** To whom correspondenceshould be addressed. E-mail: [email protected]
414
Chinese J. Ch. E. (Vol. 14, No3)
2 SIMULATION DETAILS 2.1 Monte Carlo made1 The MC method used in the present work involved random walk of molecules with an acceptance probability determined by the energy and temperature. The simulation was carried out in a NVT ensemble (system with N particles at temperature T and volume V), and the standard procedure of the Metropolis sampling with the periodic boundary condition was employed. The initial configuration of the system was randomly given. The molecular translation and rotation"*] were considered in the simulation. In the random movement, the maximum displacement and rotation were 0.03nm and 15", respectively. These motions were accepted by a certain probability of 25%35% in the simulation. A Monte Carlo Step (MCS) was defined as the process in which all molecules in the simulation cell finished one trial of random movement, and one MCS was assumed equal to a time duration of 1O-I4s. Link-cell-list was used in the simulation to speed up the cal~ulation['~]. 2.2 Potentials The TIP function[141was employed to describe the intermolecular interaction of methanol and acetone. The methanol and acetone molecules were represented by three interaction sites (CH3, 0, H)[I5] and four interaction sites (CH3, C, 0, CH3)[16],respectively. Each pair interaction was calculated by
(3) where ~ v a n dau are Lennard-Jones parameters, qi is the partial charge assigned to each site and e is the magnitude of electron charge. The Lennard-Jones potential was used for calculating intermolecular potential of other molecules in the system. The Lennard-Jones potential is of the form
The parameters employed in the simulation were tabulated in Table l[15--171. June, 2006
a b l e 1 The potential parameters employed in the simulation Molecule methanol
acetone n-pentane n-hexane n-hentane
Site CH3 0 H 0 CH3 C CH3
9, e 0.265 -0.700 0.435 -0.424 0.062 0.300 0
u, nm 0.3775 0.3070 0 0.2960 0.3910 0.3750 0.3905
CH2
0
0.3905
E,
J.rno1-I 869.4 714.0 0 445.2 340.2 222.6 730.8 491.4
2.3 Calculation method The MC simulation was applied to each binary system of methanol and a kind of solvent. We selected structurally identical n-pentane, n-hexane, and n-heptane as nonpolar solvent, and acetone as polar solvent. The self-diffusion constants of methanol molecule in different dilute critical solutions were calculated by analyzing the mean square displacement of methanol molecule in 70 picoseconds (lps=lOO MCS). To explore the reason resulting in the difference between the diffusivities of methanol molecule in different media, the microscopic configuration of the binary system was investigated in terms of radial distribution function (RDF). The critical parameters employed in the simulation were tabulated in Table 2[17]. Tgble 2 The critical parameters of the solvents employed in the simulation n-pentane n-hexane n-heptane acetone
494 523 556 509
0.230 0.233 0.233 0.278
3.369 3.014 2.734 4.600
3 RESULTS AND DISCUSSION 3.1 Diffusion constant of methanol in different critical media Through modeling the random walk of methanol molecule in the critical fluids, the mean square displacement (MSD) of methanol in different time can be obtained, as shown in Fig.1. In general, the relation between MSD and time follows Einstein's equation,
(5)
Monte Carlo Simulation of Methanol Diffusion in Critical Media
From Einstein's equation, the diffusion constant D can be estimated from Fig.1 and summarized in Table 3. From Table 3, the methanol diffusivities in different solvents are very different. It is found that the diffusion constant of methanol in n-hexane is much higher than those in other solvents. The high diffusivity is beneficial to the methanol desorption from catalyst surface.
I
I
415
surface reaction is compelled to the progress toward the direction for methanol synthesis. The calculated results are consistent with the experiment^'^], which suggests that n-hexane was the best medium among other investigated near- or supercritical fluids for the methanol synthesis, and the reaction achieves a maximum CO conversion near the critical point of n-hexane. The present results indicate that the diffusivities of methanol in various critical media are very different. The high diffusivity of methanol can effectively enhance the methanol desorption from the catalyst surface, and therefore increase the CO conversion of the methanol synthesis process.
Microscopic configurationsof methanol-solvent mixture The importance of the local structure in SCFs has been noticed p r e v i o ~ s l y " ~ .To ~ ~explain ~. the reason resulting in the difference of methanol diffusivities in different solvents, local structures of methanol-solvent mixtures under their critical conditions were investigated. The different distributions of solvent molecules around methanol at their critical points are shown in Fig.2. The local solvent density around methanol molecule is far higher than their corresponding bulk density. This indicates that there exists an obvious enrichment or clustering of solvent molecules around methanol. The most evident enrichment is found in the critical n-pentane, then n-hexane, n-heptane and acetone in sequence. The diffusion behavior of solute in different supercritical fluids has been explained in terms of the local solute-fluid density a~gmentation"~]. It is thus reasonable to suppose that the increased local solvent density would diminish the diffusivity of methanol molecule in the solvents. Therefore, the radial distribution functions (RDFs) of different solutions are analyzed to understand the influence of solvent-solute clustering on the methanol diffusivity. By integrating the RDFs, the coordination number of solvent molecules around methanol can be obtained, as tabulated in Table 4. Correspondingly, the dependence of methanol diffusivity on the coordination number is shown in Fig.3. There is no obvious relationship between the coordination number and methanol diffusivity. Both n-hexane and n-pentane have large ordination numbers, but the diffusivities of methanol molecule in them are far different. Methanol molecule has a very large
3.2
2ootLLzE I00
0
1000 2000 3000 4000 5000 6000 7000 8000 Monte Carlo Step
Figure 1 The mean square displacement of methanol molecule in the critical n-pentane,n-heme, n-heptane and acetone 0 C6H14; C7H16; 7 C5Hn; W c3&0 'Igble 3 Diffusion constants of CHJOHin different critical solutions Item density, g ~ r n - ~ 0.230
DX lo8,m 2 K 1
0.29
C ~ H M GHi6 0.233 0.233 1.22 0.31
C3&0 0.278 0.032
In many heterogeneously catalytic reactions, there usually exists a liquid layer on the catalyst surface. The concentration of product in the layer is very high. The high concentration of product in the layer often prevents product molecules to desorb from the catalyst surface"*'. It is generally established that supercritical media can solubilize the liquid layer and minimize the diffusion resistance of product in the layer"81. In the methanol synthesis, methanol molecules are primarily produced on the catalyst surface via a series of elementary steps. These molecules exist on the surface as adsorbed species. Due to the competitive adsorption between methanol and solvent molecules, the existence of critical media in most cases can hasten the methanol desorption from the catalyst surface. The methanol molecule in the critical n-hexane has the largest diffusivity, and therefore is most easily desorbed from the surface. As a result, the
Chinese J. Ch. E. 14(3) 413 (2006)
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Chinese J. Ch. E. (Vol. 14, No3)
diffusivity in n-hexane, and oppositely a very small one in n-pentane. Therefore, it is suggested from the present results that the size of the solute-solvent cluster does not play a crucial role in determining the methanol diffusivity. 4t
r, nm
Figure 2 Radial distributions of solvent molecules around methanol in four systems l-cH30H-C~H12; 2-CH30H-C&14; 3-cH30H-C7H16; 4--CH;OH-C3&0
Table 4 The coordination number of solvent molecules around methanol Coordination number
14.32
8.49
4.49
4.18
1.4
1.2
-
-6
1.o
Y)
0.8
m
2 0.6 X
9 0.4
0.2
0
\
4
6
12 coordination number 8
10
14
16
Figure 3 Relationship between the methanol diffusivity and the coordination number of solvent-solute clusters
the investigated mixtures. However, the RDFs cannot give information about the survival time of these clusters. Therefore, it is still questionable how these solvent clusters influence the methanol diffusion. In this section, the stability of solvent clusters in the critical mixtures is investigated, and the concept of lifetime is employed to characterize the stability of the solvent clusters[*’]. Usually the solvent clusters under the critical conditions are kinetic, and this indicates that the molecules in each cluster are very variable. Therefore, we define the lifetime of a cluster as the time that more than 50% molecules in the cluster have moved away. To calculate the lifetime, the solvent molecules within a distance of 0.55nm to a given methanol molecule were counted[”]. The distance was given according to the position of the first peak in the RDFs shown in Fig.2. The evolution of fifty clusters containing five or more molecules in the binary system of methanol and a critical fluid was tracked, and the average lifetime of these clusters was calculated in different systems. The lifetimes of different solvent clusters were tabulated in Table 5. The lifetime of the hexane cluster is the shortest among the investigated solvents, and this indicates that the hexane cluster around methanol is the most unstable. Fig.4 shows the dependence of methanol diffusivity on the cluster lifetime. The diffusion constant of methanol declines with the increase of the lifetime. The results presented in Fig.4 may provide a clue to understand the diffusion behavior of methanol in different solvents. The solvent clustering around methanol can restrict the motion of methanol molecule, and the methanol diffusion may be achieved by continuously escaping from the solvent clusters. Obviously the methanol diffusion in unstable clusters is relatively easy. Because the hexane cluster is more unstable than others, the methanol molecule has the largest diffusivity in the critical hexane.
Table 5 The lifetime of solvent clusters Solvents lifetime, vs
CSHl2 0.90
c 6 H 1 4
0.25
C7H16 0.43
c3H60
10.0
~
3.3 Lifetimes of solvent clusters around methanol molecule Generally the radial distribution function can describe the fluid structure in a time-averaged, statistical approach. The RDFs shown in Fig.2 indicate that there exists obvious solvent clustering around methanol in June, 2006
3.4 Hydrogen-bonding analysis From Table 5, the lifetime of acetone cluster around methanol is much longer than those of other solvents. To investigate the reason, we calculated the pair distribution function of oxygen from acetone and hydrogen from methanol, as shown in F i g 5 There is a
Monte Carlo Simulation of Methanol DifPusion in Critical Media
417
found that the solvent clustering around methanol molecule is very popular under the critical conditions. The combined effect of the size and lifetime of the solvent cluster determines the methanol diffusion. However, the lifetime of the solvent cluster plays a more important role in determining the diffusivity of methanol molecule in critical media. Because the lifetime of the n-hexane cluster is the shortest, the methanol diffusion in the critical n-hexane has the largest diffusivity among the investigated solvents. On the contrary, due to the strong hydrogen-bonding between methanol and acetone, the acetone cluster is very stable, so as to result in a very low diffusivity of methanol.
lifetime, ps
Figure 4 Relationship between methanol dupusivity and the lifetimes of solvent clusters
sharp peak appearing at the position of r = 0.19nm, which is associated with the formation of hydrogen bonding between acetone and methanol. Because the hydrogen bonding between acetone and methanol is very strong, the acetone cluster is very stable and thus has a long lifetime. This is also an important reason leading to the low diffusivity of methanol in the critical acetone.
NOMENCLATURE D E e g(r) MSD PC 4i r ‘ij
r0
TC t Eij
‘t
Pc uij
diffusion constant, m2.s-’ pair interaction the magnitude of the electron charge radial distribution function mean square displacement critical pressure, MPa partial charge assigned to each site, e the positions of particals, nm the distance of two particals, nm the initial positions of particals, nm critical temperature, K calculational time, s Lennard-Jones parameter critical density, g - ~ m - ~ Lennard-Jones parameter
5
REFERENCES
0
0.2
I
1
I
I
I
0.4
0.6 r, nm
0.8
1.0
1.2
Figure 5 The 0-H pair distribution in the mixture of methanol and acetone
In summary, the diffusion behavior of methanol molecule in different critical solvents was investigated by the MC simulation in this article. From the simulation, the diffusion constant of methanol in the critical n-hexane is much higher than those in the critical n-pentane, n-heptane and acetone, and the diffusivity of methanol in the critical acetone is the lowest. By analyzing the RDFs of the critical solutions, it is
Kerler, B., Martin, A., “Partial oxidation of alkanes to oxygenates in supercritical carbon dioxide”, Catal. Today,61,-17(2OOO). Eaton, A.P., Akgerman, A., “Infinite-dilution coefficients in supercritical fluids”, Ind. Eng. Chem. Res., 36, 92393 1(1997). Tanko, J.M., Pacut, R., “Enhanced cage effects in supercritical fluid solvents. The behavior of diffusive and germinate caged-pairs in supercritical carbon dioxide”, J. Am. Chem. Soc., 123.5703-5709(2001). Ikushima,Y., Saito, N., Arai, M.,“Supercritical carbon dioxide as reaction medium: Examination of its solvent effects in the near-critical region”, J. Phys. Chem., 96, 2293-2297( 1992). Heitz, M.P., Bright, F.V.,“Probing the scale of local density augmentation in supercritical fluids: A picosecond rotational reorientation study”, J. Phys. Chem., 100, 6 8 8 H 8 9 7 ( 1996). Ludemann, H.D., Chen, L., “Transport properties of supercritical fluids and their binary mixtures”, J. Phys: Condens. Matter., 14, 11453-1 1462(2002). Zhong, B., Li, W.H., Xiang, H.W., Ma, Y.G, Ning, J.B.,
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8
9
10
11
12 13 14
Chinese J. Ch. E. (Vol. 14, No.3) Peng, S.Y., “A method for methanol synthesis”, China Pat., ZL 95115889.9 (2000). Liu, J.G, Qin, Z.F., Wang, J.G, “Methanol synthesis under supercritical conditions: Calculations of equilibrium conversions by using the soave-redlich-kwong equation of state”, Ind. Eng. Chem. Res., 40.3801-3805(2001). Zhang, X.G. Guo, X.Y., Zhong, B., Peng, S.Y., “Monte Carlo simulation in supercritical methanol-hexane system”, J. Chem. Ind. Eng., 49, 735-739( 1998). (in Chinese) Zhang, X.G. Han, B.X., Li, Y.W., Zhong, B., Peng, S.Y., “Monte Carlo simulation for methanol synthesis on metal catalyst in supercritical n-hexane”, J. Supercn’t. Fluids, 23, 169-177(2002). Jia, Y.X., Guo, X.Y., Zhong, B., “Effects of n-hexane on methanol desorption from copper surface: Monte Carlo simulation”,J. Supercnr. Fluids, 32, 177-182(2004). Leach, A.R., Molecular Modelling: Principles and Applications, Prentice Hall (2001). Allen, M.P., Tildesley. D.J., Computer Simulation of Liquids, Oxford University Press, Oxford (1987). Jorgenson, W.L., “Transferable intermolecular potential functions. Application to liquid methanol including internal rotation”, J. Am. Chem. SOC., 103, 341 3 4 3 1981).
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16
17
18
19
20
21
Wright, D., EI-Shall. M.S., “A Monte Carlo study of methanol clusters (CH30H), , N=5 -256”. J. Chem. Phys., 105, 11199-1 1208(1996). Freitas, L., Cordeiro, J., Garbujo, F., “Theoretical studies of liquids by computer simulations: The water-acetone mixture”, J. Mol. Liq., 79, 1-lS(1999). Smit, B., Karaborni, S., Siepmann, J.I., “Computer simulations of vapor-liquid phase equilibria of n-alkanes”, J. Chem. Phys., 102,2126-2140(1995). Subramaniam, B., “Enhancing the stability of porous catalysts with supercritical reaction media”, Appl. Caral. A: General, 212, 199-213(2001). Tanaka, H., Shen, J.W., Nakanishi, K., Zeng, X.C., “Integral equation and Monte Carlo simulation studies of clusters in infinitely dilute supercritical solutions”, Chem. Phys. Lett., 239, 168-172( 1995). Kitao, O., Tanabe, K., Ono, S., Kumakura, S., Nakanishi, K., “Theoretical studies on the cluster structure in the supercritical area”, Fluid Phase Equil., 144, 279286( 1998). Petsche, I.B., Debenedetti, P.G, “Solute-solvent interactions in infinitely dilute supercritical mixtures: A molecular dynamics investigation”, J. Chem. Phys., 91, 7075-7084( 1989).
(2006)
RESEARCH NOTES
Monte Carlo Simulation of Methanol Diffusion in Critical Media* JIA Yuxiang( 3 ;$)a*b and GUO Xiangyun(@ I$ Z)%** 'State Key Laboratory of Coal Conversion, Institute of Coal Chemistry, CAS, Taiyuan 030001,China bSchool of Chemical and Materials Engineering, Southern Yangtze University, Wuxi 214122, China
Abstract The diffusion behavior of methanol in different critical media (n-pentane, n-hexane, n-heptane and acetone) was investigated by the Monte Carlo (MC) method. From the simulation results, the diffusion constant of methanol molecule in the critical n-hexane is much larger than those in n-pentane, n-heptane and acetone. By analyzing the microscopic configurations of the critical mixtures, it is found that the diffusion constant of methanol is related to the local solvent clustering around methanol, but it does not exhibit strong dependence on the size of solvent cluster around methanol. Moreover, the survival time of the solvent cluster plays an important role in determining the diffusion constant. Keywords critical fluids, Monte Carlo simulation, methanol, diffusion
1 INTRODUCTION Supercritical fluids (SCFs) have already been known for more than 150 years. However, only in the latest decades, they attracted considerable attention and obtained important applications"]. The impetus for the applications of SCFs originates from their unique solvent properties. Low viscosity, high diffusivity and adjustable solvency have enabled SCFs to be used as media for chemical reactions, adsorptionheparation and extraction Many studies have shown that the use of SCFs as reaction media leads to significant increase in reaction rate and obvious variation in product distrib~tiod~]. These features have attracted many interests in employing SCFs as a possible alternative instead of traditional liquid solvents. When using conventional liquid solvents, one has to employ multifarious solvents in order to obtain required physicochemical properties (e.g. solvency, density, and dielectric ~onstant)'~]. However, SCFs are substantially flexible, and thus provide a means to actually tune physicochemical properties by adjusting the fluid temperature and/or pressure. As a result, SCFs are generally considered as continuously property-tunable solvents. In addition, compared with classical organic solvents, SCFs have less health and environment riskst6]. In view of the unique properties of SCFs, Zhong et aLL7' studied the methanol synthesis process in near-critical and supercritical n-pentane, n-hexane,
n-heptane and acetone, and found that the CO conversion for one way was greatly increased up to 90% by employing supercritical n-hexane as reaction medium and the reaction achieved a maximum of CO conversion when operating the process near the critical point of n-hexane. Theoretical calculations about the methanol synthesis showed that the addition of suitable solvents such as n-hexane and n-heptane could greatly improve the CO conversion under supercritical conditions by using the Soave-Redlich-Kwong equation of state[*].Zhang et uZ.'~' studied the microscopic structures of the binary mixture of methanol-hexane under different conditions by the Monte Carlo (MC) method. The work suggested that there existed significant clustering of n-hexane among methanol molecules under the supercritical condition. They also simulated the supercritical methanol synthesis in copper catalyst with ZGB model and investigated the influence of methanol aggregation number (MAN) and the mole fraction of n-hexane on the catalyst activity"']. Previously, we studied the effect of n-hexane on methanol desorption from copper catalyst surface and found that there existed evident clustering between solvent and solute molecules under the critical condition by the MC method'"]. In this work, the diffusion behavior of methanol in different critical fluids is investigated, and the reasons resulting in the different diffusion behavior of methanol molecule are discussed.
Received 2005-08-02, accepted 2006-01-08.
* Supported by the One-Hundred-Talent Program of the Chinese Academy of Sciences.
** To whom correspondenceshould be addressed. E-mail: [email protected]
414
Chinese J. Ch. E. (Vol. 14, No3)
2 SIMULATION DETAILS 2.1 Monte Carlo made1 The MC method used in the present work involved random walk of molecules with an acceptance probability determined by the energy and temperature. The simulation was carried out in a NVT ensemble (system with N particles at temperature T and volume V), and the standard procedure of the Metropolis sampling with the periodic boundary condition was employed. The initial configuration of the system was randomly given. The molecular translation and rotation"*] were considered in the simulation. In the random movement, the maximum displacement and rotation were 0.03nm and 15", respectively. These motions were accepted by a certain probability of 25%35% in the simulation. A Monte Carlo Step (MCS) was defined as the process in which all molecules in the simulation cell finished one trial of random movement, and one MCS was assumed equal to a time duration of 1O-I4s. Link-cell-list was used in the simulation to speed up the cal~ulation['~]. 2.2 Potentials The TIP function[141was employed to describe the intermolecular interaction of methanol and acetone. The methanol and acetone molecules were represented by three interaction sites (CH3, 0, H)[I5] and four interaction sites (CH3, C, 0, CH3)[16],respectively. Each pair interaction was calculated by
(3) where ~ v a n dau are Lennard-Jones parameters, qi is the partial charge assigned to each site and e is the magnitude of electron charge. The Lennard-Jones potential was used for calculating intermolecular potential of other molecules in the system. The Lennard-Jones potential is of the form
The parameters employed in the simulation were tabulated in Table l[15--171. June, 2006
a b l e 1 The potential parameters employed in the simulation Molecule methanol
acetone n-pentane n-hexane n-hentane
Site CH3 0 H 0 CH3 C CH3
9, e 0.265 -0.700 0.435 -0.424 0.062 0.300 0
u, nm 0.3775 0.3070 0 0.2960 0.3910 0.3750 0.3905
CH2
0
0.3905
E,
J.rno1-I 869.4 714.0 0 445.2 340.2 222.6 730.8 491.4
2.3 Calculation method The MC simulation was applied to each binary system of methanol and a kind of solvent. We selected structurally identical n-pentane, n-hexane, and n-heptane as nonpolar solvent, and acetone as polar solvent. The self-diffusion constants of methanol molecule in different dilute critical solutions were calculated by analyzing the mean square displacement of methanol molecule in 70 picoseconds (lps=lOO MCS). To explore the reason resulting in the difference between the diffusivities of methanol molecule in different media, the microscopic configuration of the binary system was investigated in terms of radial distribution function (RDF). The critical parameters employed in the simulation were tabulated in Table 2[17]. Tgble 2 The critical parameters of the solvents employed in the simulation n-pentane n-hexane n-heptane acetone
494 523 556 509
0.230 0.233 0.233 0.278
3.369 3.014 2.734 4.600
3 RESULTS AND DISCUSSION 3.1 Diffusion constant of methanol in different critical media Through modeling the random walk of methanol molecule in the critical fluids, the mean square displacement (MSD) of methanol in different time can be obtained, as shown in Fig.1. In general, the relation between MSD and time follows Einstein's equation,
(5)
Monte Carlo Simulation of Methanol Diffusion in Critical Media
From Einstein's equation, the diffusion constant D can be estimated from Fig.1 and summarized in Table 3. From Table 3, the methanol diffusivities in different solvents are very different. It is found that the diffusion constant of methanol in n-hexane is much higher than those in other solvents. The high diffusivity is beneficial to the methanol desorption from catalyst surface.
I
I
415
surface reaction is compelled to the progress toward the direction for methanol synthesis. The calculated results are consistent with the experiment^'^], which suggests that n-hexane was the best medium among other investigated near- or supercritical fluids for the methanol synthesis, and the reaction achieves a maximum CO conversion near the critical point of n-hexane. The present results indicate that the diffusivities of methanol in various critical media are very different. The high diffusivity of methanol can effectively enhance the methanol desorption from the catalyst surface, and therefore increase the CO conversion of the methanol synthesis process.
Microscopic configurationsof methanol-solvent mixture The importance of the local structure in SCFs has been noticed p r e v i o ~ s l y " ~ .To ~ ~explain ~. the reason resulting in the difference of methanol diffusivities in different solvents, local structures of methanol-solvent mixtures under their critical conditions were investigated. The different distributions of solvent molecules around methanol at their critical points are shown in Fig.2. The local solvent density around methanol molecule is far higher than their corresponding bulk density. This indicates that there exists an obvious enrichment or clustering of solvent molecules around methanol. The most evident enrichment is found in the critical n-pentane, then n-hexane, n-heptane and acetone in sequence. The diffusion behavior of solute in different supercritical fluids has been explained in terms of the local solute-fluid density a~gmentation"~]. It is thus reasonable to suppose that the increased local solvent density would diminish the diffusivity of methanol molecule in the solvents. Therefore, the radial distribution functions (RDFs) of different solutions are analyzed to understand the influence of solvent-solute clustering on the methanol diffusivity. By integrating the RDFs, the coordination number of solvent molecules around methanol can be obtained, as tabulated in Table 4. Correspondingly, the dependence of methanol diffusivity on the coordination number is shown in Fig.3. There is no obvious relationship between the coordination number and methanol diffusivity. Both n-hexane and n-pentane have large ordination numbers, but the diffusivities of methanol molecule in them are far different. Methanol molecule has a very large
3.2
2ootLLzE I00
0
1000 2000 3000 4000 5000 6000 7000 8000 Monte Carlo Step
Figure 1 The mean square displacement of methanol molecule in the critical n-pentane,n-heme, n-heptane and acetone 0 C6H14; C7H16; 7 C5Hn; W c3&0 'Igble 3 Diffusion constants of CHJOHin different critical solutions Item density, g ~ r n - ~ 0.230
DX lo8,m 2 K 1
0.29
C ~ H M GHi6 0.233 0.233 1.22 0.31
C3&0 0.278 0.032
In many heterogeneously catalytic reactions, there usually exists a liquid layer on the catalyst surface. The concentration of product in the layer is very high. The high concentration of product in the layer often prevents product molecules to desorb from the catalyst surface"*'. It is generally established that supercritical media can solubilize the liquid layer and minimize the diffusion resistance of product in the layer"81. In the methanol synthesis, methanol molecules are primarily produced on the catalyst surface via a series of elementary steps. These molecules exist on the surface as adsorbed species. Due to the competitive adsorption between methanol and solvent molecules, the existence of critical media in most cases can hasten the methanol desorption from the catalyst surface. The methanol molecule in the critical n-hexane has the largest diffusivity, and therefore is most easily desorbed from the surface. As a result, the
Chinese J. Ch. E. 14(3) 413 (2006)
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Chinese J. Ch. E. (Vol. 14, No3)
diffusivity in n-hexane, and oppositely a very small one in n-pentane. Therefore, it is suggested from the present results that the size of the solute-solvent cluster does not play a crucial role in determining the methanol diffusivity. 4t
r, nm
Figure 2 Radial distributions of solvent molecules around methanol in four systems l-cH30H-C~H12; 2-CH30H-C&14; 3-cH30H-C7H16; 4--CH;OH-C3&0
Table 4 The coordination number of solvent molecules around methanol Coordination number
14.32
8.49
4.49
4.18
1.4
1.2
-
-6
1.o
Y)
0.8
m
2 0.6 X
9 0.4
0.2
0
\
4
6
12 coordination number 8
10
14
16
Figure 3 Relationship between the methanol diffusivity and the coordination number of solvent-solute clusters
the investigated mixtures. However, the RDFs cannot give information about the survival time of these clusters. Therefore, it is still questionable how these solvent clusters influence the methanol diffusion. In this section, the stability of solvent clusters in the critical mixtures is investigated, and the concept of lifetime is employed to characterize the stability of the solvent clusters[*’]. Usually the solvent clusters under the critical conditions are kinetic, and this indicates that the molecules in each cluster are very variable. Therefore, we define the lifetime of a cluster as the time that more than 50% molecules in the cluster have moved away. To calculate the lifetime, the solvent molecules within a distance of 0.55nm to a given methanol molecule were counted[”]. The distance was given according to the position of the first peak in the RDFs shown in Fig.2. The evolution of fifty clusters containing five or more molecules in the binary system of methanol and a critical fluid was tracked, and the average lifetime of these clusters was calculated in different systems. The lifetimes of different solvent clusters were tabulated in Table 5. The lifetime of the hexane cluster is the shortest among the investigated solvents, and this indicates that the hexane cluster around methanol is the most unstable. Fig.4 shows the dependence of methanol diffusivity on the cluster lifetime. The diffusion constant of methanol declines with the increase of the lifetime. The results presented in Fig.4 may provide a clue to understand the diffusion behavior of methanol in different solvents. The solvent clustering around methanol can restrict the motion of methanol molecule, and the methanol diffusion may be achieved by continuously escaping from the solvent clusters. Obviously the methanol diffusion in unstable clusters is relatively easy. Because the hexane cluster is more unstable than others, the methanol molecule has the largest diffusivity in the critical hexane.
Table 5 The lifetime of solvent clusters Solvents lifetime, vs
CSHl2 0.90
c 6 H 1 4
0.25
C7H16 0.43
c3H60
10.0
~
3.3 Lifetimes of solvent clusters around methanol molecule Generally the radial distribution function can describe the fluid structure in a time-averaged, statistical approach. The RDFs shown in Fig.2 indicate that there exists obvious solvent clustering around methanol in June, 2006
3.4 Hydrogen-bonding analysis From Table 5, the lifetime of acetone cluster around methanol is much longer than those of other solvents. To investigate the reason, we calculated the pair distribution function of oxygen from acetone and hydrogen from methanol, as shown in F i g 5 There is a
Monte Carlo Simulation of Methanol DifPusion in Critical Media
417
found that the solvent clustering around methanol molecule is very popular under the critical conditions. The combined effect of the size and lifetime of the solvent cluster determines the methanol diffusion. However, the lifetime of the solvent cluster plays a more important role in determining the diffusivity of methanol molecule in critical media. Because the lifetime of the n-hexane cluster is the shortest, the methanol diffusion in the critical n-hexane has the largest diffusivity among the investigated solvents. On the contrary, due to the strong hydrogen-bonding between methanol and acetone, the acetone cluster is very stable, so as to result in a very low diffusivity of methanol.
lifetime, ps
Figure 4 Relationship between methanol dupusivity and the lifetimes of solvent clusters
sharp peak appearing at the position of r = 0.19nm, which is associated with the formation of hydrogen bonding between acetone and methanol. Because the hydrogen bonding between acetone and methanol is very strong, the acetone cluster is very stable and thus has a long lifetime. This is also an important reason leading to the low diffusivity of methanol in the critical acetone.
NOMENCLATURE D E e g(r) MSD PC 4i r ‘ij
r0
TC t Eij
‘t
Pc uij
diffusion constant, m2.s-’ pair interaction the magnitude of the electron charge radial distribution function mean square displacement critical pressure, MPa partial charge assigned to each site, e the positions of particals, nm the distance of two particals, nm the initial positions of particals, nm critical temperature, K calculational time, s Lennard-Jones parameter critical density, g - ~ m - ~ Lennard-Jones parameter
5
REFERENCES
0
0.2
I
1
I
I
I
0.4
0.6 r, nm
0.8
1.0
1.2
Figure 5 The 0-H pair distribution in the mixture of methanol and acetone
In summary, the diffusion behavior of methanol molecule in different critical solvents was investigated by the MC simulation in this article. From the simulation, the diffusion constant of methanol in the critical n-hexane is much higher than those in the critical n-pentane, n-heptane and acetone, and the diffusivity of methanol in the critical acetone is the lowest. By analyzing the RDFs of the critical solutions, it is
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