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Evaluation of Interaction Forces between Macroparticles in Simple Fluids by Molecular Dynamics Simulation
Journal of Colloid and Interface Science 209, 79 – 85 (1999) Article ID jcis.1998.5870, available online at http://www.idealibrary.com on
Evaluation of Interaction Forces between Macroparticles in Simple Fluids by Molecular Dynamics Simulation Hiroyuki Shinto, Minoru Miyahara, and Ko Higashitani1 Department of Chemical Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto, 606-8501 Japan E-mail: [email protected] Received April 24, 1998; accepted September 17, 1998
bonding liquids (7, 8), aqueous electrolyte solutions (9 –11), liquid alkanes (12–14), and polymer melts (15–17). As the results, the forces which the DLVO theory fails to predict were discovered (1, 18, 19, 53). For example, the oscillatory solvation forces appear when two smooth surfaces approach each other closer than a few molecular diameters (3– 8, 11, 12, 14 –16), and the long-range attractive forces occur between hydrophobic surfaces in pure water (20 –22). In order to elucidate theoretically the origin of these nonDLVO forces on the molecular level, the integral equation theory (IET) based on the Ornstein–Zernike relation with various equations of closure approximation (23) has been developed and applied to various systems: A pair of either uncharged or charged large spheres (or planar walls) is immersed in a pure fluid composed of neutral hard-sphere (24 –26, 54), dipolar hard-sphere (24, 25), waterlike hard-sphere (25), or Lennard-Jones particle (27–29). Recently, in addition to the one-component fluids, the IET was extended to the multicomponent fluids of hard-sphere mixtures (29, 30) and aqueous electrolytes (25). It has been pointed out, however, that the results largely depend on the closure equations used (31). Therefore, it is necessary to compare carefully the results from the IET with those from the computer simulations based on molecular dynamics (MD) and Monte Carlo (MC) methods. MD and MC simulations (32) have been applied to relatively simple systems: two planar walls are immersed in a pure fluid of Lennard-Jones particle (33–38), waterlike hard-sphere (39), or alkane (37, 40, 41), rather than mixtures (42). The restriction comes mainly from the difficulty in specifying a chemical potential, or an alternative intensity factor, of the bulk phase that should be in equilibrium with the confined film between the walls. For this purpose, the grand canonical ensemble (33, 35, 38, 39, 41) and the test particle insertion method (42) were employed in the above simple systems. A new statistical ensemble (36) and a new MD cell (37, 40) were also developed. Nonetheless, in our opinion it seems quite difficult to apply these methods to mixtures and complex fluids other than pure simple fluids. The goal of this article is to present, to the best of our knowledge, the first study of the solvation forces between two
The present article provides the description of the solvation forces between large spheres in a fluid. The molecular dynamics (MD) method was applied to the relatively simple systems in which a pair of structureless macroparticles, either solvophobic or solvophilic, is immersed in a simple fluid of two types, either a soft-sphere or a Lennard-Jones fluid. When a pair of solvophobic macroparticles was in the attractive Lennard-Jones fluid, no dense layer of the solvent particles formed near the surface of the macroparticles and the strong attractive forces were induced between them. In the other combinations of macroparticles and fluids, the dense layers formed and the solvation forces oscillated, exhibiting the attraction and repulsion, whose periodic distance was about the diameter of solvent particles. Our results agreed well with those of the other simulation and theoretical studies with respect to the solvent density profile near a macroparticle and the force– distance profile between macroparticles. The benefit of our approach would be the simplicity in specifying or finding the bulk condition that is in equilibrium with the thin film of molecules between large surfaces. The present method can be applied straightforward to macroparticles immersed in mixtures and complex fluids described by the bead–spring model, to which the conventional grand canonical ensemble Monte Carlo (GCEMC) method is hardly accessible. © 1999 Academic Press Key Words: solvation; surface force; solvophobic attraction; molecular dynamics simulation.
1. INTRODUCTION
Both detailed understanding and exact description of interaction forces between colloidal particles immersed in fluids are the central subject of colloid science. Historically, the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory and the van der Waals theory have been frequently used to describe the interactions between colloidal particles in electrolyte solutions because they successfully explain experimental results (1). The surface force apparatus (SFA) was developed in late 1970s (1, 2) and has been used for in situ measurements of the interaction forces between cleaved or chemically modified mica surfaces in inert nonpolar liquids (3–5), polar liquids (6, 7), hydrogen1
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SHINTO, MIYAHARA, AND HIGASHITANI
large spheres in a fluid using classical MD simulations and compare our results with those from other simulations and the IET. We considered a pair of spherical structureless macroparticles immersed in a pure simple fluid. A large system was required to allow some of solvent particles to locate between the macroparticles and exchange themselves with those in the bulk reservoir. Our method presented here consumes much computation time but it can be applied straightforward to mixtures and complex fluids, e.g., those described by the bead–spring model (43, 44).
TABLE 1 Simulation Systems System
Solventa
Macroparticle
Diameter, d M
I II III IV
Soft-sphere Lennard-Jones particle Soft-sphere Lennard-Jones particle
Solvophobic Solvophobic Solvophilic Solvophilic
10d S, 20d S 10d S 10d S 10d S
Reduced density r* (5 r d 3S) and temperature T* (5k BT/ e ) of the solvent are 0.5925 and 1.2, respectively (see the text). a
2. MOLECULAR MODEL 3. CONSTRAINED MOLECULAR DYNAMICS METHOD
The interactions between solvent particles were represented by the shifted Lennard-Jones (12-6) potential with energy parameter e, core diameter d S, and cutoff radius R cut SS :
H
f ~r! 2 f ~R cut r # R cut SS !, SS , u SS~r! 5 0, r . R cut SS , f ~r! 5 4 e @~d S/r! 12 2 ~d S/r! 6#.
[1a] [1b]
We considered two different fluids: a Lennard-Jones (LJ) fluid cut with R cut SS 5 2.5d S, and a soft-sphere (SS) fluid with R SS 5 1/6 3 2 d S. Reduced density r* (5 r d S) and temperature T* (5k BT/ e ) of the fluids were chosen to be 0.5925 and 1.2, respectively, where k B is the Boltzmann’s constant. The shifted (10-4-3) potential was used for particle–macroparticle interaction (45),
H
`, r # r M, cut cut u MS~r! 5 c ~r! 2 c ~R MS!, r M , r # r M 1 R MS, cut 0, r . r M 1 R MS,
r M 5 d M/ 2, Dr 5 d S/ Î2,
A system composed of two macroparticles (A and B) and N solvent particles is considered. MD simulations of the system are performed keeping the center-to-center separation between macroparticles A and B in the same way as Ciccotti et al. first used to evaluate the mean force potential of an ion pair in a polar solvent (46). Accordingly, an additional constraint is incorporated,
j ~r A, r B! 5 ~r A 2 r B! 2 2 R 2 5 0,
[3]
where R is the fixed separation between macroparticles A and B. The forces acting on the macroparticles due to the solvent particles, FAS and FBS, are calculated. The solvation force F s(R) is evaluated as an average over the different configurations using the following expressions:
[2a] F s~R! 5
c ~r! 5 2 pe @0.4$d S/~r 2 r M!% 10 2 $d S/~r 2 r M!% 4 2 d 4S/$3Dr~r 2 r M 1 0.61Dr! 3%#,
3.1. Description of Interactions between Macroparticles
[2b] [2c]
where d M is the diameter of macroparticles and R cut MS is the cutoff distance. Note that the center of the outermost atoms in the macroparticle is located at separation r M from the center of the macroparticle. We considered two types of macroparticles with d M 5 10d S (or 20d S): a “solvophilic” macroparticle with R cut MS 5 10d S, in which the interaction includes both repulsive and attractive forces; and a “solvophobic” macroparticle with R cut MS 5 0.987d S, in which the interaction is completely repulsive. The mass of the macroparticle m M is equal to m S(d M/ d S)3, where m S is the mass of the solvent particle. In this study, we employed four systems as listed in Table 1; the solvophobic macroparticles in SS and LJ fluids (systems I and II) and the solvophilic macroparticles in SS and LJ fluids (systems III and IV).
1 ^rˆ z ~F AS 2 F BS!&, 2 AB
[4a]
rˆ AB 5 ~r A 2 r B!/ur A 2 r Bu.
[4b]
The potential W s(R) can be obtained by integrating Eq. [4a] from the large separation R 0 to a given separation R,
E
R
W s~R! 5 2
F s~R9!dR9.
[5]
R0
Supposing that the macroparticle is uniformly composed of the LJ particles of the number density r d 3S 5 1.0 and only the attraction term of 2(4 e d 6S)/r 6 in Eq. [1b] is considered, the potential of the direct macroparticle–macroparticle interaction, W d(R), is given by the analytical equation (47): W d~R! 5 2
F
S
A 2 2 s2 2 4 1 1 ln 6 s2 2 4 s2 s2
DG
,
[6a]
INTERACTIONS BETWEEN MACROPARTICLES IN FLUIDS
s5
2~R 2 d M! dS
Y S dd D 1 2, M
[6b]
S
A 5 p 2r 2~4 e d 6S! 5 4 p 2e ,
[6c]
where A is the Hamaker constant. The force F d(R) is obtained by differentiating Eq. [6a], F d~R! 5 2
1 dW d~R! 64A 52 . dR 3d M s 3~s 2 2 4! 2
[7]
Consequently, the total mean force between the macroparticles F t(R) and the potential W t(R) are described as F t~R! 5 F s~R! 1 F d~R!,
[8]
W t~R! 5 W s~R! 1 W d~R!.
[9]
In the present paper we applied the above description to the short-range potential systems as in Table 1, but this method is applicable straightforward also to long-range potential systems (e.g., Coulomb systems). 3.2. MD Simulation We implemented simulations of a system composed of two macroparticles with d M 5 10d S (or 20d S) and N 5 108,000 solvent particles in a cubic cell with periodic boundary conditions. The length of the cubic cell was selected to be 56.8d S (or 57.6d S) such that the density of the bulk fluid r* is 0.5925. It is confirmed that the dimension of this cell was large enough not to be affected by the neighboring image cells. In order to compute interactions between a large number of solvent particles efficiently, we employed the layered link cell (LLC) procedure connecting with the Verlet neighbor list on the Cray T-94/4128 vector computer (48). In this procedure, the neighbor list for force calculations was built up in the following way; the pairs of interacting particles i and j were found out according to the link cell (LC) algorithm and subsequently the identity numbers of these pairs, (i, j), were sorted systematically using the “layering algorithm” in order to vectorize all of the force loops. The neighbor list was updated automatically following Chialvo and Debenedetti (49). The constraint on the center-to-center separation of two macroparticles was imposed using the SHAKE method (50). The equations of motion for the solvent particles and the constrained macroparticles were solved using the leap-frog algorithm with a time step of Dt 5 0.00462 t 0 , where t 0 5 d S(m S/e)1/2. The temperature of the solvent was kept at T* 5 1.2 using Berendsen’s external bath method with a time constant of 0.462t0 (51), while we did not incorporate any temperature-control procedures into the macroparticles except the separation constraint. This is because it is favorable in the
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calculation of the solvation force to avoid artificial procedures imposed on the macroparticles as few as possible. The nearest separation between the macroparticles, R 2 d M, ranged from 0.0d S to 5.0d S with increments of 0.1–0.2d S. In each system, simulations were performed as follows. (i) Two macroparticles of A and B were positioned on a diagonal line of the cell such that the A–B separation was the largest separation of R 2 d M 5 5.0d S, and 108,000 solvent particles were placed at the face-centered-cubic (fcc) lattice excluded by the macroparticles. (ii) The system was equilibrated over 5.5 3 104 time steps. (iii) The instantaneous forces of FAS and FBS were computed during 2 3 104 time steps to obtain the solvation force F s(R) using Eq. [4]. (iv) Then, external forces were assigned to the macroparticles along the A–B line for 5 3 103 time steps such that the A–B separation was reduced by 0.1– 0.2d S. (v) In order to equilibrate the solvent particles around the macroparticles, the system was allowed to evolve for 5 3 103 time steps with keeping the new A–B separation. The procedure from (iii) to (v) was repeated to obtain the force– distance profile. The statistical errors of the solvation forces were evaluated by the block averaging method (55) and found to be less than 62.0, 61.4, 62.8, and 62.5 (in units of k BT/d s) for the systems I to IV, respectively. The potential of W s(R) was calculated by integrating the values of F s(R), where the trapezoidal rule was used without any smoothing procedures. 4. RESULTS AND DISCUSSION
4.1. Influence of Macroparticle Diameter on Solvation Force How the macroparticle diameter d M affects the solvation forces was examined simulating the system I with d M 5 10d S and 20d S. The results are given in Fig. 1, where the positive and negative values mean the repulsive and attractive forces, respectively. Figure 1 indicates that the force– distance profiles were qualitatively similar although the amplitude of the oscillatory behavior was larger in the case of d M 5 20d S. After both the profiles were multiplied by the factor of d S/ p d M on the basis of the Derjaguin approximation (52), they were almost identical to each other in all the range of the separations between macroparticles. We believe, therefore, that the succeeding results of the solvation force obtained using d M 5 10d S give the quantitative behavior within the model used here if the force– distance profiles are normalized by the value of d M. 4.2. Solvation Forces between “Solvophobic” Macroparticles The solvophobic macroparticles immersed in SS and LJ fluids, that is, the systems I and II, were considered. We calculated the density profile of solvent next to the macroparticle surface ( g MS), the solvation force, and the potential as shown in Figs. 2, 3, and 4, respectively. In the case of the system I, Fig. 2 illustrates that the dense
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SHINTO, MIYAHARA, AND HIGASHITANI
FIG. 1. Solvation force profiles for the system I (see Table 1). The solid and dotted lines represent the profiles for d M 5 10d S and 20d S, respectively. The inset gives the profiles normalized by the Derjaguin approximation (see the text).
layers of solvent particles formed adjacent to the surface of the large sphere even in such a repulsive potential system in which particle–particle and particle–macroparticle interactions were modeled as the short-range repulsion. This “packing efficiency” of particles near the surface results from the minimization of the free energy of the system, that is, the maximization of the entropy because the enthalpy is trivial in the repulsive potential system. We emphasize that the packing
FIG. 2. Reduced density profiles of solvent particles near the surface of the solvophobic macroparticle with d M 5 10d S. The dotted and solid lines represent the profiles for soft-sphere (SS) and Lennard-Jones (LJ) fluids, that is, the systems I and II, respectively.
FIG. 3. Solvation force profiles between the solvophobic macroparticles with d M 5 10d S in fluids. The dashed line shows Eq. [7], and the rest are the same as in Fig. 2.
efficiency that is entirely attributable to the entropy effect is, more or less, potentially intrinsic to any systems. When these two surfaces approached each other, the solvation force and the potential oscillated around the zero value, exhibiting the attraction and repulsion, and their periodic distances were about d S, that is, the solvent diameter as shown in Figs. 3 and 4. This oscillatory behavior is caused by the packing effect. After the solvent particles in the only one layer between the nearest surfaces of macroparticles started to be pushed out into the bulk at R 2 d M , 1.9d S, a strong attraction acted between the macroparticles and they came into contact with each other, which was the most stable state with regard to the free energy
FIG. 4. Potentials of solvation force corresponding to Fig. 3. The dashed line shows Eq. [6a], and the rest are the same as in Fig. 2.
INTERACTIONS BETWEEN MACROPARTICLES IN FLUIDS
as expected from Fig. 4. This is explained by the following mechanism. In the system I where the enthalpy is negligible, the behavior of the solvent particles shown in Fig. 2 are most favorable for the overall entropy of the system, in which a single macroparticle exists in the solvent. Nonetheless, the solvent particles in the dense layers near the surface of the macroparticle are less advantageous in terms of the entropy than those in the bulk. Hence, if two macroparticles attach to each other, the solvent particles are pushed out into the bulk, which leads to the increase of the overall entropy of the system, that is, the decrease of the total free energy. Thus the strong attractive force is induced even in a repulsive potential system such as the system I. The above behavior was also derived from the IET studies on the similar systems: hard-spherical macroparticles were immersed in a hard-sphere fluid (25, 54). The force– distance profiles predicted from these simulation and theoretical studies coincide qualitatively with those from the SFA experiments: the interaction forces were measured between the molecularly smooth surfaces of a cleaved mica in the liquid of quasispherical nonpolar molecules such as octamethylcyclotetrasiloxane (OMCTS), tetrachloromethane, cyclohexane, and benzene (1, 3–5, 18). On the other hand, in the case of the system II where the attractive forces prevailed for solvent–solvent but not for solvent–macroparticle, the different behavior was observed as in Figs. 2, 3, and 4; no dense layer of the solvent particles formed near the surface, i.e., the “dewetting,” and the solvation force profile exhibited a strong attraction without oscillating, i.e., the “solvophobic attraction.” The similar results have been also reported in the other studies (29, 35) and are interpreted as follows. When a large solvophobic particle intrudes into an attractive potential fluid, favorable bonds between the solvent
83
FIG. 6. Solvation force profiles between the solvophilic macroparticles with d M 5 10d S in fluids. The dashed line shows Eq. [7], and the rest are the same as in Fig. 5.
particles are unavoidably disrupted. Accordingly the particles near the macroparticle surface are reorganized at the sacrifice of the packing efficiency such that the bonds are prevented from breaking, which leads to the dewetting in this case. This behavior is advantageous to the bonds, that is, the enthalpy, but disadvantageous to the packing efficiency, that is, the entropy. These contradictory contributions to the free energy are compromised to minimize it. Nevertheless, the solvent particles near the surface are too unstable to form any dense layers. If the surfaces approach each other, these unstable particles are pushed out into the bulk, by which the free energy of the system decreases consequently. Thus the strong and monotonous attraction is generated between solvophobic surfaces in an attractive potential fluid. Figure 3 represents that the attraction caused by the presence of solvent was stronger than the direct macroparticle–macroparticle interaction given by Eq. [7] in the range of R 2 d M . 0.8d S. 4.3. Solvation Forces between “Solvophilic” Macroparticles
FIG. 5. Reduced density profiles of solvent particles near the surface of the solvophilic macroparticle with d M 5 10d S. The dotted and solid lines represent the profiles for soft-sphere (SS) and Lennard-Jones (LJ) fluids, that is, the systems III and IV, respectively.
The solvophilic macroparticles immersed in SS and LJ fluids, that is, the systems III and IV, were considered and the results are shown in Figs. 5, 6, and 7. It is noted that the results in the systems III and IV were almost the same in spite of the difference of the fluids: SS and LJ fluids. Comparison between Fig. 5 for the systems III and IV and Fig. 2 for the system I indicates that the solvent density near the solvophilic surface was higher than that near the solvophobic surface, though the behavior of these density profiles was qualitatively similar. Figure 6 shows that the solvation forces oscillated with a periodicity of about d S due to the packing effect and were steeply repulsive in the short range of R 2 d M , 1.9d S, where a layer of solvent particles between the surfaces began to be
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to which the conventional grand canonical ensemble MC (GCEMC) method is hardly accessible. In the near future the present method will be extended to long-range potential systems with the advance of computational power. ACKNOWLEDGMENT Computation time of the Cray T-94/4128 was provided by the Supercomputer Laboratory, Institute for Chemical Research, Kyoto University.
REFERENCES
FIG. 7. Potentials of solvation force corresponding to Fig. 6. The dashed line shows Eq. [6a], and the rest are the same as in Fig. 5.
pushed out into the bulk. This repulsion was still stronger than the attraction F d(R) given by Eq. [7]. The potentials of W s(R) shown in Fig. 7 suggest that the net solvation forces were repulsive from the viewpoint of free energy and the surfaces preferred being separated from each other with interposing more than two layers of solvent particles, which is attributable to a strong affinity of the macroparticle for solvent. The above results of the system IV agree well with those of the other studies (27–29, 33, 34). It is worth noting that the sign of solvation forces at the small separation for the systems III and IV was opposite from that for the system I as shown in Figs. 3 and 6, although their solvent density profiles were the same qualitatively as described above. This discrepancy comes from the difference in the solvent affinity between the solvophilic and solvophobic macroparticles. 5. CONCLUSION
In the present paper we first provided the description of the solvation forces between large spheres in a fluid using classical MD simulations. It was applied to the relatively simple systems in which a pair of spherical structureless macroparticles, either solvophobic or solvophilic, is immersed in a fluid of two types, either an SS or an LJ fluid. Our results agreed well with those of the other simulation and theoretical studies with respect to the solvent density profile near the surface of the macroparticles and the solvation force profile between them. The present simulation method requires much computation time because a large system is indispensable to allow some of solvent particles to locate between the macroparticles and exchange themselves with those in the bulk reservoir. However, our method can be applied straightforward to the large spheres immersed in mixtures and complex fluids described by the bead–spring model,
1. Israelachvili, J. N., “Intermolecular and Surface Forces,” 2nd ed. Academic Press, New York, 1991. 2. Israelachvili, J. N., and Adams, G. E., J. Chem. Soc., Faraday Trans. 1 74, 975 (1978). 3. Horn, R. G., and Israelachvili, J. N., J. Chem. Phys. 75, 1400 (1981). 4. Christenson, H. K., J. Chem. Phys. 78, 6906 (1983). 5. Christenson, H. K., J. Phys. Chem. 90, 4 (1986). 6. Christenson, H. K., and Horn, R. G., Chem. Phys. Lett. 98, 45 (1983). 7. Christenson, H. K., J. Chem. Soc., Faraday Trans. 1 80, 1933 (1984). 8. Christenson, H. K., and Horn, R. G., J. Colloid Interface Sci. 103, 50 (1985). 9. Pashley, R. M., J. Colloid Interface Sci. 83, 531 (1981). 10. Pashley, R. M., and Israelachvili, J. N., J. Colloid Interface Sci. 97, 446 (1984). 11. Pashley, R. M., and Israelachvili, J. N., J. Colloid Interface Sci. 101, 511 (1984). 12. Christenson, H. K., Gruen, D. W. R., Horn, R. G., and Israelachvili, J. N., J. Chem. Phys. 87, 1834 (1987). 13. Herder, C. E., Ninham, B. W., and Christenson, H. K., J. Chem. Phys. 90, 5801 (1989). 14. Gee, M. L., and Israelachvili, J. N., J. Chem. Soc., Faraday Trans. 86, 4049 (1990). 15. Horn, R. G., and Israelachvili, J. N., Macromolecules 21, 2836 (1988). 16. Horn, R. G., Hirz, S. J., Hadziioannou, G., Frank, C. W., and Catala, J. M., J. Chem. Phys. 90, 6767 (1989). 17. Israelachvili, J. N., and Kott, S. J., J. Chem. Phys. 88, 7162 (1988). 18. Israelachvili, J. N., Acc. Chem. Res. 20, 415 (1987). 19. Israelachvili, J. N., and Wennerstro¨m, H., Nature (London) 379, 219 (1996). 20. Christenson, H. K., and Claesson, P. M., Science 239, 390 (1988). 21. Tsao, Y.-H., Yang, S. X., Evans, D. F., and Wennerstro¨m, H., Langmuir 7, 3154 (1991). 22. Kurihara, K., and Kunitake, T., J. Am. Chem. Soc. 114, 10927 (1992). 23. Hansen, J. P., and McDonald, I. R., “Theory of Simple Liquids,” 2nd ed. Academic Press, London, 1986. 24. Attard, P., Be´rard, D. R., Ursenbach, C. P., and Patey, G. N., Phys. Rev. A 44, 8224 (1991). 25. Kinoshita, M., Iba, S., and Harada, M., J. Chem. Phys. 105, 2487 (1996). 26. Go¨tzelamann, B., and Dietrich, S., Phys. Rev. E 55, 2993 (1997). 27. Kjellander, R., and Sarman, S., Mol. Phys. 70, 215 (1990). 28. Sarman, S., J. Chem. Phys. 92, 4447 (1990). 29. Kinoshita, M., Iba, S., Kuwamoto, K., and Harada, M., J. Chem. Phys. 105, 7177 (1996). 30. Kinoshita, M., Iba, S., Kuwamoto, K., and Harada, M., J. Chem. Phys. 105, 7184 (1996). 31. Attard, P., and Patey, G. N., J. Chem. Phys. 92, 4970 (1990). 32. Allen, M. P., and Tildesley, D. J., “Computer Simulation of Liquids.” Clarendon Press, Oxford, 1987. 33. Snook, I. K., and van Megen, W., J. Chem. Phys. 72, 2907 (1980). 34. Magda, J. J., Tirrell, M., and Davis, H. T., J. Chem. Phys. 83, 1888 (1985).
INTERACTIONS BETWEEN MACROPARTICLES IN FLUIDS 35. Be´rard, D. R., Attard, P., and Patey, G. N., J. Chem. Phys. 98, 7236 (1993). 36. Bordarier, P., Rousseau, B., and Fuchs, A. H., J. Chem. Phys. 106, 7295 (1997). 37. Gao, J., Luedtke, W. D., and Landman, U., J. Phys. Chem. B 101, 4013 (1997). 38. Schoen, M., and Diestler, D. J., Phys. Rev. E 56, 4427 (1997). 39. Luzar, A., Bratko, D., and Blum, L., J. Chem. Phys. 86, 2955 (1987). 40. Wang, Y., Hill, K., and Harris, J. G., J. Chem. Phys. 100, 3276 (1994). 41. Dijkstra, M., J. Chem. Phys. 107, 3277 (1997). 42. Somers, S. A., McCormick, A. V., and Davis, H. T., J. Chem. Phys. 99, 9890 (1993). 43. Smit, B., Phys. Rev. A 37, 3431 (1988). 44. Smit, B., Hilbers, P. A. J., and Esselink, K., Tenside Surf. Det. 4, 287 (1993).
85
45. Steele, W. A., Surf. Sci. 36, 317 (1973). 46. Ciccotti, G., Ferrario, M., Hynes, J. T., and Kapral, R., Chem. Phys. 129, 241 (1989). 47. Hamaker, H. C., Physica 4, 1058 (1937). 48. Grest, G. S., Du¨nweg, B., and Kremer, K., Comput. Phys. Commun. 55, 269 (1989). 49. Chialvo, A. A., and Debenedetti, P. G., Comput. Phys. Commun. 60, 215 (1990). 50. Ryckaert, J.-P., Ciccotti, G., and Berendsen, H. J. C., J. Comput. Phys. 23, 327 (1977). 51. Berendsen, H. J. C., Postma, J. P. M., van Gunsteren, W. F., DiNola, A., and Haak, J. R., J. Chem. Phys. 81, 3684 (1984). 52. Derjaguin, V. B., Kolloid Z. 69, 155 (1934). 53. Christenson, H. K., J. Disp. Sci. Technol. 9, 171 (1988). 54. Henderson, D., J. Colloid Interface Sci. 121, 486 (1988). 55. Flyvbjerg, H., and Petersen, H. G., J. Chem. Phys. 91, 461 (1989).
Evaluation of Interaction Forces between Macroparticles in Simple Fluids by Molecular Dynamics Simulation Hiroyuki Shinto, Minoru Miyahara, and Ko Higashitani1 Department of Chemical Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto, 606-8501 Japan E-mail: [email protected] Received April 24, 1998; accepted September 17, 1998
bonding liquids (7, 8), aqueous electrolyte solutions (9 –11), liquid alkanes (12–14), and polymer melts (15–17). As the results, the forces which the DLVO theory fails to predict were discovered (1, 18, 19, 53). For example, the oscillatory solvation forces appear when two smooth surfaces approach each other closer than a few molecular diameters (3– 8, 11, 12, 14 –16), and the long-range attractive forces occur between hydrophobic surfaces in pure water (20 –22). In order to elucidate theoretically the origin of these nonDLVO forces on the molecular level, the integral equation theory (IET) based on the Ornstein–Zernike relation with various equations of closure approximation (23) has been developed and applied to various systems: A pair of either uncharged or charged large spheres (or planar walls) is immersed in a pure fluid composed of neutral hard-sphere (24 –26, 54), dipolar hard-sphere (24, 25), waterlike hard-sphere (25), or Lennard-Jones particle (27–29). Recently, in addition to the one-component fluids, the IET was extended to the multicomponent fluids of hard-sphere mixtures (29, 30) and aqueous electrolytes (25). It has been pointed out, however, that the results largely depend on the closure equations used (31). Therefore, it is necessary to compare carefully the results from the IET with those from the computer simulations based on molecular dynamics (MD) and Monte Carlo (MC) methods. MD and MC simulations (32) have been applied to relatively simple systems: two planar walls are immersed in a pure fluid of Lennard-Jones particle (33–38), waterlike hard-sphere (39), or alkane (37, 40, 41), rather than mixtures (42). The restriction comes mainly from the difficulty in specifying a chemical potential, or an alternative intensity factor, of the bulk phase that should be in equilibrium with the confined film between the walls. For this purpose, the grand canonical ensemble (33, 35, 38, 39, 41) and the test particle insertion method (42) were employed in the above simple systems. A new statistical ensemble (36) and a new MD cell (37, 40) were also developed. Nonetheless, in our opinion it seems quite difficult to apply these methods to mixtures and complex fluids other than pure simple fluids. The goal of this article is to present, to the best of our knowledge, the first study of the solvation forces between two
The present article provides the description of the solvation forces between large spheres in a fluid. The molecular dynamics (MD) method was applied to the relatively simple systems in which a pair of structureless macroparticles, either solvophobic or solvophilic, is immersed in a simple fluid of two types, either a soft-sphere or a Lennard-Jones fluid. When a pair of solvophobic macroparticles was in the attractive Lennard-Jones fluid, no dense layer of the solvent particles formed near the surface of the macroparticles and the strong attractive forces were induced between them. In the other combinations of macroparticles and fluids, the dense layers formed and the solvation forces oscillated, exhibiting the attraction and repulsion, whose periodic distance was about the diameter of solvent particles. Our results agreed well with those of the other simulation and theoretical studies with respect to the solvent density profile near a macroparticle and the force– distance profile between macroparticles. The benefit of our approach would be the simplicity in specifying or finding the bulk condition that is in equilibrium with the thin film of molecules between large surfaces. The present method can be applied straightforward to macroparticles immersed in mixtures and complex fluids described by the bead–spring model, to which the conventional grand canonical ensemble Monte Carlo (GCEMC) method is hardly accessible. © 1999 Academic Press Key Words: solvation; surface force; solvophobic attraction; molecular dynamics simulation.
1. INTRODUCTION
Both detailed understanding and exact description of interaction forces between colloidal particles immersed in fluids are the central subject of colloid science. Historically, the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory and the van der Waals theory have been frequently used to describe the interactions between colloidal particles in electrolyte solutions because they successfully explain experimental results (1). The surface force apparatus (SFA) was developed in late 1970s (1, 2) and has been used for in situ measurements of the interaction forces between cleaved or chemically modified mica surfaces in inert nonpolar liquids (3–5), polar liquids (6, 7), hydrogen1
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SHINTO, MIYAHARA, AND HIGASHITANI
large spheres in a fluid using classical MD simulations and compare our results with those from other simulations and the IET. We considered a pair of spherical structureless macroparticles immersed in a pure simple fluid. A large system was required to allow some of solvent particles to locate between the macroparticles and exchange themselves with those in the bulk reservoir. Our method presented here consumes much computation time but it can be applied straightforward to mixtures and complex fluids, e.g., those described by the bead–spring model (43, 44).
TABLE 1 Simulation Systems System
Solventa
Macroparticle
Diameter, d M
I II III IV
Soft-sphere Lennard-Jones particle Soft-sphere Lennard-Jones particle
Solvophobic Solvophobic Solvophilic Solvophilic
10d S, 20d S 10d S 10d S 10d S
Reduced density r* (5 r d 3S) and temperature T* (5k BT/ e ) of the solvent are 0.5925 and 1.2, respectively (see the text). a
2. MOLECULAR MODEL 3. CONSTRAINED MOLECULAR DYNAMICS METHOD
The interactions between solvent particles were represented by the shifted Lennard-Jones (12-6) potential with energy parameter e, core diameter d S, and cutoff radius R cut SS :
H
f ~r! 2 f ~R cut r # R cut SS !, SS , u SS~r! 5 0, r . R cut SS , f ~r! 5 4 e @~d S/r! 12 2 ~d S/r! 6#.
[1a] [1b]
We considered two different fluids: a Lennard-Jones (LJ) fluid cut with R cut SS 5 2.5d S, and a soft-sphere (SS) fluid with R SS 5 1/6 3 2 d S. Reduced density r* (5 r d S) and temperature T* (5k BT/ e ) of the fluids were chosen to be 0.5925 and 1.2, respectively, where k B is the Boltzmann’s constant. The shifted (10-4-3) potential was used for particle–macroparticle interaction (45),
H
`, r # r M, cut cut u MS~r! 5 c ~r! 2 c ~R MS!, r M , r # r M 1 R MS, cut 0, r . r M 1 R MS,
r M 5 d M/ 2, Dr 5 d S/ Î2,
A system composed of two macroparticles (A and B) and N solvent particles is considered. MD simulations of the system are performed keeping the center-to-center separation between macroparticles A and B in the same way as Ciccotti et al. first used to evaluate the mean force potential of an ion pair in a polar solvent (46). Accordingly, an additional constraint is incorporated,
j ~r A, r B! 5 ~r A 2 r B! 2 2 R 2 5 0,
[3]
where R is the fixed separation between macroparticles A and B. The forces acting on the macroparticles due to the solvent particles, FAS and FBS, are calculated. The solvation force F s(R) is evaluated as an average over the different configurations using the following expressions:
[2a] F s~R! 5
c ~r! 5 2 pe @0.4$d S/~r 2 r M!% 10 2 $d S/~r 2 r M!% 4 2 d 4S/$3Dr~r 2 r M 1 0.61Dr! 3%#,
3.1. Description of Interactions between Macroparticles
[2b] [2c]
where d M is the diameter of macroparticles and R cut MS is the cutoff distance. Note that the center of the outermost atoms in the macroparticle is located at separation r M from the center of the macroparticle. We considered two types of macroparticles with d M 5 10d S (or 20d S): a “solvophilic” macroparticle with R cut MS 5 10d S, in which the interaction includes both repulsive and attractive forces; and a “solvophobic” macroparticle with R cut MS 5 0.987d S, in which the interaction is completely repulsive. The mass of the macroparticle m M is equal to m S(d M/ d S)3, where m S is the mass of the solvent particle. In this study, we employed four systems as listed in Table 1; the solvophobic macroparticles in SS and LJ fluids (systems I and II) and the solvophilic macroparticles in SS and LJ fluids (systems III and IV).
1 ^rˆ z ~F AS 2 F BS!&, 2 AB
[4a]
rˆ AB 5 ~r A 2 r B!/ur A 2 r Bu.
[4b]
The potential W s(R) can be obtained by integrating Eq. [4a] from the large separation R 0 to a given separation R,
E
R
W s~R! 5 2
F s~R9!dR9.
[5]
R0
Supposing that the macroparticle is uniformly composed of the LJ particles of the number density r d 3S 5 1.0 and only the attraction term of 2(4 e d 6S)/r 6 in Eq. [1b] is considered, the potential of the direct macroparticle–macroparticle interaction, W d(R), is given by the analytical equation (47): W d~R! 5 2
F
S
A 2 2 s2 2 4 1 1 ln 6 s2 2 4 s2 s2
DG
,
[6a]
INTERACTIONS BETWEEN MACROPARTICLES IN FLUIDS
s5
2~R 2 d M! dS
Y S dd D 1 2, M
[6b]
S
A 5 p 2r 2~4 e d 6S! 5 4 p 2e ,
[6c]
where A is the Hamaker constant. The force F d(R) is obtained by differentiating Eq. [6a], F d~R! 5 2
1 dW d~R! 64A 52 . dR 3d M s 3~s 2 2 4! 2
[7]
Consequently, the total mean force between the macroparticles F t(R) and the potential W t(R) are described as F t~R! 5 F s~R! 1 F d~R!,
[8]
W t~R! 5 W s~R! 1 W d~R!.
[9]
In the present paper we applied the above description to the short-range potential systems as in Table 1, but this method is applicable straightforward also to long-range potential systems (e.g., Coulomb systems). 3.2. MD Simulation We implemented simulations of a system composed of two macroparticles with d M 5 10d S (or 20d S) and N 5 108,000 solvent particles in a cubic cell with periodic boundary conditions. The length of the cubic cell was selected to be 56.8d S (or 57.6d S) such that the density of the bulk fluid r* is 0.5925. It is confirmed that the dimension of this cell was large enough not to be affected by the neighboring image cells. In order to compute interactions between a large number of solvent particles efficiently, we employed the layered link cell (LLC) procedure connecting with the Verlet neighbor list on the Cray T-94/4128 vector computer (48). In this procedure, the neighbor list for force calculations was built up in the following way; the pairs of interacting particles i and j were found out according to the link cell (LC) algorithm and subsequently the identity numbers of these pairs, (i, j), were sorted systematically using the “layering algorithm” in order to vectorize all of the force loops. The neighbor list was updated automatically following Chialvo and Debenedetti (49). The constraint on the center-to-center separation of two macroparticles was imposed using the SHAKE method (50). The equations of motion for the solvent particles and the constrained macroparticles were solved using the leap-frog algorithm with a time step of Dt 5 0.00462 t 0 , where t 0 5 d S(m S/e)1/2. The temperature of the solvent was kept at T* 5 1.2 using Berendsen’s external bath method with a time constant of 0.462t0 (51), while we did not incorporate any temperature-control procedures into the macroparticles except the separation constraint. This is because it is favorable in the
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calculation of the solvation force to avoid artificial procedures imposed on the macroparticles as few as possible. The nearest separation between the macroparticles, R 2 d M, ranged from 0.0d S to 5.0d S with increments of 0.1–0.2d S. In each system, simulations were performed as follows. (i) Two macroparticles of A and B were positioned on a diagonal line of the cell such that the A–B separation was the largest separation of R 2 d M 5 5.0d S, and 108,000 solvent particles were placed at the face-centered-cubic (fcc) lattice excluded by the macroparticles. (ii) The system was equilibrated over 5.5 3 104 time steps. (iii) The instantaneous forces of FAS and FBS were computed during 2 3 104 time steps to obtain the solvation force F s(R) using Eq. [4]. (iv) Then, external forces were assigned to the macroparticles along the A–B line for 5 3 103 time steps such that the A–B separation was reduced by 0.1– 0.2d S. (v) In order to equilibrate the solvent particles around the macroparticles, the system was allowed to evolve for 5 3 103 time steps with keeping the new A–B separation. The procedure from (iii) to (v) was repeated to obtain the force– distance profile. The statistical errors of the solvation forces were evaluated by the block averaging method (55) and found to be less than 62.0, 61.4, 62.8, and 62.5 (in units of k BT/d s) for the systems I to IV, respectively. The potential of W s(R) was calculated by integrating the values of F s(R), where the trapezoidal rule was used without any smoothing procedures. 4. RESULTS AND DISCUSSION
4.1. Influence of Macroparticle Diameter on Solvation Force How the macroparticle diameter d M affects the solvation forces was examined simulating the system I with d M 5 10d S and 20d S. The results are given in Fig. 1, where the positive and negative values mean the repulsive and attractive forces, respectively. Figure 1 indicates that the force– distance profiles were qualitatively similar although the amplitude of the oscillatory behavior was larger in the case of d M 5 20d S. After both the profiles were multiplied by the factor of d S/ p d M on the basis of the Derjaguin approximation (52), they were almost identical to each other in all the range of the separations between macroparticles. We believe, therefore, that the succeeding results of the solvation force obtained using d M 5 10d S give the quantitative behavior within the model used here if the force– distance profiles are normalized by the value of d M. 4.2. Solvation Forces between “Solvophobic” Macroparticles The solvophobic macroparticles immersed in SS and LJ fluids, that is, the systems I and II, were considered. We calculated the density profile of solvent next to the macroparticle surface ( g MS), the solvation force, and the potential as shown in Figs. 2, 3, and 4, respectively. In the case of the system I, Fig. 2 illustrates that the dense
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SHINTO, MIYAHARA, AND HIGASHITANI
FIG. 1. Solvation force profiles for the system I (see Table 1). The solid and dotted lines represent the profiles for d M 5 10d S and 20d S, respectively. The inset gives the profiles normalized by the Derjaguin approximation (see the text).
layers of solvent particles formed adjacent to the surface of the large sphere even in such a repulsive potential system in which particle–particle and particle–macroparticle interactions were modeled as the short-range repulsion. This “packing efficiency” of particles near the surface results from the minimization of the free energy of the system, that is, the maximization of the entropy because the enthalpy is trivial in the repulsive potential system. We emphasize that the packing
FIG. 2. Reduced density profiles of solvent particles near the surface of the solvophobic macroparticle with d M 5 10d S. The dotted and solid lines represent the profiles for soft-sphere (SS) and Lennard-Jones (LJ) fluids, that is, the systems I and II, respectively.
FIG. 3. Solvation force profiles between the solvophobic macroparticles with d M 5 10d S in fluids. The dashed line shows Eq. [7], and the rest are the same as in Fig. 2.
efficiency that is entirely attributable to the entropy effect is, more or less, potentially intrinsic to any systems. When these two surfaces approached each other, the solvation force and the potential oscillated around the zero value, exhibiting the attraction and repulsion, and their periodic distances were about d S, that is, the solvent diameter as shown in Figs. 3 and 4. This oscillatory behavior is caused by the packing effect. After the solvent particles in the only one layer between the nearest surfaces of macroparticles started to be pushed out into the bulk at R 2 d M , 1.9d S, a strong attraction acted between the macroparticles and they came into contact with each other, which was the most stable state with regard to the free energy
FIG. 4. Potentials of solvation force corresponding to Fig. 3. The dashed line shows Eq. [6a], and the rest are the same as in Fig. 2.
INTERACTIONS BETWEEN MACROPARTICLES IN FLUIDS
as expected from Fig. 4. This is explained by the following mechanism. In the system I where the enthalpy is negligible, the behavior of the solvent particles shown in Fig. 2 are most favorable for the overall entropy of the system, in which a single macroparticle exists in the solvent. Nonetheless, the solvent particles in the dense layers near the surface of the macroparticle are less advantageous in terms of the entropy than those in the bulk. Hence, if two macroparticles attach to each other, the solvent particles are pushed out into the bulk, which leads to the increase of the overall entropy of the system, that is, the decrease of the total free energy. Thus the strong attractive force is induced even in a repulsive potential system such as the system I. The above behavior was also derived from the IET studies on the similar systems: hard-spherical macroparticles were immersed in a hard-sphere fluid (25, 54). The force– distance profiles predicted from these simulation and theoretical studies coincide qualitatively with those from the SFA experiments: the interaction forces were measured between the molecularly smooth surfaces of a cleaved mica in the liquid of quasispherical nonpolar molecules such as octamethylcyclotetrasiloxane (OMCTS), tetrachloromethane, cyclohexane, and benzene (1, 3–5, 18). On the other hand, in the case of the system II where the attractive forces prevailed for solvent–solvent but not for solvent–macroparticle, the different behavior was observed as in Figs. 2, 3, and 4; no dense layer of the solvent particles formed near the surface, i.e., the “dewetting,” and the solvation force profile exhibited a strong attraction without oscillating, i.e., the “solvophobic attraction.” The similar results have been also reported in the other studies (29, 35) and are interpreted as follows. When a large solvophobic particle intrudes into an attractive potential fluid, favorable bonds between the solvent
83
FIG. 6. Solvation force profiles between the solvophilic macroparticles with d M 5 10d S in fluids. The dashed line shows Eq. [7], and the rest are the same as in Fig. 5.
particles are unavoidably disrupted. Accordingly the particles near the macroparticle surface are reorganized at the sacrifice of the packing efficiency such that the bonds are prevented from breaking, which leads to the dewetting in this case. This behavior is advantageous to the bonds, that is, the enthalpy, but disadvantageous to the packing efficiency, that is, the entropy. These contradictory contributions to the free energy are compromised to minimize it. Nevertheless, the solvent particles near the surface are too unstable to form any dense layers. If the surfaces approach each other, these unstable particles are pushed out into the bulk, by which the free energy of the system decreases consequently. Thus the strong and monotonous attraction is generated between solvophobic surfaces in an attractive potential fluid. Figure 3 represents that the attraction caused by the presence of solvent was stronger than the direct macroparticle–macroparticle interaction given by Eq. [7] in the range of R 2 d M . 0.8d S. 4.3. Solvation Forces between “Solvophilic” Macroparticles
FIG. 5. Reduced density profiles of solvent particles near the surface of the solvophilic macroparticle with d M 5 10d S. The dotted and solid lines represent the profiles for soft-sphere (SS) and Lennard-Jones (LJ) fluids, that is, the systems III and IV, respectively.
The solvophilic macroparticles immersed in SS and LJ fluids, that is, the systems III and IV, were considered and the results are shown in Figs. 5, 6, and 7. It is noted that the results in the systems III and IV were almost the same in spite of the difference of the fluids: SS and LJ fluids. Comparison between Fig. 5 for the systems III and IV and Fig. 2 for the system I indicates that the solvent density near the solvophilic surface was higher than that near the solvophobic surface, though the behavior of these density profiles was qualitatively similar. Figure 6 shows that the solvation forces oscillated with a periodicity of about d S due to the packing effect and were steeply repulsive in the short range of R 2 d M , 1.9d S, where a layer of solvent particles between the surfaces began to be
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to which the conventional grand canonical ensemble MC (GCEMC) method is hardly accessible. In the near future the present method will be extended to long-range potential systems with the advance of computational power. ACKNOWLEDGMENT Computation time of the Cray T-94/4128 was provided by the Supercomputer Laboratory, Institute for Chemical Research, Kyoto University.
REFERENCES
FIG. 7. Potentials of solvation force corresponding to Fig. 6. The dashed line shows Eq. [6a], and the rest are the same as in Fig. 5.
pushed out into the bulk. This repulsion was still stronger than the attraction F d(R) given by Eq. [7]. The potentials of W s(R) shown in Fig. 7 suggest that the net solvation forces were repulsive from the viewpoint of free energy and the surfaces preferred being separated from each other with interposing more than two layers of solvent particles, which is attributable to a strong affinity of the macroparticle for solvent. The above results of the system IV agree well with those of the other studies (27–29, 33, 34). It is worth noting that the sign of solvation forces at the small separation for the systems III and IV was opposite from that for the system I as shown in Figs. 3 and 6, although their solvent density profiles were the same qualitatively as described above. This discrepancy comes from the difference in the solvent affinity between the solvophilic and solvophobic macroparticles. 5. CONCLUSION
In the present paper we first provided the description of the solvation forces between large spheres in a fluid using classical MD simulations. It was applied to the relatively simple systems in which a pair of spherical structureless macroparticles, either solvophobic or solvophilic, is immersed in a fluid of two types, either an SS or an LJ fluid. Our results agreed well with those of the other simulation and theoretical studies with respect to the solvent density profile near the surface of the macroparticles and the solvation force profile between them. The present simulation method requires much computation time because a large system is indispensable to allow some of solvent particles to locate between the macroparticles and exchange themselves with those in the bulk reservoir. However, our method can be applied straightforward to the large spheres immersed in mixtures and complex fluids described by the bead–spring model,
1. Israelachvili, J. N., “Intermolecular and Surface Forces,” 2nd ed. Academic Press, New York, 1991. 2. Israelachvili, J. N., and Adams, G. E., J. Chem. Soc., Faraday Trans. 1 74, 975 (1978). 3. Horn, R. G., and Israelachvili, J. N., J. Chem. Phys. 75, 1400 (1981). 4. Christenson, H. K., J. Chem. Phys. 78, 6906 (1983). 5. Christenson, H. K., J. Phys. Chem. 90, 4 (1986). 6. Christenson, H. K., and Horn, R. G., Chem. Phys. Lett. 98, 45 (1983). 7. Christenson, H. K., J. Chem. Soc., Faraday Trans. 1 80, 1933 (1984). 8. Christenson, H. K., and Horn, R. G., J. Colloid Interface Sci. 103, 50 (1985). 9. Pashley, R. M., J. Colloid Interface Sci. 83, 531 (1981). 10. Pashley, R. M., and Israelachvili, J. N., J. Colloid Interface Sci. 97, 446 (1984). 11. Pashley, R. M., and Israelachvili, J. N., J. Colloid Interface Sci. 101, 511 (1984). 12. Christenson, H. K., Gruen, D. W. R., Horn, R. G., and Israelachvili, J. N., J. Chem. Phys. 87, 1834 (1987). 13. Herder, C. E., Ninham, B. W., and Christenson, H. K., J. Chem. Phys. 90, 5801 (1989). 14. Gee, M. L., and Israelachvili, J. N., J. Chem. Soc., Faraday Trans. 86, 4049 (1990). 15. Horn, R. G., and Israelachvili, J. N., Macromolecules 21, 2836 (1988). 16. Horn, R. G., Hirz, S. J., Hadziioannou, G., Frank, C. W., and Catala, J. M., J. Chem. Phys. 90, 6767 (1989). 17. Israelachvili, J. N., and Kott, S. J., J. Chem. Phys. 88, 7162 (1988). 18. Israelachvili, J. N., Acc. Chem. Res. 20, 415 (1987). 19. Israelachvili, J. N., and Wennerstro¨m, H., Nature (London) 379, 219 (1996). 20. Christenson, H. K., and Claesson, P. M., Science 239, 390 (1988). 21. Tsao, Y.-H., Yang, S. X., Evans, D. F., and Wennerstro¨m, H., Langmuir 7, 3154 (1991). 22. Kurihara, K., and Kunitake, T., J. Am. Chem. Soc. 114, 10927 (1992). 23. Hansen, J. P., and McDonald, I. R., “Theory of Simple Liquids,” 2nd ed. Academic Press, London, 1986. 24. Attard, P., Be´rard, D. R., Ursenbach, C. P., and Patey, G. N., Phys. Rev. A 44, 8224 (1991). 25. Kinoshita, M., Iba, S., and Harada, M., J. Chem. Phys. 105, 2487 (1996). 26. Go¨tzelamann, B., and Dietrich, S., Phys. Rev. E 55, 2993 (1997). 27. Kjellander, R., and Sarman, S., Mol. Phys. 70, 215 (1990). 28. Sarman, S., J. Chem. Phys. 92, 4447 (1990). 29. Kinoshita, M., Iba, S., Kuwamoto, K., and Harada, M., J. Chem. Phys. 105, 7177 (1996). 30. Kinoshita, M., Iba, S., Kuwamoto, K., and Harada, M., J. Chem. Phys. 105, 7184 (1996). 31. Attard, P., and Patey, G. N., J. Chem. Phys. 92, 4970 (1990). 32. Allen, M. P., and Tildesley, D. J., “Computer Simulation of Liquids.” Clarendon Press, Oxford, 1987. 33. Snook, I. K., and van Megen, W., J. Chem. Phys. 72, 2907 (1980). 34. Magda, J. J., Tirrell, M., and Davis, H. T., J. Chem. Phys. 83, 1888 (1985).
INTERACTIONS BETWEEN MACROPARTICLES IN FLUIDS 35. Be´rard, D. R., Attard, P., and Patey, G. N., J. Chem. Phys. 98, 7236 (1993). 36. Bordarier, P., Rousseau, B., and Fuchs, A. H., J. Chem. Phys. 106, 7295 (1997). 37. Gao, J., Luedtke, W. D., and Landman, U., J. Phys. Chem. B 101, 4013 (1997). 38. Schoen, M., and Diestler, D. J., Phys. Rev. E 56, 4427 (1997). 39. Luzar, A., Bratko, D., and Blum, L., J. Chem. Phys. 86, 2955 (1987). 40. Wang, Y., Hill, K., and Harris, J. G., J. Chem. Phys. 100, 3276 (1994). 41. Dijkstra, M., J. Chem. Phys. 107, 3277 (1997). 42. Somers, S. A., McCormick, A. V., and Davis, H. T., J. Chem. Phys. 99, 9890 (1993). 43. Smit, B., Phys. Rev. A 37, 3431 (1988). 44. Smit, B., Hilbers, P. A. J., and Esselink, K., Tenside Surf. Det. 4, 287 (1993).
85
45. Steele, W. A., Surf. Sci. 36, 317 (1973). 46. Ciccotti, G., Ferrario, M., Hynes, J. T., and Kapral, R., Chem. Phys. 129, 241 (1989). 47. Hamaker, H. C., Physica 4, 1058 (1937). 48. Grest, G. S., Du¨nweg, B., and Kremer, K., Comput. Phys. Commun. 55, 269 (1989). 49. Chialvo, A. A., and Debenedetti, P. G., Comput. Phys. Commun. 60, 215 (1990). 50. Ryckaert, J.-P., Ciccotti, G., and Berendsen, H. J. C., J. Comput. Phys. 23, 327 (1977). 51. Berendsen, H. J. C., Postma, J. P. M., van Gunsteren, W. F., DiNola, A., and Haak, J. R., J. Chem. Phys. 81, 3684 (1984). 52. Derjaguin, V. B., Kolloid Z. 69, 155 (1934). 53. Christenson, H. K., J. Disp. Sci. Technol. 9, 171 (1988). 54. Henderson, D., J. Colloid Interface Sci. 121, 486 (1988). 55. Flyvbjerg, H., and Petersen, H. G., J. Chem. Phys. 91, 461 (1989).