^{1,a)}and Gary W. Slater

^{1,b)}

### Abstract

Using simple theoretical arguments and exact numerical lattice calculations,Hickey *et al.* [J. Chem. Phys.124, 204903 (2006)] derived and tested an expression for the effective diffusion coefficient of a probe molecule in a two-phase medium consisting of a hydrogel with large gel-free inclusions. Although providing accurate predictions, this expression neglects important characteristics that such two-phase systems can present. In this article, we extend the previously derived expression in order to include local interactions between the gel and the analyte, interfacial effects between the main phase and the inclusions, and finally a possible incomplete separation between the two phases. We test our new, generalized expressions using exact numerical calculations. These generalized equations should be a useful tool for the development of novel multiphase systems for specific applications, such as drug-delivery platforms.

The authors would like to thank Derick Rousseau and his group for useful discussions and suggestions and for providing Figs. 1(a) and 7(a). This work was supported by a scholarship from the Ontario Graduate Scholarship Program (OGS) to C.K. and a Research Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) to G.W.S. We are also grateful to the High Performance Virtual Computational Laboratory (HPCVL) for providing access to computational resources. We finally would like to thank the Advanced Foods and Materials network (AFMnet) for conference opportunities.

I. INTRODUCTION

II. NUMERICAL METHOD

A. Diffusion: Homogeneous viscosity

B. Including viscosity heterogeneities

III. ATTRACTIVE GEL-PROBE MOLECULE INTERACTIONS

A. Theory

B. Results

IV. INTERFACIAL EFFECTS: OBSTACLE SHELLS

A. Theory

B. Results

V. GENERALIZATION FOR PHASES

VI. CONCLUDING REMARKS

### Key Topics

- Gels
- 64.0
- Diffusion
- 37.0
- Viscosity
- 34.0
- Phase separation
- 7.0
- Boundary value problems
- 4.0

## Figures

(a) Micrograph of a hydrogel continuous phase, 10% (w/w) gelatin containing roughly circular viscous inclusions [12% (w/w) maltodextrin (MD)]. Image courtesy of Derick Rousseau. (b) Example of a square lattice representation of the hydrogel shown in (a). MD inclusions are shown in gray. The gelatin phase is represented by a continuous phase (white) with a random distribution of impenetrable obstacles (black dots) representing the gel fibers. The gel concentration is modeled by the obstacle density.

(a) Micrograph of a hydrogel continuous phase, 10% (w/w) gelatin containing roughly circular viscous inclusions [12% (w/w) maltodextrin (MD)]. Image courtesy of Derick Rousseau. (b) Example of a square lattice representation of the hydrogel shown in (a). MD inclusions are shown in gray. The gelatin phase is represented by a continuous phase (white) with a random distribution of impenetrable obstacles (black dots) representing the gel fibers. The gel concentration is modeled by the obstacle density.

(a) Example of a 2D lattice with an obstacle concentration of 10%. Lattice sites are colored according to their potential energy, as explained in the legend. The section with a thick perimeter is represented in 3D on the right hand side of the figure. (b) Schematic representation of the well model for the section of the lattice shown in (a). The attraction felt by the probe particle is represented by the depth of the wells surrounding the obstacles.

(a) Example of a 2D lattice with an obstacle concentration of 10%. Lattice sites are colored according to their potential energy, as explained in the legend. The section with a thick perimeter is represented in 3D on the right hand side of the figure. (b) Schematic representation of the well model for the section of the lattice shown in (a). The attraction felt by the probe particle is represented by the depth of the wells surrounding the obstacles.

Normalized diffusion coefficient vs gel concentration for different values of the interaction parameter . The data points were obtained from exact 2D calculations and were averaged over 50 independent realizations; the error bars are smaller than the symbols. The system (see inset) is a 2D lattice of size with a single viscous inclusion and periodic boundary conditions. For the data points, the exact values of were calculated for each randomly generated system. Solid lines correspond to Eq. (8) with given by the approximation (9).

Normalized diffusion coefficient vs gel concentration for different values of the interaction parameter . The data points were obtained from exact 2D calculations and were averaged over 50 independent realizations; the error bars are smaller than the symbols. The system (see inset) is a 2D lattice of size with a single viscous inclusion and periodic boundary conditions. For the data points, the exact values of were calculated for each randomly generated system. Solid lines correspond to Eq. (8) with given by the approximation (9).

Schematic representation of the formation of the inclusions in the gelatin continuous phase. While the small maltodextrin bubbles coalesce to form large inclusions, the gelatin is slowly solidifying, making it harder to maintain a uniform gel concentration—especially around the perimeter of the inclusions. The second and third drawings show high-concentration gelatin shells around the large inclusions.

Schematic representation of the formation of the inclusions in the gelatin continuous phase. While the small maltodextrin bubbles coalesce to form large inclusions, the gelatin is slowly solidifying, making it harder to maintain a uniform gel concentration—especially around the perimeter of the inclusions. The second and third drawings show high-concentration gelatin shells around the large inclusions.

Normalized diffusion coefficient vs the gel concentration in the shell surrounding the inclusion for different values of the shell fractional volume . The data points were obtained from exact 2D calculations. The lattice system was similar to the third schematics in Fig. 4; each result is an average over 25 independent realizations and the error bars are smaller than the symbols. The viscosities were and . The lines were computed using Eq. (13). For the solid lines, the exact free volume fraction was used. For the dashed lines, the free volume fraction was estimated using the expression .

Normalized diffusion coefficient vs the gel concentration in the shell surrounding the inclusion for different values of the shell fractional volume . The data points were obtained from exact 2D calculations. The lattice system was similar to the third schematics in Fig. 4; each result is an average over 25 independent realizations and the error bars are smaller than the symbols. The viscosities were and . The lines were computed using Eq. (13). For the solid lines, the exact free volume fraction was used. For the dashed lines, the free volume fraction was estimated using the expression .

The difference between the calculated and predicted normalized diffusion coefficients vs the gel concentration in the obstacle barrier. Data points for two different system sizes are shown (these systems are similar to those used for Fig. 5, except for the size). The data points were obtained for all the values of reported in Fig. 5. Calculations for were also done and give results between those for and , as expected, but were not shown in order to improve clarity.

The difference between the calculated and predicted normalized diffusion coefficients vs the gel concentration in the obstacle barrier. Data points for two different system sizes are shown (these systems are similar to those used for Fig. 5, except for the size). The data points were obtained for all the values of reported in Fig. 5. Calculations for were also done and give results between those for and , as expected, but were not shown in order to improve clarity.

(a) Micrograph of a hydrogel composed of gelatin and maltodextrin clearly showing a case of incomplete separation. Small gelatin inclusions can be found in the MD zones, and vice versa. Image courtesy of Derick Rousseau. (b) The 2D lattice we used to test Eq. (14). Some gelatin inclusions with obstacles are found inside the main MD inclusion. Small MD inclusions can also be found throughout the gelatin main phase. Every gelatin phase has a different concentration and every maltodextrin phase has a different viscosity.

(a) Micrograph of a hydrogel composed of gelatin and maltodextrin clearly showing a case of incomplete separation. Small gelatin inclusions can be found in the MD zones, and vice versa. Image courtesy of Derick Rousseau. (b) The 2D lattice we used to test Eq. (14). Some gelatin inclusions with obstacles are found inside the main MD inclusion. Small MD inclusions can also be found throughout the gelatin main phase. Every gelatin phase has a different concentration and every maltodextrin phase has a different viscosity.

Graph of the normalized diffusion coefficient vs the main phase gel concentration for different values of the fractional volume of the main viscous inclusion . The following values were used: , , , , , , and . Solid lines correspond to Eq. (14). Calculations were done for a lattice. Data points are averages over 25 independent realizations. Error bars are smaller than symbol size.

Graph of the normalized diffusion coefficient vs the main phase gel concentration for different values of the fractional volume of the main viscous inclusion . The following values were used: , , , , , , and . Solid lines correspond to Eq. (14). Calculations were done for a lattice. Data points are averages over 25 independent realizations. Error bars are smaller than symbol size.

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