Langmuir 1991, 7, 525-531
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Evaporation Characteristics of Droplets Coated with Immiscible Layers of Nonvolatile Liquids A. K. Ray,* B. Devakottai, A. Souyri, and J. L. Huckaby Department of Chemical Engineering, University of Kentucky, Lexington, Kentucky 40506-0046 Received April 6, 1990. I n Final Form: August 10, 1990 Evaporation rates of single glycerol core droplets coated with immiscible layers of dioctyl phthalate have been measured as a function of layer thickness. Experiments were conducted in an electrodynamic balance where a charged droplet was suspended in the path of a vertically polarized laser beam in a stream of dry air with precisely controlled temperature. Two light-scattering techniques based on angular and fixed angle scattered intensity data have been used to obtain the evaporation rate of the droplet. The intensities (Le., resonant peaks) observed in the experimental scattering data at a fixed angle as a function of time have been interpreted by using elastic scattering theory to obtain the core droplet size and layer thickness with high precision. The experimental data show that the evaporation rate of a core droplet increases with increasing layer thickness. A theoretical model based on the diffusion of glycerol molecules through the dioctyl phthalate layer has been developed to interpret the experimental results. A parametric study based on the model is presented. The study shows that the liquid phase diffusion coefficient and the miscibility limit of glycerol in dioctyl phthalate greatly influence the evaporation rate of the core droplet.
Introduction Many physical and industrial processes such as spray drying and combustion involve the evaporation of droplets containing immiscible components. Although evaporation and growth of single phase droplets containing one or more components have been examined theoretically and experimentally in numerous studies (see refs 1-7 for reviews), only a very limited number of studies have been devoted to droplets containing immiscible components. Recently, Avedisian and Fatehis have examined evaporation characteristics of water-hydrocarbon emulsion droplets. For water-heptane emulsion droplets, they observed that the slightly less volatile component, water, preferentially evaporated from the droplets. The coalescence of internal microdroplets in an emulsified droplet leads to the formation of a core droplet coated with an immiscible layer. Hitherto, the evaporation characteristics of such droplets have not been examined, except for droplets covered with adsorbed monolayers. Rubel and GentrygJo have measured transfer rates of water and ammonia to droplets covered with hexadecanol monolayers. They observed that the transfer rate decreases as the coverage of the droplet surface by hexadecanol molecules increases, and a t a critical coverage, when the hexadecanol surface phase goes from liquid to solid phase, the transfer rate reduces dramatically. Similar reductions in evaporation rates of water droplets covered with sodium dodecyl sulfate films were observed by Taflin et a1.6
* Author to whom correwondence should be sent.
(1)Fuchs, N. A. Evaporatidn and Droplet Growth in Gaseous Media; Pergamon: New York, 1959. (2) Rubel, G. 0. J. Colloid Interface Sci. 1981,81, 188. (3)Richardson, C.B.; Lin, H. B.; McGraw, R.; Tang, I. N. Aerosol Sci. Technol. 1986,5,103. (4)Sageev, G.; Flagan, R. C.; Seinfeld, J. H.; Arnold, S. J. Colloid Interface Sci. 1986,113,421. (5)Ray, A. K.;Johnson, R. D.; Souyri, A. Langmuir 1989,5, 133. ( 6 )Taflin, D. C.; Zhang, S. H.; Allen, T.; Davis, E. J. AIChE J. 1989, 34,1310. (7)Huckaby, J. L.;Ray, A. K. Chem. Eng. Sci. 1989,12,2797. (8)Avedisian, C.T.; Fatehi, M. Int. J.Heat Mass Transfer 1988,31, 1587. (9)Rubel, G. 0.; Gentry, J. W. J . Phys. Chem. 1984,88, 3142. (10) Rubel, G.0.; Gentry, J. W. J . Aerosol Sci. 1985,16,571.
0743-7463/91/2407-0525$02.50/0
It has been demonstrated in numerous ~tudiesl'-'~that compressed monolayers of insoluble surfactants forming rigid surface films provide high interfacial resistance to mass transfer. The reduction in the transfer rate of mass through such a surface film cannot be explained by the Fickian diffusion model.'s It has also been observed16-18 that soluble or expanded surfactant films offer little or no interfacial resistance to mass transfer. These studies indicate that the formation of rigid, immobile, somewhat ordered interfaces provide surface barriers to mass transfer, whereas expanded, liquidlike mobile surface layers offer little or no barrier to mass transfer. The above literature survey indicates that mass transfer through monolayers has been studied extensively. However, mass transfer through a thick layer existing as a macroscopic phase has received very little attention. For a core droplet covered with a thick immiscible layer, two counteractive effects are associated with the transfer of a species between the phase lying outside the layer and the core droplet. As the thickness of the layer increases, the retardation of transfer rate due to the increase in the diffusional resistance is counteracted by the increase in the surface area between the layer and outer phase. The overall transfer rate is expected to be governed by the size of the core droplet, layer thickness, diffusion coefficients, and miscibility limit of the transferring species in the layer phase. The purpose of this study is to examine the effects of various physical, thermodynamic, and transport parameters on the transfer rate through a layer covering a core droplet. In this paper, we shall report our experimental data on evaporation of core glycerol droplets covered with immiscible layers of nonvolatile dioctyl (11)La Mer, V. K. Retardation of Evaporation by Monolayers; Academic: New York, 1962. (12)Davis, J. T.; Rideal, E. K. Interfacial Phenomena; Academic: New York, 1963. (13)Sherwood, T. K.;Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975. (14)Adamson, A. W.Phys,ical Chemistry of Surfaces; Wiley-Interscience: New York, 1982. (15)Quickenden, T. I.; Barnes, G. T. J. Colloid Interface Sci. 1978, 67,415. (16)Plevan, R. E.;Quinn, J. A. AIChE J . 1966,12,894. (17)Springer, T. G.;Pigford, R. L. Ind. Eng. Chem. Fundam. 1970, 9, 458. (18)Frey, D. D.; King, C. J. AIChE J. 1986,32,437.
0 1991 American Chemical Society
526 Langmuir, Vol. 7, No. 3, 1991
Ray et al.
Gas Phase
'Ab)
'O
'ECO
'O
radial position r , and a is the outer radius of the droplet. The following boundary conditions apply to eq 2 a t r = a,, XA = 'Am (3) at r = a , x A = x & (4) where X A ~is the mole fraction of A in the shell phase a t the droplet-gas interface a t r = a, and is the miscibility limit A in B which is related to x ' h by the following equilibrium condition: T'AmX'Am = TAm'Am (5) In the above equation YA,, is the activity coefficient of A a t a composition x h . The solution ofeq 2 for the boundary conditions in eqs 3 and 4 is given by
The gas phase concentration distribution is given by CA /S
Figure 1. Physical description of the problem.
phthalate. We shall also present a theoretical model based on diffusion of glycerol through immiscible layers and interpret our experimental results on the basis of that model.
Theory Let us consider the physical situation shown in Figure 1. A stationary droplet composed of two components A and B, that are partially miscible is suddenly exposed to a stagnant gas phase that is devoid of A and B but contains an inert nontransferable species C. The partially miscible components form two distinct phases consisting of the core and shell of the droplet. Initially, both phases of the droplet are in equilibrium, and the compositions are given by the miscibility limits. The droplet evaporates because the gas phase does not contain any vapor molecules. We shall denote the principal components in the core phase as A and the shell phase as B. The volatility of component B is significantly lower than that of A. During the time required for the complete evaporation of A from the droplet, the evaporation of B can be considered negligible. In the present study, the densities of the two components are significantly different; thus, the densities of the two phases differ accordingly. However, the miscibility of one component in the other is so small that the variation of the density in any phase due to the variation of composition can be neglected. The droplet evaporates relatively slowly, and the system remains nearly isothermal. After a short transient period, the evaporation process attains a quasi steady state. Under the above conditions, no concentration gradient exists in the core phase, and the concentration is given by for 0 Ir 5 a, (1) where X'A is the mole fraction of A in the core phase, (1 - x ' A ~ )is the miscibility limit of B in A, and a, is the radius of the core. The concentration profile in the shell phase is given by the following ordinary differential equation x'A
= x'A,,,
(7) where CAis the molar concentration of A in the gas phase a t the radial position r . For the problem under consideration, the molar concentrations of A a t the droplet surface and far from the droplet are given by
as r - m , CA=CA,=O (9) where P A O is the vapor pressure of A, R is the universal gas constant, and Tis the absolute temperature. Equation 8 implies that the gas phase behaves ideally, and an equilibrium exists at the droplet-gas interface. The solution of eq 7 along with the boundary conditions in eqs 8 and 9 is the following:
To obtain the droplet surface composition x h , involved in the concentration distributions of A given by eqs 6 and 10, we need to apply the following continuity equation a t the droplet-gas interface dXA dCA at r = a , D C - = (11) drD dr where DL and DG are the liquid and gas phase diffusion coefficients of A, respectively, and CL is the total molar concentration in the shell phase. Due to the low miscibility limit of A in B, we can neglect the variation of the shell phase diffusion coefficient and total molar concentration with composition. Using eqs 6,10, and 11, we obtain the following relation for the surface composition:
DLCLm= ('As
- 'Am)
YAaxdAo
RT
(12)
\a, a / To solve for X A ~ we , need a relation between the activity coefficient YA and the composition X A . In the present study, we will use the following van Laar equations:
..
where X A is the mole fraction of A in the shell phase a t the
When the miscibility limits,
and x ' A ~ are , known, the
Langmuir, Vol. 7, No. 3, 1991 527
Droplets Coated with Immiscible Liquids constants A and B in the above equations can be obtained by satisfying the equilibrium condition given in eq 5. The dependence of Y A on ~ X A ~makes eq 12 a nonlinear algebraic equation, and no analytical solution for the surface composition xAScan be obtained from the equation from its present form. However, when the surface composition X A ~differs by a small amount from the equilibrium composition X A ~ ,the activity of A a t the surface of the droplet may be linearized by the following relation:
t 0-60 V d . c .
+---T-I
---+ T
o-‘200
T 0-60Vd.c.
Figure 2. Schematic of a two-ring electrodynamic balance.
such a situation, the volume of the shell will remain almost constant during the evaporation period, i.e. a3 -a:
where
= a, 3 -ac, 3
(21) where a0 and acoare the initial outer and inner radii of the droplet, respectively. Equation 21 implies that C L X A
