Interaction Forces between Colloidal Particles in Alcoholâ Water

Langmuir 2000, 16, 3361-3371

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Interaction Forces between Colloidal Particles in Alcohol-Water Mixtures Evaluated by Simple Model Simulations Hiroyuki Shinto, Minoru Miyahara, and Ko Higashitani* Department of Chemical Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto, 606-8501 Japan Received May 6, 1999. In Final Form: November 10, 1999 The interaction force between hydrophilic macroparticles in an alcoholic (or amphiphilic) solution is investigated using a classical molecular dynamics simulation with coarse-grained models. When the fluid employed is a pure amphiphilic liquid, (i) amphiphiles are adsorbed vertically on the surface to form an amphiphile bilayer, and (ii) the surface force becomes strongly repulsive at the separation less than about the thickness of the bilayer. When the fluid is an amphiphile-water mixture, (iii) a bilayer of hydrated amphiphiles is formed near the surface, (iv) the surface force becomes attractive suddenly at the separation less than the length of the amphiphile molecule, and (v) the adhesion force has a maximum at a higher concentration of amphiphiles and the magnitude of this maximum is larger as an amphiphile has a longer hydrophobic tail. It is worth noting that result iv originates from a liquid-to-liquid-phase separation of the amphiphile-water mixture between the surfaces, in which amphiphiles are pushed out into the bulk and water molecules enter to bridge the gap between the surfaces instead. These simulation results are in fair agreement with recent experimental results by an atomic force microscope.

1. Introduction An understanding of the various forces between surfaces, particles, or molecules in a fluid is essential for understanding, at the molecular level, not only the behavior of colloidal and biological systems but also the efficient manipulation of technological and industrial processes such as lubrication, flotation, particulate material processing, and enhanced oil discovery. Historically, the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory successfully explained experimental results, considering only the attractive van der Waals and repulsive electric double-layer forces.1,2 A surface force apparatus (SFA)3 and an atomic force microscope (AFM)4,5 were developed in the last 2 decades and have allowed us to measure in situ the interaction forces between surfaces in media of liquid and vapor. This led to the discovery of the forces that the DLVO theory fails to predict (i.e., nonDLVO forces); for example, the oscillatory structural force, the repulsive hydration force, the attractive hydrophobic force, the steric repulsion in polymeric systems, and the capillary and adhesion forces.1,2,6,7 To elucidate theoretically the origin of the non-DLVO forces at the molecular level, the integral equation theory (IET) based on the Ornstein-Zernike relation with various * To whom correspondence should be addressed. Phone: +81-75-753-5562. Fax: +81-75-753-5913. E-mail: higa@ cheme.kyoto-u.ac.jp. (1) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1991. (2) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain: where physics, chemistry, and biology meet, 2nd ed.; Wiley-VCH: New York, 1999. (3) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (4) Binning, G.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930. (5) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Langmuir 1992, 8, 1831. (6) Israelachvili, J. N. Acc. Chem. Res. 1987, 20, 415 and references therein. (7) Christenson, H. K. J. Dispersion Sci. Technol. 1988, 9, 171 and references therein.

closure approximations8 has been applied to the pure fluid systems: A pair of large spheres or planar walls is immersed in a pure fluid composed of a Lennard-Jones (LJ) particle,9,10 a hard sphere that is neutral,11-14 dipolar,12,14 or waterlike,14 or an atomistic model of SPC/E water.15 Following these one-component fluids, the IET was applied to an aqueous electrolyte solution14 and binary hard-sphere fluids which mimic a cyclohexane-octamethylcyclotetrasiloxane mixture10 and a water-nonpolar component mixture.16 It has been pointed out, however, that the reliability of the analyzed results depends largely on the closure approximations used.17 This indicates that 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. Using MD and MC simulations,18 the interaction force between planar walls across a fluid has been calculated, in which the fluid usually employed is a pure simple liquid of an LJ particle,19-24 a dipolar LJ particle,25 a waterlike (8) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic Press: London, 1986. (9) (a) Kjellander, R.; Sarman, S. Mol. Phys. 1990, 70, 215. (b) Kjellander, R.; Sarman, S. Mol. Phys. 1991, 74, 665. (c) Sarman, S. J. Chem. Phys. 1990, 92, 4447. (10) Kinoshita, M.; Iba, S.; Kuwamoto, K.; Harada, M. J. Chem. Phys. 1996, 105, 7177. (11) Henderson, D. J. Colloid Interface Sci. 1988, 121, 486. (12) Attard, P.; Be´rard, D. R.; Ursenbach, C. P.; Patey, G. N. Phys. Rev. A 1991, 44, 8224. (13) Go¨tzelamann, B.; Dietrich, S. Phys. Rev. E 1997, 55, 2993. (14) Kinoshita, M.; Iba, S.; Harada, M. J. Chem. Phys. 1996, 105, 2487. (15) Kinoshita, M.; Hirata, F. J. Chem. Phys. 1996, 104, 8807. (16) (a) Kinoshita, M.; Iba, S.; Kuwamoto, K.; Harada, M. J. Chem. Phys. 1996, 105, 7184. (b) Kinoshita, M. Mol. Phys. 1998, 94, 485. (c) Kinoshita, M. Mol. Phys. 1999, 96, 71. (17) Attard, P.; Patey, G. N. J. Chem. Phys. 1990, 92, 4970. (18) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987. (19) (a) Snook, I. K.; van Megen, W. J. Chem. Phys. 1980, 72, 2907. (b) van Megen, W.; Snook, I. K. J. Chem. Phys. 1981, 74, 1409. (20) Magda, J. J.; Tirrell, M.; Davis, H. T. J. Chem. Phys. 1985, 83, 1888. (21) Be´rard, D. R.; Attard, P.; Patey, G. N. J. Chem. Phys. 1993, 98, 7236.

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hard sphere,26 or an ellipsoidal Gay-Berne particle,27 rather than a binary fluid mixture28 and a fluid of chain molecules like alkane.29-31 In these simulations, a molecular film confined between walls must be in equilibrium with the bulk fluid. For this purpose, the grand canonical ensemble and the test particle insertion method were employed in most of the studies, whereas a new statistical ensemble22 and a new MD cell23,29,30 were also developed. In our opinion, however, it seems rather difficult to apply these methods to a multicomponent fluid and a complex fluid other than a pure simple fluid. To overcome this difficulty, we worked out not only the new computational method to evaluate the interaction force between large spheres in a fluid using a classical MD simulation32 but also the simple description of water, oil, and surfactant molecules and macroscopic colloidal particles.33 A combination of our method and model enables us to calculate the interaction force between colloidal particles in a complex fluid composed of water, oil, and surfactants.34 Recently, Kanda et al. measured the interaction forces between hydrophilic surfaces in water-alcohol (methanol, ethanol, n-propanol, ..., and n-hexanol) mixtures using an AFM and reported as follows:35,36 (i) The force-distanse profile becomes steplike in the short range of the surface separation when the alcohol weight fraction wA exceeds 0.9. This steplike profile will be attributable to the vertical adsorption of alcohol molecules on the surface, as shown in Figure 1a. (ii) The adhesion force becomes most attractive at wA ≈ 0.9. The magnitude of this attraction increases with increasing molecular weight of alcohol. The strong attraction originates possibly from the water bridging between the surfaces, as shown in Figure 1b. These experimental results are very significant and suggestive but insufficient and indirect to support the mechanisms proposed in Figure 1. The aim of the present study is to elucidate the origin of the strong adhesion force between hydrophilic surfaces in an alcohol-water mixture. Using our simulation method described above, we investigate systematically the interaction force between smooth hydrophilic macroparticles in an alcohol-water mixture and show the following results: (i) structure of the solution near the macroparticle surface; (ii) force-distance profile between the macroparticles; (iii) adhesion force as a function of the alcohol concentration. These results are discussed in detail and compared with recent experimental AFM results.35,36 (22) (a) Bordarier, P.; Rousseau, B.; Fuchs, A. H. Mol. Simul. 1996, 17, 199. (b) Bordarier, P.; Rousseau, B.; Fuchs, A. H. J. Chem. Phys. 1997, 106, 7295. (23) Gao, J.; Luedtke, W. D.; Landman, U. Phys. Rev. Lett. 1997, 79, 705. (24) Schoen, M.; Diestler, D. J. Phys. Rev. E 1997, 56, 4427. (25) Han, K. K.; Cushman, J. H.; Diestler, D. J. Mol. Phys. 1993, 79, 537. (26) Luzar, A.; Bratko, D.; Blum, L. J. Chem. Phys. 1987, 86, 2955. (27) (a) Gruhn, T.; Schoen, M. Phys. Rev. E 1997, 55, 2861. (b) Gruhn, T.; Schoen, M. Mol. Phys. 1998, 93, 681. (c) Gruhn, T.; Schoen, M. J. Chem. Phys. 1998, 108, 9124. (28) Somers, S. A.; McCormick, A. V.; Davis, H. T. J. Chem. Phys. 1993, 99, 9890. (29) Wang, Y.; Hill, K.; Harris, J. G. J. Chem. Phys. 1994, 100, 3276. (30) Gao, J.; Luedtke, W. D.; Landman, U. J. Phys. Chem. B 1997, 101, 4013. (31) Dijkstra, M. J. Chem. Phys. 1997, 107, 3277. (32) Shinto, H.; Miyahara, M.; Higashitani, K. J. Colloid Interface Sci. 1999, 209, 79. (33) Shinto, H.; Tsuji, S.; Miyahara, M.; Higashitani, K. Langmuir 1999, 15, 578. (34) Shinto, H. Ph.D. Thesis, Kyoto University, Kyoto, Japan, 1999. (35) Kanda, Y.; Nakamura, T.; Higashitani, K. Colloids Surf. A 1998, 139, 55. (36) Kanda, Y.; Iwasaki, S.; Higashitani, K. J. Colloid Interface Sci. 1999, 216, 394.

Shinto et al.

Figure 1. Schematic of molecules adsorbed on the interface in alcohol-water solutions, after Kanda et al.35 (a) The structural change of the adsorbed layer with alcohol weight fraction wA. (b) The proposed model for the attractive force at wA ) 0.9, where the water bridging occurs between surfaces in the alcohol-rich medium.

We emphasize that the application of the simulation method employed is not restricted to the present system.34 2. Models 2.1. Alcohol-Water Mixtures. Water and alcohol (or amphiphile) molecules are represented using the simple model developed by Telo da Gama and Gubbins37 and Smit and co-workers.38 We consider three types of particles: waterlike, headlike, and taillike particles, which are referred to as w, h, and t particles, respectively. The w and h particles are hydrophilic, whereas the t particle is hydrophobic. Figure 2 illustrates a water molecule represented by one w particle and two types of amphiphiles represented by ht and ht3, which are different in hydrophobic chain length but the same in hydrophilic headgroup. One can consider ht and ht3 as coarse-grained models of linear monohydric alcohol molecules. Pairs i and j between three types of particles (w, h, and t) interact with each other via a shifted LJ potential with energy parameter S, core diameter dS, and cutoff radius Rcut ij :

uij(r) )

{

cut φ(r) - φ(Rcut ij ) r e Rij r > Rcut 0 ij

φ(r) ) 4S

(i, j ) w, h, t) (1a)

[( ) ( ) ] dS r

12

-

dS r

6

(1b)

where r is the distance between particles. To represent the hydrophilicity and hydrophobicity of the particles, we 1/6 choose the different values of Rcut ij ) 2.5dS and 2 dS. These values depend on the types of pairs: the intraspecies interaction contains the repulsive and attractive forces (37) Telo da Gama, M. M.; Gubbins, K. E. Mol. Phys. 1986, 59, 227. (38) (a) Smit, B. Phys. Rev. A 1988, 37, 3431. (b) Smit, B.; Schlijper, A. G.; Rupert, L. A. M.; van Os, N. M. J. Phys. Chem. 1990, 94, 6933.

Interaction Forces between Colloidal Particles

Langmuir, Vol. 16, No. 7, 2000 3363

- ∆r at volume density Fvol ) 1/dS3, where ∆r ) Fsurf/Fvol ) 2-1/3dS. The particle-macroparticle potential is derived after the analytical calculation (see Appendix):

Figure 2. Illustration of the simple models of water and alcohol (or amphiphile) molecules. Symbols w, h, and t denote waterlike, headlike, and taillike particles, respectively. Two amphiphiles used are displayed.

ψ(r) )

[( ) ( { [( ) ( [( ) (

2πFsurfdS2rMS 2 dS r 5 r - rM

10

-

)]

dS r - rM

4

+

)] ) ]}

dS 9 dS dS 8 πFvoldS3rin M S 1 8 - in 3r 30 r - rin rM r - rin M M dS 3 dS dS 2 2 - in r - rin rM r - rin M M

(3)

where r is the center-to-center separation between the particle and macroparticle. On the right-hand side of eq 3, the first and second terms indicate the interactions of a particle with the outermost surface and the inner sphere of a macroparticle, respectively. Figure 3b shows the potential energy profile given by eq 3. This potential has a minimum of -2.72S at r ′ ) 0.992dS, where r ′ indicates the distance from the macroparticle surface, r - rM. The mass of the macroparticle mM is equal to mS(dM/dS)3, where mS is the mass of the solvent particle. To represent a hydrophilic macroparticle, we truncate and shift the particle-macroparticle potential of eq 3:

Figure 3. (a) Schematic of the particle and macroparticle. (b) Potential energies for particle-particle and particle-macroparticle interactions represented by eqs 1b and 3, respectively. r ′ indicates the distance from the macroparticle surface, r rM. The macroparticle is composed of Lennard-Jones (LJ) particles at a number density of 1/dS3; the LJ particles are uniformly located on the spherical surface of radius rM ) 5dS at density Fsurf ) 2-1/3/dS2 and in the inner sphere of radius rin M ) rM - ∆r at density Fvol ) 1/dS3, where ∆r ) Fsurf/Fvol ) 2-1/3dS (see the text).

using Rcut ij ) 2.5dS; the interspecies interaction is com) 21/6dS (see Figure 3b). pletely repulsive using Rcut ij Neighboring particles in an amphiphile are connected together by a harmonic potential with spring length l0 and spring constant k:

1 ubond (r) ) k(r - l0)2 ij 2

(i, j ) h, t)

(2)

where l0 ) dS and k ) 200S/dS2 are used. Note that the model amphiphile ht3 is flexible compared with a real amphiphile whose conformation is restricted by the bending and torsional angles. Although the potentials of these angles can be incorporated into the model amphiphile, this spoils the advantage of the simple model, that is, the simplicity. The thermodynamic state of a fluid is F* ) FdS3 ) 0.7 and T* ) kBT/S ) 1.0, where F is the number density of particles, T is the absolute temperature, and kB is Boltzmann’s constant. The average separation of the connected particles is found to be 1.06dS in the present systems. 2.2. Hydrophilic Macroparticle. Suppose that the macroparticle M of diameter dM ) 10dS ()2rM) is composed of LJ particles at a number density of 1/dS3, as shown in Figure 3a; that is, the LJ particles are uniformly located on the spherical surface of radius rM at surface density Fsurf ) 2-1/3/dS2 and in the inner sphere of radius rMin ) rM

{

r e rM ∞ cut cut uMj(r) ) ψ(r) - ψ(RMj ) rM < r e rM + RMj cut r > rM + RMj 0 (j ) w, h, t) (4) cut is the cutoff distance between the macroparwhere RMj ticle surface and particle center.32-34 When the macroparticle M and particle j are of the same kind (i.e., j ) w, h), their interaction contains the repulsive and attractive cut ) 5dS; otherwise (i.e., j ) t), the forces using RMj cut ) 0.992dS. interaction is completely repulsive using RMj The direct macroparticle-macroparticle interaction is neglected here except for the excluded-volume effect:

Wdir(R) )

{

∞ R < dM + dS 0 R g dM + dS

(5)

where R is the center-to-center separation between the macroparticles. This treatment is reasonable and hardly influences the results presented below, because the direct interaction is nonsignificant compared with the interaction caused by a fluid medium, as pointed out in our previous study.32 3. Simulation Methods 3.1. Description of Interactions between Macroparticles. The computational method to evaluate the interaction force between macroparticles in a fluid is given elsewhere32 but briefly explained here. A system composed of two macroparticles (C and D) and N particles is considered. During an MD simulation of the system, the center-to-center separation between the macroparticles is kept at constant value R. For this purpose, the following constraint on the macroparticles at rC and rD is incorporated into the equations of motion:

ξ(rC,rD) ) |rC - rD|2 - R2 ) 0

(6)

The forces on macroparticles C and D caused by the solvent, FCS and FDS, are calculated during a simulation

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Table 1. Systems of Constrained Molecular Dynamics Simulations system fluida I II III IV V

w ht3 ht ht3-w ht-w

XAb 0 1 1 0.70-0.98 0.50-0.95

macroparticle diameter box length pair dM L hydrophilic hydrophilic hydrophilic hydrophilic hydrophilic

10dS 10dS 10dS 10dS 10dS

53.8dS 53.8dS 53.8dS 53.8dS 53.8dS

a The total number of particles N is 108 000, and the thermodynamic state is F* ) FdS3 ) 0.7 and T* ) kBT/S ) 1.0. b Volume fraction of amphiphiles in a fluid; that is, XA ) nsiteNA/N ) (N NW)/N, where NA is the number of amphiphiles with nsite sites and NW is the number of waterlike particles.

of the specific value of R. The solvent contribution to the interaction force between the macroparticles, that is, the solvation force Fsolv(R) is calculated by the following expression:

1 Fsolv(R) ) 〈rˆ CD‚(FCS - FDS)〉 2

(7a)

rˆ CD) (rC - rD)/|rC - rD|

(7b)

The total interaction force FMM(R) is given by

FMM(R) ) Fsolv(R) + Fdir(R)

(8)

i.e.

FMM(R) ) Fsolv(R)

for R g dM + dS

(8′)

where the direct interaction of Fdir(R) ) -dWdir(R)/dR equals zero because of eq 5. 3.2. Systems. Table 1 shows five systems employed, a pair of hydrophilic macroparticles immersed in water (I), an amphiphilic liquid of ht3 or ht (II, III), or an amphiphile-water mixture (IV, V). These systems are composed of two macroparticles of dM ) 10dS and 108 000 particles in a cubic cell with three-dimensional periodicity. The length of the cell is selected to be 53.8dS such that the density of the bulk fluid F* is 0.7. We confirm that the basic cell has enough dimensions to be unaffected by the neighboring image cells. The volume fraction of amphiphiles in a fluid XA is defined as XA ) nsiteNA/N ) (N - NW)/N, where NA is the number of amphiphiles with nsite sites and NW is the number of waterlike particles. 3.3. Algorithms. The constraint on a macroparticle pair by eq 6 is imposed using the SHAKE method.39 The equations of motion for the constrained macroparticles and the particles are integrated by the leapfrog algorithm with one time step of ∆t ) 0.00464τ0, where τ0 ) dS(mS/ S)1/2.18,40 The temperature of the fluid is kept at T* ) 1.0 using Berendsen’s heat bath method with a time constant of τT ) 0.928τ0,40 whereas no temperature-control procedure except the separation constraint is incorporated into the macroparticles. This is because it is favorable in the calculation of the solvation force to use as few artificial procedures as possible, which are imposed on the macroparticles. For efficient computation of short-range interactions between a large number of particles, the layered link cell method with the Verlet neighbor list is employed on the (39) Ryckaert, J.-P.; Ciccotti, G.; Berendsen, H. J. C. J. Comput. Phys. 1977, 23, 327. (40) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684.

Cray T-94/4128 vector computer.41 The neighbor list is updated automatically following Chialvo and Debenedetti.42 Intramolecular interactions of amphiphiles given by eq 2 are computed in the vector fashion using the multicolor method.43 3.4. Preparation of Dense Amphiphilic Liquids. When the fluid employed is a simple fluid, it is easy to prepare an equilibrated sample starting from an arbitrary configuration (e.g., the face-centered-cubic lattice). This method is inapplicable to a dense complex fluid including chain molecules. For this reason, the modified technique was taken in our previous study.33 However, in the present study we develop the more sophisticated method that is similar to the method proposed by McKechnie et al.44 3.4.1. Pure Amphiphilic Liquids. First of all, a pure amphiphile fluid is considered. The method for the sample preparation consists of three steps: (I) the random generation of initial coordinates of amphiphiles; (II) the short relaxation of the system, in which all of the amphiphiles are treated as chains of soft spheres; (III) the long equilibration of the system after replacing the chains by the amphiphiles. For the sake of convenience, the sites in a linear amphiphile are numbered 1, ..., nsite from the end h to the other end t. Step I. A macroparticle pair with a given separation is placed on a diagonal line of the cell. After that, the position of the site 1 in an amphiphile is determined at random and then the rest sites 2, ..., nsite are generated following the line from site 1 with the random direction; these sites are placed in the space excluding the macroparticle regions, and the separations between the neighboring sites are set as 1.0dS. The same procedure is repeated such that NA amphiphiles are constructed. Thus, all excludedvolume effects are ignored except for the space of macroparticles. This will lead to a large number of overlaps between the particle sites. Step II. To implement the stable computation, one has to gently introduce the excluded-volume effects into the amphiphile sites. Therefore, all of the sites are treated as 1/6 soft spheres using eq 1 with Rcut ij ) 2 dS. The interaction force between these soft spheres is constrained to be a constant value of Ftr only when they are within a critical separation Rtr:44

-

(

)

dum dφ(Rtr) ij (r) ≡ Ftr ) dr dr

for r e Rtr

(9)

The modified form of the shifted LJ potential, um ij (r), is then

{

tr (Rtr - r)Ftr + φ(Rtr) - φ(Rcut ij ) r e R cut Rtr < r e Rcut um ij ij (r) ) φ(r) - φ(Rij ) r > Rcut 0 ij (10)

where Rtr must be sufficiently small such that soft spheres are hardly within this separation after the short relaxation (41) (a) Rapaport, D. C. Comput. Phys. Rep. 1988, 9, 1. (b) Grest, G. S.; Du¨nweg, B.; Kremer, K. Comput. Phys. Commun. 1989, 55, 269. (42) Chialvo, A. A.; Debenedetti, P. G. Comput. Phys. Commun. 1990, 60, 215. (43) Mu¨ller-Plathe, F.; Brown, D. Comput. Phys. Commun. 1991, 64, 7. (44) (a) McKechnie, J. I.; Brown, D.; Ckarke, J. H. R. Macromolecules 1992, 25, 1562. (b) Clarke, J. H. R. In Monte Carlo and Molecular Dynamics Simulations in Polymer Science; Binder, K., Ed.; Oxford University Press: New York, 1995; Chapter 5.


but not so small that the large value of Ftr causes the unstable computation.44 The macroparticles are also treated as large soft spheres. The interaction between the macroparticle and particle is set to be completely repulsive using eqs 3 and 4 with cut RMj ) 0.992dS. The short relaxation is performed using eqs 9 and 10 as follows. The value of Rtr is decreased from 0.9dS to 0.7dS for 5 × 103 time steps using ∆t ) 0.00464τ0. The strong spring of k ) 1000S/dS2 is used such that no bonds connecting the sites are broken during the introduction process of the excluded volume. The temperature is kept at T* ) 1.0 by scaling the velocity of particles at each step to remove a large amount of the thermal energy released. Note that the velocity scaling at every step is identical with the use of the heat bath with τT ) 0.00464τ0 ()∆t). Step III. After the short relaxation of step II, all of the chains of soft spheres are treated as the amphiphiles with k ) 200S/dS2 and the large soft spheres are treated as the hydrophilic macroparticles. The subsequent relaxation of the amphiphilic liquids, ht and ht3, is performed for 2.5 × 104 and 6 × 104 time steps, respectively. 3.4.2. Amphiphile-Water Mixtures. When an amphiphile-water mixture is prepared, all of the sites in a specific number of amphiphiles are replaced by w particles using an equilibrated sample of the pure amphiphilic liquid, such that the desired number of w particles are generated. Thereafter, the system is sufficiently equilibrated over 1 × 105-4 × 105 time steps. 3.5. Simulation Procedures. Force-Distance Profile. The nearest separation between macroparticles, R ′ ) R - dM, ranges from 1.0dS to 6.0-8.0dS with increments of 0.1dS and 0.2dS. In each system, a series of constrained MD simulations is performed as follows:32 (i) The separation between macroparticles C and D is chosen to be the largest value of R ′ and the equilibrated fluid is prepared as described in section 3.4. (ii) The instantaneous forces of FCS and FDS are computed during 6 × 104-8 × 104 time steps, and the force FMM(R) is calculated using eqs 7 and 8′. (iii) Then, external forces are assigned to the macroparticle pair along the C-D line for 5 × 103 time steps to reduce the C-D separation by 0.1dS or 0.2dS. (iv) For equilibration of the fluid around the macroparticles, the system is allowed to evolve for another 5 × 103-4.5 × 104 time steps, keeping the new C-D separation. Repeating the procedure from steps ii to iv, we can obtain the force-distance profile. This profile is evaluated for systems I-III and for systems IV and V of XA ) 0.90. The statistical errors of FMM are calculated by the blockaveraging method47 and found to be almost within 1.4, 1.4, 1.4, 1.1, and 1.3 in units of kBT/dS for systems I-IV, respectively. Adhesion Force. The force profile of systems IV and V exhibits the maximum attraction at the surface separation of R ′ ≈ 2dS, as is shown in section 4.3. This attraction is named an adhesion force, -Fad, in the present study. The adhesion forces in systems IV and V of various concentrations of XA given in Table 1 are computed following the above procedure ii, but the total number of simulation steps exceeds 1.8 × 105. It is found that the statistical errors of Fad range from 0.59 to 0.87 in units of kBT/dS. (45) Hamaker, H. C. Physica 1937, 4, 1058 (46) Steele, W. A. Surf. Sci. 1973, 36, 317. (47) Flyvbjerg, H.; Petersen, H. G. J. Chem. Phys. 1989, 91, 461.

Langmuir, Vol. 16, No. 7, 2000 3365

Figure 4. Reduced density profile of waterlike particles near the surface of the hydrophilic macroparticle in system I.

Figure 5. Interaction force between the hydrophilic macroparticles in system I.

4. Results and Discussion In sections 4.1-4.3, systems I-V are investigated mainly in terms of (i) the amphiphile and water densities near the macroparticle surface, gMA and gMW, and (ii) the interaction force between the surfaces, FMM. Section 4.4 deals with the problem (iii) how the adhesion force is influenced by the concentration and molecular architecture of amphiphiles. 4.1. Surface Force in Pure Water. The density profile of waterlike particles near the hydrophilic surface and the interaction force between the surfaces are shown in Figures 4 and 5, respectively. Figure 4 indicates that about three layers of waterlike particles are formed next to the surface. Figure 5 shows that the surface force oscillates with the periodicity of the water diameter and is steeply repulsive at the small separation. The behavior of these oscillation and repulsion originates from the water layers formed on the hydrophilic surface. The similar results have been reported in other studies.9,10,19,20,22-24,32 4.2. Surface Forces in Pure Amphiphilic Liquids. Pure amphiphile liquids of ht3 and ht are employed in following sections 4.2.1 and 4.2.2, respectively. 4.2.1. Pure ht3 Liquid. Density Profile. Figure 6 shows the density profiles of the center-of-mass (com) and sites of amphiphiles near the hydrophilic surface. The profiles of sites 1-4 have a peak at r ′ ) 0.99dS, 1.96dS, 2.84dS, and 3.63dS, respectively, whose neighboring gaps are roughly equal to the average bond length of 1.06dS; hence, amphiphiles are adsorbed almost vertically on the surface with their heads and tails pointing toward and away from the surface, respectively. This vertical adsorption of amphiphiles coincides with the adsorption model by the AFM measurements (see Figure 1a).35 The profiles of sites 1-4 have another peak at r ′ ) 5.56dS, 4.53dS, 3.86dS, and 3.01dS, respectively, whose order is opposite to that in the above case; therefore, amphiphiles are adsorbed on the first layer of amphiphiles, pointing their tails toward the layer, to form an amphiphile bilayer. This bilayer formation is explained also by two peaks of the com profile.

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Figure 6. Reduced density profile of amphiphile molecules near the surface of the hydrophilic macroparticle in system II.

Shinto et al.

Figure 8. Same as in Figure 6 but in system III.

Figure 9. Same as in Figure 7 but in system III. Figure 7. Interaction force between the hydrophilic macroparticles in system II.

The average thickness of the bilayer, H1, is defined as the separation between two major peaks of the site 1 profile plus dS which represents the size of two hemispheres of h particles. The value of H1 is evaluated as 5.6dS. It is worth noting that the sites 4 of amphiphiles in the second layer, at r ′ ) 3.01dS, are located closer to the surface than those in the first layer, at r ′ ) 3.63dS. This indicates that the amphiphiles in the second layer penetrate into the first layer. Force-Distance Profile. The interaction force between the hydrophilic macroparticles across amphiphiles is shown in Figure 7. The force is strongly attractive around R ′ ) 6.8dS but becomes repulsive at R ′ ≈ 5.9dS in which a well-ordered bilayer is formed between the surfaces. The latter distance is referred to as a stable separation in the present study. The thickness of the stable film, H2, is defined as the stable separation minus dS which represents the net width excluded by two surfaces. The value of H2 ) 4.9dS is smaller than that of H1 ) 5.6dS, indicating that the tail chains of amphiphiles in one layer of the bilayer film enter deeply into the other layer, compared with the case of the single surface. This behavior is attributable to the presence of the second surface. When the surfaces approach closer at R ′ < 5.9dS, the repulsion occurs to become stronger because of the steric hindrance between the tails of adsorbed amphiphiles. However, this repulsion decreases in the range of R ′ < 3.5dS, where because of the narrow space between the surfaces, amphiphiles are no longer adsorbed vertically on the surface and tend to be pushed out into the bulk, pointing their heads and tails toward the surface and the bulk, respectively. Figure 7 shows no clear step that was found in force curves of the AFM measurements,35,36 despite the agreement in vertical adsorption of amphiphiles on the surface as described above. This disagreement in force curves is probably because the model amphiphiles used are flexible, as pointed out in section 2.1, or an AFM has certain instability regions in which no forces may be measured (see Appendix 2 of ref 3).

Therefore, the more extensive studies of simulations and experiments are necessary to clarify the reason for the disagreement. 4.2.2. Pure ht Liquid. Density Profile. Figure 8 shows that the site profile has two major peaks at r ′ ) 0.99dS and 3.91dS for site 1 and at r ′ ) 1.94dS and 2.84dS for site 2. These peaks appear in the same way as described in the case of the pure ht3 liquid. The com profile has two peaks at r ′ ) 1.44dS and 3.24dS. These results indicate that an amphiphile bilayer with a thickness of H1 ) 3.9dS is formed at the surface. Force-Distance Profile. Figure 9 shows that the steric repulsion occurs at the short separation and is weak in strength, compared with that for ht3 shown in Figure 7; this behavior is attributable to a tail of ht shorter than that of ht3. The result coincides well with the experimental results.35,36 The value of H2 is evaluated to be 3.3dS and smaller than that of H1 ) 3.9dS. This result is similar to that in the case of ht3 as described above. In the range of R ′ < 4dS, the force oscillates with the periodicity of dS, because the ht dumbbells form a molecularly ordered film between the surfaces. 4.3. Surface Forces in Amphiphile-Water Mixtures. Amphiphile solutions of ht3 and ht including 10 vol % water (i.e., XA ) 0.90) are employed in sections 4.3.1 and 4.3.2, respectively. 4.3.1. ht3 Solution of 90 Vol %. Density Profile. Figure 10 displays the density profiles of amphiphile and water molecules near the surface. The amphiphile profile is similar to that for the pure ht3 liquid shown in Figure 6, indicating that the amphiphile bilayer is formed at the surface even in the presence of water. Water molecules are remarkably concentrated next to the surface and also located around the point of r ′ ) 6.8dS, which is about dS away from the peak of the site 1 density at r ′ ) 5.7dS; however, they hardly exist in the region of r ′ ) 2.2-5.0dS. These results manifest that water molecules are excluded from the hydrophobic interior of the bilayer and located near the hydrophilic outer surfaces of the bilayer. For an amphiphile-water mixture, H1 is defined as the separation between two major peaks of the water profile


Langmuir, Vol. 16, No. 7, 2000 3367

Figure 10. Reduced density profile of (a) amphiphile and (b) water molecules near the surface of the hydrophilic macroparticle in system IV of XA ) 0.90.

Figure 11. Interaction force between the hydrophilic macroparticles in system IV of XA ) 0.90. The inset represents the net force caused by the water bridging between the macroparticle surfaces (see the text).

plus dS. H1 represents the average thickness of a bilayer of hydrated amphiphiles and is evaluated to be 6.8dS. This value is larger than that of H1 ) 5.6dS for the anhydrous amphiphile bilayer. It is worth noting that the com, site 1, and site 2 profiles have a shoulder at the point about dS away from the first peak next to the surface, although no shoulder is observed in Figure 6. This indicates that a specific number of amphiphiles are adsorbed vertically on the surface, interposing a water molecule between their heads and the surface, as shown in the inset of Figure 10b. This adsorption is called solvent-separated adsorption. The number ratio of water and amphiphile molecules adsorbed on the surface, Psurf W/A, is roughly calculated using the heights of the first maxima of water and amphiphile profiles:

Psurf W/A )

1st maximum of gMW Pbulk 1st maximum of gMA for site 1 W/A Pbulk W/A

1 - XA ) XA/nsite

(11a) (11b)

where Pbulk W/A represents the water/amphiphile ratio in the bulk bulk. We obtain Psurf W/A ) 8/9 and PW/A ) 4/9, in which the absolute number of amphiphiles on the surface is different from that in the bulk. These values manifest that water molecules are adsorbed on the surface almost as many as amphiphiles, although the ratio in the bulk Pbulk W/A is less than a half. This behavior indicates that the mediumaveraged affinity of the surface for the w particle is stronger than that for the ht3 amphiphile, whereas in the absence of the medium these affinities are the same as expected from the particle-macroparticle interactions. Force-Distance Profile. The force-distance profile of XA ) 0.90 is shown in Figure 11, where the profiles in pure liquids of water (XA ) 0) and amphiphile (XA ) 1) are displayed for comparison. The profile of XA ) 0.90 is attractive around R ′ ) 7.2dS but repulsive in the region of R ′ ) 4.4-6.3dS. This feature is similar to that in the profile of XA ) 1. The value of H2 is calculated to be 5.3dS and greater than that of H2 ) 4.9dS for XA ) 1. This behavior agrees with that of the AFM measurements35,36

Figure 12. Contour maps of (a) the amphiphile center-of-mass density and (b) the water density near the hydrophilic macroparticle pair, C and D, for the separation of R′ ) 4.0dS in system IV of XA ) 0.90. Values in each panel are normalized such that the value of 1 indicates the density in the bulk.

and is explained by the formation of the thick hydrated bilayer near the surface as mentioned above. The point to note in Figure 11 is that the surface force drops steeply at R ′ ) 4.4dS and becomes attractive at R ′ ) 3.5dS. Why does the force change suddenly? To answer the question, we calculate the densities of amphiphile and water molecules near the macroparticle pair for the separations of R ′ ) 4.0dS and 4.4dS and display the contour maps in Figures 12 and 13. Figure 12 shows that amphiphiles between the macroparticles are pushed out into the bulk and water molecules enter to bridge the gap between the surfaces instead. This behavior is a capillaryinduced phase separation2 and is attributable to the size of an ht3 amphiphile larger than that of the w particle. No similar feature is observed in Figure 13, where the surface distance is increased by only 0.4dS. These results are the satisfactory evidence for the prediction that the strong adhesion force at wA ≈ 0.9 originates from the water bridging between surfaces, as pictured in Figure 1b.35

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Figure 13. Same as Figure 12 but for the separation of R′ ) 4.4dS.

Figure 14. Same as in Figure 10 but in system V of XA ) 0.90.

Thus, the critical feature of the surface force in Figure 11 is closely related to the nature of the amphiphilewater liquid-to-liquid-phase separation that leads to the formation of a water lens between the surfaces. This water lens appears at R ′ ≈ 4.2dS and then has a minimum thickness of 3.2dS. Remember that the water layer on the surface has a thickness of dS only, as in Figure 10b. These results indicate that the water bridging occurs before the surfaces with a water layer make the direct layer-to-layer contact. Another point to note in Figure 11 is that the surface force in the range of R ′ < 3.7dS oscillates with the periodicity of dS, because of the structural hindrance between the surfaces across water molecules; however, this force is more attractive than that in pure water. We roughly calculate the net bridging force Fbr by the following expression:

Fbr(R;XA) ) FMM(R;XA) - FMM(R;XA ) 0)

(12)

where FMM(R;XA) represents the surface force at separation R in a solution of XA. The result shown in the inset of Figure 11 reveals that Fbr becomes more attractive as the surfaces approach closer. This feature reminds us of the capillary force profile given by the analytical expressions such as eq 15.34 in ref 1 and eq 5.4.11 in ref 2. Main contributions to the strong attraction are considered to be (i) the Laplace pressure, resulting from the difference between the inner and outer pressures of the concave meniscus; (ii) the force arising from the tangent component of the interfacial tension; and (iii) the depletion force, resulting from the difference in the osmotic pressures between the bulk region and the confined water-rich region, in which amphiphiles can hardly enter because of their large size. Unfortunately, we have no suitable expression to evaluate these quantities using simulation data. The surface force becomes most attractive at R ′ ) 2.1dS, where the surfaces remain noncontact with each other and interpose one layer of water molecules. Although this feature differs from that in Figure 1b, the existence of a few water molecules between surfaces is not completely denied.36 The maximum attraction is investigated in section 4.4 extensively.

Figure 15. Same as in Figure 11 but in system V of XA ) 0.90.

4.3.2. ht Solution of 90 Vol %. Density Profile. Figure 14 displays the amphiphile and water densities near the surface and exhibits the following features: (i) A bilayer of hydrated amphiphiles is formed near the single surface, but water molecules are hardly excluded from the hydrophobic interior of the bilayer because of a short tail of the ht amphiphile. (ii) The hydrated bilayer of H1 ) 5.0dS is thicker than the anhydrous bilayer of H1 ) 3.9dS. (iii) Water molecules are concentrated next to the surface, in which the number ratio of Psurf W/A ) 7.3/9 is about 4 times larger than that of Pbulk W/A ) 2/9. Force-Distance Profile. The force profile is shown in Figure 15 and the characteristics are as follows: (i) The value of H2 ) 3.6dS is greater than that of H2 ) 3.3dS for pure ht liquid, because the hydrated bilayer near the surface is thicker than the anhydrous bilayer. (ii) The surface force drops at R ′ ) 3.5dS, but this decrease is even less pronounced than that for ht3 shown in Figure 11. (iii) The force profile in the range of R ′ ) 2.4-3.5dS is almost identical with that for pure water.


Langmuir, Vol. 16, No. 7, 2000 3369

Figure 16. Adhesion force between the hydrophilic macroparticles, -Fad, in systems IV and V of various concentrations of XA.

Figure 18. Same as Figure 17 but near the hydrophilic macroparticle pair for the separation of R′ ) 2.0dS in system V of XA ) 0.80 (a), 0.85 (b), and 0.90 (c).

Figure 17. Contour maps of the water density near the hydrophilic macroparticle pair, C and D, for the separation of R′ ) 2.1dS in system IV of XA ) 0.85 (a), 0.90 (b), and 0.95 (c). Values in each panel are normalized such that the value of 1 indicates the density in the bulk.

(iv) Fbr is attractive in the short range and weak in strength, compared with that for ht3 shown in the inset of Figure 11. (v) The surface force becomes most attractive at R ′ ) 2.0dS. Our careful observation of the contour maps of water and amphiphile densities manifests that the water bridging occurs at R ′ ) 3.5dS, but the confined film does not completely separate into two phases until R ′ e 3.1dS. This nonsimultaneity of the bridging and the phase separation explains the lower degree of the force dropping. 4.4. Adhesion Forces. The surface forces in amphiphile-water mixtures become most attractive at R ′ ≈ 2dS, as shown in Figures 11 and 15. These adhesion forces

of -Fad at various concentrations of XA are computed as given in Figure 16. The adhesion force has a maximum at XA ) 0.90 for ht3 and at XA ) 0.85 for ht. As an amphiphile has a longer hydrophobic tail, the adhesion force exhibits a maximum at a larger value of XA and the magnitude of this maximum becomes larger. This feature coincides with that of the AFM measurements.35,36 To investigate the mechanism of the adhesion, we focus on the water molecules adsorbed near the macroparticle pair. The contour maps of the water density at around a maximum of -Fad are displayed in Figure 17 for ht3 and Figure 18 for ht. The water density near the macroparticles is reduced from panels a to c of Figures 17 and 18, where panel b corresponds to the water density at a maximum of -Fad. This indicates that the adhesion forces become maximal when the macroparticle pair is moderately wetted by water. Comparison in Figures 17 and 18 manifests that the interface between the confined water lens and the amphiphiles of ht3 is clearer than that for ht. This behavior is expected from comparison in Figures 10 and 14. These results explain the reason why the maximum adhesion force in an ht3-w mixture is stronger than that in an ht-w mixture. 5. Conclusion In the present study, the interaction force between hydrophilic macroparticles in an alcoholic (or amphiphilic) solution is investigated systematically using a classical MD simulation with coarse-grained models and the following conclusions are drawn. Pure Amphiphilic Liquids. (i) Amphiphile molecules are adsorbed vertically on the hydrophilic surface to form an amphiphile bilayer.

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(ii) The surface force becomes strongly repulsive when the separation falls below about the thickness of the bilayer, because of the steric hindrance between adsorbed amphiphiles. Amphiphile-Water Mixtures. (iii) In the presence of 10 vol % water, a bilayer of hydrated amphiphiles is formed near the hydrophilic surface, in which waterlike particles are remarkably concentrated. (iv) When the surfaces approach each other, their interaction force becomes repulsive, whose profile is similar to that in the pure amphiphilic liquid. However, the surface force becomes attractive suddenly when the separation falls below the length of the amphiphile molecule. This behavior originates from a liquid-to-liquid phase separation of the amphiphile-water mixture between the surfaces, in which amphiphiles are pushed out into the bulk and waterlike particles enter to bridge the gap between the surfaces instead. (v) The adhesion force of -Fad has a maximum at a higher concentration of XA (≈0.9), and the magnitude of this maximum is larger as an amphiphile has a longer hydrophobic tail. The adhesion is largely influenced by the wetting degree of the macroparticle pair and the sharpness of the interface between the confined water lens and the amphiphile. The wetting degree and the interface sharpness depend on the concentration and molecular architecture of amphiphiles. Most of our simulation results are in fair agreement with recent experimental AFM results and their interpretations, whereas our force curves do not completely agree with those of the AFM measurements.35,36 It is probable that this disagreement arises from either the simplicity of our simulation models or the force-measuring instability of an AFM, as explained in section 4.2.1; hence, the more extensive studies of simulations and experiments are required to clarify the reason for the disagreement. Nonetheless, we believe firmly that the present simulation method is a good tool to investigate surface forces in complex fluids. Acknowledgment. Computation time of the Cray T-94/4128 was provided by the Supercomputer Laboratory, Institute for Chemical Research, Kyoto University, Kyoto, Japan. Appendix The potential functions for particle-macroparticle interaction are derived following Hamaker, who presented the van der Waals attraction between two macroparticles.45 The particle-particle interaction is given by an LJ potential: 12

6

[( ) ( ) ]

σ u(r) ) 4 r

σ r

(A.1)

Consider a sphere of radius rM and center O and a particle P outside at a distance OP ) r, as shown in Figure 19. Two types of spheres are considered: (I) the sphere containing the LJ particles uniformly at volume density Fvol and (II) the empty sphere whose surface is uniformly composed of the LJ particles at surface density Fsurf. Sphere I. The sphere around O will cut out from a second sphere of radius l around P a surface ABC, whose area is then

surface (ABC) )

∫0

θ0

(2πl sin θ) l dθ ) 2πl2(1 - cos θ0) (A.2)

Figure 19. Schematic of the particle and macroparticle.

where θ0 is given by

rM2 ) r2 + l2 - 2rl cos θ0

(A.3)

Equation A.2 yields

surface (ABC) )

πl 2 [r - (r - l)2] r M

(A.4)

The potential energy of an LJ particle at P is written down as

uP(r) )

r+r πl Fvolu(l) [rM2 - (r - l)2] dl ∫r-r r M

(A.5)

M

Carrying out the integration, we get up as a function of r:

uP(r) )

[

{[

πFvolσ4 1 8rMσ8 8rMσ8 + + 3r 30 (r + r )9 (r - r )9 M M

]

σ8 σ8 8 (r + rM) (r - rM)8 2

2

]}

2rMσ σ2 σ2 + + (r + rM)3 (r - rM)3 (r + rM)2 (r - rM)2 2rMσ

(A.6)

When the sphere is sufficiently large (rM J 4σ), eq A.6 is approximated by

uP(r) )

{ [ ( ) ( )] [ ( ) ( ) ]}

πFvolσ3rM 1 9 8 σ σ σ 8 3r 30 r - rM rM r - r M 3 2 σ σ σ 2 (A.7) r - rM rM r - r M

If the sphere becomes infinitely large (rM ) ∞) and the distance of P from the sphere surface, r - rM, is kept at constant z, eqs A.6 and A.7 are found to be

uP(z) )

2πFvolσ3 2 σ 9 σ 3 15 z z

3

[ ( ) ( )]

(A.8)

which is identical with the 9-3 potential for particlewall interaction.46 Sphere II. The surface ABC shown in Figure 19 has an edge circumference with a length of 2πrM sin φ, where φ is given by

l2 ) rM2 + r2 - 2rMr cos φ

(A.9)

Differentiating eq A.9 with respect to φ, we get

l dφ ) dl rMr sin φ

(A.10)

The potential energy of an LJ particle at P is written down


Langmuir, Vol. 16, No. 7, 2000 3371

When rM J 4σ, eq A.12 is approximated by

as

uP(r) )

∫0πFsurfu(l) (2πrM sin φ)rM dφ ) 2πr l ∫rr-+rr Fsurfu(l) r M dl M

M

uP(r) ) (A.11)

where eq A.10 is used for transformation. Carrying out the integration, we get up as a function of r: 2

{ [( [(

) ( ) (

)] ) ]}

2πFsurfσ rM 2 10 10 σ σ uP(r) ) r 5 r - rM r + rM 4 4 σ σ (A.12) r - rM r + rM

[(

)]

) (

2πFsurfσ2rM 2 σ r 5 r - rM

10

-

σ r - rM

4

(A.13)

If rM ) ∞ and r - rM ≡ z ()constant), eqs A.12 and A.13 are found to be

uP(z) ) 2πFsurfσ2

[52(σz) - (σz) ] 10

4

(A.14)

which is identical with the 10-4 potential for particlewall interaction.46 LA990554J

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