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Anti-solvent crystallization of a ternary Lennard–Jones mixture performed by molecular dynamics
Journal of Molecular Liquids 209 (2015) 1–5
Contents lists available at ScienceDirect
Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Anti-solvent crystallization of a ternary Lennard–Jones mixture performed by molecular dynamics Kouji Maeda ⁎, Takumi Miki, Kazuhiro Itoh, Koji Arafune, Takuji Yamamoto, Keisuke Fukui Department of Chemical Engineering, University of Hyogo, 2167 Shosha, Himeji, Hyogo 671-2201, Japan
a r t i c l e
i n f o
Article history: Received 8 January 2015 Received in revised form 5 May 2015 Accepted 7 May 2015 Available online 20 May 2015 Keywords: Molecular dynamics Anti-solvent crystallization Lennard–Jones potential Local composition
a b s t r a c t Anti-solvent crystallization from a ternary mixture was examined by an NpT ensemble molecular dynamics simulation. The co-solvent and anti-solvent effects were represented by the Lennard–Jones interaction energy parameter, εij. The homogeneous binary solution of the solute and solvent was achieved at a constant temperature and pressure. Anti-solvent crystallization was introduced by changing some co-solvent molecules to anti-solvent molecules, immediately. The configuration of solute molecules was investigated by using the radial distribution function, g(r), and the local composition, xL. The value of εij affected the configuration of the solute molecules significantly; the decrease in εij provided the localization and crystallization of the solute molecules. The composition of the anti-solvent molecules in the solution also affected the configuration of the solute molecules; the increase in the anti-solvent composition produced the crystal structure of the solute molecules more rapidly. These qualitative results corresponded well to anti-solvent crystallization. The radial distribution function represented the crystal structure for solute molecules, and the local composition of the solute was increased from the bulk composition as the effect of the anti-solvent increased. We proposed that the time variation of the local composition of the solute represents the temporal development of the crystal structure and the time to crystallization well. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Crystallization is a ubiquitous molecular process widely used in science and engineering. The main properties of crystals as industrial products are the purity, defects, crystal morphology, and crystal size distribution [1]. All of these properties depend on the molecular configuration in solution. However, there are no experimental methods to investigate the dynamic processes of crystallization on the molecular scale. Molecular dynamics simulations were first created by Alder and Wright [2,3] in the 1950s using a high-speed computer. Many techniques or ensembles have been developed to simulate real phenomena [4–7]. We have also studied the solubility properties of crystals with the Lennard–Jones solution, which is considered to be a dilute solution consisting of Lennard–Jones molecules [7]. We reported the solid–liquid phase transition of the Lennard–Jones solution [8] considering the work published by Anwar and Boateng [9]. The Lennard–Jones parameters were slightly changed from the values obtained from the complete solid–liquid phase diagram of the Lennard–Jones binary systems by Gibbs–Duhem integration [10]. After we defined the solid–liquid phase equilibrium by molecular dynamics [8], we first performed homogeneous nucleation of cooling crystallization in Lennard–Jones solutions, which represented supercooling [11]. We also considered crystal ⁎ Corresponding author. E-mail address: [email protected] (K. Maeda).
http://dx.doi.org/10.1016/j.molliq.2015.05.019 0167-7322/© 2015 Elsevier B.V. All rights reserved.
growth in a pseudo-NpT ensemble with a temperature gradient, and proposed the distribution rate instead of the growth rate for the ambiguous solid–liquid interface [12]. The second method of crystallization is anti-solvent crystallization. This is crystallization induced by the interactions between solute and solvents rather than by cooling. Therefore, supercooling and supersaturation are not easily defined in this method. The third method of crystallization is high-pressure crystallization. This occurs via the Alder transition of non-interacting molecules [13], and the effect of the Lennard–Jones interaction energy parameters on the phase transition pressure on highpressure crystallization will be considered in our future works. In this paper, we focused on the effect of the Lennard–Jones interaction energy parameters between anti-solvent and solute, and the composition of anti-solvent in the ternary solutions on anti-solvent crystallization phenomena for ternary Lennard–Jones solutions. In order to quantify the molecular configurations of solute in the solutions during anti-solvent crystallization as the significant objectives in this study, the radial distribution function for crystal structure and the local composition of solute for liquid–liquid phase separation were discussed at the same time. 2. Simulation method The NpT ensemble molecular dynamics proposed by Andersen [4] is the most convenient procedure for representing crystallization of
2
K. Maeda et al. / Journal of Molecular Liquids 209 (2015) 1–5
Lennard–Jones solutions. We have previously simulated homogeneous nucleation by cooling [11]. We used the same method for anti-solvent crystallization. Fig. 1 shows the schematic diagram of anti-solvent crystallization. Before the anti-solvent crystallization, a stable homogeneous Lennard–Jones binary solution containing the solute and solvent was prepared at 500 K. The Lennard–Jones size parameter, σ, and interaction energy parameter, ε/kB, for the solute and solvent were set as listed in Table 1. In this study, size parameter, σ, and molecular weight, Mw, were the same for all molecules, and only ε/kB was changed for different types of molecule. Anti-solvent crystallization at 500 K was immediately carried out after some co-solvent molecules converted to the antisolvent molecules by the change in energy parameters listed in Table 1. The properties of the anti-solvent were adjusted by the interaction energy parameters and the composition of the anti-solvent molecules in the solution. Table 2 shows the conditions for the molecular dynamics simulations for anti-solvent crystallization. 3. Results and discussion In this work, the solute composition of 10% is set to be uncrystallized in the pure co-solvent composition of 90%, which does not contain the anti-solvent, at an average temperature of T = 500 K according to the phase transition of Lennard–Jones solutions [8]. When an anti-solvent is added to the solution, the solute molecules can be localized. Fig. 2 shows the typical configurations of the solute molecules as a function of the anti-solvent Lennard–Jones interaction energy parameter, ε/kB, when the anti-solvent and solvent composition was 40% and 50% respectively at T = 500 K. Green particles indicate the solute molecules and red particles indicate the solution of anti-solvent molecules. The anti-solvent molecules with εij/kB = 600 K did not affect the molecular configuration of the solute because the difference in the intermolecular force between the anti-solvent molecules and solute molecules is the same as that between the co-solvent and solute molecules. When the anti-solvent εij/kB decreased, the solute molecules localized gradually. The decrease in εij/kB corresponds to the reduction in binding energy between the anti-solvent and solute molecules. The solute molecules crystallized for the smallest εij/kB value of 100 K. Here, the εij/kB value of 600 K between solute (εij/kB = 800 K) and co-solvent (εij/kB = 400 K) exactly represented the homogeneous solution, but the εij/kB values of 400 K, 300 K, 200 K, 100 K less than Lorentz–Berthelot rule between solute (εij/kB = 800 K) and anti-solvent (εij/kB = 400 K) obviously showed the heterogeneity of the solution. The difference between cosolvent and anti-solvent was the εij/kB value of Lennard–Jones interaction energy parameter to solute. The typical configurations of the solute molecules are shown in Fig. 3 as a function of the anti-solvent composition with εij/kB fixed at 100 K. The highly localized solute molecules gradually dispersed as the antisolvent composition decreased, and then the solute molecules could no longer maintain their crystal structure with an anti-solvent composition of 10%, even though the solute molecules were localized.
Initial solution
Table 1 Lennard–Jones parameters for ternary systems. ε/kB [K]
σ [A]
Mw [g mol]
Pure Solute Solvent Anti-solvent
800 400 400
4 4 4
40 40 40
Pairs Solute–co-solvent Solute–anti-solvent Co-solvent–anti-solvent
600 100–400 400
4 4 4
– – –
The radial distribution function, g(r), shows the behavior of the solute molecules and solvent molecules in the solution. The average g(r) was calculated by the following equation from the number of molecules, N, as a function of intermolecular distance, r in the controlled volume, VNPT.
g ðr Þ ¼
* + N X 1 V NPT ΔN ð r Þ : i 4πr 2 Δr NðN−1Þ i¼1
Fig. 4 shows g(r) between the anti-solvent and co-solvent molecules for the present analysis including anti-solvent crystallization. We did not show the dependence of g(r) on εij/kB or the composition of the anti-solvent because εij/kB was identical between the anti-solvent and co-solvent, the anti-solvent and anti-solvent, and the co-solvent and co-solvent (Table 1). All solvent molecules started with a liquid structure under any conditions at T = 500 K because the second peak of the g(r) distribution around r ~ 8 is weak and broad. g(r) of the solute is shown in Fig. 5 at different εij/kB values with an anti-solvent composition of 40% at T = 500 K. The first peak increased when εij/kB decreased, and the typical peak for a face-centered cubic (FCC; r = 1.4–1.6σ) structure was observed at εij/kB = 100 K. This showed that decreasing εij/kB induced both the localization and crystallization of the solute molecules, even at temperatures above the melting point (Tm = 360–370 K) [12]. Therefore, εij/kB represented the anti-solvent effect accurately. Fig. 6 shows g(r) of the solute at different antisolvent compositions with εij/kB = 100 K at T = 500 K. When the anti-solvent composition deceased, the typical FCC peak of g(r) disappeared and the first peak of g(r) became smaller. The change in the g(r) peak distribution was consistent with the dissipation of the crystal structure of the solute molecules. This showed that the composition of the anti-solvent in the solution was also an important variable for controlling the solute crystal structure. Therefore, the ratio of the antisolvent to the solute molecules should be kept large to maintain the crystal structure of the solute molecules even though εij/kB was lower. We have already shown that the εij/kB and the composition of the anti-solvent played an important role in the localization and
Anti-solvent crystallization Co-solvent molecules are exchanged to antisolvent suddenly.
solute co-solvent anti-solvent
NpT MD for the L-J binary solution at 500 K
ð1Þ
NpT MD for the L-J ternary solution at 500 K
Fig. 1. Schematic of anti-solvent crystallization by NpT molecular dynamics.
K. Maeda et al. / Journal of Molecular Liquids 209 (2015) 1–5
3
Table 2 Conditions of molecular dynamics simulations for anti-solvent crystallization. Ensemble Temperature Pressure Cell condition Time step Molecules Composition Solute Anti-solvent Solvent
NpT 500 K 1 bar 3D periodic boundary 2 fs/step for 1 × 105 to 1 × 107 steps Solute 100, solvent 900 10% 10%–40% 90%–60%
kB = 100 K Anti-solvent=40%
kB = 100 K Anti-solvent=30%
kB = 100 K Anti-solvent=20%
kB = 100 K Anti-solvent=10%
crystallization of solute molecules. Next, we focus on the temporal development of the solute molecule structure from the localized state to the crystallized state. We introduce the local composition of the solute as another variable for evaluating the time evolution of the localization and crystallization of solute molecules. The local composition of the solute, xL, was calculated by the following equation from number of molecules, N, as a function of intermolecular distance, r. * xL ¼
NX solute i¼1
Z
+ N i ðr Þ : r¼0 N ðr Þ o
7A
ð2Þ
Fig. 7 shows the change in the local composition as a function of time step during anti-solvent crystallization at different εij/kB. For the completely homogeneous binary solution with εij/kB = 600 K, shown by the dark blue line in Fig. 7, the local composition was almost constant as the time step increased. However, for ternary solutions containing an anti-solvent, the value of the local composition increased considerably. The rate of increase and the value at the plateau of the local composition increased with a decrease in εij/kB. When the value of εij/kB of the antisolvent was close to that of the co-solvent, the induced phase separation force of the two liquid phases was weak, and the energy difference between the anti- and co-solvent could not crystallize the solute molecules. Fig. 8 shows the variation in the local composition as a function of time step at different anti-solvent compositions with εij/kB = 100 K. Even though anti-solvent has a weak interaction energy of εij/kB = 100 K, the local composition of the solute did not increase at lower compositions. An anti-solvent composition of 30% was required to crystallize solute molecules. The nucleation needs a flocculation of solute
Fig. 3. Solute molecular configurations at different anti-solvent compositions.
molecules before crystallization. The local composition is an essential value to see the solute structure in the solution. The local composition is the same as the bulk composition for an ideal solution, but the real solution that causes liquid–liquid phase separation or crystallization forms clusters of the localized solutes in the solution. As seeing Figs. 7 and 8, the homogeneous solution still shows the different local composition from the bulk composition, and states a certain value. When the nucleation occurs, the local composition is stable at large value [11]. We may recognize whether the solute molecules form crystal phase. As seeing Figs. 5 and 6, the solute molecules in the ternary solutions containing small interaction energy εij/kB = 100 K, 200 K, 300 K between solute and anti-solvent with 40% anti-solvent and large composition of anti-solvent more than 30% with interaction energy εij/kB =
kB = 600 K Anti-solvent=40%
kB = 400 K Anti-solvent=40%
kB = 200 K Anti-solvent=40%
kB = 100 K Anti-solvent=40%
kB = 300 K Anti-solvent=40%
Fig. 2. Molecular configurations of solute at different anti-solvent eij/kB values.
4
K. Maeda et al. / Journal of Molecular Liquids 209 (2015) 1–5
14
14 kB = 600 K, Anti-solvent=40% kB = 400 K, Anti-solvent=40% ij kB = 300 K, Anti-solvent=40% ij kB = 200 K, Anti-solvent=40% ij kB = 100 K, Anti-solvent=40% ij kB = 100 K, Anti-solvent=30% ij kB = 100 K, Anti-solvent=20% ij kB = 100 K, Anti-solvent=10% ij
12
kB = 100 K, Anti-solvent=40% kB = 100 K, Anti-solvent=30% ij kB = 100 K, Anti-solvent=20% ij kB = 100 K, Anti-solvent=10% ij
12
ij
8 6
10
g(r) [-]
g(r) [-]
10
ij
8 6
4
4
2
2
0
0 0
2
4
6
8
10
12
14
0
2
4
r [A] Fig. 4. Radial distribution function of co-solvent and anti-solvent molecules under different MD conditions.
14 kB = 600 K, Anti-solvent=40% kB = 400 K, Anti-solvent=40% ij kB = 300 K, Anti-solvent=40% ij kB = 200 K, Anti-solvent=40% ij kB = 100 K, Anti-solvent=40% ij
12
ij
g(r) [-]
10 8 6 4 2 0 0
2
4
6
8
10
12
14
r [A]
6 r [A]
100 K could be crystallized when the local composition of solute increased more than 0.4 and the value of the local composition was stable as shown in Figs. 7 and 8.
ij
xL [-]
14
The anti-solvent interaction energy value εij/kB was more effective than the anti-solvent composition for localizing solute molecules as two liquid phases, and the anti-solvent composition was more significant than εij/kB for the crystallization of solute molecules, even though both anti-solvent parameters affected each other. Consequently, the supersaturation of anti-solvent crystallization by molecular dynamics could be found qualitatively. The ternary solutions containing small interaction energy εij/kB = 100 K, 200 K, and 300 K between solute and anti-solvent with 40% anti-solvent and large composition of anti-solvent more than 30% with interaction energy εij/kB = 100 K were supersaturated solutions. However, the other ternary solutions were under saturation even though the solute molecules were localized. The quantitative index to present the supersaturation for antisolvent crystallization of the Lennard–Jones mixtures will be considered in the further study.
Molecular dynamics simulations of the anti-solvent crystallization of a ternary solution were performed with the NpT ensemble. The simulations determined the effects of the anti-solvent εij/kB and composition on the localization and crystallization of the solute molecules at a constant temperature and pressure. The anti-solvent εij/kB and composition
kB = 100 K, Anti-solvent=40%
ij kB = 200 K, Antisolvent=40%
ij kB = 300 K, Antisolvent=40%
0.5 0.4 0.3
ij
0.2 ij
kB = 400 K, Anti-solvent=40%
kB = 600 K, Anti-solvent=40%
0.1 0
12
Fig. 6. Radial distribution function of solute molecules at different anti-solvent compositions.
0.7
0
10
4. Conclusion
Fig. 5. Radial distribution function of solute molecules at different εij/kB.
0.6
8
0.2
0.4
0.6 t [ns]
0.8
1.0
1.2
Fig. 7. Local composition as a function of solute at different anti-solvent εij/kB.
K. Maeda et al. / Journal of Molecular Liquids 209 (2015) 1–5
5
0.7 ij
0.6
kB = 100 K, Anti-solvent=40%
xL [-]
0.5 ij
0.4 0.3
ij
kB = 100 K, Anti-solvent=30% kB = 100 K, Anti-solvent=20%
0.2 ij
kB = 100 K, Anti-solvent=10%
0.1 ij
0
0
kB = 600 K, Anti-solvent=40% 0.2
0.4
0.6 t [ns]
0.8
1.0
1.2
Fig. 8. Local composition of solute as a function of time at different anti-solvent compositions.
can control anti-solvent crystallization of solute in relation to which of three combinations is selected solute and co-solvent, solute and antisolvent, and co-solvent and anti-solvent. As for the radial distribution functions of solute, small anti-solvent εij/kB or large anti-solvent composition induced crystallization of solute. As for the local composition of solute, the local composition behavior described when solute molecules were localized and crystallized. References [1] J.M. Mullin, Crystallization, 3rd ed. Butterworth-Heinemann, Oxford, 1993. [2] B.J. Alder, T.E. Wainwright, J. Chem. Phys. 27 (1957) 1208.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
B.J. Alder, T.E. Wainwright, J. Chem. Phys. 31 (1959) 459. H.C. Andersen, J. Chem. Phys. 72 (1980) 2384. M. Parrrinello, A. Rahman, J. Appl. Phys. 52 (1981) 7182. S. Nose, M.L. Klein, Mol. Phys. 50 (1983) 1055. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford Science Pub., Oxford, 1987. K. Maeda, W. Matsuoka, T. Fuse, K. Fului, S. Hirota, J. Mol. Liq. 102 (2003) 1. J. Anwar, P.K. Boateng, J. Am. Chem. Soc. 120 (1998) 9600. M.C. Lamm, C.K. Hall, AIChE J. 47 (2001) 1664. K. Maeda, Y. Asakuma, K. Fukui, J. Mol. Liq. 122 (2005) 43. K. Maeda, Y. Asakuma, K. Fukui, J. Chem. Phys. 128 (2008) 044716. B.J. Alder, T.E. Wainwright, Phys. Rev. 127 (1962) 359.
Contents lists available at ScienceDirect
Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Anti-solvent crystallization of a ternary Lennard–Jones mixture performed by molecular dynamics Kouji Maeda ⁎, Takumi Miki, Kazuhiro Itoh, Koji Arafune, Takuji Yamamoto, Keisuke Fukui Department of Chemical Engineering, University of Hyogo, 2167 Shosha, Himeji, Hyogo 671-2201, Japan
a r t i c l e
i n f o
Article history: Received 8 January 2015 Received in revised form 5 May 2015 Accepted 7 May 2015 Available online 20 May 2015 Keywords: Molecular dynamics Anti-solvent crystallization Lennard–Jones potential Local composition
a b s t r a c t Anti-solvent crystallization from a ternary mixture was examined by an NpT ensemble molecular dynamics simulation. The co-solvent and anti-solvent effects were represented by the Lennard–Jones interaction energy parameter, εij. The homogeneous binary solution of the solute and solvent was achieved at a constant temperature and pressure. Anti-solvent crystallization was introduced by changing some co-solvent molecules to anti-solvent molecules, immediately. The configuration of solute molecules was investigated by using the radial distribution function, g(r), and the local composition, xL. The value of εij affected the configuration of the solute molecules significantly; the decrease in εij provided the localization and crystallization of the solute molecules. The composition of the anti-solvent molecules in the solution also affected the configuration of the solute molecules; the increase in the anti-solvent composition produced the crystal structure of the solute molecules more rapidly. These qualitative results corresponded well to anti-solvent crystallization. The radial distribution function represented the crystal structure for solute molecules, and the local composition of the solute was increased from the bulk composition as the effect of the anti-solvent increased. We proposed that the time variation of the local composition of the solute represents the temporal development of the crystal structure and the time to crystallization well. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Crystallization is a ubiquitous molecular process widely used in science and engineering. The main properties of crystals as industrial products are the purity, defects, crystal morphology, and crystal size distribution [1]. All of these properties depend on the molecular configuration in solution. However, there are no experimental methods to investigate the dynamic processes of crystallization on the molecular scale. Molecular dynamics simulations were first created by Alder and Wright [2,3] in the 1950s using a high-speed computer. Many techniques or ensembles have been developed to simulate real phenomena [4–7]. We have also studied the solubility properties of crystals with the Lennard–Jones solution, which is considered to be a dilute solution consisting of Lennard–Jones molecules [7]. We reported the solid–liquid phase transition of the Lennard–Jones solution [8] considering the work published by Anwar and Boateng [9]. The Lennard–Jones parameters were slightly changed from the values obtained from the complete solid–liquid phase diagram of the Lennard–Jones binary systems by Gibbs–Duhem integration [10]. After we defined the solid–liquid phase equilibrium by molecular dynamics [8], we first performed homogeneous nucleation of cooling crystallization in Lennard–Jones solutions, which represented supercooling [11]. We also considered crystal ⁎ Corresponding author. E-mail address: [email protected] (K. Maeda).
http://dx.doi.org/10.1016/j.molliq.2015.05.019 0167-7322/© 2015 Elsevier B.V. All rights reserved.
growth in a pseudo-NpT ensemble with a temperature gradient, and proposed the distribution rate instead of the growth rate for the ambiguous solid–liquid interface [12]. The second method of crystallization is anti-solvent crystallization. This is crystallization induced by the interactions between solute and solvents rather than by cooling. Therefore, supercooling and supersaturation are not easily defined in this method. The third method of crystallization is high-pressure crystallization. This occurs via the Alder transition of non-interacting molecules [13], and the effect of the Lennard–Jones interaction energy parameters on the phase transition pressure on highpressure crystallization will be considered in our future works. In this paper, we focused on the effect of the Lennard–Jones interaction energy parameters between anti-solvent and solute, and the composition of anti-solvent in the ternary solutions on anti-solvent crystallization phenomena for ternary Lennard–Jones solutions. In order to quantify the molecular configurations of solute in the solutions during anti-solvent crystallization as the significant objectives in this study, the radial distribution function for crystal structure and the local composition of solute for liquid–liquid phase separation were discussed at the same time. 2. Simulation method The NpT ensemble molecular dynamics proposed by Andersen [4] is the most convenient procedure for representing crystallization of
2
K. Maeda et al. / Journal of Molecular Liquids 209 (2015) 1–5
Lennard–Jones solutions. We have previously simulated homogeneous nucleation by cooling [11]. We used the same method for anti-solvent crystallization. Fig. 1 shows the schematic diagram of anti-solvent crystallization. Before the anti-solvent crystallization, a stable homogeneous Lennard–Jones binary solution containing the solute and solvent was prepared at 500 K. The Lennard–Jones size parameter, σ, and interaction energy parameter, ε/kB, for the solute and solvent were set as listed in Table 1. In this study, size parameter, σ, and molecular weight, Mw, were the same for all molecules, and only ε/kB was changed for different types of molecule. Anti-solvent crystallization at 500 K was immediately carried out after some co-solvent molecules converted to the antisolvent molecules by the change in energy parameters listed in Table 1. The properties of the anti-solvent were adjusted by the interaction energy parameters and the composition of the anti-solvent molecules in the solution. Table 2 shows the conditions for the molecular dynamics simulations for anti-solvent crystallization. 3. Results and discussion In this work, the solute composition of 10% is set to be uncrystallized in the pure co-solvent composition of 90%, which does not contain the anti-solvent, at an average temperature of T = 500 K according to the phase transition of Lennard–Jones solutions [8]. When an anti-solvent is added to the solution, the solute molecules can be localized. Fig. 2 shows the typical configurations of the solute molecules as a function of the anti-solvent Lennard–Jones interaction energy parameter, ε/kB, when the anti-solvent and solvent composition was 40% and 50% respectively at T = 500 K. Green particles indicate the solute molecules and red particles indicate the solution of anti-solvent molecules. The anti-solvent molecules with εij/kB = 600 K did not affect the molecular configuration of the solute because the difference in the intermolecular force between the anti-solvent molecules and solute molecules is the same as that between the co-solvent and solute molecules. When the anti-solvent εij/kB decreased, the solute molecules localized gradually. The decrease in εij/kB corresponds to the reduction in binding energy between the anti-solvent and solute molecules. The solute molecules crystallized for the smallest εij/kB value of 100 K. Here, the εij/kB value of 600 K between solute (εij/kB = 800 K) and co-solvent (εij/kB = 400 K) exactly represented the homogeneous solution, but the εij/kB values of 400 K, 300 K, 200 K, 100 K less than Lorentz–Berthelot rule between solute (εij/kB = 800 K) and anti-solvent (εij/kB = 400 K) obviously showed the heterogeneity of the solution. The difference between cosolvent and anti-solvent was the εij/kB value of Lennard–Jones interaction energy parameter to solute. The typical configurations of the solute molecules are shown in Fig. 3 as a function of the anti-solvent composition with εij/kB fixed at 100 K. The highly localized solute molecules gradually dispersed as the antisolvent composition decreased, and then the solute molecules could no longer maintain their crystal structure with an anti-solvent composition of 10%, even though the solute molecules were localized.
Initial solution
Table 1 Lennard–Jones parameters for ternary systems. ε/kB [K]
σ [A]
Mw [g mol]
Pure Solute Solvent Anti-solvent
800 400 400
4 4 4
40 40 40
Pairs Solute–co-solvent Solute–anti-solvent Co-solvent–anti-solvent
600 100–400 400
4 4 4
– – –
The radial distribution function, g(r), shows the behavior of the solute molecules and solvent molecules in the solution. The average g(r) was calculated by the following equation from the number of molecules, N, as a function of intermolecular distance, r in the controlled volume, VNPT.
g ðr Þ ¼
* + N X 1 V NPT ΔN ð r Þ : i 4πr 2 Δr NðN−1Þ i¼1
Fig. 4 shows g(r) between the anti-solvent and co-solvent molecules for the present analysis including anti-solvent crystallization. We did not show the dependence of g(r) on εij/kB or the composition of the anti-solvent because εij/kB was identical between the anti-solvent and co-solvent, the anti-solvent and anti-solvent, and the co-solvent and co-solvent (Table 1). All solvent molecules started with a liquid structure under any conditions at T = 500 K because the second peak of the g(r) distribution around r ~ 8 is weak and broad. g(r) of the solute is shown in Fig. 5 at different εij/kB values with an anti-solvent composition of 40% at T = 500 K. The first peak increased when εij/kB decreased, and the typical peak for a face-centered cubic (FCC; r = 1.4–1.6σ) structure was observed at εij/kB = 100 K. This showed that decreasing εij/kB induced both the localization and crystallization of the solute molecules, even at temperatures above the melting point (Tm = 360–370 K) [12]. Therefore, εij/kB represented the anti-solvent effect accurately. Fig. 6 shows g(r) of the solute at different antisolvent compositions with εij/kB = 100 K at T = 500 K. When the anti-solvent composition deceased, the typical FCC peak of g(r) disappeared and the first peak of g(r) became smaller. The change in the g(r) peak distribution was consistent with the dissipation of the crystal structure of the solute molecules. This showed that the composition of the anti-solvent in the solution was also an important variable for controlling the solute crystal structure. Therefore, the ratio of the antisolvent to the solute molecules should be kept large to maintain the crystal structure of the solute molecules even though εij/kB was lower. We have already shown that the εij/kB and the composition of the anti-solvent played an important role in the localization and
Anti-solvent crystallization Co-solvent molecules are exchanged to antisolvent suddenly.
solute co-solvent anti-solvent
NpT MD for the L-J binary solution at 500 K
ð1Þ
NpT MD for the L-J ternary solution at 500 K
Fig. 1. Schematic of anti-solvent crystallization by NpT molecular dynamics.
K. Maeda et al. / Journal of Molecular Liquids 209 (2015) 1–5
3
Table 2 Conditions of molecular dynamics simulations for anti-solvent crystallization. Ensemble Temperature Pressure Cell condition Time step Molecules Composition Solute Anti-solvent Solvent
NpT 500 K 1 bar 3D periodic boundary 2 fs/step for 1 × 105 to 1 × 107 steps Solute 100, solvent 900 10% 10%–40% 90%–60%
kB = 100 K Anti-solvent=40%
kB = 100 K Anti-solvent=30%
kB = 100 K Anti-solvent=20%
kB = 100 K Anti-solvent=10%
crystallization of solute molecules. Next, we focus on the temporal development of the solute molecule structure from the localized state to the crystallized state. We introduce the local composition of the solute as another variable for evaluating the time evolution of the localization and crystallization of solute molecules. The local composition of the solute, xL, was calculated by the following equation from number of molecules, N, as a function of intermolecular distance, r. * xL ¼
NX solute i¼1
Z
+ N i ðr Þ : r¼0 N ðr Þ o
7A
ð2Þ
Fig. 7 shows the change in the local composition as a function of time step during anti-solvent crystallization at different εij/kB. For the completely homogeneous binary solution with εij/kB = 600 K, shown by the dark blue line in Fig. 7, the local composition was almost constant as the time step increased. However, for ternary solutions containing an anti-solvent, the value of the local composition increased considerably. The rate of increase and the value at the plateau of the local composition increased with a decrease in εij/kB. When the value of εij/kB of the antisolvent was close to that of the co-solvent, the induced phase separation force of the two liquid phases was weak, and the energy difference between the anti- and co-solvent could not crystallize the solute molecules. Fig. 8 shows the variation in the local composition as a function of time step at different anti-solvent compositions with εij/kB = 100 K. Even though anti-solvent has a weak interaction energy of εij/kB = 100 K, the local composition of the solute did not increase at lower compositions. An anti-solvent composition of 30% was required to crystallize solute molecules. The nucleation needs a flocculation of solute
Fig. 3. Solute molecular configurations at different anti-solvent compositions.
molecules before crystallization. The local composition is an essential value to see the solute structure in the solution. The local composition is the same as the bulk composition for an ideal solution, but the real solution that causes liquid–liquid phase separation or crystallization forms clusters of the localized solutes in the solution. As seeing Figs. 7 and 8, the homogeneous solution still shows the different local composition from the bulk composition, and states a certain value. When the nucleation occurs, the local composition is stable at large value [11]. We may recognize whether the solute molecules form crystal phase. As seeing Figs. 5 and 6, the solute molecules in the ternary solutions containing small interaction energy εij/kB = 100 K, 200 K, 300 K between solute and anti-solvent with 40% anti-solvent and large composition of anti-solvent more than 30% with interaction energy εij/kB =
kB = 600 K Anti-solvent=40%
kB = 400 K Anti-solvent=40%
kB = 200 K Anti-solvent=40%
kB = 100 K Anti-solvent=40%
kB = 300 K Anti-solvent=40%
Fig. 2. Molecular configurations of solute at different anti-solvent eij/kB values.
4
K. Maeda et al. / Journal of Molecular Liquids 209 (2015) 1–5
14
14 kB = 600 K, Anti-solvent=40% kB = 400 K, Anti-solvent=40% ij kB = 300 K, Anti-solvent=40% ij kB = 200 K, Anti-solvent=40% ij kB = 100 K, Anti-solvent=40% ij kB = 100 K, Anti-solvent=30% ij kB = 100 K, Anti-solvent=20% ij kB = 100 K, Anti-solvent=10% ij
12
kB = 100 K, Anti-solvent=40% kB = 100 K, Anti-solvent=30% ij kB = 100 K, Anti-solvent=20% ij kB = 100 K, Anti-solvent=10% ij
12
ij
8 6
10
g(r) [-]
g(r) [-]
10
ij
8 6
4
4
2
2
0
0 0
2
4
6
8
10
12
14
0
2
4
r [A] Fig. 4. Radial distribution function of co-solvent and anti-solvent molecules under different MD conditions.
14 kB = 600 K, Anti-solvent=40% kB = 400 K, Anti-solvent=40% ij kB = 300 K, Anti-solvent=40% ij kB = 200 K, Anti-solvent=40% ij kB = 100 K, Anti-solvent=40% ij
12
ij
g(r) [-]
10 8 6 4 2 0 0
2
4
6
8
10
12
14
r [A]
6 r [A]
100 K could be crystallized when the local composition of solute increased more than 0.4 and the value of the local composition was stable as shown in Figs. 7 and 8.
ij
xL [-]
14
The anti-solvent interaction energy value εij/kB was more effective than the anti-solvent composition for localizing solute molecules as two liquid phases, and the anti-solvent composition was more significant than εij/kB for the crystallization of solute molecules, even though both anti-solvent parameters affected each other. Consequently, the supersaturation of anti-solvent crystallization by molecular dynamics could be found qualitatively. The ternary solutions containing small interaction energy εij/kB = 100 K, 200 K, and 300 K between solute and anti-solvent with 40% anti-solvent and large composition of anti-solvent more than 30% with interaction energy εij/kB = 100 K were supersaturated solutions. However, the other ternary solutions were under saturation even though the solute molecules were localized. The quantitative index to present the supersaturation for antisolvent crystallization of the Lennard–Jones mixtures will be considered in the further study.
Molecular dynamics simulations of the anti-solvent crystallization of a ternary solution were performed with the NpT ensemble. The simulations determined the effects of the anti-solvent εij/kB and composition on the localization and crystallization of the solute molecules at a constant temperature and pressure. The anti-solvent εij/kB and composition
kB = 100 K, Anti-solvent=40%
ij kB = 200 K, Antisolvent=40%
ij kB = 300 K, Antisolvent=40%
0.5 0.4 0.3
ij
0.2 ij
kB = 400 K, Anti-solvent=40%
kB = 600 K, Anti-solvent=40%
0.1 0
12
Fig. 6. Radial distribution function of solute molecules at different anti-solvent compositions.
0.7
0
10
4. Conclusion
Fig. 5. Radial distribution function of solute molecules at different εij/kB.
0.6
8
0.2
0.4
0.6 t [ns]
0.8
1.0
1.2
Fig. 7. Local composition as a function of solute at different anti-solvent εij/kB.
K. Maeda et al. / Journal of Molecular Liquids 209 (2015) 1–5
5
0.7 ij
0.6
kB = 100 K, Anti-solvent=40%
xL [-]
0.5 ij
0.4 0.3
ij
kB = 100 K, Anti-solvent=30% kB = 100 K, Anti-solvent=20%
0.2 ij
kB = 100 K, Anti-solvent=10%
0.1 ij
0
0
kB = 600 K, Anti-solvent=40% 0.2
0.4
0.6 t [ns]
0.8
1.0
1.2
Fig. 8. Local composition of solute as a function of time at different anti-solvent compositions.
can control anti-solvent crystallization of solute in relation to which of three combinations is selected solute and co-solvent, solute and antisolvent, and co-solvent and anti-solvent. As for the radial distribution functions of solute, small anti-solvent εij/kB or large anti-solvent composition induced crystallization of solute. As for the local composition of solute, the local composition behavior described when solute molecules were localized and crystallized. References [1] J.M. Mullin, Crystallization, 3rd ed. Butterworth-Heinemann, Oxford, 1993. [2] B.J. Alder, T.E. Wainwright, J. Chem. Phys. 27 (1957) 1208.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
B.J. Alder, T.E. Wainwright, J. Chem. Phys. 31 (1959) 459. H.C. Andersen, J. Chem. Phys. 72 (1980) 2384. M. Parrrinello, A. Rahman, J. Appl. Phys. 52 (1981) 7182. S. Nose, M.L. Klein, Mol. Phys. 50 (1983) 1055. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford Science Pub., Oxford, 1987. K. Maeda, W. Matsuoka, T. Fuse, K. Fului, S. Hirota, J. Mol. Liq. 102 (2003) 1. J. Anwar, P.K. Boateng, J. Am. Chem. Soc. 120 (1998) 9600. M.C. Lamm, C.K. Hall, AIChE J. 47 (2001) 1664. K. Maeda, Y. Asakuma, K. Fukui, J. Mol. Liq. 122 (2005) 43. K. Maeda, Y. Asakuma, K. Fukui, J. Chem. Phys. 128 (2008) 044716. B.J. Alder, T.E. Wainwright, Phys. Rev. 127 (1962) 359.