Langmuir 2005, 21, 7057-7060
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Modeling of Drying in Films of Colloidal Particles Yuri Reyes† and Yurko Duda*,‡ Facultad de Quı´mica, Universidad Nacional Autonoma de Mexico, Me´ xico D.F., Me´ xico, Programa de Ingenierı´a Molecular, Instituto Mexicano del Petro´ leo, 07730, Me´ xico D.F., Me´ xico, and Institute for Condensed Matter Physics, National Academy of Science of Ukraine, 79011 Lviv, Ukraine Received January 20, 2005. In Final Form: March 31, 2005 The process of film formation on a solid substrate from polymer colloid dispersion during solvent evaporation has been investigated by means of the Monte Carlo simulation method. Colloid particles are modeled as hard spheres. Time evolution of the colloid density distribution and coverage of the solid substrate are studied. Both density and structure of colloid film is shown to depend strongly on the evaporation rate. At a low evaporation rate, the coexistence of hexagonal and tetragonal domains of dried colloid monolayer has been observed. The results of monolayer structure are in good agreement with the confocal scanning laser microscopy observations of Dullens et al. (2004).
Well-ordered latex coatings have been an active area of research for a long time.1-20 Their use in most applications is based on the formation of a continuous film during drying. Theoretical and experimental works have been devoted to study the fundamental mechanism of the film formation. However, how the solvent evaporation process affects the structure of the film is still not well-understood. Up to now, it is accepted that three kinds of interparticle forces govern the packing of latex films during solvent evaporation: lateral capillary forces (LCF), flotation forces, and convection forces.1-6 * Address for correspondence: Yurko Duda, Programa de Ingenierı´a Molecular, Instituto Mexicano del Petro´leo, Eje Central L. Ca´rdenas 152, 07730, Me´xico D. F., Mexico. Tel.: (55) 9175 8269. Fax: (55) 9175 6239. E-mail:
[email protected]. † UNAM. ‡ Instituto Mexicano del Petro ´ leo and NAS of Ukraine. (1) Winnik, M. A. In Emulsion Polymerization and Emulsion Polymers;, Lovell, P. A., El-Aasser, M. S., Eds.; John Wiley & Sons, Ltd.: New York, 1997; Chapter 14, p 467, and references therein. (2) Wallin, M.; Glover, P. M.; Hellgren, A.-C.; Keddie, J. L.; McDonald, J. P. Macromolecules 2000, 33, 8443. (3) Salamanca, J. M.; Ciampi, E.; Faux, D. A.; Glover, P. M.; McDonald, J. P.; Routh, A. F.; Peters, A. C. I. A.; Satguru, R.; Keddie, J. L. Langmuir 2001, 17, 3202. (4) Routh, A.; Zimmerman, W. B. Chem. Eng. Sci. 2004, 59, 2961. (5) Wang, Y.; Kats, A.; Juhue´, D.; Winnik, M. Langmuir 1992, 8, 1435. (6) Ko, H.-Y.; Park, J.; Shin, H.; Moon, J. Chem. Mater. 2004, 16, 4212. (7) Tsige, M.; Grest, G. S. Macromolecules 2004, 37, 4333. (8) Rakers, S.; Chi, L. F.; Fuchs, H. Langmuir 1997, 13, 7121. (9) Stirniman, M.; Gui, J. J. Phys. Chem. B 2002, 106, 5967. (10) Dullens, R. P. A.; Claesson, E. M.; Derks, D.; van Blaaderen, A.; Kegel, W. K. Langmuir 2003, 19, 5963. (11) Dullens, R. P. A.; Claesson, E. M.; Kegel, W. K. Langmuir 2004, 20, 658. (12) Dushkin, C. D.; Lazarov, G. S.; Kotsev, S. N.; Yoshimura, H.; Nagayama, K. Colloid Polym. Sci. 1999, 277, 914. (13) Cardoso, A. H.; Paula Leite, C. A.; Darbello Zaniquelli, M. E.; Galembeck, F. Colloids Surf., A 1998, 144, 207. (14) van Blaaderen, A.; Wiltzius, P. Science 1995, 270, 1177. (15) Kim, K.; Shin, K.; Kim, H.; Kim, C.; Byun, Y. Langmuir 2004, 20, 5396. (16) Yamaguchi, K.; Taniguchi, T.; Kawaguchi, S.; Nagai, K. Colloid Polym. Sci. 2004, 282, 684. (17) Soto, N.; Reyes, Y.; Sanmiguel, M.; Dominguez, M. A.; Duda, Y.; Va´zquez, F. Int. J. Polym. Mater. 2005, accepted. (18) Dijkstra, M. Curr. Opin. Colloid Interface Sci. 2001, 6, 372. (19) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Ivanov, I. B.; Yoshimura, H.; Nagayama, K. Langmuir 1992, 8, 3183. (20) Liao, Q.; Chen, L.; Qu, X.; Jin, X. J. Colloid Interface Sci. 2000, 227, 84.
The objective of our work is twofold. The first goal is to present a new simple modeling approach for a film drying process and the simulation results that indicate the influence of the film drying rate on the latex ordering on a solid substrate. On the other hand, and more importantly, we wish to show that a well-ordered model latex film can be obtained without considering colloid agglomeration20 and intercolloidal attraction due to, for example, LCF.19 Ignoring the above-mentioned effects is just a first step of our simulation study of latex ordering. However, it should be noted here that LCF acts solely on particles protruding out of the solvent surface (due to the menisci formed around the particles). Unlike the previous works,3,6,12,19 where lateral drying in the thin films has been studied, we consider here the vertical, with respect to the substrate, drying front. Therefore, we can expect that the LCF effect on the film formation is only weak. The irreversible nature of the evaporation process necessitates the use of nonstandard methods, and analysis of even simple models is nontrivial. Computer simulations of colloidal suspensions are prohibited by slow equilibration as very different length and time scales are involved for various species. This is the reason that most simulations involve some degree of coarse-graining, whereby the degree of freedom of the microscopic particles is traced out, and the mesoscopic particles interact with an effective potential, resulting in a coarse-grained, effective onecomponent description of the dispersion.18 That is why unlike the recent simulation works of evaporation,2,4,7 where the two-component mixtures (polymer + solvent) have been considered, in this work we consider a onecomponent colloidal fluid under the external field of a solvent liquid-vapor interface. Here, we propose the simplest model and simulation method in which colloidal film growth through solvent evaporation can be dealt with. The starting point for this task is the assumption that the hard-sphere model is suitable for particles of confined fluids and colloids. This assumption is quite reasonable and is based on physical intuition and experience that we have learned from our research efforts in this area as well as from studies carried out by other authors.21-25 For (21) Dinsmore, A. D.; Yodh, A. G.; Pine, D. J. Phys. Rev. E 1995, 52, 4045. (22) Leone, R.; Odriozola, G.; Mussio, L.; Schmitt, A.; Hidalgo-Alvarez, R. Eur. Phys. J. E 2002, 7, 153.
10.1021/la050167b CCC: $30.25 © 2005 American Chemical Society Published on Web 06/11/2005
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Figure 1. Two-dimensional cartoon of the simulation box of length L. Curve K1-K0 schematically depicts the soft repulsive potential, βUi(zr), between a particle and the solvent liquidvapor interface, eq 1. Point K1 indicates the location where βUi(zr) ) 1000. The range of soft repulsion is limited by point K0, and the distance zr between points L and K0 is equal to 3d. The interface moves in the direction indicated by the arrow. The colloid particle denoted by A1 is no longer mobile (“frozen”).
example, three commonly used colloidal hard sphere model systems are silica spheres in refractive index matched solutions with high ionic strength,14 sterically stabilized poly(methyl methacrylate) (PMMA) particles in apolar solvent,11 and charged polystyrene microspheres.21 The kinetic Monte Carlo (MC) simulation process26 was carried out on a square section prism of side Lx ) Ly and length L; see Figure 1 and Supporting Information. Inside, N identical hard spheres of diameter d were randomly placed avoiding overlapping among them. Their diameter is chosen to be the length unit, d ) 1. Because particle deposition and solvent evaporation processes are simulated, there are two contributions to the particle displacement: one corresponds to the Brownian motion (modeled by usual MC translation26 with maximum particle displacement lp) and the other to the effective external field of surface tension force due to the solvent liquid-vapor interface front. We assume that drying is homogeneous; that is, no difference in the drying behavior could be noted between any two points of the (x, y) plane. The effective external field due to the liquid-vapor interface, βUi(zr), is modeled through the soft repulsion26,27 between a colloid particle and the right-hand wall of the simulation cell (see Figure 1)
βUi(zr) )
{
(d/zr)n, zr < 3d zr > 3d 0,
(1)
where zr is the distance along normal of a colloidal particle with respect to the right-hand wall, β ) 1/kT, and n is a parameter of softness which is related to temperature and humidity;27 that is, a higher value of n corresponds to lower temperature and humidity. The potential (1) is cut and shifted at zr )3d. In the Figure 1, this repulsion is depicted schematically by the K0 - K1 curve. The right(23) Batina, N.; Huerta, A.; Pizio, O.; Sokolowski, S.; Trokhymchuk, A. J. Electroanal. Chem. 1998, 450, 213. (24) Wasan, D.; Nikolov, A. D. Nature 2003, 423, 156; Langmuir 1992, 8, 2985. (25) Duda, Y.; Henderson, D.; Trokhymchuk, A.; Wasan, D. J. Phys. Chem. B 1999, 103, 7495. (26) Landau, D. P.; Binder, K. A guide to Monte Carlo simulations in statistical physics; Cambridge University Press: Cambridge, U.K., 2000. (27) Such potential qualitatively resembles the density distribution profile of liquid-vapor interfaces: it can be considered as an effective potential, which is related to temperature (see, for example, Alejandre, J.; Duda, Y.; Sokolowski, S. J. Chem. Phys. 2003, 118, 329.)
hand wall moves with constant “velocity” reducing the volume of the cell. When the value of the repulsion between a particle and the right-hand wall, βUi(zr), becomes higher than 1000 (point K1 in Figure 1), such a particle (for example, particle A1 in Figure 1) is no longer mobile. Colloid particles which stop moving are regarded as frozen particles. Similar to the percolation-type model of Croll,28 such a trick reflects the situation when an evaporation front is envisaged to move into coating, leaving behind a “dry” porous layer, containing no continuous solvent. The algorithm terminates when the solvent film thickness has reached 0 and all the colloidal particles are immobilized. The left-hand wall is considered to be structureless and impermeable for colloidal particles: the particle-wall interaction potential, βUw(z), is defined as
βUw(z) )
{
∞, z < 0 0, z > 0
(2)
where z is the distance along the normal of a colloidal particle with respect to the left-hand wall. Periodic boundary conditions were established for the directions x and y only. Hence, the system may be understood as a small portion of a macroscopic one. However, in the deposition direction, z (we define the prism left side as z ) 0 and the prism right side as z ) L), no periodic boundary condition was imposed to naturally obtain a change in film properties with the prism length (film thickness) and particle adsorption at the prism lefthand wall. The parameters employed for the present simulations are the following: a maximum MC displacement lp ) 0.05, vapor-liquid interphase displacement li ) 0.01, initial prism length L ) 12, section side Lx ) Ly ) 30, number of particles N ) 3000, and parameter of repulsive potential n ) 8. By modifying the MC parameters one mimics changes of the experimental conditions. For example, high dispersion viscosity and low temperature correspond to the relatively low acceptance of MC displacement, which depends on the parameter lp.26 Similarly, the slow evaporation at high humidity and/or low temperature is modeled by relatively small values of the parameter li. Parameters Lx, Ly, L, and N define the initial concentration of colloids. Their values are chosen is such a way to guarantee the formation of at least a three-layer final film. In this work we present only the study of the evaporation rate effect. Detailed discussion of the effect of humidity, temperature, and solvent viscosity deserves a separate paper in which more considerations about drying will appear.29 We have used the MC step as a time parameter to be able to control the solvent evaporation rate in a qualitative way. The whole time period of drying was divided into discrete intervals and all not “frozen” particles were tested ν times to be displaced during each time interval. In other words, each particle is allowed to try to move ν times according to a Metropolis algorithm before the vaporliquid interphase translation. Thus, the inverse value of parameter ν defines the rate of evaporation, that is, the “velocity” of the interface moving. In this work we have considered three different rates, ν ) 2, 8, and 60. Note that for higher values of ν the slower a film is drying and the longer the time available for local structural reorganization of colloids. (28) Croll, S. G. J. Coat. Technol. 1986, 58, 41. (29) Reyes, Y.; Va´zquez, F.; Duda, Y. Manuscript in preparation.
Drying in Films of Colloidal Particles
Figure 2. Simulation results of the evolution of the first (open symbols) and the last peak values (filled symbols) of the density profile layers for evaporation rates ν ) 8 (circles) and ν ) 60 (squares). The first and the last (fourth) peaks are the nearest to the solid substrate and the vapor-liquid interface, respectively. The inset presents the time evolution of the density profiles, F(z, t), at an evaporation rate ν ) 8 and t ) 200, 500, 750, and 1000.
This entire evaporation process simulation has been repeated no less than three times to obtain a series of independent “experiments”, and the mean values of the density profiles, F(z), as well as a dynamic of film layer formation, have been determined for each time step. In the inset of Figure 2, we present the evolution of the density distribution of colloids when the evaporation rate is equal to ν ) 8. Here, decreasing of the film thickness leads to pronounced oscillation of density and its increase on the vapor-liquid interface. As can be seen from the first and fourth peak value evolution presented in the Figure 2, at ν ) 8 up to t ) 750, the particle adsorption on the vapor-liquid interface is more intensive if compared with adsorption on the solid surface. At t ) 750, the colloid density at the vapor-liquid interface reaches its maximum value, Fz)3.3 ) 1.7. It is interesting to note that, at that time, Fz)0 ≈ Fz)3.3; that is, particle adsorption on the vaporliquid interface is equal to the adsorption on the solid surface. These observations are similar to the picture of the drying process for a latex film proposed by van Tent.1 According to van Tent, as water is lost, the particle density in the bulk remains constant, but the layer at the airliquid surface becomes denser until it reaches the substrate. When the evaporation process is slower, ν ) 60, the particles have more time for equilibration during the film thinning, and the value of the last density peak never is higher than the value of the first peak. Figure 3 shows the particle density profiles of a totally dried film (t ) 1150, all particles are immobilized). The film prepared at rapid solvent evaporation, ν ) 2, is the thickest one (up to z ) 8), but with the smallest density and lowest structural ordering: one can observe only two poorly ordered layers. At slower evaporation, ν ) 8 and ν ) 60, there are dense and well-ordered films. One important point to notice is the location of density peaks: their locations suggest the hexagonal domain structure in the z direction because the distance between the peaks is approximately 0.7-0.8. Next, we analyze the dynamic of these peak formations in Figure 4. As expected, in the first stage of the film organization process the number of particles in each layer is almost constant (up to approximately t ) 800 for ν ) 2). As the evaporation proceeds in time, the first layer begins to gain particles due to increasing film density. For the slow evaporation rate, ν ) 60, the first layer growing starts before the other film layers do, because of the “equilibrium” adsorption of
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Figure 3. Density profiles, F(z), of totally dried colloid particle films prepared at different evaporation rates: ν ) 60 (solid line), 8 (dotted-dashed line), and 2 (dotted line).
Figure 4. Evolution of i-layer formation, 〈ni(t)〉, of all layers presented in Figure 3: symbols (first layer), solid (second), dashed-dotted line (third), and dashed line (fourth).
particles on the substrate. Meanwhile, when ν ) 8, the first layer growing is correlated with the other layer densities. Such oscillations for ν ) 60 exist only between the second, the third, and the fourth layers: each sublayer gains particles from the right neighbor region. It can be seen, for example, that the second layer maximum appears when the third and fourth layers reach their maximum and start to decay. Interesting features can be observed in Figure 4 at ν ) 2. After the initial stage of evaporation (t ≈ 900) the increasing of the particle population on the wall surface, 〈n1(t)〉, has neither linear nor asymptotic power behavior:22 one can see two inflections on that curve at t ≈ 980 and 1060. Similar inflections have been detected also for other evaporation rates; however, they were not so clearly manifested after data averaging. Besides, pronounced and long steps have been observed during the simulation of film formation from the dispersion of colloids with soft repulsion.17,29 We think that the possible reason
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Figure 6. Two-dimensional radial distribution function, g(r), of particles of the first layer at evaporation rates ν ) 8 (gray line) and ν ) 60 (black line). The inset is a typical configuration of particles of the first layer, ν ) 60.
Figure 5. Two-dimensional radial distribution function, g(r), of particles of the first layer at evaporation rate ν ) 8. Solid line and circles depict MC and experimental results,10,11 respectively. Part a presents the structure of the colloid dispersion, t ) 900; part b presents the structure of the dried colloid film, t ) 1150.
for these steps or inflections is the film layers structural reorganization, similar to the film thickness transitions studied by Nikolov and Wasan.24 As can be seen, at the same time of the step formation on the 〈n1(t)〉 curve, the next second layer, 〈n2(t)〉, reaches its maximum. Recently, Dullens et al.10 have employed fluorescent cross-linked PMMA particles to study the dynamics and structure of colloidal dispersions using confocal scanning laser microscopy (CSLM). These polymeric particles have been dispersed in a mixture of tetralin, cis-decalin, and carbon tetrachloride, which matches the mass density as well as the refractive index following the hard-sphere model. Before imaging, the particles were dried on a glass slide. Using CSLM, Dullens et al.10 have determined the two-dimensional radial distribution function. The same research group has determined the two-radial distribution of the same PMMA particles but dispersed in toluene at a volume fraction η ) 0.59. Besides, in this case11 the latex dispersion was not dried. The CSLM image of the first particle layer was taken at the glass wall of the container. In Figure 5, both experimental radial distribution functions obtained by Dullens et al. are compared with our simulation data. Namely, part a of this figure presents a comparison of the adlayer structure of the dispersion when the colloid film is still not dried, t ) 900. As expected, the simulation results cannot predict the correct behavior of g(r) when r < 1 because of the restriction of hard-sphere potential.17,29 Meanwhile, the comparison is sufficiently good for the distances larger than the colloid diameter. Besides, the volume fraction of the simulated system is quite close to the experimental one, ηMC ) 0.58. In the case of the dried film, part b of Figure 5, there is some disagreement between the experiment and the simulation results, which could be attributed to the combination of experimental uncertainty and simplification in the modeling. First of all, modeling of intercolloid pair interaction with soft sphere potential would improve the quality of structure predictions. Dushkin et al.12 studied the effect of the water evaporation rate, liquid meniscus at the boundary, particle size and concentration, and so forth on circular-shaped crystals formed from a thin layer of a latex suspension. It has been shown that the crystal structure and its quality are mainly affected by the rate of crystallization, which is determined
in turn by the evaporation rate. The authors mentioned that a hexagonal lattice from monodispersed colloids prevails in the formed crystal, but a square lattice can be observed in the transition regions between various hexagonal lattices. The crystal containing such inclusions is produced at a very low crystallization rate (high humidity of the air). A similar coexistence of different crystallographic arrangements has also been observed by Cardoso et al.,13 who reported the morphology of the films formed by drying the latex dispersions (poly[styrene-co(2-hydroxyethyl methacrylate)]). Our simulation results at moderate evaporation rate ν ) 8, neither radial distribution functions (Figure 5) nor snapshots of MC configurations, do not signal about the coexistence of the two different structure arrangements. However, when evaporation is slowed to ν ) 60, the particle in-layer radial distribution function, g(r), at t ) 1150 indicates the tetrahedral domain formation (peak growing at r ≈ x2).24 Besides, in a typical snapshot of the first layer of the film (inset of Figure 6) one can observe two types of domains: there are particles with four and six nearest neighbors. Therefore, we can conclude that latex crystal films of good quality and structure are grown at a reasonably low evaporation rate, which also has been observed in experiment.12,13 Moreover, because our simplified simulation model does not consider LCF, hydrodynamic effects, and colloid aggregations, the packing effect due to short-range intercolloid repulsion may be considered as an important factor of latex film formation. In summary, we proposed kinetic MC simulation study of the structure of colloid films during solvent evaporation. The results clearly demonstrate that the present method is a versatile and simple way of modeling of the film formation on a flat surface. Although the model is highly simplified in comparison to the experimental situation, it is able to reproduce some trends of the latex film formation. We believe that the sophistication of the interactions between particles may put the model somewhat closer to the experiment. In our next work we plan to include the effect of an intercolloid soft repulsion, an effective attraction due to a capillary force, and a colloid-surface short-range interaction.29 Acknowledgment. We thank Flavio Va´zques and Jose´ Campos for helpful discussions. We also thank Roel Dullens for kindly supplying his experimental data. Supporting Information Available: Visualization of colloid film formation on a flat surface. This material is available free of charge via the Internet at http://pubs.acs.org. LA050167B
