The role of mass balance equations in growth mechanics illustrated in surface and volume dissolutions

Abstract

Growth mechanics problems require the solution of mass balance equations that include supply terms and account for mass exchanges among constituents of a mixture. Though growth may often be accompanied by a variety of concomitant phenomena that increase modeling complexity, such as solid matrix deformation, evolving traction-free configurations, cell division, and active cell contraction, it is important to distinguish these accompanying phenomena from the fundamental growth process that consists of deposition or removal of mass from the solid matrix. Therefore, the objective of this study is to present a canonical problem of growth, namely, dissolution of a rigid solid matrix in a solvent. This problem illustrates a case of negative growth (loss of mass) of the solid in a mixture framework that includes three species, a solid, a solvent, and a solute, where the solute is the product of the solid dissolution. By analyzing both volumetric and surface dissolutions, the two fundamental modes of growth are investigated within the unified framework of mixture theory.

Figure 2.1

A singular interface Γ (t) separates a region V (t) into subregions V⁺ and V^–. v_Γ is the velocity of Γ and n is the unit outward normal to the ’+’ side.

Figure 3.1

In a dissolution problem, $\mathcal{B}$ represents the dissolving body, ${\mathcal{B}}_{\ast }$ represents the bath in which the body dissolves, and Γ is the evolving interface between $\mathcal{B}$ and ${\mathcal{B}}_{\ast }$.

Figure 3.2

Volume dissolution (${\widehat{\rho }}^s<0$) may be viewed as a macroscopic manifestation of surface dissolution at the microscopic level. In this illustration, struts of a solid scaffold in a representative elemental volume dV are shown to become progressively thinner (left to right) as they dissolve. In the absence of deformation (constant dV), the loss of solid mass dm^s produces a decrease in apparent solid density ρ^s.