= Boundary-density calibration for erosion fronts
\b[Boundary-density calibration for erosion fronts] determines the local <density> needed to convert a surface mass-loss rate into a <polymer erosion front speed>. The relevant density is that of the material crossing the boundary, conditioned consistently with the mass and dimensional measurements. Whole-specimen mass divided by whole-specimen volume gives an average, not automatically the required boundary value.
For a spherical, inward-moving boundary of radius $R$, negligible interior density change gives
$$
v=-\frac{M'}{4\pi R^2\rho_s},\qquad \rho_s=\rho(R).
$$
The local-density requirement follows from <nonuniform-density erosion inference>. NIST defines mass density as mass divided by volume; the experimental volume and sampled region must therefore be specified.<NIST Guide to the SI>[Chapter 8, §8.6.7]{ref}
An original calibration construction removes or samples a shell of thickness $\delta$. Its mean density is
$$
\rho_\delta=\frac{\Delta M}{(4\pi/3)[R^3-(R-\delta)^3]}
=\frac{\int_{R-\delta}^{R}\rho(r)r^2\,dr}{\int_{R-\delta}^{R}r^2\,dr}.
$$
A finite shell measures a weighted layer average. For a smooth radial profile, expansion gives $\rho_\delta=\rho(R)-\rho'(R)\delta/2+O(\delta^2)$. Thus a thinner shell approaches the boundary value, but finite-layer bias depends on the density gradient. <Finite-shell bias in boundary-density calibration> is distinct from random weighing error.
For the stipulated profile $\rho(r)=\rho_0[1+(r/R_0)^2]$ at current radius $R=R_0$, a shell with $\delta=0.1R_0$ gives
$$
\rho_\delta=\rho_0\left[1+\frac35\frac{1-0.9^5}{1-0.9^3}\right]
\approx1.9067\rho_0,
$$
whereas the actual boundary density is $2\rho_0$. Substituting the shell average into the instantaneous speed conversion overestimates speed by about 4.9%. These are original profile calculations, not measured material properties.
Very thin layers have their own uncertainty cost. For measured layer mass $m$ and volume $V$, first-order propagation gives
$$
\frac{u^2(\rho)}{\rho^2}\simeq\frac{u^2(m)}{m^2}+\frac{u^2(V)}{V^2}
-\frac{2\operatorname{cov}(m,V)}{mV}.
$$
Shrinking the layer can increase relative mass and thickness uncertainty. Shared dimensional corrections can correlate volumes and successive layer estimates; the GUM requires relevant covariance terms.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, Eq. 13]{ref} A smaller sampling thickness therefore trades spatial averaging against measurement uncertainty.
A proposed experiment uses matched specimens or recorded sequential sectioning, measures removed material and layer geometry, and compares several layer thicknesses. Distinguish wet, dried, porous-envelope and solid-material volumes. Cutting, drying or extracting a layer can change its composition, so independent spatial observations should test whether the sampled material represents the original boundary.
Kureha's <polyglycolic acid> degradation test filters and dries residual solids after water exposure.<Kuredux Polyglycolic Acid Resin technical brochure by Kureha>[PDF p. 3, Degradability test method]{ref} It supplies retained dry mass; it does not supply local boundary density or prove the existence of a sharp erosion front. The additional calibration construction above makes those missing measurements explicit.
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