Mass transport mechanism for the formation of latex-modified epoxy coatings by evaporation from aqueous dispersions.

The transport mechanism for the evaporation of dispersing liquid during the solidification of an epoxy dispersion that had been stabilized to prevent crack formation with a latex dispersion was studied. Aqueous dispersions consisting of an experimentally determined ratio of epoxy resin and nitrile latex were evaporated at 35[degrees]C. When the dispersion was evaporated under controlled conditions without forced air flow, a flexible and adherent polymer material formed. The mechanism for coalescence was related to the loss in weight of dispersing liquid during an initial zero order kinetics stage. This was followed by a rate-controlled Fick's law diffusion through the developing coating with subsequent evaporation to the atmosphere. Experimental measurements are compared with theoretical predictions. The rate constant for the zero order time frame is 0.086 [+ or -] 0.02 h[r.sup.-1]. In the second time frame, Ficks's law evaporation rate constant is 0.046 [+ or -] 0.017 cm*h[r.sup.-1] with a diffusion coefficient of 0.00092 [+ or -] 0.00051 [cm.sup.2]*h[r.sup.-1] at 35 [+ or -] 1[degrees]C and RH 35 [+ or -] 7%. Applications for evaporation kinetics are discussed.

Keywords: Epoxy resins, drying, heat and mass transfer, waterborne, aerospace, aircraft, Fick's law

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The objective of this work is to examine the solution to Fick's law diffusion equation for the formation of polymer coatings by evaporation from water or other aqueous dispersions. Deposition of polymers from waterborne dispersions by dispersing liquid evaporation has application to coating substrates. (1-13) The overall deposition process consists of a number of steps including water transport from the dispersion to the atmosphere, (1-10) coalescing of the dispersed polymer, (4,6,8) evaporation of dispersing water (1,2,4) or dispersing liquid, and solid formation by particle movement within the coalescing coating. (4,6,9-11) The mechanism for water transport is initially surface evaporation, followed by diffusion-controlled mass transport (1-6) through the coating. The initial evaporation is manifested by a linear increase in the amount of liquid evaporated over a relatively short time frame. This is followed by a nonlinear increase where the mass of water evaporated decreases for a longer time.

Croll (1) studied heat and mass transfer during drying by measuring the weight loss of latex coatings versus time with a digital balance connected to a microcomputer. Evaporation was controlled at 22 [+ or -] 1[degrees]C and 50 [+ or -] 2% RH; air velocity was 1.8 m*se[c.sup.-1] or ambient condition. The kinetic data were interpreted as a mechanism consisting of two drying stages: water loss due to evaporation from the surface, followed by additional water loss that is limited by transport of water through a coalescing coating. In a subsequent publication, Croll (2) described an initial stage with a short time frame when the flux is 0.85 of the flux of pure water. In the second stage, water evaporation is limited by the concentration in a wet reaction layer. A derived equation reproduced the drying curves. Adesanya, Nanda, and Beard (3) monitored the moisture weight by a computer system during drying of samples of yellow poplar wood with precise dimensions at temperatures from 100-150[degrees]C with controlled humidity. The data on moisture content were analyzed for three drying stages. First, the decrease in moisture content with time was plotted, resulting in a curve that fell smoothly in an overall nonlinear fashion from about 0.95 to zero moisture content. A second plot had an initial linear portion identified as the constant rate time frame, when the fraction of moisture removed was plotted versus time. When the moisture fraction removed was plotted as a function of the square root of time, a linear portion was identified and described as the falling rate time interval. A second square root linear portion was identified in the falling rate interval, immediately after the first. The data analysis was interpreted as consistent with a mechanism involving three drying time frames. Fick's diffusion equation was applied to describe the moisture concentration profile as a function of time. Winnik and Feng (4) measured water weight loss with an analytical balance of 0.300 g samples ~50 m[micro] thick films of water and two latex coatings at 22 [+ or -] 1[degrees]C and 55 [+ or -] 10% RH. Plots of the water evaporated were linear over two-thirds of the drying time for all three. Raghavan, Tulasidas, Sablani, and Ramaswamy (5) applied Fick's diffusion partial differential equation to the transport of moisture on drying grapes at 50[degrees]C. The hot air flow was 2.0 m*se[c.sup.-1]; samples were weighed with an electronic balance. An experimental drying curve of the decrease in moisture ratio has a shape similar to that in Adesanya, Nanda and Beard. (3) They solved Fick's equation for spherical geometry approximately by using an infinite series relating the moisture ratio to the function exp (-[n.sup.2][[pi].sup.2]Dt/[R.sup.2]), where R is the radius of a sphere. A computer program evaluated diffusivity and accounted for shrinkage. Duineveld, Lilja, Johansson, and Inganas (10) interpreted the diffusion of ethanol, n-propanol, and 1-butanol solvents in elastomers for micromolding in capillaries with Fick's equation. The solvent concentration followed an erfc [x/(2(Dt)[.sup.1/2])] function. Blandin, David, Vergnaud, Illien, and Malizewicz (13) used Fick's law equation for mass transport of solvent in a thin layer. Christie, Lauer, and Osteryoung (14) derived the Fick's law expression for the quantity of charge in rate-controlled electrochemical reactions. Their plot of charge vs [t.sup.1/2] has a shape identical to that in the second stage for Adesanya et al. (3) Delahay (15,16) derived the concentration dependence for both the oxidized and reduced forms in an electrochemical reaction by solving Fick's diffusion equation for semi-infinite mass transport with the Laplace transform. Crank (17) has a section on evaporation kinetics consisting of solutions to Fick's diffusion equation, and figures depicting the concentration gradient in a semi-in-finite medium, and the concentration of the evaporated form at the medium surface. Galus (18) has summarized the equivalent Fick diffusion equation for the concentration, flux, and quantity of charge for both rate-controlled and diffusion-controlled mass transport in electrochemistry.

[FIGURE 1 OMITTED]

Dispersion samples that provide thick coatings for use in sealing aircraft fuel tanks are used in the experimental portion. Toughened epoxy coatings were prepared without curing agents. Interpretation of the kinetics for dispersing liquid transport in terms of polymer reactions as coalescence proceeds is treated elsewhere. (1,6,8,9) Evaporation with heat transfer from the atmosphere is discussed by Croll. (2) Evaporation and condensation kinetics from the atmosphere to a solid without subsequent diffusion was studied by Chaix, van den Bergh, and Rossi. (19) Water is generally used here in the theoretical portion, while dispersing liquid implying a water mixture is used for the experimental portion.

THEORY

The evaporation mass transport model considers two surface forms of liquid water, Figure 1A and 1B. There is liquid water, [H.sub.2][O.sub.evap], in the bulk of the dispersion that transports towards the surface where it becomes condensed water. The condensed water, [H.sub.2][O.sub.cond], is present as a surface layer at the dispersion-atmosphere interface, and can diffuse into the bulk of the dispersion. Formation of water vapor, [H.sub.2][O.sub.vap], by evaporation of condensed water is assumed to be instantaneous. This model is depicted in Figure 1B when the dispersion has an impermeable substrate at one surface, defined as x = [infinity]. The farthest surface at x = 0 is the dispersion-atmosphere interface where the final reaction product is water vapor. When evaporation begins, the surface water reacts to form water vapor, causing coalescence of the dispersed particles to form a solid material. It is convenient to treat water transport during the beginning of evaporation separately, in the next section.

[FIGURE 2 OMITTED]

Zero Order Kinetics

Freshly prepared dispersions have surface water [H.sub.2][O.sub.cond] in direct contact with the atmosphere, Figure 1A. When evaporation begins, the surface layer vaporizes to form [H.sub.2][O.sub.vap] according to equation (1). The double arrows in equation (1) show that water vapor can also condense. However, the condensation rate of water vapor is taken as negligible compared with the evaporation rate. Evaporation of surface water proceeds steadily until the concentration is depleted. Since the surface concentration is unchanged,

[H.sub.2][O.sub.cond] [left and right arrow] [H.sub.2][O.sub.vap] (1)

the rate of evaporation is zero order (20) expressed by equation (2). Here, dy/dt is the rate of evaporation in units of g*[cm.sup.-2]*h[r.sup.-1]; y is the amount of water evaporating from the surface to the atmosphere, g*[cm.sup.-2]; t is time in hours; k is the rate constant, h[r.sup.-1]; [y.sup.0] is the total amount of water evaporated, g*[cm.sup.-2]; [y.sub.[infinity]] is the concentration of water in equilibrium with water vapor in the atmosphere according to equation (1), g*[cm.sup.-2]. It is seen from equation (2), the rate of evaporation is highest when [y.sub.[infinity]] = 0, while the rate is zero when [y.sup.0] = [y.sub.[infinity]]. If [y.sub.[infinity]] is greater than [y.sup.0], water vapor condenses onto the surface.

dy/dt = k ([y.sup.0] - [y.sub.[infinity]]) (2)

Integration of equation (2) leads to equation (3). According to equation (3), a plot of the amount of water evaporated as a function of time is linear, with a slope equal to k ([y.sup.0] - [y.sub.[infinity]]). The slope increases with increasing k for a constant ([y.sup.0] - [y.sub.[infinity]]). The evaporation rate constant k is obtained from the slope when [y.sup.0] and [y.sub.[infinity]] are known.

y = k ([y.sup.0] - [y.sub.[infinity]])t (3)

Figure 2 is a plot of y for k = 0.01 h[r.sup.-1]; area = 40 [cm.sup.2]; [y.sub.[infinity]] = 0, and [y.sup.0] = 0.25, 0.50, 0.75 g*[cm.sup.-2]. Dividing each slope in Figure 2 by [y.sup.0] results in the intrinsic rate constant, k = 0.01 h[r.sup.-1] for each line. A linear rising portion was obtained for pure water by Winnik and Feng. (4)

Fick's Law Kinetics

The model for evaporation at a coalescing surface is Figure 1B and equation (4). Water that will be evaporated from the bulk of the dispersion diffuses through a coalescing coating to form condensed water at the surface. The coating thickness decreases slightly, but is assumed constant. The double arrows indicate the reverse reaction where condensed water is absorbed by the coalescing coating may occur. The rates of the two reactions differ. Condensed water subsequently evaporates to form water vapor. The double arrows express that both forward and back reactions occur. Only water condensed on the surface is available for absorption and subsequent diffusion into the coalescing dispersion. The mode of mass transfer of water from the coalescing dispersion to the surface is governed by diffusion.

[FIGURE 3 OMITTED]

[H.sub.2][O.sub.evap] [left and right arrow] [H.sub.2][O.sub.cond] [left and right arrow] [H.sub.2][O.sub.vap] (4)

Before evaporation begins, the concentration of evaporating water, [C.sub.evap](x,t), is its mass per unit volume in the dispersion as expressed by equation (5) for semi-infinite diffusion. In equation (5) t is evaporation time:

t = 0, 0 < x < [infinity]: [C.sub.evap] (x,0) = [C.sup.0], [C.sub.cond] (x,0) = 0 (5)

x is linear distance within the coalescing dispersion perpendicular to the coating surface; [C.sub.evap](x,0) is the distance-dependent concentration of water within the coalescing coating; and [C.sup.0] is the initial water concentration in the dispersion. The distance x is bounded by the atmosphere-coating surface at x = 0, and a distance receding from this surface toward the impermeable substrate-dispersion interface where there is no evaporation, x [right arrow] [infinity]. This receding distance is sufficiently large so that the initial condition [C.sub.evap]([infinity],0) = [C.sup.0] always applies: this is the boundary condition for semi-infinite diffusion.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

CASE 1: When evaporation begins at t > 0, the rate is so high that the concentration of water is zero at x = 0, as specified by equation (6). However, at a large distance from the surface, x [right arrow] [infinity], the

t > 0, x = 0: [C.sub.evap] (0,t) = 0 (6)

concentration of water is unchanged from the initial value according to equation (7). As water evaporates from the surface, additional

t > 0, x [right arrow] [infinity]: [C.sub.evap]([infinity], t) [right arrow] [C.sup.0], [C.sub.cond] ([infinity],t) = 0 (7)

water transport is driven by a concentration gradient through the coalescing coating to the surface at x = 0 by diffusion. The rate at which water transports is expressed by Fick's equation for linear diffusion to a stationary plane, equation (8), when the diffusion coefficient, [D.sub.evap], is constant.

[FIGURE 6 OMITTED]

[partial derivative][C.sub.evap](x,t)/[partial derivative]t = [D.sub.evap][[[partial derivative].sup.2][C.sub.evap](x,t)/[partial derivative][x.sup.2]] (8)

Equation (8), with initial and boundary conditions [equations (5-7)], is solved by the Laplace transform method. Briefly, both sides of equation (8) are multiplied by exp(-st) dt, then integrated by parts from 0 to [infinity]. The [[integral].sub.0.sup.[infinity]] parameter s is a Laplace transform variable. The integration constants are calculated from the boundary conditions and the differential equation is inversely transformed (15,16,18) to obtain equation (9). In equation (9), erf is the error function, and x/(2([D.sub.evap])[.sup.1/2]) is the argument of the error function. More details on the Laplace transform with application to equation (8) are found in Delahay (15,16) and Galus. (18)

[C.sub.evap](x,t) = [C.sup.0] erf [x/(2([D.sub.evap]t)[.sup.1/2])] (9)

Equation (9) mathematically describes the water concentration profile within the coating so long as equations (5-7) are applicable. At x = 0, erf(0) = 0, and for x > 2.7, erf([xi]) = 1. A table of erf([xi]), exp([[xi].sup.2]), erfc([xi]), and exp([[xi].sup.2]) erfc ([xi]) for [xi] from 0 to 5.5 was developed in this work.

The water flux and the rate of water loss by surface evaporation, d[M.sub.t]/dt, are obtained by differentiating equation (9) at x = 0, and multiplying by the diffusion coefficient [D.sub.evap] to obtain equation (10). Integration of equation (10) results in equation (11),

d[M.sub.t]/dt = [D.sub.evap][([partial derivative][C.sub.evap](0,t)/[partial derivative]x)][.sub.x=0] = [C.sup.0][D.sub.evap]/([D.sub.evap][pi]t)[.sup.1/2] (10)

where the mass [M.sub.t] of diffusing water per unit area that has evaporated is a [t.sup.1/2] function. (17,18) A plot of [M.sub.t] versus [t.sup.1/2] is linear, with a slope equal to 2[C.sup.0]([D.sub.evap]/[pi])[.sup.1/2] as shown in Figure 3. The effect of the magnitude

[M.sub.t] = 2[C.sup.0]([D.sub.evap]t/[pi])[.sup.1/2] (11)

of the water diffusion coefficient, [D.sub.evap], is shown in Figure 3. Here, the lower line is calculated with [D.sub.evap] = 0.02 [cm.sup.2]*h[r.sup.-1], [C.sup.0] = 1 g*[cm.sup.-3]; the upper line with [D.sub.evap] = 0.04 [cm.sup.2]*h[r.sup.-1] and [C.sup.0] = 1 g*[cm.sup.-3] has a larger slope. An inspection of equation (11) shows that a larger slope is also obtained with larger values of [C.sup.0].

CASE 2: Diffusing water loss is now limited by rate of evaporation at the atmosphere interface, Figure 1B. The initial conditions are stated in equation (5). When evaporation occurs at t > 0 the rate is sufficiently low that surface water is present, and equation (6) no longer is applicable. Instead, the rate of evaporation depends on other factors such as the rate of heat transfer, breaking of surface bonds, or other retarding factors. The result is that while water diffuses through the coalescing coating, immediate evaporation does not take place. When this surface water does completely evaporate, then the rate is diffusion controlled. When water evaporation starts, t > 0, equations (12) and (13) define the water flux at x = 0. (15,18)

t > 0, x = 0: [[D.sub.evap]([partial derivative][C.sub.evap](0,t)/[partial derivative]x)][.sub.x=0] = [k.sub.evap][C.sub.evap](0,t) - [k.sub.cond][C.sub.cond](0,t) (12)

[[D.sub.evap]([partial derivative][C.sub.evap](0,t)/[partial derivative]x)][.sub.x=0] + [[D.sub.cond]([partial derivative][C.sub.cond](0,t)/[partial derivative]x)][.sub.x=0] = 0 (13)

Equation (12) equates the net flux of dispersing water to the difference between the rate of evaporation and the rate of condensation. Equation (13) is a material balance that specifies the sum of the fluxes is zero. In these equations, [D.sub.evap] is the diffusion coefficient of water diffusing out of the coalescing dispersion, and [D.sub.cond] is the diffusion coefficient for water diffusing into the coalescing dispersion. The rate constant for evaporation is [k.sub.evap], and [k.sub.cond] for condensation. At the substrate interface where evaporation does not occur, the boundary condition is:

t > 0, x [right arrow] [infinity]: [C.sub.evap]([infinity],t) [right arrow] [C.sup.0], [C.sub.cond]([infinity],t) [right arrow] 0 (14)

Equation (14) requires that as the concentration profile of dispersing water recedes from x = 0 during evaporation, the initial concentrations of evaporating water and condensed water are eventually unchanged. The dispersion will not completely dry during the experiment.

Fick's equations relating the change in [C.sub.evap](x,t) and [C.sub.cond](x,t) for planar diffusion as a function of time and distance in the case of constant diffusion coefficients and thickness are equations (15) and (16). Equations (15) and (16) are solved with the Laplace transform to convert the partial differential equations into ordinary differential equations. This is followed by integration from 0 to [infinity], calculating the integration constants, and then obtaining the inverse transforms. (15,16,18) The Laplace solution derived

[partial derivative][C.sub.evap](x,t)/[partial derivative]t = [D.sub.evap][[[partial derivative].sup.2][C.sub.evap](x,t)/[partial derivative][x.sup.2]] (15)

[partial derivative][C.sub.cond](x,t)/[partial derivative]t = [D.sub.cond][[[partial derivative].sup.2][C.sub.cond](x,t)/[partial derivative][x.sup.2]] (16)

by Delahay (16) after redefinition of electrochemical terms for application to aqueous dispersions is equation (17) for evaporating water, and equation (18) for condensed water, with Q defined by equation (19). (16)

[C.sub.evap](x,t) = [C.sup.0]([k.sub.evap]/([D.sub.evap.sup.1/2]Q)) [erf (x/(2([D.sub.evap]t)[.sup.1/2]) + exp (Qx/[D.sub.evap.sup.1/2] + [Q.sup.2]t) erfc (Q[t.sup.1/2] + x/(2([D.sub.evap]t)[.sup.1/2])] (17)

[C.sub.cond](x,t) = [C.sup.0] - ([D.sub.evap]/[D.sub.cond])[.sup.1/2] [C.sub.evap](x,t) (18)

Q = [k.sub.evap]/[D.sub.evap.sup.1/2] + [k.sub.cond]/[D.sub.cond.sup.1/2] (19)

In equation (17), erf is the error function, exp is the exponent, and erfc = 1 - erf. When [k.sub.cond] is zero, Q = [k.sub.evap]/[D.sub.evap.sup.1/2] and the term in front of the brackets is [C.sup.0]. If [k.sub.evap] is infinite, equation (17) becomes equation (9), the diffusion-controlled case. At the coalescing surface where x = 0, terms in x are zero so that equation (17) becomes equation (20).

[C.sub.evap](0,t) = [C.sup.0] ([k.sub.evap]/([D.sub.evap.sup.1/2]Q)) [exp ([Q.sup.2]t) erfc (Q[t.sup.1/2])] (20)

[C.sub.cond](0,t) = [C.sup.0] - ([D.sub.evap]/[D.sub.cond])[.sup.1/2] [C.sub.evap](0,t) (21)

According to equation (21), the quantity of condensed water on the coalescing polymer surface that subsequently evaporates is equal to the initial water concentration in the dispersion minus the concentration of evaporating water, if the ratio of diffusion coefficients is one. If [k.sub.cond] is zero, then Q = [k.sub.evap]/[D.sub.evap.sup.1/2], and in equation (20) the term before the brackets as expressed by equation (22) depends only on [C.sup.0]. Values of [C.sub.cond](0,t) depend on the ratio of the diffusion coefficients, and [C.sub.evap](0,t).

[C.sub.evap](0,t) = [C.sup.0] exp ([Q.sup.2]t) erfc (Q[t.sup.1/2]) (22)

Combining equation (22) with equation (21) results in equation (23), the concentration of [C.sub.cond](0,t) at x = 0. At t = 0 the

[C.sub.cond](0,t) = [C.sup.0] [1 - ([D.sub.evap]/[D.sub.cond])[.sup.1/2] exp ([Q.sup.2]t) erfc (Q[t.sup.1/2])] (23)

exp ([Q.sup.2]t) erfc (Q[t.sup.1/2]) = 1, and [C.sub.cond] (0,t) = 0. At long times, exp ([Q.sup.2]t) erfc (Q[t.sup.1/2]) = 0, and [C.sub.cond] (0,t) = [C.sup.0]. For [xi] > 3, the exp ([[xi].sup.2]) erfc ([xi]) function is calculated from the series (17) in equation (24), where [xi] = Q[t.sup.1/2].

exp ([[xi].sup.2]) erfc ([xi]) = 1/([[pi].sup.1/2]){1/[xi] - 1/(2[[xi].sup.3]) + (1 X 3)/([2.sup.2][[xi].sup.5]) ...} (24)

According to equation (12), the net flux of [H.sub.2][O.sub.evap] at x = 0 is the difference between the rate of evaporation and the rate of condensation. If [k.sub.cond] = 0, then the water flux to the atmosphere is limited by the rate of evaporation. Conversely, if [k.sub.evap] = 0, the water flux from the atmosphere is limited by the rate of condensation. Since water vapor is generally present, the effect of the rate of condensation can be estimated from equation (22) when [D.sub.evap] = [D.sub.cond] with Q = (1/[D.sub.evap.sup.1/2]) ([k.sub.evap] + [k.sub.cond]). In Figure 4, [C.sub.evap] is calculated at x = 0 from equation (22) with [k.sub.cond] = 0, [C.sup.0] = 1 g*[cm.sup.-3], [k.sub.evap] = 0.1 cm*h[r.sup.-1], [D.sub.evap] = [D.sub.cond] = 0.005 [cm.sup.2]*h[r.sup.-1] giving the uppermost curve in Figure 4. Curves with [k.sub.cond] = 0, 0.01, 0.05, 0.1, 0.2, and 0.3 cm*h[r.sup.-1] are plotted successively in Figure 4. The shape of each curve with increasing [k.sub.cond] is similar over the entire time range. Increased values of [k.sub.cond] act to decrease [C.sub.evap], and cause a reduction in the amount of water evaporated. For example, at 24 hr [C.sub.evap]/[C.sup.0] = 0.09 when [k.sub.cond] = 0, compared to [C.sub.evap]/[C.sup.0] = 0.02 when [k.sub.cond] = 0.3 cm*h[r.sup.-1].

The quantity of water evaporated, [C.sub.cond](0,t), at the coating-atmosphere interface is calculated by equation (23) in Figure 5 for [k.sub.evap] = 0.1 cm*h[r.sup.-1]; [D.sub.evap] = [D.sub.cond] = 0.005 [cm.sup.2]*h[r.sup.-1]; [C.sup.0] = 1 g*[cm.sup.-3]; [k.sub.cond] = 0, 0.01, 0.05, 0.1, 0.2 cm*h[r.sup.-1]. The amount of water available to evaporate increases with increasing values of [k.sub.cond] at all portions of the curves. There is an initial steep rising portion from zero to two hours, then a rapid flattening at longer times. The curves in Figure 5 were calculated with equation (23) and have the same shape as those in other publications. (1-4) The slopes for the rising portion are a measure of the drying rate. (1-3)

The quantity of water diffusing out of the coalescing dispersion per unit area per hour at x = 0, d[M.sub.t]/dt, is obtained by combining equation (12) with [k.sub.cond] = 0 and equation (22), to give equation (25). (16,17) Equation (25) contains the exp ([Q.sup.2]t) erfc (Q[t.sup.1/2]) term, and differs from equation (22) by only the rate constant [k.sub.evap].

d[M.sub.t]/dt = [[D.sub.evap]([partial derivative][C.sub.evap](0,t)/[partial derivative]x)][.sub.x=0] = [k.sub.evap][C.sup.0] exp ([Q.sup.2]t) erfc (Q[t.sup.1/2]) (25)

Integration of equation (25) with respect to time results in equation (26), (14,16) written here in dimensionless form.

[k.sub.evap][M.sub.t]/[C.sup.0][D.sub.evap] = exp ([k.sub.evap.sup.2]t/[D.sub.evap]) erfc [[k.sub.evap](t/[D.sub.evap])[.sup.1/2]] - 1 + 2 [k.sub.evap](t/[pi][D.sub.evap])[.sup.1/2] (26)

CASE 3: Rate constant for evaporation equals the rate constant for condensation. The boundary condition at the coalescing surface is written such that [k.sub.evap] = [k.sub.cond] = [k.sub.ec], [equation (27)].

[FIGURE 7 OMITTED]

[[D.sub.evap]([partial derivative][C.sub.evap](0,t)/[partial derivative]x)][.sub.x=0] = [k.sub.ec][C.sub.evap](0,t) - [k.sub.ec][C.sub.cond](0,t) (27)

In equation (27), [k.sub.ec] is the rate constant in units of cm*h[r.sup.-1]; [C.sub.evap] (0,t) and [C.sub.cond](0,t) have been defined. When [k.sub.ec][C.sub.evap](0,t) > [k.sub.ec][C.sub.cond](0,t), surface water evaporates; when [k.sub.ec][C.sub.evap](0,t) < [k.sub.ec][C.sub.cond], water vapor is taken up by the coalescing dispersion. The rate constant [k.sub.ec] has the same value for both evaporation and condensation. Equation (19) with equal values for the forward and reverse rates is written as equation (28). The magnitude of Q depends on [k.sub.ec] and the sum of the reciprocals of the two diffusion coefficients. When [D.sub.evap] = [D.sub.cond], Q = 2[k.sub.ec]/[D.sub.evap.sup.1/2]. A plot of [C.sub.evap](0,t) versus time is in Figure 4, with the symbol, [square], [k.sub.evap] = [k.sub.cond] = 0.1 cm*h[r.sup.-1] and [D.sub.evap] = [D.sub.cond] = 0.005 [cm.sup.2]*h[r.sup.-1]. The shape of this plot is unchanged, while the weight loss is less than that when [k.sub.evap] > [k.sub.cond], but more than that for [k.sub.evap] < [k.sub.cond]. Similar considerations apply to the curves in Figure 5, plots of [C.sub.cond](0,t) versus time.

Q = [k.sub.ec] [1/[D.sub.evap.sup.1/2] + 1/[D.sub.cond.sup.1/2]] (28)

Crank (17) equates the flux for rate controlled evaporation as proportional to the difference between the surface concentration of evaporating liquid, and its concentration at large distances from the surface, [C.sub.0]. The proportionality constant is [alpha], and the constant has the same value for both evaporation and for condensation. The derived equation for the quantity of dispersing liquid evaporated, [M.sub.t], is equation (29). Since h = [alpha]/D, the argument in equation 29 is [alpha](t/D)[.sup.1/2] and, equation (26) is taken as identical to equation (29) because [alpha] = [k.sub.ec], except for the ([C.sub.0] - [C.sub.2]) concentration term. [C.sub.2] is the initial dispersion concentration.

[M.sub.t] = ([C.sub.0] - [C.sub.2])/h [exp ([h.sup.2]Dt) erfc [h(Dt)[.sup.1/2]] - 1 + 2h(Dt/[pi])[.sup.1/2] (29)

When the higher terms of exp ([[xi].sup.2]) erfc ([xi]) in equation (24) for large [xi] are small, for example at long times, equation (26) can be written as equation (30).

[M.sub.t] = - [C.sup.0][D.sub.evap]/[k.sub.evap] + (2/[[pi].sup.1/2])[C.sup.0]([D.sub.evap]t)[.sup.1/2] (30)

Equation (30) predicts that a plot of the quantity of water that has evaporated versus [t.sup.1/2] is linear, with an intercept equal to -[C.sup.0][D.sub.evap]/[k.sub.evap], and slope equal to (2/[[pi].sup.1/2])[C.sup.0]([D.sup.1/2]). This is shown by the dotted straight line in Figure 6. Equation (30) has been applied to electrochemical experimental data to calculate a diffusion coefficient and a rate of electron transfer. (14)

Equation (26) is plotted in the dimensionless form [k.sub.evap][M.sub.t]/[C.sup.0][D.sub.evap] versus [k.sub.evap](t/[D.sub.evap])[.sup.1/2] in Figure 6. There is an initial slowly rising portion between 0 and about 1.0 units of [k.sub.evap](t/[D.sub.evap])[.sup.1/2], corresponding to the constant rate time frame, followed by an apparent linear increase. Equation (30), the dotted straight line, has an intercept of -1 at [k.sub.evap](t/[D.sub.evap]) = 0, and a slope of 2[C.sup.0]([D.sub.evap]/[pi])[.sup.1/2]. The theoretical slope in Figure 6 is 2/[[pi].sup.1/2] = 1.1283, and the intercept is -1 when the exp ([k.sub.evap.sup.2]t/[D.sub.evap]) erfc [[k.sub.evap](t/[D.sub.evap])[.sup.1/2]] term is zero. For the solid line plot in Figure 6 the term is not yet zero so the slope and intercept differ.

MATERIALS, DISPERSION PROPERTIES, AND INSTRUMENTS

The polymer dispersions were commercially available, and were used as received. The epoxy dispersion was Shell Chemical EPI-REZ 3551-WY-43 waterborne resin, a modified high molecular weight bisphenol [alpha]-epichlorohydrin resin dispersed in a liquid comprised of 80.5% water and 19.5% 2-propoxyethanol. The 2-propoxyethanol (ethanol, 2-propoxy), CAS 2807-30-9, has a boiling point of 149.8[degrees]C, and is soluble in water. The dispersion weight is 9.0 lb*ga[l.sup.-1] (1.08 g*[cm.sup.-3]). The measured pH 8.0 [+ or -] 0.1 compared exactly with the manufacturer's pH 8.0, and a measured 45.5 wt% nonvolatile content (105[degrees]C, 24 hr) compared with 43% in the dispersion specifications. The measured Brookfield #3 viscosity at 100 rpm was 589 mPa*s (25[degrees]C), compared with the manufacturer's 600 mPa*s (25[degrees]C). The latex required to stabilize the epoxy coating was BFGoodrich Hycar 1561 high acrylonitrile copolymer dispersed in water with a 39.8%w nonvolatile content (105[degrees]C, 24 hr). The manufacturer lists 41% total solids and 8.35 lb*ga[l.sup.-1] (1.001 g*[cm.sup.-3]). The measurements of two different latex dispersions were pH 9.2 and pH 9.8, compared with a typical pH 10.8 and a pH range 8.5-11.0 in the technical data from the manufacturer. The Brookfield #1 viscosity at 50 rpm was 35 mPa*s (24[degrees]C). All chemicals were reagent grade. The aqueous dispersions used here were prepared by thoroughly mixing 400 g of epoxy and 600 g of latex (40/60), usually in a 1000 mL stainless steel container, with a Teflon coated magnetic stirring bar. For an electrodeposition study, the metal container serves as a cathode. The 40/60 formulation had a 40.8 %w nonvolatile content (105[degrees]C, 24 hr), and a pH 9.1 [+ or -] 0.3 (Cole-Parmer pHTestr3, microprocessor controlled). The pH probe was calibrated with standards at pH 4.0, 7.0, and 10.0. The Brookfield #3 viscosity at 100 rpm of the 40/60 dispersion was 302 [+ or -] 12 mPa*s (17 [+ or -] 2[degrees]C). Titrations of known volumes of the epoxy, the latex and the 40/60 dispersions with 0.1 M HCl showed a break in each pH-mL HCl curve at pH 4.4. For the 40/60 weight ratio, this represents a change of almost 5 pH units from a mildly basic to a mildly acidic solution. The 40/60 dispersion mixture contained additives, e.g., stabilizer, thickener, dispersing agent, that are in the latex or epoxy dispersions. Infrared FTIR spectra recorded with a Nicolet instrument (Herguth Laboratories, Vallejo, CA) show a nitrile band at 2240 [cm.sup.-1] due to the latex that persists after drying at 23[degrees], 52[degrees], and 110[degrees]C. The coalescing polymer thickness and diameter (Mitutoyo Digimatic Micrometer, 0 to 25.4 [+ or -] 0.001 mm) and weight (Scientech SA 120 Electronic Analytical Balance, 0 to 120 [+ or -] 0.0001 g) were recorded. Known weights of dispersions were placed into an oven at 35[degrees]C (Lindberg/Blue Gravity Convection Oven Model GO1320A, room temperature to 260 [+ or -] 1[degrees]C) to determine the reduction in weight in an atmosphere without forced air flow.

EXPERIMENTAL

Dried epoxy coatings prepared by electrodeposition from aqueous dispersions are not stable: they quickly develop cracks and spontaneously exfoliate. A coating is stable when a polymer modifier such as a toughener (21,22) is mixed with the epoxy. Experimental data for determining the optimum weight ratios of dispersed epoxy/latex were obtained by electrodeposition from 200 g of dispersions onto 99.9% zinc anodes, typically 38 X 38 X 0.7 mm squares, at 22 [+ or -] 1[degrees]C. The cathodes were 6.35 X 200 mm stainless steel rods. Desired properties for the electrodeposited coatings included a 1-mm thickness, absence of pores, and high adhesion to the zinc substrate. Known ratios of epoxy/latex dispersions selected for examination were prepared by weighing into 250 mL plastic beakers and thoroughly mixing. The weights were g epoxy/g latex: 140/60, 120/80, 100/100, 80/120 (40/60), 60/140, and 40/160. Duplicate samples were electrocoated from unstirred dispersions for each ratio at a constant voltage of 14.0 V, applied for 180 sec, at 22 [+ or -] 1[degrees]C. The decrease in Faradaic current with time during the electrodeposition was recorded. Both the shape of the resulting current-time curves and the magnitude of the currents were found to correlate with the epoxy content in the coating dispersions. The initial shape of the current-time curves becomes less steep with increasing epoxy, while the current is almost 25 times larger at 180 sec for the highest epoxy ratio, compared to the lowest ratio. The electrocoated samples were dried in an oven at 35[degrees]C. The coatings were visually inspected during drying, and typical photographs are shown in Figure 7. Samples with a 140/60 ratio were initially free from cracking when removed from the electrocoating dispersion shortly after the deposit was obtained. However, the coating developed cracks and demonstrated a lack of adhesion by spontaneously exfoliating within 90 min of evaporation. The next highest epoxy content, 120/80, was soft when touched lightly, but began to develop cracks after about 60 min drying. Between 60 min and 90 min, cracks were clearly visible along all four edges. The cracks continued to cover the surface, and cracking was virtually complete after about 135 min, after which there was no apparent significant change from 3.5 to 24 hr. Slight cracking occurred for the 100/100 ratio when dried up to 150 min, and then moderate cracking developed after 24 hr. The epoxy/latex ratios of 60/140 (30/70) and 80/120 (40/60) did not develop cracks. When the epoxy/latex ratio was 40/160 the electrodeposited coating was not uniform and was not continuous in coverage of the zinc surface. Both the weight and thickness of all the coatings correlated with the epoxy content. Based on these experiments, it was concluded that the weight ratios 30/70 and 40/60 were optimum for producing a continuous toughened epoxy coating free from cracks. The 40/60 ratio was selected for further study in this work because thicker coatings were obtained, and thick coatings were preferred for a fuel tank sealant material.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

The experimental measurements centered on the weight decrease of the dispersion liquid from circular containers during evaporation. The proper geometry is diffusion for cylindrical mass transfer, since the level of the dispersions is below the top of the circular aluminum weighing vessels. However, because the radius is large (3.5 cm) a planar geometry was taken as a reasonable approximation to the cylindrical geometry. When evaporation begins at t = 0, the 40/60 dispersion-atmosphere interface is at the surface, Figure 1A. When coalescence begins, the surface is at x = 0, and the aluminum substrate-dispersion interface for semi-infinite diffusion is at x = [infinity], as shown in Figure 1B. At x = 0 the dispersion liquid evaporates to become the vapor form of the liquid; for example, water vapor.

Selected weights of the 40/60 dispersion were added to five circular 1.9 g aluminum containers with dimensions 6.99 cm diameter X 1.5 cm height X 0.009 cm thickness. The container samples had different initial dispersion weights, and are identified as 448, 449, 450, 451, and 452. These weights were the initial values of [C.sup.0] at 5 g intervals in the range 10-30 g, and are recorded as [w.sub.0] in Table 1. The five containers were then placed inside the evaporation oven at 35[degrees]C. At selected time intervals between 24 hr and 624 or 1272 hr, the containers were removed from the oven to measure the reduction in weight, the decrease in diameter, and the decrease in thickness of the coalescing material. The decrease in weight as a function of evaporation time from 0-336 hr is recorded in Table 1 for each sample. The last weight in the rows of each column, [w.sub.[infinity]], was obtained by removing the dried coating from the aluminum container after 336 hr, and permitting the samples to dry to a constant value in the oven at 35[degrees]C. These values of [w.sub.[infinity]] were taken as the weight of each specimen after all the liquid had evaporated at 35[degrees]C and 35 [+ or -] 7% RH. A constant weight was obtained for the 10, 15, and 20 g samples after 624 hr, while 1272 hr were required for the 25 and 30 g samples. Since the values of [w.sub.[infinity]] are all less than the weights at 336 hr, it is concluded that the samples contained dispersing liquid from 0 to 336 hr, and that the semi-infinite diffusion boundary condition is applicable. The analytical balance used reads to [+ or -] 0.0001 g in the range 0-120 g. This uncertainty was tripled to [+ or -] 0.0003 g to take into consideration water loss during weighing, if any, and is assigned to the weights [w.sub.0] and [w.sub.[infinity]]. For oven samples, a temperature effect between 35[degrees]C and room temperature is assigned an uncertainty of [+ or -] 0.0009 g by tripling the room temperature estimation from 24 to 336 hours.

The weight, thickness, and diameter decreases during evaporation were measured near room temperature by removing the samples from the oven. A significant effect of this temperature difference on the evaporation rate is expected to be negligible because the loss in weight at a constant 25[degrees]C is significantly lower than at the constant 35[degrees]C temperature. A comparison of this set of samples with two other samples of higher weights showed about 75 g more liquid was evaporated at 35[degrees]C than at 25[degrees]C after 50 hr, with a difference of about 90 g after 100 hr. A second comparison of these two drying curves in the range 0-336 hours showed that the curve for 25[degrees]C decreased almost linearly to about 250 hr. Conversely, in the time span 0-65 hr, the curve at 35[degrees]C fell more rapidly. The oven in which the samples were placed was heavy so that it was not easily moved, and the samples were not otherwise disturbed while in the oven, to ensure liquid transport was by diffusion.

RESULTS AND DISCUSSION

The formation of a polymeric coating by evaporation of the dispersing liquid reduces the thickness, diameter, and weight. These properties were measured during evaporation.

Decrease in Thickness

The coating samples formed a concave up geometric surface. The circumference of the surfaces in contact with the aluminum substrate container was always slightly higher by a small, but measurable, amount compared with the middle of the circular surfaces. For consistency, the thickness was obtained at the middle of the surfaces. Measurements of the coating thickness were made at the same times (hr) as those obtained for the change in weight in Table 1. These values were fitted to a second degree polynomial, and are plotted in Figure 8. The curves show a comparatively rapid decrease from 24 up to about 80 hr, then approach a constant value for each sample. The fitted thickness at 0 hr was taken as the thickness of the samples at the beginning of coalescence, and is recorded for each sample in Table 2. The uncertainty ([+ or -]) in thickness (h) values in Table 2 were taken as the difference between each measured thickness and fitted equation thickness from 24-336 hr. The average difference was calculated for each sample and was multiplied by five to obtain the uncertainty.

Decrease in Diameter

The diameters of the five samples decreased by a small but measurable amount during evaporation, and the changes with time are plotted in Figure 9. The diameters decrease most rapidly between 24 and 200 hr, with the largest change occurring in the three samples with the highest initial weights. The measurements for each sample were fitted to a second degree polynomial to obtain the estimated diameters when coalescence begins; these are tabulated in Table 2. This value of the diameters was used to calculate the area and volume for each sample. The uncertainty ([+ or -]) in diameter values in Table 2 was obtained from the difference between the measured diameter and fitted equation diameter from 24-336 hours. The average difference was calculated for each sample and multiplied by five to obtain the uncertainty.

[FIGURE 11 OMITTED]

Initial Conditions

The initial thickness is taken as the value at t = 0 from the plots in Figure 8. The data plotted are the measurements of coalescent thickness, during times when the dispersion was solidifying. The thickness at t = 0 is obtained from a computer fitting of the experimental data to a second degree polynomial, and is interpreted as the thickness when coalescing has begun. Similarly, diameter measurements plotted in Figure 9 are interpreted as the diameter when the dispersions begin to coalesce. The initial values for both thickness and diameter are in Table 2. The areas were calculated from [pi][r.sup.2] and the volumes from [pi][r.sup.2]h, using the diameter and thickness in Table 2. The initial concentrations for each sample were calculated from the liquid content of each dispersion, divided by the volumes in Table 2. The liquid concentration was taken as the difference between the weight of the freshly prepared dispersion, [w.sub.0], and the dried weight, [w.sub.[infinity]], in Table 1. The liquid weight was then divided by the sample volume in Table 2. These concentrations are tabulated in Table 4 as [C.sup.0], g*[cm.sup.-3].

[FIGURE 12 OMITTED]

Zero Order Rate

The weight decrease for each of the five samples in Table 1 was plotted as a function of increasing evaporation time from 0-336 hr. The curves for each plot were a computer-drawn smoothing of the experimental measurements in Table 1. The weights of interest are between zero hours and a time when equation (3) applies. The weight loss of evaporated liquid was calculated at two-hour intervals from the difference in the weight at t = 0 and the estimated weight with an uncertainty of [+ or -] 0.1 g at each two-hour interval up to 24 hr. Straight lines were obtained from 0-12 hr. The weight of dispersing liquid evaporated was obtained by subtracting the seven weights from t = 0 to t = 12 hr from the initial weight for each dispersion. The units, g, were converted to g*[cm.sup.-2] by dividing the values by the areas, in Table 2. The increase in surface concentration of evaporating liquid and the amount evaporated g*[cm.sup.-2], are plotted versus time for each dispersion in Figure 10, with straight lines obtained for the samples. Rate constants were calculated with equation (3) by dividing the slopes, g*[cm.sup.-2]*h[r.sup.-1], in Table 3 by the initial liquid concentration, [y.sup.0], of the five dispersions. Initial concentrations, g*[cm.sup.-2], in Table 3 were obtained from the total amount of liquid evaporated at 12 hr divided by the surface area. These are also the values in Figure 10 at t = 12 hr. The values for the five heterogeneous rate constants, k, for liquid evaporation at the dispersion-atmosphere interface were calculated by dividing the slope by the initial concentration, and are recorded in Table 3. The average and standard deviation are 0.086 [+ or -] 0.001, the standard deviation was multiplied by 20 to give 0.086 [+ or -] 0.02. The uncertainty in the rate constant, k, [+ or -] 0.02 h[r.sup.-1], mainly reflects an uncertainty in the weight data used in the plots.

Fick's Law Rate

Equations (22), (23), and (26) are solutions to Fick's diffusion equation for the decrease in sample weight ([C.sub.evap]), the increase in the weight of evaporated dispersing liquid ([C.sub.cond]), and the total weight of evaporated liquid ([M.sub.t]). The shapes of the theoretical curves for this second time frame are shown in Figures 4-6. In this section, the experimental data are plotted according to the three equations and are compared with the shapes predicted by the three equations. Agreement between prediction and experiment is interpreted as validation of the Fick's diffusion equation model.

Weights estimated from 14-22 hr from the smoothed plots for each dispersion sample were combined with those from 0-12 hr and the weights in Table 1. The combined weights were calculated as [M.sub.t], g*[cm.sup.-2], and plotted versus [t.sup.1/2] in accordance with equation (26). A linear rising portion was found for each sample between 3.7 and 4.7 h[r.sup.1/2] units of time. The slopes, intercepts, and [R.sup.2] values are in Table 4. The diffusion coefficients were calculated from the slopes with equation (30), and the evaporation rate constants from the intercept with equation (30). The standard deviation of the average for [D.sub.evap] was [+ or -] 0.00017; this was tripled to an uncertainty of [+ or -] 0.00051, as recorded in Table 4.

Fick's Law Diffusion Controlled Liquid Transport

A third time frame where mass transfer is by diffusion is anticipated following the rate controlled time frame. (14,15,16,18) If this is a diffusion controlled region, then plots of the flux according to equation (10) should be linear when plotted with [t.sup.-1/2]. Plots of weight loss in Table 1 from 24-336 hr were smoothed by fitting to straight lines, and slopes, intercepts, and [R.sup.2] values are given in Table 5. The linear functions were used to compute weight data at selected times between 24 and 96 hours. These were combined to form a database with the experimental measurements in Table 1, and the fitting to equation (30) for 14-22 hr in Table 4. The data were plotted as the weight during drying in Figure 11. These experimental plots have the form calculated with equation (22) in Figure 4 from zero to about 100 hours. The weight loss on evaporating was calculated from this database, and plotted versus drying time in Figure 12. A comparison with the curves in Figure 5 plotted according to equation (23) shows the predicted shape. The evaporation weight loss was next plotted versus h[r.sup.1/2] in Figure 13. The shape of the initial slow rise followed by the more rapid increase compares with the shape in Figure 6 predicted by equation (26). The quantity of dispersing liquid evaporated was plotted versus [t.sup.1/2] in Figure 13, using the experimental values in Table 1, and the estimated data from 2-12 hr from Figure 10. Linear regions with values for calculations with equation 30 were found by fitting to straight lines. The coefficient of determination, [R.sup.2], was a criterion for the best fit. Slopes, intercepts, and [R.sup.2] values for each sample are recorded in Table 4. The rate constant, [k.sub.evap], and diffusion coefficient, [D.sub.evap], were obtained from the slope and intercept with equation (30). These are recorded in Table 4. The average [D.sub.evap] for the dispersing liquid in the coalescing coating is 9.2 X [10.sup.-4] [cm.sup.2]*h[r.sup.-1]; the uncertainty, [+ or -] 0.00051, is triple the standard deviation of the average. The average rate constant for evaporation into the atmosphere is 0.046 [+ or -] 0.017 cm*h[r.sup.-1]; the uncertainty is taken as the standard deviation of the average.

CONCLUSION AND IMPLICATIONS FOR DISPERSING LIQUID TRANSPORT IN COATINGS TECHNOLOGY

The transport mechanism for drying latex-toughened epoxy coatings in ovens without convective air flow at 35[degrees]C has three major stages: (1) Zero order kinetics proceeding at a rapid rate when a surface of dispersing liquid is present. Between 0 and 12 hr the fractional weight loss exceeds 0.55 of the total; (2) Evaporation kinetics limited by the rate at which surface water evaporates when thin layers are present on the surface. The second stage subsequently involves lower weight loss by diffusion through a coalescing coating; (3) A third stage where evaporation is limited by the rate of diffusion of dispersing liquid through a more developed coalescent material than in stage (2). The first two stages are quantitatively characterized by two rate constants and a diffusion coefficient. A time frame for the first stage is established by an initial linear region when liquid weight loss is plotted versus evaporation time; a second by a linear rising portion when the weight loss is plotted versus [t.sup.1/2]. The third stage is identified by a straight line when liquid weight loss is plotted versus [t.sup.-1/2].

[FIGURE 13 OMITTED]

Agreement Between Theory and Experiment

Fick's diffusion equation solved with the Laplace transform in this work provides diffusing liquid-time relations in various forms for computing evaporation rate constants and diffusion coefficients. Equation (22) relates the weight decrease of dispersing liquid as a function of evaporation time, as depicted in Figure 4. Experimental data in Table 1 and estimated values plotted in Figure 11 have a similar shape to Figure 4. Adesanya, Nanda, and Beard's (3) plot of moisture content versus time also has a similar shape. Their experiments involved convective hot air at an average velocity of 11 m*se[c.sup.-1] impinging onto both sides of a wood sample at elevated temperatures. Since the experimental methods differ, only a qualitative comparison is made. Raghavan et al. (5) used a Fourier series solution to Fick's diffusion equation for spherical symmetry. They used a computer program to account for a concentration dependent diffusion coefficient, and for geometrical shrinkage during drying. Their experimental data were obtained at 50[degrees]C, with 2.0 m*se[c.sup.-1] convective air flow. A qualitative comparison of their moisture ratio versus time plot has a shape similar to Figure 11, predicted according to equation (22). In electrochemistry (15,16,18) and data in Table 2.1 in Crank, (17) the shape of the curves follows the exp ([[xi].sup.2]) erfc ([xi]) function, as shown in Figure 4.

Equation (23) is the predicted increase in evaporated liquid weight with time, as plotted in Figure 5. The experimental data and estimated values are plotted in Figure 12. Both sets of curves have identical shapes. Experimental data in Croll, (1,2) Adesanya et al., (3) Winnik and Feng, (4) and Blandin et al. (13) have similar shapes. The slope of an initial linear portion is calculated and recorded as the evaporation rate. (1-4) An advantage of the Fick's equation is that both rate constants and a diffusion coefficient can be computed.

Equation (26) relates the concentration of evaporated liquid to [t.sup.1/2], and is plotted in Figure 6. Experimental data in Table 1 and estimated values are plotted in Figure 13, and have the predicted shape between 0 and 4.5 h[r.sup.1/2]. Adesanya et al., (3) Crank, (17) and Christie et al. (14) show identical shapes when compared to those in Figures 6 and 13.

Implications for Drying Coatings

The theoretical treatment in equations (17) and (18) permits calculation of the concentration gradient of evaporating water within the coalescing dispersion, and the concentration gradient of condensing water. This could be correlated with the timeline during formation of polyurethane coatings (9) and in latex drying. (1,2,4,8) Equations (20) and (21) calculate the shape of the curves at x = 0 for the weight decrease due to evaporation, Figure 11, and for the increase in weight evaporated, Figure 12. Equation (12) equates the net flux to the difference between the rate of evaporation and the rate of condensation, so that the rate constants, [k.sub.evap] and [k.sub.cond], are not assumed equal. This is an advantage compared to equations (28) and (29) where the rates are assumed equal, and the effect of condensation rate is not immediately obvious. Figure 4 depicts the effect of condensed water on the rate of evaporation, for selected values of [k.sub.cond]. The weight evaporated, [C.sub.cond], is obtained from equation (23).

The model for semi-infinite diffusion in Figure 1B at x = 0 interfaces the aqueous dispersion and the humid atmosphere. Evaporation occurs at this boundary and condensation would also take place. This is consistent with the interface for evaporation and condensation kinetics without diffusion (19) onto or off from an ice surface. This is also the interface for the loss or gain of moisture by an absorbing solid in contact with air (17) with diffusion into or out of the solid. Chaix, van den Bergh, and Rossi (19) studied the evaporation and condensation of [D.sub.2][.sup.18]O water vapor on ice without coupled diffusion and utilized a rate equation with both kinetic constants. The equations developed here could be used to check for diffusion of [D.sub.2][.sup.18]O in the ice, if any. Similarly, water uptake in coatings (12) might be amenable to Fick's law application, where now [k.sub.evap] = 0, and [k.sub.cond] would be obtained.

The plots in Figures 4 and 11 have initial linear portions from 0-12 hr that were fitted to a zero order chemical reaction. Theoretical plots are shown in Figure 2 for three different concentrations. These linear plots compare with experimental data in Winnik and Feng, (4) the experimental data from this work in Figure 10, and in Adesanya et al. (3) The units for equation (3) are h[r.sup.-1], rather than a conventional g*[cm.sup.-2]*h[r.sup.-1]. An advantage is that the reciprocal time units conform to physicochemical data where a rate constant can be expressed as a function of temperature by a rate equation. (20) The rate of a chemical reaction, dN/dt, is defined as the number of moles reacting (N) per unit time to form a product, rather than grams. In this work, the concentration (moles or grams) does not appear in the rate constants or the diffusion coefficient; grams are used to be consistence with other publications.

When aqueous dispersions are dried, surface evaporation occurs with or without forced airflow. In both cases, Fick's equation plots of grams remaining versus time, grams evaporated versus time, and quantity evaporated versus tim[e.sup.1/2] have similar shapes. However, weight evaporated should be higher in the zero order kinetics time frame because the surface water is in direct contact with the atmosphere. In the Fick's equation time frame, the liquid (water) evaporated is limited by the rate of evaporation, and then by diffusion through the coalescing dispersion. The weight of liquid evaporated should be independent of the airflow velocity rate. Both the diffusion rate and evaporation rate should be temperature dependent, but not to the same degree.

An important criterion for sealant specifications is the completeness of solidification. Weights can be measured accurately at all stages of solidification, and the measurements then used to predict the temperature and length of time required for complete solidification. These data are needed in computer assisted predictive modeling of the solidification process. An improvement in equipment would be a constant temperature-constant humidity digital moisture weight recorder, with software to plot and compute diffusion coefficients and rate constants. Information on the surface liquid and vapor composition, for example, FTIR, would provide information on chemical aspects of the transport mechanism.

Table 1 -- Weight of 40/60 Dispersion Samples at 35 [+ or -] 1[degrees]C
and RH 35 [+ or -] 7% added to 6.99 cm Diameter, 1.5 cm Height, 0.009 cm
Thick Aluminum Weighing Containers. Area of Containers 38.4 [cm.sup.2],
Volume is 57.6 [cm.sup.3]. Estimated Uncertainty: [w.sub.0] and
[w.sub.[infinity]], [+ or -]0.0003 g; 24-336 hr, [+ or -] 0.0009 g

                             Sample Weight, g
Evaporation
Time, hr                    448      449      450      451      452

   0, [w.sub.0]           10.0118  15.0145  20.0071  25.0099  30.0128
  24                       5.0181   8.6539  12.7328  17.4825  22.1315
  96                       4.5163   7.1145   9.9086  13.2660  16.6256
 120                       4.4610   7.0137   9.7076  12.7948  15.9088
 144                       4.4222   6.9415   9.5782  12.5121  15.4478
 168                       4.3939   6.8823   9.4795  12.3176  15.1269
 192                       4.3668   6.8209   9.3885  12.1521  14.8804
 264                       4.3295   6.7041   9.2068  11.8732  14.4856
 288                       4.3205   6.6735   9.1547  11.8001  14.3873
 312                       4.3140   6.6479   9.1093  11.7333  14.3038
 336                       4.3073   6.6263   9.0707  11.6825  14.2378
 624, [w.sub.[infinity]]   4.2320   6.4093   8.5767  --       --
1272, [w.sub.[infinity]]  --       --       --       10.6772  12.8223
solid, %w                 42.27    42.69    42.87    42.69    42.72
evaporated, %w            57.73    57.31    57.13    57.31    57.28

Table 2 -- Dimensions of Coalescing Polymer for Each Sample When
Coalescence Begins. Surface Area Calculated from [pi][r.sup.2], Volume
from [pi][r.sup.2]h.

Sample  Thickness (h), cm (a)   Diameter, cm (b)

448     0.13 [+ or -] 0.04      7.03 [+ or -] 0.01
449     0.19 [+ or -] 0.03      7.08 [+ or -] 0.15
450     0.25 [+ or -] 0.02      7.25 [+ or -] 0.13
451     0.27 [+ or -] 0.03      7.28 [+ or -] 0.07
452     0.33 [+ or -] 0.02      7.44 [+ or -] 0.12

Sample  Area, [cm.sup.2]    Volume, [cm.sup.3]

448     38.8 [+ or -] 0.21   5.04 [+ or -] 0.21
449     39.4 [+ or -] 1.7    7.49 [+ or -] 1.7
450     41.3 [+ or -] 1.5   10.3 [+ or -] 1.5
451     41.6 [+ or -] 0.8   11.2 [+ or -] 0.8
452     43.5 [+ or -] 1.4   14.3 [+ or -] 1.4

(a) Intercept for t = 0, Figure 8.
(b) Intercept for t = 0, Figure 9.

Table 3 -- Zero Order Kinetics Rate Constant, 0-12 hr. Slope and
Intercept are from Figure 10

        [y.sup.0],     Slope,                     k,
Sample  g*[cm.sup.-2]  g*[cm.sup.-2]*h[r.sup.-1]  h[r.sup.-1]

448     0.089          0.0076                     0.085
449     0.112          0.0096                     0.086
450     0.128          0.011                      0.086
451     0.130          0.011                      0.085
452     0.136          0.012                      0.088
                       avg [+ or -] uncertainty   0.086 [+ or -] 0.02

Table 4 -- Slope, Intercept, [D.sub.evap] and [k.sub.evap], from Rising
Portion of Figure 13, 14-22 hr

        [C.sup.0],          Slope,                  Intercept,
Sample  g*[cm.sup.-3]  g*[cm.sup.-2]*h[r.sup.1/2]  g*[cm.sup.-2]

448     1.15             0.0317                      -0.0207
449     1.15             0.0407                      -0.0294
450     1.11             0.0399                      -0.0150
451     1.27             0.0455                      -0.0344
452     1.20             0.0436                      -0.0282

                      [D.sub.evap],             [k.sub.evap],
Sample  [R.sup.2]     [cm.sup.2]*h[r.sup.-1]    cm*h[r.sup.-1]

448     0.999         0.00061                   0.034
449     0.993         0.00098                   0.039
450     0.993         0.0010                    0.075
451     0.993         0.0010                    0.037
452     0.993         0.0010                    0.044
        avg [+ or -]  0.00092 [+ or -] 0.00051  0.046 [+ or -] 0.017
        uncertainty

Table 5 -- Coefficients for Slope and Intercept for Smoothing Function
from Fitting Sample Weight Data in Table 1 from [24.sup.-1/2] to
[336.sup.-1/2] h[r.sup.-1/2] Used to Calculate Weights from 25-94 hr

        Slope,           Intercept,             Uncertainty in
Sample  g*h[r.sup.-1/2]  g           [R.sup.2]  Calculated Weight, g(a)

448      4.7967           4.0316     0.998      [+ or -] 0.0021
449     13.510            5.8436     0.992      [+ or -] 0.0093
450     24.416            7.6296     0.988      [+ or -] 0.0228
451     39.034            9.3976     0.995      [+ or -] 0.0273
452     53.388           11.153      0.997      [+ or -] 0.0253

(a) Difference between weight in Table 1 minus weight calculated from
fitting equation.

ACKNOWLEDGMENT 

This work was performed under SBIR Contract No. N62269-96-C-0035, U.S. Department of Defense, Naval Air Warfare Center (NAWC), Aircraft Division. Stanley R. Brown, NAWC, Patuxent River, MD, provided guidance for the properties required of a polymeric material to seal the inner surfaces of a jet fuel tank.

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Sidney L. Phillips, ([dagger]) M. Troy Davis, and Daniel J. Phillips -- Camatx Basic Chemistry, Inc.*

* 171 El Toyonal, Orinda, CA 94563.

([dagger]) Author to whom correspondence should be addressed.