= Nonuniform-density erosion inference
\b[Nonuniform-density erosion inference] estimates a <polymer erosion front speed> when local <density> varies through a specimen. Total mass divided by total volume gives an average density; the material being removed at the boundary can have a different density. Substituting the average into a surface-removal formula can therefore misestimate the speed.
For an original spherically symmetric model, let $R(t)$ be the surviving radius and $\rho(r,t)$ the local mass density. Then
$$
M(t)=4\pi\int_0^{R(t)}\rho(r,t)r^2\,dr.
$$
Differentiating the integral and defining positive inward speed $v=-R'$ gives
$$
-M'=4\pi R^2\rho(R,t)v
-4\pi\int_0^R\frac{\partial\rho}{\partial t}(r,t)r^2\,dr.
$$
This radial form of the <Reynolds transport theorem> separates loss at a moving boundary from density change inside the surviving volume. All quantities must refer to the same material and conditioning state; a water-filled wet volume and a subsequently dried polymer mass do not automatically define the local density used here.
If the interior density profile is stationary in spatial coordinates over the measurement interval, the integral term vanishes. The appropriate boundary conversion is then $v=-M'/(4\pi R^2\rho_s)$, with $\rho_s=\rho(R,t)$. Writing average density as $\overline\rho=3M/(4\pi R^3)$, a uniform-density substitution instead gives
$$
v_{\rm uniform}=\frac{\rho_s}{\overline\rho}v.
$$
The ratio is measurable information about the profile, not a shape correction already supplied by knowing the outer radius.
An illustrative fixed profile is $\rho(r)=\rho_0[1+(r/R_0)^2]$, with $R_0$ a fixed initial radius, not the changing current radius. At $R=R_0$, the boundary density is $2\rho_0$ and the mean is $\rho_0(1+3/5)=1.6\rho_0$. The uniform-density inference overestimates the instantaneous front speed by 25%. For $R_0=1$ cm, $\rho_0=1\,\mathrm{g\,cm^{-3}}$ and true $v=0.001\,\mathrm{cm\,day^{-1}}$, mass leaves at $0.008\pi\approx0.02513\,\mathrm{g\,day^{-1}}$; dividing by the mean-density surface factor gives $0.00125\,\mathrm{cm\,day^{-1}}$. These are stipulated profile values, not measured properties of a named <polymer>.
Time-dependent hydration, porosity or chemical change can make the interior integral nonzero. One mass slope cannot independently identify both that contribution and boundary speed. <Boundary-density calibration for erosion fronts> therefore needs spatial or compositional evidence in addition to a bulk mass measurement. NIST's mass-density definition is mass divided by volume; it does not equate a whole-specimen average with an unknown boundary value.<NIST Guide to the SI>[Chapter 8, §8.6.7]{ref}
A discriminating experiment measures outer dimensions, conditioned mass and density or composition at several depths, using matched specimens or accounting for destructive sampling. Kureha's <polyglycolic acid> protocol filters and dries the remaining solids after water exposure.<Kuredux Polyglycolic Acid Resin technical brochure by Kureha>[PDF p. 3, Degradability test method]{ref} That observation supplies retained dry mass, not a radial density field. Establish an inward-moving degradation boundary before interpreting the inferred speed as <polymer erosion>, rather than another cause of material loss.
Back to article page
