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Research Papers

# The Role of Mass Balance Equations in Growth Mechanics Illustrated in Surface and Volume Dissolutions

[+] Author and Article Information
Gerard A. Ateshian

Department of Mechanical Engineering, and Department of Biomedical Engineering, Columbia University, 500 West 120th Street, MC4703, New York, NY 10027ateshian@columbia.edu

The apparent density may be represented as $ρα=dmα/dV$, where $dV$ is an elemental volume of the mixture and $dmα$ is the mass of constituent $α$ in $dV$.

In this canonical problem, we do not consider the more complex case where $B$ may be initially a dry porous solid (with air-filled pores, for example) later wetted by the bathing solution.

In other problems of continuum mechanics, the singular interface $Γ$ need not be defined on a solid boundary; for example, in compressible fluid mechanics, $Γ$ may represent a shock wave within the fluid.

The true density of constituent $α$ may be represented as $ρTα=dmα/dVα$, where $dVα$ is the elemental volume of constituent $α$ in the elemental mixture volume $dV$. It follows that $0≤ρα≤ρTα$.

Thus, $φα=dVα/dV$.

A similar phenomenon occurs, for example, when ice melts into liquid water. Since ice has a lower true density than liquid water, the resulting volume of the ice-water mixture will decrease with melting.

The molar concentration $cu$ of the solute is related to the apparent density via $Mucu=ρu/(1−φs)$, where $Mu$ is the solute molecular weight.

J Biomech Eng 133(1), 011010 (Dec 23, 2010) (6 pages) doi:10.1115/1.4003133 History: Received October 20, 2010; Revised November 16, 2010; Posted November 29, 2010; Published December 23, 2010; Online December 23, 2010

## 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.

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## Figures

Figure 1

A singular interface Γ(t) separates a region V(t) into subregions V+ and V−. vΓ is the velocity of Γ, and n is the unit outward normal to the “+” side.

Figure 2

In a dissolution problem, B represents the dissolving body, B∗ represents the bath in which the body dissolves, and Γ is the evolving interface between B and B∗

Figure 3

Volume dissolution (ρ̂s<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 dms produces a decrease in apparent solid density ρs.

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