Langmuir 1994,10, 4403-4408
4403
Monte Carlo Simulation of Inlayer Structure Formation in Thin Liquid Films X.L. Chu, A. D. Nikolov, and D. T. Wasan* Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Illinois 60616 Received May 31, 1994. In Final Form: September 23, 1994@ The structure formation in free liquid films containing surfactant micelles or other colloidal particles is of fundamental importancein colloid and interfacescience and its applications. Experiments on thinning of single flat or curved symmetrical liquid films and asymmetrical or pseudoemulsion films formed from colloidal dispersions show that these fdms thin in a stepwise manner. We present here results of our grand-canonical-ensemble Monte Carlo simulations using both the hard sphere and Leonard-Jones potentials, which not only verify the presence of particle layering inside the free thinning films but also reveal for the first time that, depending on the film thickness and particle volume fraction, there exists within the particle layers, in the direction parallel to the film surfaces,an ordered 2-D hexagonal structure. The calculated,in-layer radial distribution functions show that the in-layer structure transition depends on the film thickness and the position of the layer inside the film. The sequence of the formation of microstructures in a thin film of fixed thickness with increasing particle concentration is found to be disorder layering inlayer ordering for the surface layers inlayer ordering for middle layers -bulk type ordering. This ordering phenomenon in free liquid film is new. The formation of ordered structures inside the lavers has practical implications for both rheology of such films and the film stability and thereby the dispersion stability.
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I. Introduction There is an increasing interest in the structuring phenomenon of submicrometer particles or surfactant micelles inside free liquid films such as those associated with foams, emulsions, and particle suspensions, and a connection between an observed stepwise thinning film and the structuring phenomena inside the film has been proposed.'-14 The most convincing evidence for the existence of micellar or submicrometer particle layering structure inside a liquid film is the experimental observations that the thinning films, formed from aqueous, concentrated dispersions of spherical latex particles, as well as surfactant miceIlar solutions, change thickness in a stepwise manner.'-14 The height of the steps is approximately constant and nearly corresponds to the 'effectiven diameter of the micelles or submicrometer particle^.^-^ In order to explain the experimental findings, it has been proposed7~*J2J3 that particles must organize themselves and form an ordered structure inside a thin film. The stepwise thickness transition occurs as the result of formation and expansion of a black (thinner)
* To whom all correspondence should be directed. Abstract published in Advance ACS Abstracts, November 1, 1994. (1)Johnnott, E.S. Philos. Mag. 1906,70,1339. (2)Perrin, J. Ann. Phys. (Paris) 1918,10, 160. (3)Bruil, H. G.;Lyklema, J. Nature (London),Phys. Sci. 1971,233, 19. (4)Kenskemp,J.W.; Lyklema,J.AdsorptionatInterfaces;ACS Symp. Ser. 8; American Chemical Society: Washington, DC, 1975;191 pp. (5)Friberg, S.; Linden, St. E.; Saito, H. Nature 1974,251,494. (6)Kruglyakov, P. M. Kolloid. Zh. 1974,36, 160. (7)Nikolov, A.D.; Wasan, D. T.; Kralchevsky, P. A.; Ivanov, I. B. In Ordering and Organization in Ionic Solutions-Proceedingsof Yamada Conference Xrx; h e , N., Sogami, I., Eds.; World Scientific: Singapore, @
19AR.
(8)Nikolov, A. D.; Kralchevasky, P. A.; Ivanov, I. B.; Wasan, D. T.
J. Colloid Interface Sei. 1989,133,13. (9)Kralchevsky, P.A,; Nikolov, A. D.; Wasan, D. T.; Ivanov, I. B. Langmuir 1990,6,1180. (10)Nikolov, A. D.;Wasan, D.; Denkov, N. D.; Kralchevsky, P. A.; Ivanov, I. B. Prog. Colloid Polym. Sci. 1990,82,87. (11)Basheva. E.S.:Nikolov. A. D.: Kralchevskv, P. A.: Ivanov. I. B.: Wasan, D. T. Surfactants Solution (Proc. Int. Sjmp.) 1991,11; 467: (12)Wasan, D. T.;Nikolov, A. D.; Kralchevsky, P. A.; Ivanov, I. B. Colloid Surf.1992,67,139. (13)Nikolov, A.D.;Wasan, D. T. Langmuir 1992,8,2985. (14)Wasan, D.T.;Nikolov, A. D. In Particulate Two Phase Flow; Roco, M. C., Ed.; Bethernorth-Heineman: 1993;325 pp.
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spot inside the film, caused by the condensation (aggregation) of vacancies in a colloid crystal-like structure.l3 Our paper is concerned with the grand-canonicalensemble Monte Carlo simulations of a model film consisting of film surfaces and micelles/particles. Both the particle layering phenomenon inside free liquid films and the in-layer particle structure formation, as well as their relations with the film thickness stability, are studied versus film thickness and particle/micellar concentration. The sequence of structural transitions from disordered to ordered particle structures inside a thin film is discussed.
11. Model and Method
A direct, theoretical approach to the study of particle structuring phenomenon inside a liquid film is to use a numerical simulation,16 either the Monte Carlo16J7or molecular dynamics'* method. Both methods have previously been used to predict the formation of layering structure. Analytically, the existence of the film surfaces can be regarded as an external field, which affects the arrangement of particles confined in between. This idea has been realized in various approaches: Percus' shielding theory;lgthe generalized mean spherical approximation;20 the anisotropic PY approximation.21 For a review we refer to Rickayzen et a1.22More recently, Schmid and Schickz3 studied this problem in the frame of the Landau theory of phase transition. In free liquid films containingmicelles or submicrometer particles, the nature ofthe interactions between particles can be classified as follows: (1)electrostatic-stearic forces; (2) van der Walls forces; (3) structural forces caused by (15)Hirtzel, C. S.;Rajagopalan, R. Computer Experiments for Structure and Thermodynamic and Transport Properties of Colloidal Fluids. In Micellar Solutions a n d Microemulsions; Chen, S. H., Rajagopalan, R., Eds.; Springer-Verlag: New York, 1990. (16)Snook, I. K.;Henderson, D. J. Chem. Phys. 1978,68, 2134. (17)Snook, I. K.; Megen, W. V. J. Chem. Phys. 1980,72,2907. (18)Henderson, J. R.;Van Swol, F. MoE. Phys. 1984,51,991. (19)Percus, J . K. J. Stat. Phys. 1980,23,657. (20)Waisman, E.;Henderson, D.; Lebowitz, J. L. Mol. Phys. 1976, 32, 1373. (21)Kjellander, R.; Sarman, S. Chem. Phys. Lett. 1988,149, 102. Richmond, P. in Thin Liquid Films; Ivanov, I. B. (22)Rickayzen, G.; Ed.; Marcel Dekker, Inc.: New York, 1988;131 pp. (23)Schmid, F.; Schick, M. Phys. Reu. E 1993,48,1882.
0743-746319412410-4403$04.50/0 0 1994 American Chemical Society
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4404 Langmuir, Vol. 10, No. 12, 1994
solvent and surfactant molecules.24 The total effect of these forces has been recognized in general as a strong repulsive interaction acting over a short distance around the effectiveparticle diameter and a weak attractive force acting over a relatively long distance. We first consider the repulsive interactions, the most essential part, and regard a particle as a hard sphere of effective diameter “d‘,i.e., the actual micellar or particle diameter plus a distance in which the repulsive electrostatic interactionneeds to be taken into account(i.e.,Debye length). This “effective”diameter can be obtained independently from the diffusion coefkient or by dynamic light s ~ a t t e r i n g . Similarly, ~ ~ . ~ ~ we model the film surfaces as “hard walls”. Thus, we study here a model of many hard spheres confined between two narrowly separated, parallel hard walls (HSHW). A more realistic model should include both repulsive and attractive interactions. Here as the first step of our study, we use the 12-6Leonard-Jones (L-J)potential with both repulsive and attractive interactions included in an simple analytic expression. For the pair interaction between two particles, we have
where Vu is the interaction energy between two particles i andj, separated by a distance rc, and o is the effective diameter of the particle; E is the depth of the “attractive well” and scaled by the temperature kT. Assuming that the interaction between the particles in the film and the molecules on the film surface obey the same 12-6L-J potential, in the continuum film surface limit, we can integrate all over the film surface and obtain the “10-4” interaction between a particles and a film surface
where esis the material density of the surface and taken to be 1 in our calculations. In the case of large, ionic colloid particles, where other interparticle potentials (e.g., electrostearic, Hamaker, and depletion potentials) may be more appropriate, the usage of L-J potential can be still regarded as a reasonable approximation as long as the parameters E and o in L-J potential are chosen to fit the effective particle size and the depth of the attractive well. The Monte Carlo method is a random sampling technique to estimate the statistical averages. By weighting the randomly generated configurations according to the given ensemble, it provides exact “experimental” conditions and an unbiased test for a model system.26 There has been much discussion about the choice of a proper statistical ensemble for the system of particles confined in a thin ~ h a n n e l . l ~The 9 ~ ~bulk phase plays the role of a reservoir of constant concentrationand chemicalpotential p. Thus a good approximation for our film system is a grand canonical ensemble @-ensemble)of fmed @,V,T). V =H*L*L is the volume of a portion of the film as shown in Figure 1, where H is the film thickness, ranging from 1.0010to 100 used in our calculations,andL is the length of two other directions parallel to the film surfaces, which (24) Vold, R. D.; Vold, M. J. Colloid and Interface Chemistry;AddisonWesley, Inc.: London, 1983. (25) Reiss-Husson, F.;Luzzati, V. J . Phys. Chem. 1964,68, 3504. (26) Valleau, J. P.; Whittington, S. G. A Guide to Monte Carlo for Statistical Mechanics. In Modern Theoretical Chemistry;Berne, B. J., Ed.; Plenum Press: New York, 1977. (27) Wertheim, M. S.;Blum, L.; Bratko, D. Statistical mechanics of confined systems: The solvent-induced force between smooth parallel plates. In Micellar Solutions and Microemulsions; Chen, S . H., Rajagopalan, R., Eds.; Springer-Verlag: New York, 1990.
Film Surface
t‘
L
4
r
/ I
ParticlelMicelle
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Figure 1. Schematic of the model film. The particles are sandwiched between two film surfaces. The thickness of the film, H, is taken between 1.001 and 10 particle diameters in our calculations,and the size in the two other directions, L,is chosen to be between 6 and 50 times larger than H. is chosen to be 6 to 50 times larger than the thickness H. Periodic boundary conditions are used in two parallel directions. It should be pointed out that though the choice of the model and p-ensemble simplifiesthe system substantially, it can still provide us with much useful information about the film: the presence of micellar structuring in the normal and parallel directions of the film surfaces, the oscillatory disjoiningpressure, the number of observable layers, etc. It does not, however, tell us about the dynamic properties and behavior such as the formation and expansion of a black spot, which are not within the scope of this paper. More realistic models and the corresponding ensembles need to be used if these are the targets of the study. We follow the numerical procedure for a p-ensemble described by Valleau and Whittington.26We initially place N particles inside the film. Each run consists of three possible attempts upon the current configuration: (1) Randomly moving a randomly chosen particle with probability 1-2p; (2)adding a particle in a randomly chosen position with probabilityp; (3) deleting a randomly chosen particle with probabilityp. In the present study, we choosep = 0.15as suggested by Wertheim et aZ.27The acceptance of a moving attempt depends on the change of the configurational potential energy which is exactly the same as in the canonical case.28 The acceptance probability of an attempted addition or deletion f i j is
fi,i+l = 1, fi+l,i = l/r,
for r > 1
where the first subscript indicates the current state to be changed and the second subscriptrepresents the new state to be achieved; thus fi,i+l is the acceptance probability of adding a particle upon a state of i particles, and fi+l,i the acceptance probability of deleting a particle from a state of i 1 particles. r is the dimensionless quantity here
+
+
r = Q/@i+l exp[(Ap - Ui+l Ui)/kZ‘l
p is the effective particle concentration in the bulk (in volume fraction),ei+lis the effectiveparticle concentration of state i 1 and Ui and Ui+l are the configurational potential energy of state i and i 1, respectively. The excess chemical potential Ap/kT can be obtained either from the Carnahan-Starling equation of state27
+
AplkT = Q(8 - 9Q
+
+ 3e2)/(1-
(28) Metropolis, N.;Metropolis, A. W.; Rosebluth, M. N.; Teller, A. H.; Teller, E. J. Chem. Phys. 1953,21, 1087.
Letters
Langmuir, VoZ.10,No.12, 1994 4405
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Figure 2. Normal density distribution calculated from the HSHW model. Three peaks indicate the formation of three particle layers. The average particle volume fractions are 15%, 25%, and 35% and the film thickness is 3.00.
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Here, Qw is the particle density at a hypothetical surface in the bulk phase. We plot calculatedstructural disjoining pressure against the film thickness H at different bulk concentrations in Figure 3. It can be seen that the structural disjoining pressure exerted by particles in a thin liquid film oscillates around zero and that it can be positive or negative, depending on the thickness of the film. The period of the oscillation is 0,the effective diameter of the particles. Note that when the film thickness is about an integer multiple of the particle diameter, la, 2a,3a, ...,the disjoiningpressure is positive, and when the film thickness is around 1.50,2.50,3.50,..., the disjoining pressure is negative. The states with negative particle disjoining pressure are intrinsically unstable from a thermodynamic point of view; thus film changes its thickness during a thinning process by stepwisetransitions and stays at thicknesses with positive particle disjoining pressure. As seen in Figure 3, the amplitude of the oscillation of the particle disjoining pressure becomes weaker as the film thickness increases, and beyond a certain thickness, there will be no more
I
Figure 3. Structural disjoining pressure ll versus film thickness H . Three curves are the results for three different bulk concentrations: 20%, 30%, and 40%. The values for disjoining pressure are for a system with the particle of 10 nm effective diameter, at a temperature of 300 K.
or by Adam's method,29both giving identical results. We normally skip 1million to 10 million runs before we start to record the configurations,thus the initial conditions do not affect the final resu1ts.l' After initial relaxation, we start to calculate the averages of accepted configurations. Normally 2 million to 10 million configurations are generated from the procedure described above. 111. Numerical Results and Comparison with Recent Experiments Typical density distributions of micelles or particles across a film of thickness 30 are shown in Figure 2, for the averageparticle concentrationin the film ranging from 15 vol % to 35 ~ 0 1 % The . three peaks (at -1, 0, and +1> indicate that particles inside the film form three layers parallel to the film surfaces. Also the figure shows that, at a given film thickness, as the particle concentration inside the film increases, the particle-layering becomes more pronounced. The pressure exerted by the particles on the surfaces of the film of the HSHW model is gwkT,where gwis the contact particle density at the film surfaces. Only the excess part of this pressure over that at the bulk phase, the disjoining pressure, is relevant to the film stability
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Figure 5. Structural disjoining pressure versus film thickness: comparison of theory and experiment. The experiments30 are performed with aqueous solution of cetyltrimethylammonium bromide (CTAB) at micellar volume fraction of 0.073 (corresponding to the effective concentration of 50 vol % a t the effective micellar diameter 9 nm). The theoretical results are calculated from a HSHW model with effectivemicellar diameter of 9 nm and concentration 50 ~ 0 1 % .
oscillations. Therefore, a system of given particle concentration has only a finite number of maxima in the particle disjoining pressure isotherm and one should expect to observethe same correspondingnumber of steptransitions during actual film drainage experiments. A comparison of the number of stepwise transitions, as the function of particle concentration, observed experimentally13and those predicted by the HSHW model is given in Figure 4. We also compare in Figure 5 the disjoining
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4406 Langmuir, Vol. 10,No. 12, 1994
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Figure 6. (a, top left) Surface plot of inlayer RDF versus inlayer distance and concentration for HSHW model. At very low concentrations, there is only one small peak near la. At higher concentrations (20-43 vol %), there are peaks near every integer distance: la, 20,30, ..., i.e., typical dense liquid RDF. In the high concentration limit ( ' 4 3 vol %), two new peaks near J3 and 2/7 can be seen, indicating the hexagonal packing. The film thickness is 20. (b, top right)Surface plot of inlayer RDF versus inlayer distance and concentrationfor J-L ( E = 0.5KT) model. The film thickness is 20. (c, bottom left) Contour plot of inlayer RDF versus inlayer distance and concentrationfor HSHW model same as part a. This figure shows with broken line the concentration near which the new peak branches out. (d, bottom right) Contour plot of inlayer RDF versus inlayer distance and concentration for L-J ( E = 0.5kT) model same as part b.
pressure isotherms, in the presence of micelles inside the film, calculated from our HSHW model and measured in a surfaceforce experimentby Kekicheffet al.30 One notices the good agreement between our Monte Carlo simulation results and the experimental observations. In order to find the particle structure inside the layers parallel to the film surfaces, the radial distribution function (RDF) of the particles in a layer has been examined. The RDF gives the probability of finding particles around a reference particle in a shell of radius r. When the particle concentration is low, the particles inside a layer pack randomly to form a liquid-like structure. Only when the concentration is over a certain value do the particles in a layer start to form a more ordered structure. This value for the inlayer structure transition depends on the film thickness and the position of the layer inside the film. In parts a and b of Figure 6 , we plot inlayer RDF of a surface layer versus the average particle concentration in the film and the inlayer distance for HSHW and L-J model, respectively. At the average effective particle concentration of 40 vol % (a solution of 0.052 mom of Enordet AE 1215-30corresponds to such a concentration)or lower the inlayer RDF for both models showstypical liquid-like2-D structure without order, that is, damped peaks near the integers, 1,2,3, ... (in the unit of particle diameter). When the average effective concentration increases to somewhere between 40 and 45 vol %, new peaks begins to appear in the RDF especially near r = 2/3 (the actual value is always somewhat larger than (30) Kbkicheff, P.; Richetti, P. Prog. Colloid Polym. Sci. 1992,88,8.
d3 due to nonvanishing spacing between particles) and r = d7,which indicates the formation of a 2-D hexagonal structure inside the layer. Notice that this peak grows gradually as the concentration increases, indicating a smooth structural transition from a liquid-like, inlayer structure to the colloid crystal-like,inlayer structure. From the contour plot of RDF in parts c and d of Figure 6 , we can see the development of the new peaks near d 3 and 2/7, which gives the transition point. Both the HSHW and L-J models predict that the disorder-order transition for the surface layers for a film of thickness 20 starts at about 43 vol %, the same as that observed experimentally by Pieranski et aL31in a wedged film. It should be pointed out that the structural transitions we studied here are different from the Kirkwood-Alder This difference type transition for the bulk can be seen not only in the transition value (Alder's prediction is about 50 ~ 0 1 %but ) also in the manner and sequenceof the occurrence of transitions. In our case, the structure transition occurs inside the layer and layerby-layer, starting from the two layers adjacent to the film surfaces, then to the next layers at higher concentration, and finally the layer in the middle of the film at even higher concentration. In Figure 7 we plotted the inlayer (31) Pieranski, P.; Strzlecki, L.; Pansu, B. Phys. Rev. Lett. 1983,50, 900. (32)Kirkwood, J.G.; Maun, E. K.; Alder, B. J. J. Chem. Phys. 1960, 18,1040. (33) Alder, B.J.; Wainwright, T. E. J . Chem. Phys. 1957,27,1208. (34) Courtemanch, D. J.; Van Swol, F. Phys. Rev. Lett. 1992,69, 2078.
Letters Surface Layer
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, 6 shell, hexagonal structure
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Langmuir, Vol. 10,No. 12, 1994 4407
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Figure 9. Inlayer RDF for a film of thickness of 1.70 at concentration of 46%. A film of such a thickness can on1 be realized in a system with two solid walls. A peak near is the indication of square type inlayer structure.
h
The theoretical prediction of inlayer 2-D hexagonal particle structure has been verified by our recent transmitted light diffraction experiments.13J4 The formation of such an ordered inlayer structure may result in the increase in shear visc0sity3~and, consequently, in the decrease in the film drainage velocity. Indeed, Basheva et aZ.ll observed experimentally that the thinner the film is, the lower the drainage velocity will be. Our focus here is to study free liquid films, and thus only the films of thickness of la, 20, ... are of our main interest. In the case of thin liquid films confined by two solid walls, the film thickness can be controlled at an arbitrary value. Pieranski et aL31observed square-type inlayer packing structure for such a film of thickness between na and (n 1)a (it should be noted that for a liquid film with fluid surfaces, such a thickness does not exist). Such a square type inlayer structure is predicted by our Monte Carlo simulation. Figure 9 is the inlayer RDF for a film of thickness of 1.70 and the particle concentration of 46 ~ 0 1 % .Contrary to the inlayer RDF for free liquid films (Figures 6-8), there is no 4 3 peak in this case, but there is a peak instead at d2, indicating a 2-D square structure.
+
0
1
2
3
DISTANCE (Unit: Particle Diameter, 6 )
Figure 8. Inlayer radial distribution function for the surface layer of HSHW films of the average particle concentration of 46vol %, but at differentthicknesses. It shows that the thinner the film is, the better the orderedinlayer 2-D structure is formed.
RDFs for different layers in a HSHW film of particle concentration of 46 ~ 0 1 % .One can see a peak near d3 for the surface layer which indicates the formation of 2-D hexagonal packing. This peak becomes weaker in the next layer and disappears in the middle layer. The difference in the inlayer RDFs of different layers illustrates that there exists, in a single film, more ordered structure in the surface layer and disordered structure in the middle layer. The sequence of the transitions in a thin film of fixed thickness with increasing particle concentration is identified as: disorder layering inlayer ordering for surface layers -.inlayer ordering for middle layers -.bulk type ordering. The inlayer structure depends not only on the position of the layer in the film but also on the film thickness. In Figure 8 we plotted the inlayer RDF of surface layers for different film thicknesses, 2a,3a, 40, and 50, at the same concentration of 46 vol %. The degree of ordered 2-D hexagonal structure can be detected by the height of the peak near d3. It is clearly seen that the inlayer particles in thinner films of thicknesses 20 and 30 are better organized than those in the thicker films of thickness 40 and 50.
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W. Summary We studied the structuring phenomenon inside liquid films formed from surfactant micellar solutions or suspensions of Brownian particles by using grand-canonicalensemble Monte Carlo simulations. We modeled the film by hard-sphere-hard-wall and Leonard-Jones systems. The micelles or particles inside the film tend to be layered parallel to the film surfaces and the particle density profile becomes oscillatory across the film. Such a layering structure becomes more and more pronounced as the particle concentrationincreases. The structural disjoining pressure due to the layering structures of particles has been calculated, and we find that it oscillates around zero, positive when the film thickness is near an integer multiple of the particle diameter and negative when the thickness is between two consecutive integer numbers. Those states of negative disjoining pressure are thermodynamically unstable; thus we expectthat the film changes its thickness in a stepwise fashion, and the number of the steps (transitions) corresponds to the number of oscillations in the disjoining pressure. The theoretical predictions of the film thickness transitions are in good agreementwith experimental data. (35) Rhykerd, C. L., Jr.; Schoen, M.; Diestler, D. J.; Cushman, J. H. Nature 1987,330,461.
4408 Langmuir,
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In the effective concentration ranging from about 40 to 50 vol %, there exists not only layering structure but also an ordered 2-D structure inside a layer. We studied this inlayer particle structure by examining the inlayer radial distribution function and found the transition from a disordered, liquid-like 2-D structure to an ordered, crystallike 2-D hexagonal structure. This finding is confirmed by our recent observations obtained by using the lowangle transmitted light diffraction technique in a film formed from latex suspensions containing two layers of 156-nm particles.13J4 Our numerical model is for the free liquid film, therefore the particle structures, especially the structure in the direction parallel to the surface, are different from those reported by Rhykerd et al.35and Schoen et al .,36 which are primarily the extension of the structure of the solid surfaces. Our results, on the one hand, are supported by (36) Schoen,M.;Diestler,D. J.; Cushman,J. H. J. Chem.Phys. 1987, 87,5464.
Letters the experiments ofthin films confined between two smooth solid s u r f a ~ e s ~ and, l ~ on ~ ~the * ~other ~ hand, can interpret the peculiar phenomenon of stepwise thickness transitions in free liquid The restrictive geometry in the thin film condition forces the micelles or particles inside the film to be stratified and organized and, eventually, affects the rheology and the thickness stability ofthe film.39 Such a relationship should be revealed explicitly in a more realistic model system with flexible film surfaces, instead of nondeformable walls, in an isobaric ensemble. Work in this direction is in progress.
Acknowledgment. This project is supported in part by the National Science Foundation and by the U.S. Department of Energy. (37) Richetti, P.;Kekicheff, P. Phys. Rev. Lett. 1992, 68, 1951. (38)Van Winkle, D. H.; Murray, C. A. Phys. Rev. A 1986,34,562. (39) Wasan, D.T.Chem. Eng. Ed. 1992, Spring Issue, 589.
