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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 givesThe 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.^[ref]

An original calibration construction removes or samples a shell of thickness $\delta$. Its mean density isA finite shell measures a weighted layer average. For a smooth radial profile, expansion gives ${\rho }_{\delta }=\rho (R)-{\rho }^{\prime }(R)\delta /2+O({\delta }^2)$. 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 $\delta =0.1R_0$ giveswhereas 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 givesShrinking 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.^[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.^[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.

Finite-shell bias in boundary-density calibration is the difference between a density averaged through a layer of nonzero thickness and the local density at an erosion boundary. Making a layer thinner reduces spatial averaging, but can increase relative uncertainty in the small removed mass and volume. Boundary-density calibration for erosion fronts must manage both effects.

For an original spherical model with outer radius $R$, sampled thickness $\delta$ and smooth radial density $\rho (r)$,Write $s=R-r$. Expanding $\rho (R-s)=\rho (R)-s{\rho }^{\prime }(R)+O(s^2)$ and the volume weight $(R-s{)}^2$ givesThe leading bias is set by the local density gradient. Curvature changes higher-order terms; a uniform density has no such averaging bias, even for a thick shell.

A first-order polymer erosion front speed conversion uses $v=-M^{\prime }/(4\pi R^2{\rho }_s)$, with ${\rho }_s=\rho (R)$. Replacing ${\rho }_s$ by ${\rho }_{\delta }$ givesA density increasing toward the surface therefore leads to overestimated speed when the sampled layer has lower mean density. This calculation assumes a genuine moving front and negligible independent interior density change.

An original local planning model makes the thickness tradeoff explicit. Approximate the sampled area by $A$, the density-gradient magnitude by $|g|$, and the standard uncertainty of removed mass by a thickness-independent $u_M$. Neglect other uncertainties. Squared bias plus variance is approximatelyMinimization givesFor stipulated $A=1\mathrm{c}{\mathrm{m}}^2$, $|g|=0.2\mathrm{g}\mathrm{c}{\mathrm{m}}^{-4}$ and $u_M=0.0001$ g, ${\delta }_{\ast }\approx 0.0316$ cm, or 0.316 mm. At that thickness, bias magnitude and measurement standard deviation are each about $0.00316\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$. This is a conditional planning result, not a universal required slice thickness.

Layer-thickness tradeoff in erosion-density sampling also depends on uncertain layer thickness, geometry, conditioning and the accuracy of the gradient model. Shared balance corrections can correlate successive mass differences; the GUM requires those covariance contributions to be included.^[ref]

A proposed check compares several layer thicknesses and looks for a stable extrapolation toward zero thickness, with matched drying and dimensional protocols. A changing gradient or damage caused by sectioning can invalidate the extrapolation. Kureha's polyglycolic acid experiment measures filtered, dried retained solids after water exposure.^[ref] A local shell-density calibration would be an additional measurement, not a quantity already determined by that residue curve.

Geometry-dependent erosion-rate inference converts a measured polymer mass-loss rate into a polymer erosion front speed using an explicit model of specimen shape, exposed surfaces and density. The conversion is justified only if an inward-moving boundary actually describes the specimen. IUPAC's definition of polymer erosion concerns degradation progressing from the surface inward; loss of mass alone does not establish that spatial mechanism.^[ref]

An original constant-density calculation shows the geometric dependence. Write $q=-M^{\prime }/M$ for positive relative mass-loss rate and $v$ for positive inward boundary speed. For a sphere of radius $R$, a long cylinder of radius $R$ whose ends are protected, and a sheet of full thickness $h$ attacked on both large faces with negligible edge effects, respectively,These follow by differentiating $M=4\pi \rho R^3/3$, $M=\pi \rho R^2\ell$ with fixed length $\ell$, and $M=\rho Ah$ with fixed face area $A$. The radial boundaries obey $R^{\prime }=-v$; the two sheet boundaries give $h^{\prime }=-2v$. Exposing the cylinder ends or the sheet edges requires including their motion as well.

For stipulated $q=0.03\mathrm{da}{\mathrm{y}}^{-1}$ and current $R=1\mathrm{mm}$, the sphere implies $v=0.01\mathrm{mm}\mathrm{da}{\mathrm{y}}^{-1}$, whereas the end-protected cylinder implies $0.015\mathrm{mm}\mathrm{da}{\mathrm{y}}^{-1}$. A sheet with current $h=1\mathrm{mm}$ gives the latter speed under its two-face assumption. The identical relative mass slope therefore does not identify a unique front speed without geometry.

The retained mass fraction $f=M/M_0$ similarly impliesFor $f=0.729$, the respective radius ratios are 0.900 and approximately 0.854. These are original model predictions, not measurements of a particular polymer. A fit to retained mass cannot by itself select between the shapes or demonstrate a sharp front.

Even the relation between mass and volume needs checking. If a specimen remains spatially uniform but its density changes with time, $M=\rho V$ givesA density change contributes to the inferred relative mass slope independently of boundary motion. Nonuniform-density erosion inference requires the further spatial information hidden by this uniform-density equation. Swelling, fragment detachment and an extended reaction zone also challenge a sharp-front interpretation.

A discriminating experiment combines timed dry-mass measurements with radius or thickness measurements, imaging of cut specimens and chemical observations at different depths. Specify which faces were exposed, how drying was performed and whether dimensions were measured wet or dry. Kureha's polyglycolic acid test holds material in water, then filters and dries it before measuring retained solids.^[ref] That apparatus supplies a retained-mass curve; direct front or dimensional observations would be an additional measurement, rather than information already contained in that curve.

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. ThenDifferentiating the integral and defining positive inward speed $v=-R^{\prime }$ givesThis 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 $v=-M^{\prime }/(4\pi R^2{\rho }_s)$, with ${\rho }_s=\rho (R,t)$. Writing average density as $\overline{\rho }=3M/(4\pi R^3)$, a uniform-density substitution instead givesThe 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, ${\rho }_0=1\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$ and true $v=0.001\mathrm{cm}\mathrm{da}{\mathrm{y}}^{-1}$, mass leaves at $0.008\pi \approx 0.02513\mathrm{g}\mathrm{da}{\mathrm{y}}^{-1}$; dividing by the mean-density surface factor gives $0.00125\mathrm{cm}\mathrm{da}{\mathrm{y}}^{-1}$. 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.^[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.^[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.

Polymer erosion is degradation that begins at a polymer surface and progresses inward. This is the specific IUPAC usage: it identifies where degradation proceeds, rather than serving as a synonym for every reduction in specimen mass. The recommendation discourages the expression “bulk erosion” in this terminology.^[ref]

A surface-controlled process can occur when a degrading reagent reacts faster than it penetrates the interior by diffusion. Conversely, reagent penetration before substantial reaction permits internal degradation without establishing the surface-inward mechanism. The relative time scales depend on the material, reagent and specimen geometry. A measured loss of dry mass alone cannot decide between these possibilities.^[ref]

An original ideal front model gives a clear quantitative prediction. Suppose a spherical specimen has constant density, initial radius $R_0$, no swelling and a sharp boundary receding at constant speed $v$. ThenThe polymer erosion front speed is constant, but the mass-loss rate decreases as the exposed area shrinks. For stipulated $R_0=1\mathrm{mm}$, $v=0.01\mathrm{mm}\mathrm{da}{\mathrm{y}}^{-1}$ and $t=10\mathrm{days}$, the retained mass fraction is $0.9^3=0.729$.

A flat sheet of initial thickness $h_0$, attacked on both large faces with negligible edge effects, instead gives $M(t)/M(0)=1-2vt/h_0$ before the fronts meet. At the same stipulated speed and time with $h_0=1\mathrm{mm}$, the fraction is 0.8. Geometry alone changes the mass curve; neither calculation is a measured degradation rate for a particular polymer.

Experimentally, combine timed mass measurements with evidence about depth and structure, such as cross-sectional imaging or measurements of chemical change at different depths. Retained mass, dissolved fragments and chain scission answer different questions. Swelling, fragment detachment and density changes can invalidate the ideal front model.

Polyglycolic-acid chain scission and mass loss supplies a useful comparison: a molecular chain can break many times before material leaves the specimen. Establish the spatial mechanism before calling that change erosion in the IUPAC sense. A good mass fit alone is insufficient proof of an inward-moving front.