Diffusion coefficients of water and leachables in methacrylate-based crosslinked polymers using absorption experiments

Abstract

The diffusion of water into dentin adhesive polymers and leaching of unpolymerized monomer from the adhesive are linked to their mechanical softening and hydrolytic degradation. Therefore, diffusion coefficient data are critical for the mechanical design of these polymeric adhesives. In this study, diffusion coefficients of water and leachables were obtained for sixteen methacrylate-based crosslinked polymers using absorption experiments. The experimental mass change data was interpreted using numerical solution of the two-dimensional diffusion equations. The calculated diffusion coefficients varied from 1.05 × 10⁻⁸ cm²/sec (co-monomer TMTMA) to 3.15 × 10⁻⁸ cm²/sec (co-monomer T4EGDMA). Correlation of the diffusion coefficients with crosslink density and hydrophilicity showed an inverse trend (R ² = 0.41). The correlation of diffusion coefficient with crosslink density and hydrophilicity are closer for molecules differing by simple repeat units (R ² = 0.95). These differences in the trends reveal mechanisms of interaction of the diffusing water with the polymer structure.

Appendix

The diffusion of leachables is explained in a similar way to water diffusion. The governing equations for leaching are given in terms of the local flux $(\vec{j})$, the diffusion coefficient $(E)$ and the concentration of leachable $\nu$ as follows:

$$ \overrightarrow {j} = - D_{\text{L}} \overrightarrow {\nabla } v $$

(21)

$$ \frac{\partial v}{\partial t} + \overrightarrow {\nabla } .\overrightarrow {j} = 0 $$

(22)

$$ \nabla^{2} v \approx \frac{{v_{i + 1,j}^{t + 1} - 2v_{i,j}^{t + 1} + v_{i - 1,j}^{t + 1} }}{{\left( {\Updelta x} \right)^{2} }} + \frac{{v_{i,j + 1}^{t + 1} - 2v_{i,j}^{t + 1} + v_{i,j - 1}^{t + 1} }}{{\left( {\Updelta x} \right)^{2} }} $$

(23)

$$ \frac{\partial v}{\partial t} \approx \frac{{v_{i,j}^{t + 1} - v_{i,j}^{t} }}{\Updelta t} $$

(24)

$$ - \beta_{x} \left[ {v_{i - 1,j}^{t + 1} + v_{i + 1,j}^{t + 1} } \right] + \left[ {1 + 2\beta_{x} + 2\beta_{y} } \right]v_{{_{i,j} }}^{t + 1} - \beta_{y} \left[ {v_{i,j - 1}^{t + 1} + v_{i,j}^{t + 1} } \right] = v_{i,j}^{t} $$

(25)

$$ \beta_{x} = \frac{{D_{\text{L}} \Updelta t}}{{\left( {\Updelta x} \right)^{2} }} $$

(26)

$$ \beta_{y} = \frac{{D_{\text{L}} \Updelta t}}{{\left( {\Updelta y} \right)^{2} }} $$

(27)

Assembling the system of discretized equations gives the form shown below.

$$ KD_{{{\text{L}}_{ij} }} v_{j}^{t + 1} = v_{j}^{t} $$

(28)

KDL_ij are the diffusion (leachable) multipliers obtained from Eq. 25.