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Langmuir 1997, 13, 3900-3901
Model for the Convective Transport of Particles in a Two-Dimensional Cluster V. A. Kuz Instituto de Fı´sica de Lı´quidos y Sistemas Biolo´ gicos, IFLYSIB (UNLP-CONICET-CIC), c.c 565, 1900 La Plata, Argentina Received January 2, 1996. In Final Form: April 7, 1997X
Introduction The formation of a two-dimensional (2D) colloidal crystal on a solid substrate of micron-size particles was recently reported.1,2 The (2D) crystallization is done by controlling the water evaporation from a suspension of latex particles. From a direct observation of the dynamics of this process a two-stage mechanism of 2D crystallization is suggested:1 (a) Nucleus formation governed by attractive capillary forces (lateral immersion forces3). There is no presumption that the repulsive electrostatic forces control the formation of the (2D) crystal as happens in the (3D) colloidal crystal because the 2D crystal begins to assemble just when the layer depth is approximately equal to the particle diameter (1.7 µm).1,4 Capillary more than disjoining forces may play the most important role.5 (b) Crystal growth by a convective flow of particles caused by the evaporation of the wetting film and by the uneven rate of evaporation at the periphery of this ordered array. It is clear from the experiments that lateral immersion forces are responsible for the 2D seed crystal formation, but the convective flow of water (and particles) is the main driving force that causes the 2D clustering. An attempt to explain this phenomenon, taking into account only some of these experimental facts, has been done by using a kinetic model.2 We consider evaporation as a sink in the mass balance equation, and we assume that the velocity of restitution of the thickness and shape of the wetting film is the same as the evaporation velocity found by solving the mass balance equation. Then, by using this velocity together with the conservation of flow of volume of the system particles-water (particles flow to the center and water flows to the periphery), we found an exponential growth of the area with time. This equation also shows, in agreement with the experiments, that the area grows faster by increasing the evaporation rate. Nucleation and Growth Process Let us consider a suspension of nanometric latex particles in water over a glass plate encircled in a ring. Initially, the particles perform a Brownian motion (see Figure 1). Water mass diminishes, by evaporation, with time, and capillary attractive forces between the particles initiate, at the center of the cell, the formation of a (2D) crystal6 (see Figures 1b and 2a). The evaporation of the film spread on the latex particles7 (see Figure 2b) and at (1) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Ivanov, I. B.; Yoshimura, H.; Nagayama, K. Nature 1993, 361, 26. (2) Dushkin, C. D.; Yoshimura, H.; Nagayama, K. Chem. Phys. Lett. 1993, 455, 204. (3) Kralchevsky, P. A.; Nagayama, K. Langmuir 1994, 10, 23. (4) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Ivanov, I. B.; Yoshimura, H.; Nagayama, K. Langmuir 1992, 8, 3138. (5) Sweeney, J. B.; Davis, T.; Scriven, L. E.; Zasadzinzki, J. A. Langmuir 1993, 9, 1551. (6) Kralchevsky, P. A.; Denkov, N. D.; Paunov, V. N.; Velev, O. D.; Ivanov, I. B.; Yoshimura, H.; Nagayama, K. J. Phys.: Condens. Matter. A 1994, 6, 396. (7) Wayner, P. C., Jr.; Kao, Y. K.; LaCroix, L. V. Int. J. Heat. Transfer 1976, 19, 487-492.
S0743-7463(96)00016-9 CCC: $14.00
Figure 1. Schematic representation of the submicrometric particles in a liquid layer.
Figure 2. Two spheres partially immersed in a liquid layer on a horizontal substrate.
the interline8 of the 2D crystal-water layer induces a water flow in order to preserve the shape of the wedge and to restore the equilibrium thickness of the film wetting the particles. The convective flow of water carries the particles toward the center, forming a a 2D cluster. First we write the mass conservation law for the solvent (the water)
div(Fvr) ) -Cm ˘
(1)
where F is the density of the water and m ˘ is the evaporated mass (a sink) at the liquid-vapor interface. C is a constant having dimensions of inverse volume. The water density remains constant ∂F/∂t ) 0. (8) Wayner, P. C.; Jr.; Schonberg, J. Colloid Interface Sci. 1992, 507, 152.
© 1997 American Chemical Society
Notes
Langmuir, Vol. 13, No. 14, 1997 3901
Convective Flow and Conclusions Let us analyze here the effect of evaporation of the wetting film. The spherical wetting shell has a wedge structure (see Figure 3). Due to evaporation, a continuous restitution of the wedge’s thickness and shape is produced at the expense of the bulk liquid. The velocity of this restitution is related to the velocity of diffusion of the vapor into the whole cell. A diminution of the cell volume or an increase of humidity of the gas leads to a complete stopping of the process of ordering.4 The restitution flow in the wedge drags by shear the nanometric particles toward the center. Let us estimate this velocity vo.
d2vo dΠ )µ 2 dr dz Figure 3. Time dependence of the area for a (2D) crystal on a solid surface. The solid points represent the experimental data (see ref 2), and the line, the theoretical curve given by eq 6 with A1 + A2 ) 0, A1 ) 1.69 mm2, and β ) 6.193 × 10-2 min-1.
The center of the cell r ) 0 has minimum curvature and is a stagnation point. For a frictionless potential flow,9 the vertical (the evaporation velocity vr) and the horizontal velocities (the convective flow velocity) are proportional. The integration of eq 1 gives this velocity
vr ) vo - ar
∂ARvr + Rvr grad(ARvr) ) 0 ∂t
(4)
(5)
where β ) aR. The solution of this equation is
A ) A1eβt + A2
(6)
where A1 + A2 is the area for t ) 0. The area grows exponentially with time. This equation is compared with the experimental data2 in Figure 3. (9) Schlichting, H. Boundary-Layer Theory; McGraw Hill: New York, 1978.
(8)
where K and l are parameters. This empirical expression for Π holds for h > l and is applicable to films having a thickness between 10 nm > h > 2 nm. The solution of eq 7 gives the velocity distribution in the film
(3)
The substitution of eq 2 into eq 4 leads to
∂A - βA ) 0 ∂t
( hl)
Π(h) ) K exp -
vo )
where A indicates the area of the latex particles and Rvr is the effective velocity of the latex particles. As the particles are dragged by shear, R is a factor less than 1 that rescales the water velocity vr. If the water flow is of Couette type, the rescale factor is R ) y/h. The area A, measured from the center of the cell at a fixed position, only depends on time A ) A(t). In the steady state situation (∂vr/∂t ) 0), eq 3 reduces to
∂vr ∂A + AR )0 ∂t ∂r
Here it has been assumed that there is a constant gradient of pressure, disjoining pressure gradient,8 along the film. Π and µ are the disjoining pressure and shear viscosity, respectively. The pressure Π depends on the effect of a great variety of surface forces. In the case of a water film, the most important contribution to this force is the structural component of the disjoining pressure10 and is given by
(2)
˘ /F are constants for a given where vo and a ) Cm temperature. Evaporation via a convective flow of water induces the motion of the latex particles from the periphery to the center. Simultaneously, water flow in the reverse sense and the net flow of volume remain constant. The substantial derivative of this flow of volume is
(7)
(
)
1 dΠ z2 - hz µ dr 2
(9)
and the substitution of eq 8 into eq 9 gives
vo )
1 h2 h dh K exp 2µ l l dr
( )
(10)
Let us estimate this velocity; the viscosity of water µ ) 10-3 (N × s)/m2, the constant K ≈ 106 N/m2, h ≈ 8 × 10-9 m, l ≈ 9 × 10-10 m, ∂h/∂r ≈ 10-2. With these data, the velocity of restitution of the wetting film is vo ≈ 5 × 10-5 m/s. On the other hand, the average rate of crystallization calculated from th experiments2 is 8.5 × 10-6 m/s. As we see, these two quantities have the same order of magnitude. Finally, let us make some comments about eq 6. This equation clearly shows an exponential growth of the area with time (Figure 3). The increase of the evaporation ˘ /F)) or the lowering of the rate2 (greater values of β()CRm solvent density causes a faster growth of the area. Acknowledgment. I acknowledge R. Gianotti for making me aware of this interesting problem. LA960016W (10) Derjaguin, B. V.; Churaev, N. V. In Properties of Water Layers Adjacent to Interfaces, in Fluid Interfacial Phenomena; Croxton, C. A.,Ed.; J. Wiley & Sons: Chichester, U.K., New York, 1986.
