The coarse-grained model for a water/oil/solid system: based on the correlation of water/air and water/oil contact angles

Abstract

The coarse-grained molecular dynamics (CG MD) simulation has become an important tool for studying water/oil/solid systems which are closely related to a broad range of scientific issues, such as nanotechnology, industrial applications, and environmental sciences. However, the coarse-grained force field (CG FF) for these systems, which plays a crucial role in molecular dynamic simulation, has not been accurately established. In this work, a novel method is provided to build a coarse-grained model for molecular simulation of water/oil/solid systems. The water/oil CG FF of a pre-selected analytical form was parameterized to match thermo-dynamic quantities in combination with all-atom (AA) simulation results. Upon tuning the CG FF parameters between water and solid, solid surfaces of different wettability were obtained. Then CG FF between oil and solid of different wettability was obtained based on the correlation between water/air and water/oil contact angles. We used our CG model to simulate the water/oil spontaneous capillary displacement, which demonstrates the consistency between theory and CG MD simulation. The novel method provided here is expected to promote the development of CG FF, and the study of multiphase systems.

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1. Introduction

Coarse-grained molecular dynamics (CG MD) simulation is a computational approach with cheaper potential calculation, larger time step, and smoother free energy landscape than the all-atom molecular dynamics (AA MD) approach. It is powerful in studying large-scale systems containing surfactants,^1,2 biomolecules,^3,4 membrane proteins,⁵ and other soft materials.⁶ In recent years, this simulation method has been widely used to study systems involving multiphases (water, oil and solid), such as water displacement in petroleum recovery,⁷ nanoparticle assembly and migration at water–oil interface.^8–10

Force field (FF) plays a crucial role in MD simulation. For CG MD simulation, developing reasonable CG FF is vital for the reliability of the results, which has received a huge amount of attention in the past decades. Generally, there are two types of methods for developing CG FF. One is bottom-up approach, in which CG FF is built to reproduce microscopic quantities obtained from fine-grained simulations, such as multiscale coarse-grained (MS-CG) approach.^4,11–13 The other is top-down approach like Martini coarse-grained approach,^3,14 in which CG FF is developed to reproduce macroscopic thermo-dynamic quantities. The two methods are used extensively in a wide variety of soft materials. Thorough reviews on the development of CG FF can be found by Brini,⁶ by Marrink,¹⁵ by Ingólfsson¹⁶ amongst others.

However, for some complex systems, such as water/oil/solid systems in studying water/oil displacement in the capillary,^17–19 nanoparticles adsorption at the water/oil interface,^8–10 etc., we still cannot get reasonable interaction parameters only through reproducing the microscopic and/or macroscopic data. This is because some correlations exist in these systems, and these correlations also should be included in the CG FF. Although a large number of CG MD studies relating to water/oil/solid have been conducted, hardly any of which took the correlation between water/solid and oil/solid into account. Stukan et al.¹⁹ simulated the spontaneous imbibition in nanopores and recovery of crude oils by surfactants, in which the water/solid and oil/solid interactions were tuned depending on random guessing. Chen et al.^17,18 studied forced and spontaneous displacements in capillary respectively, and during these simulations water/solid and oil/solid interactions were tuned unrelated. These works neglected the intrinsic correlation between oil/solid and water/solid interactions, thus leading to some superfluous or unreasonable FF parameters. Alberto Striolo et al.⁹ applied the corresponding relation between two kinds of interactions to study nanoparticles adsorption at the water/oil interface. The nanoparticle/solvent interactions were parameterized to reproduce the contact angles via AA MD simulations of one silica nanoparticle at the decane/water interface. The results were very concise and avoided redundant simulations. However, this coarse model is only suitable for silica nanoparticle covered by methyl (–CH₃) groups and hydroxyl (–OH) groups. Therefore, developing CG FF for common oil/water/solid systems of different wettabilities with taking into account the correlation between oil/solid and water/solid interactions is still needed.

Sorbie et al.²⁰ theoretically demonstrated relationships between water/air and water/oil contact angles in water/oil/solid system. Starting with the Bartell–Osterhof equation²¹ (eqn (1)) which was derived by solving the Young's²² equations for air/water, oil/water, and oil/air contact angles, combining with the end-point conditions for strongly water-wet and oil-wet pores, van Dijke and Sorbie²⁰ applied the minimum assumption for that region and derived the relationship between water/air and water/oil contact angles, and the linear assumption as shown in eqn (2)

σ_wa cos θ_wa = σ_wo cos θ_wo + σ_oa cos θ_oa (1)

where σ_wa, σ_wo and σ_oa are interfacial tensions and θ_wa, θ_wo, θ_oa are contact angles and subscripts w, a and o denote the water, air, and oil, respectively. Then Grate et al.²³ experimentally verified this equation for diverse surfaces, and stated that it is the most extensive correlation of water/air and water/oil contact angles for clean and silanized surfaces.

As well known, the water/air contact angle can be tuned by the water/solid interactions, the water/oil contact angle can be tuned by water/solid and oil/solid interactions. Therefore, the intrinsic relationship between water/air and water/oil contact angles could be certainly used to describe the correlation between water/solid and oil/solid interactions. Based on this idea, we developed a CG FF for water/oil/solid systems with different wettabilities of solid surface, in which the correlation between oil/solid and water/solid has been included. The novel method for CG FF development provided here is expected to promote the development of CG FF and the study of multiphase systems.

The rest of this paper is organized as follows. The next section was devoted to the description of the model and computational details. Then, in Section 3, the CG FF was developed, herein the correlation between oil/solid and water/solid has been included. Subsequently we used our CG model to simulate the water oil spontaneous capillary displacement, which demonstrated the usefulness and conciseness of our CG FF. Finally, concluding remarks are presented in Section 5.

2. Methods

Before developing the CG FF, the models and simulation details were introduced, which were used to define the wetting property of the solid and identify the water/solid interactions and oil/solid interactions. Fig. 1 shows the model for calculating static state contact angles, which was set according to systems reported by Cupelli et al.²⁴ and Chen et al.:²⁵ a half-sealing capillary partially filled with a fluid. First of all, the solid beads are arranged in a 10 × 10 × 20 nm³ simulation box of a group of 21 500 homogeneous solid beads. The system is relaxed for 5 ns to guarantee other beads can't infiltrate the solid in the following simulation. Secondly, beads in the inner with radius of 4 nm are deleted, and rest beads form a capillary which are fixed in the simulation (in yellow). Finally, a group of cylindrical liquid (water or oil) beads with 4 nm radius and 10 nm length is put in the left side of capillary (in purple). Among them, 2 nm-long beads in the left region are the sealing beads (in red).

Fig. 1 Simulation system for static contact angle calculations. (a) Side view of the capillary at the initial state of simulation. (b) Cross-section view of the system.

The mesocite model and the mesostructure model of commercial software package named Materials Studio (Accelrys Inc.) were used to conduct the simulations. The models were calculated with the time step of 10 fs in the NVT ensemble. The temperature was maintained at 298 K by means of NHL²⁶ method with periodic boundary conditions along all directions. The cutoff distance for van der Waals interaction is 12.5 Å.

To obtain static contact angles, the water or oil were subdivided into 16 concentric cylindrical shells with same thickness (Fig. 2), as done by Cupelli et al.²⁴ and Martic et al.²⁷ For each shell, we computed the density of particles as a function of the distance z into the pore, and then the densities of particles were fitted using the following hyperbolic tangent function that has been used for the liquid–vapor or liquid–liquid interface.^28–30

where ρ_i is the density, z₀ is the position of the Cibbs dividing surface, and d is the adjustable parameter related to the interfacial thickness. And the extremity positions of each shell at the distance where the density falls below a cutoff value of 0.5 times the liquid density were found. A circular fit to these extremity positions gives us the meniscus curvature and the contact angle at the pore wall.

Fig. 2 The typical meniscus curvature and the contact angle at the pore wall.

Fig. 3 is typical equilibrium process of dodecane systems with different surface wettability. For dodecane, the system quickly achieves its equilibrium in 0.2 ns, no matter which kind of surface wettability is. For other liquid, water and alkanes, equilibrium processes are same as dodecane. So, the simulations were carried out for 4 ns and the last 2 ns procedure was used for static contact angle analysis.

Fig. 3 The typical equilibrium process of dodecane systems for different surface wettability.

3. CG FF development and result

3.1 CG mapping

In this work, the CG models were based on a three-to-one mapping, i.e., three heavy atoms and their accompanying hydrogen atoms were represented by a single interaction site, which is the same as Shinoda force field.³¹ Three atomistic H₂O molecules were united into a single W bead. N-Alkanes were chosen to be the oil phase as widely reported.^32–34 Two CG interaction sites, CT (CH₃–CH₂–CH₂–) and CM (–CH₂–CH₂–CH₂–), were used to build up n-alkane chains. The fixed S (solid) beads were used for constructing the capillary as introduced above.

3.2 Pre-selected CG FF form

For intramolecular interactions, bonded interactions between beads were calculated by a universal weak harmonic potential U_Bonded,where U_stretch and U_bend represent the bond stretch and angle bend potentials, respectively. K_stertch and K_bend are spring constants. R₀ and θ₀ are equilibrium constants. Dihedral potentials, which are often used in AA MD, are omitted from our model because this simple CG FF form fulfilled our needs.

The following two shifted Lennard-Jones potential energy functions were used to characterize the non-bonded interactions according to the report of Shinoda,³¹

D₀ denotes the equilibrium well depth, and R₀ denotes the equilibrium distance. The LJ12-4 function is used only for describing the pairs involving water, and the LJ9-6 function is used for other pairs. The non-bonded interactions exclude 1-2, 1-3 bonded interactions. In the parameterize process, the D₀ for water/solid, D_WS, and alkanes/solid, D_TS & D_MS, were tunable and R₀ was calculated by Lorentz combining rules.³⁵

3.3 CG FF parameters

3.3.1 Water/alkanes interaction. A coarse-grained model for water/alkanes system were parameterized, in which the values of bond and angle were adjusted by the Gaussian distribution of corresponding bond and angle obtained from atomistic simulations, respectively. The non-bond interactions of water–water (W–W) and alkanes (CT–CT, CT–CM, CM–CM) are fitted through experimental density and surface/interfacial tension data. And the non-bond interactions of water-alkanes (W–CT, W–CM) are fitted through interfacial tension data. The detailed procedure is shown in ESI.†

3.3.2 Water/solid interaction. The water/solid interaction (D_WS) determines the wettability of solid surface and vice versa. Generally, the wettability of solid surface is evaluated by the water/air contact angle, which is defined as hydrophilic if water/air contact angle is from 0° to 90°, and hydrophobic if angle is above 90°,³⁶ respectively. Based on the model shown in Fig. 1, the water/air contact angles were calculated at different D_WS (Fig. 4). When D_WS increases from 0.05 to 0.9 kcal mol⁻¹, the cosine of contact angle changes from −1 to 1; in other word, the wetting property of solid surface changes from hydrophobic to hydrophilic, which is in good agreement with other literatures reported.¹⁸ The wettability of solid surface changes with mutual interaction which could be explained by Young's law:²² cos θ₀ = (σ_sa − σ_ws)/σ_wa. The water/solid tension reduces as the water/solid interaction increases, while the solid/air tension and the water/air tension remain the same. So the cosine of static contact angle increased with the increasing of water/solid interaction, and this relationship has been quantitatively proven by other simulation.²⁴ Therefore, wettability of solid surfaces could be tuned by changing the water/solid interaction parameters. The range of water/air static contact angles experimentally is 0° to 120° ^37,38 for smooth surfaces, so the range of the water/solid mutual interaction parameter is 0.35 to 0.9 kcal mol⁻¹ correspondingly. In summary, to describe solids with different wettability, D_WS is tunable between 0.35 and 0.9 kcal mol⁻¹ in our study.

Fig. 4 Relationship between the water/solid interaction parameter (D_WS) and the equilibrium static contact angle obtained by CG MD simulations.

3.3.3 Alkanes/solid interaction. Alkanes/solid interaction, which includes CT/solid interaction (D_TS) and CM/solid interaction (D_MS), should be parameterized for solid with different wettability. To simply the procedure of parameters calibrated, the relationship between water/air and water/alkanes static contact angles (eqn (2)), were translated to the relationship between water/air and alkanes/air contact angles for different n-alkanes (eqn (8)), via combining eqn (2) with the Bartell–Osterhof equation (eqn (1), which has been verified by CG MD simulation in Fig. S1 in ESI†). This relationship is designed to determine the alkanes/solid interactions.

In the above section, the relationships between water/air contact angles and D_WS were obtained. Likewise, the relationships between alkanes/air contact angles and D_TS & D_MS were established by simulation (Fig. 5) and the greater details of establishment was shown in the following. These relationships were combined and transformed into the corresponding relationship between D_WS and D_TS & D_MS.

Fig. 5 Relationships between the CT&CM/Solid interaction parameters (D_MS & D_TS) and the equilibrium static contact angles (cos θ) obtained by CG MD simulations, (a) nonane/solid system, (b) dodecane/solid system.

Based on Fig. 4 and eqn (8), the relationships between alkanes/air contact angles and D_WS for n-alkanes were obtained quantitatively (Table 1). The hexane was taken as example here to show the procedure to obtain the relationships between hexane/air contact angles (cos θ_ha) and D_WS. Firstly, these relationships between water/air contact angles (cos θ_wa) and D_WS in the first two columns in Table 1 were quantitatively got according to Fig. 4; secondly, the σ_wa, σ_wo and σ_oa in eqn (8) are replaced by water/air, water/hexane and hexane/air interfacial tensions, respectively, the cos θ_wa in eqn (8) was substituted by the exact values of water/air contact angles (cos θ_wa) (Table 1, column 2); finally, the cos θ_oa in eqn (8), that is cos θ_ha (Table 1, column 3), was generated via eqn (8). In Table 1, the cosine of hexane/air contact angle are above 1, which illustrates that the theory used for non-spreading oil²⁰ is not appropriate for hexane system. In addition, when the number of unknown parameters is smaller than the number of target properties, the parameters are fixed uniquely in the final process. Therefore, to establish the relationship between alkanes/air contact angles and D_TS & D_MS, the nonane/solid contact angles (cos θ_na) and the dodecane/solid contact angles (cos θ_da) were chosen.

Table 1 Relationships between water/solid interactions and alkanes/air contact angles for n-alkanes

Water/solid interaction (DWS, kcal mol^−1)	Water/air contact angle (cos θwa)	Alkanes/air contact angle
		Hexane (cos θha)	Nonane (cos θna)	Dodecane (cos θda)	Pentadecane (cos θpa)
0.35	−0.36	1.00	0.99	0.99	0.99
0.40	−0.24	1.02	0.98	0.97	0.97
0.45	−0.11	1.03	0.97	0.95	0.94
0.50	0.01	1.04	0.96	0.93	0.91
0.55	0.14	1.05	0.95	0.91	0.89
0.60	0.28	1.06	0.94	0.89	0.86
0.65	0.42	1.08	0.92	0.86	0.82
0.70	0.57	1.09	0.91	0.84	0.79
0.75	0.71	1.10	0.897	0.82	0.76
0.80	0.84	1.11	0.88	0.79	0.73
0.85	0.94	1.12	0.87	0.78	0.71
0.90	0.98	1.12	0.87	0.77	0.70

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The cos θ_na and cos θ_da were obtained via altering the D_TS and D_MS on the simulation, as shown in Fig. 5. It illustrated how did cos θ_na and cos θ_da change with D_TS and D_MS. Each of them contained 12 × 12 cases, so 288 individual simulations were carried out in all to plot Fig. 5. And then, several equations were chosen to fit these points. A binary quadratic equation (eqn (9)) was the most reasonable one with the correlation coefficient 0.99849 for the nonane/solid system and 0.99875 for the dodecane one.

here aⁿ₁, aⁿ₂, …, aⁿ₉ and a^d₁, a^d₂, …, a^d₉ are the fitting coefficients and n, d denotes the nonane/solid system and the dodecane/solid system, and the coefficients display in Table 2.

Table 2 The fitting coefficients for eqn (9)

a^n1	a^n2	a^n3	a^n4	a^n5	a^n6	a^n7	a^n8	a^n9
3.26	1.07	−1.08	−5.23	4.35	0.25	6.97	−4.15	1.25

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a^d1	a^d2	a^d3	a^d4	a^d5	a^d6	a^d7	a^d8	a^d9
2.24	0.96	−1.09	−4.56	3.41	0.53	6.00	−4.51	2.21

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For different D_WS, definite D_TS and D_MS were got by solving the two equations (eqn (9)). As show in Fig. 6, following two features can be found. Firstly, D_TS increases along with D_WS, while D_MS decreases with the increase of D_WS. This difference is qualitatively understandable according to Table 1, which indicates that the spreadability of hexane get better and the others alkanes get worse with the increase of water/solid interaction parameters. Secondly, the ranges of D_TS and D_MS are about 0.448 to 0.470 kcal mol⁻¹ and 0.480 to 0.580 kcal mol⁻¹, which are far smaller than the ranges of D_WS 0.350 to 0.900 kcal mol⁻¹. It illustrates that the change of water/oil contact angles is mainly driven by the D_WS, which is in perfect accordance with our early publication³³ that the adsorption capability of interfacial water have a profound effect on the adsorption behavior of the oil droplet.

Fig. 6 Relationships between D_TS & D_MS and D_WS.

The D_TS & D_MS and D_WS interaction parameters from Fig. 6 are employed to simulate the values of pentadecane/solid contact angles (red points in Fig. 7) and the simulation results were compared with the results by theory (black points in Fig. 7). Consistent values show that these interaction parameters are also valid for water/pentadecane/solid system.

Fig. 7 Comparison of the theory relationships (squares) and the result of CG MD simulation (circles) for the pentadecane/solid contact angles (cos θ_pa).

4. Application

The water/oil spontaneous displacement in capillaries with different wettability is closely related to many natural process and industrial applications, such as groundwater remediation³⁹ and oil recovery. As shown in Fig. 8(a), the capillary was built as mentioned above (in Section 2). A water reservoir with totally 22 260 water beads was connected to the left entrance of the capillary; a part of an oil reservoir with totally 3566 n-dodecane molecules prefilled the capillary and the rest was connected to the right exit of the capillary. Once the system is ready, D_WS was set as 0.5 kcal mol⁻¹ and D_TS & D_MS were obtained from the curve in Fig. 6 directly. In this condition, three-phase contact angle is around 90° while water was prevented from imbibing into the capillary. The system was equilibrated for 2 ns. Thereafter, D_WS were tuned from 0.5 to 0.9 kcal mol⁻¹, and D_TS & D_MS were got from Fig. 6. Each spontaneous capillary displacement process was carried out for 5 ns.

Fig. 8 Side view of the water/oil displacement simulation system at (a) the initial state, (b) 2 ns, and (c) 4 ns. The dimension of capillary is same as Fig. 1.

In general, the dynamics of processes in CG systems is faster in comparison to the process AA MD. And the simulation time have to be scaled previously before interpreting the results obtained via CG MD. To obtain the scale of the time axis, the diffusion coefficients of simulation and experimental values were compared.^14,40 The models used for calculation of water diffusion coefficients were adopted to measure the mean squared displacement, 〈r_D²〉, which was closely related with the diffusion coefficients according to the Einstein's equation for diffusion (〈r_D²〉 = 6Dt). Comparing the diffusion coefficients 2.27 × 10⁻⁹ m² s⁻¹ obtained by simulation with the experimental values 2.30 × 10⁻⁹ m² s⁻¹,⁴¹ the standard conversion factor we calculate is a factor of one. So in this reports, the simulation time is equal to real time.

For the larger water/solid interaction, water was imbibed into the capillary and oil was displaced out from capillary (Fig. 8(b) and (c)). We calculated displacement level and contact angles with different D_WS (Fig. 9). With the increase of D_WS, the spontaneous capillary displacement process became faster, and the average cosine of contact angle increased. The cosine contact angles were also calculated according to theory¹⁷ (eqn (10)).

where r represents radius of capillary, η represents the viscosity of fluid, b represents the slid length.

Fig. 9 Displacement process, (a) displacing curves and (b) the change of dynamic contact angle. Interaction parameter D₀ varied among 0.5, 0.6, 0.7,0.8, 0.9 and the CT & CM/solid parameter adjusted according to their relationships correspondingly.

Obviously, the simulation results obtained with our CG FF show good agreement with the theory results. By studying water/oil spontaneous displacement in capillaries, we demonstrated the conciseness and effectiveness of our CG FF.

5. Conclusion

In this article, we supplied a new method to build a coarse-grained model for studying water/oil/solid systems, in which the correlation of water/solid and oil/solid interaction parameters was included. It was also found that there is a one-to-one relationship between water/solid interaction and oil/solid interaction, that the CT/solid interactions increase and the CM/solid interactions decrease with increasing the water/solid interactions. At last, we used our model to study the water/oil spontaneous capillary displacement, which demonstrated the conciseness and effectiveness of our CG FF. This work is expected to promote the development of CG FF, and the study of multiphase systems.

Conflict of interest

The authors declare no competing financial interest.

Acknowledgements

This work was supported by the National Program on Key Basic Research Project of China (2015CB250904), National Science Fund for Distinguished Young Scholars (no. 51425406), Natural Science Foundation of Shandong Province (2014ZRE28048), the Fundamental Research Funds for the Central Universities (14CX06158A, 14CX02222A, 15CX08003A), and the PetroChina Innovation Foundation (2013D-5006-0206).

References

1. B. Smit, K. Esselink and P. A. J. Hilbers, et al., Langmuir, 1993, 9, 9–11.
2. M. Velinova, D. Sengupta and A. V. Tadjer, et al., Langmuir, 2011, 27, 14071–14077.
3. S. J. Marrink, A. H. de Vries and A. E. Mark, J. Phys. Chem. B, 2003, 108, 750–760.
4. S. Izvekov and G. A. Voth, J. Phys. Chem. B, 2005, 109, 2469–2473.
5. M. W. Tate, E. F. Eikenberry and D. C. Turner, et al., Chem. Phys. Lipids, 1991, 57, 147–164.
6. E. Brini, E. A. Algaer and P. Ganguly, et al., Soft Matter, 2013, 9, 2108–2119.
7. L. Lake, Enhanced Oil Recovery, Prentice Hall, Englewood Cliffs, NJ, United States, 1989.
8. H. Fan and A. Striolo, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 86, 051610.
9. X. C. Luu, J. Yu and A. Striolo, Langmuir, 2013, 29, 7221–7228.
10. X. C. Luu, J. Yu and A. Striolo, J. Phys. Chem. B, 2013, 117, 13922–13929.
11. Y. Wang, S. Izvekov and T. Yan, et al., J. Phys. Chem. B, 2005, 110, 3564–3575.
12. S. Izvekov and G. A. Voth, J. Chem. Phys., 2005, 123, 134105.
13. S. Izvekov, A. Violi and G. A. Voth, J. Phys. Chem. B, 2005, 109, 17019–17024.
14. S. J. Marrink, H. J. Risselada and S. Yefimov, et al., J. Phys. Chem. B, 2007, 111, 7812–7824.
15. S. J. Marrink and D. P. Tieleman, Chem. Soc. Rev., 2013, 42, 6801–6822.
16. H. I. Ingólfsson, C. A. Lopez and J. J. Uusitalo, et al., Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2014, 4, 225–248.
17. C. Chen, K. Lu and X. Li, et al., RSC Adv., 2014, 4, 6545–6555.
18. C. Chen, L. Zhuang and X. Li, et al., Langmuir, 2012, 28, 1330–1336.
19. M. R. Stukan, P. Ligneul and E. S. Boek, Oil Gas Sci. Technol., 2012, 67, 737–742.
20. M. I. J. van Dijke and K. S. Sorbie, J. Pet. Sci. Eng., 2002, 33, 39–48.
21. F. E. Bartell and H. J. Osterhof, Ind. Eng. Chem., 1927, 19, 1277–1280.
22. T. Young, Philos. Trans. R. Soc. London, 1805, 95, 65–87.
23. J. W. Grate, K. J. Dehoff and M. G. Warner, et al., Langmuir, 2012, 28, 7182–7188.
24. C. Cupelli, B. Henrich and T. Glatzel, et al., New J. Phys., 2008, 10, 043009.
25. C. Chen, C. Gao and L. Zhuang, et al., Langmuir, 2010, 26, 9533–9538.
26. B. Leimkuhler, E. Noorizadeh and O. Penrose, J. Stat. Phys., 2011, 143, 921–942.
27. G. Martic, F. Gentner and D. Seveno, et al., Langmuir, 2002, 18, 7971–7976.
28. J. L. Rivera, C. McCabe and P. T. Cummings, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2003, 67, 011603–011612.
29. J. L. Rivera, M. Predota and A. A. Chialvo, et al., Chem. Phys. Lett., 2002, 357, 189–194.
30. S. S. Jang, S.-T. Lin and P. K. Maiti, et al., J. Phys. Chem. B, 2004, 108, 12130–12140.
31. W. Shinoda, R. DeVane and M. L. Klein, Mol. Simul., 2007, 33(1), 27–36.
32. S. O. Nielsen, C. F. Lopez and G. Srinivas, et al., J. Chem. Phys., 2003, 119, 7043–7049.
33. J. Zhong, X. Wang and J. Du, et al., J. Phys. Chem. C, 2013, 117, 12510–12519.
34. P. Zhang, Z. Xu and Q. Liu, et al., J. Chem. Phys., 2014, 140, 164702.
35. H. A. Lorentz, Ann. Phys., 1881, 248, 127–136.
36. J. T. Gostick, M. A. Ioannidis and M. W. Fowler, et al., Electrochem. Commun., 2008, 10, 1520–1523.
37. T. Nishino, M. Meguro and K. Nakamae, et al., Langmuir, 1999, 15, 4321–4323.
38. T.-S. Wong and C.-M. Ho, Langmuir, 2009, 25, 12851–12854.
39. J. F. Pankow and J. A. Cherry, Dense Chlorinated Solvents & Other DNAPLs in Groundwater: History, Behavior and Remediation, Waterloo Educational Services Inc, Rockwood, Ontario, 1996.
40. M. R. Stukan, P. Ligneul and J. P. Crawshaw, et al., Langmuir, 2010, 26, 13342–13352.
41. D. R. Lide, CRC Handbook of Chemistry and Physics, Chapman&Halls/CRC, 2002.

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Footnote

† Electronic supplementary information (ESI) available: Comparison of Bartell–Osterhof equation and the result of CGMD simulation for the different water/solid interaction; the detailed procedure for a coarse-grained model of water/alkanes system. See DOI: 10.1039/c5ra06128h

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