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物理化学学报  2018, Vol. 34 Issue (10): 1163-1170    DOI: 10.3866/PKU.WHXB201802271
所属专题: 材料科学的分子模拟
论文     
Free Energy Change of Micelle Formation for Sodium Dodecyl Sulfate from a Dispersed State in Solution to Complete Micelles along Its Aggregation Pathways Evaluated by Chemical Species Model Combined with Molecular Dynamics Calculations
YOSHII Noriyuki1,2,*(),KOMORI Mika2,KAWADA Shinji2,TAKABAYASHI Hiroaki2,FUJIMOTO Kazushi2,OKAZAKI Susumu1,2,*()
1 Center for Computational Science, Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan
2 Department of Applied Chemistry, Nagoya University, Nagoya 464-8603, Japan
Free Energy Change of Micelle Formation for Sodium Dodecyl Sulfate from a Dispersed State in Solution to Complete Micelles along Its Aggregation Pathways Evaluated by Chemical Species Model Combined with Molecular Dynamics Calculations
Noriyuki YOSHII1,2,*(),Mika KOMORI2,Shinji KAWADA2,Hiroaki TAKABAYASHI2,Kazushi FUJIMOTO2,Susumu OKAZAKI1,2,*()
1 Center for Computational Science, Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan
2 Department of Applied Chemistry, Nagoya University, Nagoya 464-8603, Japan
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摘要:

Surfactant molecules, when dispersed in solution, have been shown to spontaneously form aggregates. Our previous studies on molecular dynamics (MD) calculations have shown that ionic sodium dodecyl sulfate molecules quickly aggregated even when the aggregation number is small. The aggregation rate, however, decreased for larger aggregation numbers. In addition, studies have shown that micelle formation was not completed even after a 100 ns-long MD run (Chem. Phys. Lett. 2016, 646, 36). Herein, we analyze the free energy change of micelle formation based on chemical species model combined with molecular dynamics calculations. First, the free energy landscape of the aggregation, ΔGi+j, where two aggregates with sizes i and j associate to form the (i + j)-mer, was investigated using the free energy of micelle formation of the i-mer, Gi, which was obtained through MD calculations. The calculated ΔGi+j was negative for all the aggregations where the sum of DS ions in the two aggregates was 60 or less. From the viewpoint of chemical equilibrium, aggregation to the stable micelle is desired. Further, the free energy profile along possible aggregation pathways was investigated, starting from small aggregates and ending with the complete thermodynamically stable micelles in solution. The free energy profiles, G(l, k), of the aggregates at l-th aggregation path and k-th state were evaluated by the formation free energy $\sum\limits_i {{n_i}\left( {l, k} \right)G_i^\dagger } $ and the free energy of mixing $\sum\limits_i {{n_i}(l, k){k_B}Tln({n_i}(l, k)/n(l, k))} $, where ni(l, k) is the number of i-mer in the system at the l-th aggregation path and k-th state, with $n\left( {l, k} \right) = \sum\limits_i {{n_i}\left( {l, k} \right)} $. All the aggregation pathways were obtained from the initial state of 12 pentamers to the stable micelle with i = 60. All the calculated G(l, k) values monotonically decreased with increasing k. This indicates that there are no free energy barriers along the pathways. Hence, the slowdown is not due to the thermodynamic stability of the aggregates, but rather the kinetics that inhibit the association of the fragments. The time required for a collision between aggregates, one of the kinetic factors, was evaluated using the fast passage time, tFPT. The calculated tFPT was about 20 ns for the aggregates with N = 31. Therefore, if aggregation is a diffusion-controlled process, it should be completed within the 100 ns-simulation. However, aggregation does not occur due to the free energy barrier between the aggregates, that is, the repulsive force acting on them. This may be caused by electrostatic repulsions produced by the overlap of the electric double layers, which are formed by the negative charge of the hydrophilic groups and counter sodium ions on the surface of the aggregates.

关键词: Free energy changeAggregation pathwaySDSMicelleMolecular dynamics calculation    
Abstract:

Surfactant molecules, when dispersed in solution, have been shown to spontaneously form aggregates. Our previous studies on molecular dynamics (MD) calculations have shown that ionic sodium dodecyl sulfate molecules quickly aggregated even when the aggregation number is small. The aggregation rate, however, decreased for larger aggregation numbers. In addition, studies have shown that micelle formation was not completed even after a 100 ns-long MD run (Chem. Phys. Lett. 2016, 646, 36). Herein, we analyze the free energy change of micelle formation based on chemical species model combined with molecular dynamics calculations. First, the free energy landscape of the aggregation, ΔGi+j, where two aggregates with sizes i and j associate to form the (i + j)-mer, was investigated using the free energy of micelle formation of the i-mer, Gi, which was obtained through MD calculations. The calculated ΔGi+j was negative for all the aggregations where the sum of DS ions in the two aggregates was 60 or less. From the viewpoint of chemical equilibrium, aggregation to the stable micelle is desired. Further, the free energy profile along possible aggregation pathways was investigated, starting from small aggregates and ending with the complete thermodynamically stable micelles in solution. The free energy profiles, G(l, k), of the aggregates at l-th aggregation path and k-th state were evaluated by the formation free energy $\sum\limits_i {{n_i}\left( {l, k} \right)G_i^\dagger } $ and the free energy of mixing $\sum\limits_i {{n_i}(l, k){k_B}Tln({n_i}(l, k)/n(l, k))} $, where ni(l, k) is the number of i-mer in the system at the l-th aggregation path and k-th state, with $n\left( {l, k} \right) = \sum\limits_i {{n_i}\left( {l, k} \right)} $. All the aggregation pathways were obtained from the initial state of 12 pentamers to the stable micelle with i = 60. All the calculated G(l, k) values monotonically decreased with increasing k. This indicates that there are no free energy barriers along the pathways. Hence, the slowdown is not due to the thermodynamic stability of the aggregates, but rather the kinetics that inhibit the association of the fragments. The time required for a collision between aggregates, one of the kinetic factors, was evaluated using the fast passage time, tFPT. The calculated tFPT was about 20 ns for the aggregates with N = 31. Therefore, if aggregation is a diffusion-controlled process, it should be completed within the 100 ns-simulation. However, aggregation does not occur due to the free energy barrier between the aggregates, that is, the repulsive force acting on them. This may be caused by electrostatic repulsions produced by the overlap of the electric double layers, which are formed by the negative charge of the hydrophilic groups and counter sodium ions on the surface of the aggregates.

Key words: Free energy change    Aggregation pathway    SDS    Micelle    Molecular dynamics calculation
收稿日期: 2017-12-25 出版日期: 2018-04-13
基金资助: This work was supported by FLAGSHIP2020, MEXT within Priority Study 5 (Development of New Fundamental Technologies for High-Efficiency Energy Creation, Conversion/Storage and Use) Using Computational Resources of the K Computer Provided by the RIKEN Advanced Institute for Computational Science through the HPCI System Research Project (hp170241). This work was also funded by MEXT KAKENHI Grant Number 17K04758 (N.Y.)
通讯作者: YOSHII Noriyuki,OKAZAKI Susumu     E-mail: yoshii@ccs.engg.nagoya-u.ac.jp;okazaki@apchem.nagoya-u.ac.jp
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YOSHII Noriyuki
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引用本文:

YOSHII Noriyuki,KOMORI Mika,KAWADA Shinji,TAKABAYASHI Hiroaki,FUJIMOTO Kazushi,OKAZAKI Susumu. Free Energy Change of Micelle Formation for Sodium Dodecyl Sulfate from a Dispersed State in Solution to Complete Micelles along Its Aggregation Pathways Evaluated by Chemical Species Model Combined with Molecular Dynamics Calculations[J]. 物理化学学报, 2018, 34(10): 1163-1170, 10.3866/PKU.WHXB201802271

Noriyuki YOSHII,Mika KOMORI,Shinji KAWADA,Hiroaki TAKABAYASHI,Kazushi FUJIMOTO,Susumu OKAZAKI. Free Energy Change of Micelle Formation for Sodium Dodecyl Sulfate from a Dispersed State in Solution to Complete Micelles along Its Aggregation Pathways Evaluated by Chemical Species Model Combined with Molecular Dynamics Calculations. Acta Phys. -Chim. Sin., 2018, 34(10): 1163-1170, 10.3866/PKU.WHXB201802271.

链接本文:

http://www.whxb.pku.edu.cn/CN/10.3866/PKU.WHXB201802271        http://www.whxb.pku.edu.cn/CN/Y2018/V34/I10/1163

Fig 1  The molecular structure of SDS. Yellow: sulfur atom, red: oxygen atom, cyan: carbon atom, gray: hydrogen atom, and blue: sodium ion. Color online.
Fig 2  Δμi0 and Δμi† as a function of aggregation number i. The value of Δμi0 obtained from MD calculation is given in Ref. 7. The black solid line was obtained by fitting the polynomials $ \sum\limits_{k = 0} {{a_k}{i^k}} $ to the calculated Δμi0 for two regions (1 ≤ i < 66 and 66 ≤ i ≤ 80). The red solid line shows Δμi† in Eq. (6), where the coefficient α was determined to reproduces the experimental CMC.
1 ≤ i ≤ 66 66 ≤ i ≤ 80
a0 4.11 × 10-21 7.11 × 10-21
a1 -5.51 × 10-21 -4.81 × 10-21
a2 3.73 × 10-22 1.21 × 10-22
a3 -1.73 × 10-23 -1.21 × 10-24
a4 4.24 × 10-25 4.00 × 10-27
a5 -5.31 × 10-27 -
a5 3.24 × 10-29 -
a7 -7.65 × 10-32 -
Table 1  Fitting coefficients, ak, of Δμi0, where Δμi0 is approximated as $\Delta \mu _i^0 = \sum\limits_{k = 0} {{a_k}{i^k}} $.
Fig 3  Aggregation number i dependence of Gi† -(i -1) kBT lnX1eq. Gi†: black line, Gi† -(I -1) kBTlnX1eq at 0.5CMC, CMC, and 30CMC are depicted by the blue, red and green lines, respectively. The black dashed line represents (I -1) kBT ln X1eq at CMC. Color online.
Fig 4  Free energy landscape, ΔGi+j†, of formation of an (i + j)-mer from i-mer and j-mer.
Fig 5  Free energy profile G(l, k) (lower plot of (a)) and Gmix(l, k) (upper plot of (a)), and ΔG(l, k) (lower plot of (b)) and ΔGmix(l, k) (upper plot of (b)) along the aggregation pathways. Note that the scales on the vertical axes are different between (a) and (b).
Fig 6  Snapshot of SDS aggregates in solution at elapsed time (a) t = 0 ns, (b) 7 ns, (c) 50 ns, and (d) 100 ns. Colors for atoms are the same as in Fig. 1. Water molecules are not depicted for clarity.
Fig 7  Time evolution of (a) maximum aggregation number, i, of aggregates in the independent six aggregation simulations, (b) four largest aggregation numbers for one trajectory, green line in Fig. 7a, and (c) free energy profile G(l, t) of the system evaluated by Eq. (14) for each trajectory. Results of the six independent MD runs are plotted in different colors in (a) and (c).
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