Drying and collapse of hollow latex

Abstract

Hollow latex particles are used as white pigments for paints and paper coatings. In the coating dispersion, each hollow particle is filled with water. As the coating dries, water vacates the latex, leaving an air-filled void sized to scatter light (~0.5 μm) within each particle. Examinations of dried coatings reveal that hollow particles can collapse, decreasing their light scattering efficiency. Cryogenic scanning electron microscopy (cryoSEM) was used to characterize the microstructure of coatings containing hollow latex during drying. Images suggest latex voids empty after air invades into the coating interstitial space and collapse occurs late in the drying process. The effects of temperature (10–60°C), humidity (20–80%), and binder concentration (0–30 wt%) on particle collapse were also studied through SEM of dried coating surfaces. High drying temperature, high humidity, and low binder concentrations promoted collapse. For hollow latex particles with porous shell walls, temperature and humidity had little effect, whereas binder increased collapse. From these results, a theoretical model is proposed. During drying, diffusion of water from the particle creates a vacuum inside the latex. The vacuum is either relieved by nucleation of a gas bubble from the dissolved air in the water-filled particle or it causes the particle to collapse by buckling.

Appendices

Appendix A1: theoretical model for the drying of a nonporous hollow latex sphere

Assuming that the radius of the particle, b, is assumed to be much bigger than the shell thickness, δ, then the water flux, $\dot{n}$, can be described by the one-dimensional equation (A1):

$$ \dot{n}_{\text{water}} = - \frac{{\wp_{\text{water}} }}{\delta }\left( {4\pi b^{2} } \right)\left( {f_{\text{water}}^{\text{void}} - f_{\text{water}}^{\text{out}} } \right) $$

(A1)

The permeability of water through polystyrene, $\wp_{\text{water}}$, has been measured using finite thicknesses of polystyrene to be 1.35 × 10⁻¹⁴ m² s⁻¹ Pa⁻¹ at room temperature.30 This quantity describes both the solubility and diffusivity of water in polystyrene.

Inside of the shell, the liquid water fugacity, f _void, is approximately the water vapor pressure, P _vap, and can be calculated by the expression:

$$ f_{\text{void}}^{\text{water}} = x_{\text{water}} P_{\text{vap}}^{\text{water}} \exp \left( {\frac{{\hat{v}_{\text{water}} }}{RT}\left( {P_{\text{void}} - P_{\text{vap}}^{\text{water}} } \right)} \right) $$

(A2)

The Poynting correction is not neglected since the pressure inside the void, P _void, may change drastically with time. Residual impurities left over from the swelling of the latex core are accounted for by the term x _water, the water molar fraction in the void. The gas constant is R, T is the temperature, and $\hat{v}$ is the molar volume of water.

Outside of the shell, the liquid water fugacity, f _out, is simply the partial pressure of water vapor in the porespace. This quantity may change with time as the coating as a whole dries.

Because the void must be completely filled by its contents at any time, the void volume must change as water exits. A volume balance can be written where the void volume change with time, $\frac{{dV_{\text{void}} }}{dt}$, is equal to the sum of the volume fluxes of every species through the particle shell wall of area A:

$$ \frac{1}{A}\frac{{dV_{\text{void}} }}{dt} = \sum\limits_{\text{species}} {\dot{n}_{i} \hat{v}_{i} } = \dot{n}_{\text{water}} \hat{v}_{\text{water}} + \dot{n}_{{{\text{O}}_{2} }} \hat{v}_{{{\text{O}}_{2} }} + \dot{n}_{{{\text{N}}_{2} }} \hat{v}_{{{\text{N}}_{2} }} $$

(A3)

where $\dot{n}$ represents the mole flow rate of a species from the shell and $\hat{v}$ is the molar volume. However, before a bubble nucleates the molar volume of dissolved oxygen and nitrogen are negligible and can be ignored, leaving the flux of water as the sole determinant of the void volume.

As the water exits the sphere and the void volume decreases, the rigid polystyrene shell resists the deformation. The pressure difference (P _void − P _out) across the shell that is necessary to compress a hollow sphere to a certain volume can be found using the equation derived by Green and Shield37,38:

$$ P_{\text{void}} - P_{\text{out}} = \frac{{4G\delta_{o} }}{{b_{o} }}\left[ {\left( {\frac{{b_{o} }}{b}} \right) - \left( {\frac{{b_{o} }}{b}} \right)^{7} } \right] $$

(A4)

This expression assumes the latex wall of thickness δ is a neo-Hookean material with a shear modulus G. Unstressed values are marked with subscript o. This equation also assumes that the sphere wall is incompressible, or:

$$ \left( {b_{o}^{3} - a_{o}^{3} } \right) = \left( {b_{{}}^{3} - a_{{}}^{3} } \right) $$

(A5)

Here a and b are the inner and outer diameters of the shell, respectively.

Equations (A1)–(A5) can be solved to determine that the pressure within the shell falls over time as the water evacuates the void. In this paper, however, the time dependence is not explored, but rather equation (A4) is used to understand the relationship between P _void and the deformation of the shell brought about by water loss (Figs. 11 and 12).

Appendix A2: predicting collapse and cavitation of a nonporous hollow latex sphere

The critical pressure for the failure of a perfect sphere with elastic modulus E and Poisson’s ratio ν has been cited by Timoshenko and Gere31:

$$ P_{\text{cr}} = P_{\text{atm}} - \frac{{2(b - a)^{2} E}}{{b^{2} \sqrt {3\left( {1 - \nu^{2} } \right)} }} $$

(A6)

However, critical pressures as predicted by equation (A6) are notoriously difficult to obtain experimentally, most likely due to the geometric perfection required of the sphere.33,39 In fact, studies have shown that the size of the imperfections of a hollow sphere determines when the sphere will buckle under a uniform pressure.32 Because none of the hollow latex particles are perfectly round or have perfectly uniform wall thicknesses, variability between imperfections may explain why all of the hollow latex particles do not collapse under the same conditions. Particles with larger imperfections will collapse under smaller pressure loads.

Predicting cavitation is also difficult. Bubble nucleation is likely heterogeneous and therefore dependent on solution and solid-state properties as well as dissolved species concentrations within the void at any given time. These properties may also differ between individual particles.

Some assumption must be made about the size of the vapor bubble once it forms. The Young–Laplace equation states that there is a large thermodynamic penalty that exists in creating the surface of a vapor bubble of small radius, r _bub:

$$ P_{\text{cavitation}} = P_{\text{void}} + \frac{2\sigma }{{r_{\text{bub}} }} $$

(A7)

In fact, experiments have subjected pure water to pressures as low as −250 atm before cavitation occurs.40,41 These experiments are performed under pristine conditions with no preexisting bubbles that would serve as nucleation sites. It can be assumed that no preexisting nucleation sites would exist within the hollow sphere, either, and the confined geometry of the shell may serve to suppress nucleation even further, analogous to studies done with animal cells.42

The first species to vaporize will be nitrogen, at a pressure as predicted by Henry’s law:

$$ P_{\text{cavitation}} = Hx_{{{\text{N}}_{2} }} $$

(A8)

The Henry’s law coefficient for nitrogen at 19°C is 8.24 × 10⁴ atm if $x_{{{\text{N}}_{2} }}$ is the mol fraction of nitrogen in solution.43 A function of temperature, it is only weakly dependant on pressure. Once a nitrogen bubble forms as a seed, water vapor and oxygen will also vaporize, relieving the vacuum within the void.