Revisit to Swelling Kinetics of Gels

Abstract

The swelling and shrinking dynamics of the gels with a wide variety of aspect ratios is experimentally investigated. As the aspect ratio becomes sufficiently larger or smaller than unity, the swelling kinetics asymptotically approaches that of infinitely long cylinders or infinitely large disks, respectively. The characteristic times of the infinitely long cylinders and infinitely large disks are compared to the predictions of the classical Li-Tanaka model and its modified models.

Appendix

We summarize the expressions for τ _1,sphere, τ _1,cylinder and τ _1,disk in each model. The expression of τ _1,sphere is common to all models, and it is given by

$$\tau _{{\rm 1}} = \frac{{a_\infty ^2 }}{{D\nu _{{\rm 1}} ^2 }}$$

(\rm A1)

where a _∞ corresponds to the equilibrium diameter and υ _1,sphere satisfies the equation (A2).

$$4R\left[ {1 - \nu _{{{\rm 1,sphere}}} \cot (\nu _{{{\rm 1,sphere}}} )} \right] - \nu _{1,{{\rm sphere}}}^2 = 0$$

(\rm A2)

The quantities υ _1,cylinder ^(YD) and υ _1,disk ^(YD) for the Yamaue-Doi[7] model are the solutions of the equations (A3) and (A4), respectively.

$$\nu _{{\rm 1,cylinder}}^{{\rm (YD)}} J^\prime_1 (\nu _{{\rm 1,cylinder}}^{{\rm (YD)}}) + \left(1 - \frac{8R}{3}\right)J_1(\nu _{{\rm 1,cylinder}}^{{\rm (YD)}}) = 0$$

(\rm A3)

$$\nu _{{{\rm 1,disk}}}^{{{\rm (YD)}}} \cos (\nu _{{{\rm 1,disk}}}^{{{\rm (YD)}}} ) - \frac{4}{3}R{{\rm sin}}(\nu _{{{\rm 1,disk}}}^{{{\rm (YD)}}} ) = 0$$

(\rm A4)

where J ₁(x) is the Bessel function. We have derived the equation (A4) on the basis of their governing equations because the solution for large disk gels was not given in the original paper [7]. The values of τ _1,cylinder ^(YD) and τ _1,disk ^(YD) are obtained from (A1) assuming a _∞ as the equilibrium diameter of cylinders and the equilibrium height of disks, respectively. The values of τ _1,cylinder ^(LT) and τ _1,disk ^(LT) for the Li-Tanaka [3] model are given by the following equations:

$$\tau _{1,{\rm cylinder}}^{{\rm (LT)}} = \frac{3}{2}\frac{a_\infty ^2}{D\nu _{{\rm 1,cylinder}}}^{{\rm (LT)} 2}$$

(\rm A5)

$$\tau _{1,{{\rm disk}}}^{{{\rm (LT)}}} = \frac{{3a_\infty ^2 }}{{D\nu _{{{\rm 1,disk}}}^{{{\rm (LT)}}2} }}$$

(\rm A6)

where υ _1,cylinder ^(LT) and υ _1,disk ^(LT) are the solutions of the equations (A7) and (A8), respectively.

$$\nu _{{\rm 1,cylinder}}^{{\rm (LT)}} J^\prime_1 (\nu_{{\rm 1,cylinder}}^{LT}) + 2(1-2R)J_1 (\nu_{1,cylinder} ^{(Lt)})=0$$

(\rm A7)

$$4R - 2 - \nu _{{{\rm 1,disk}}}^{{{\rm (LT)}}} \cot (\nu _{{{\rm 1,disk}}}^{{{\rm (LT)}}} ) = 0$$

(\rm A8)

The expressions of τ _1,cylinder ^(WLH) and τ _1,disk ^(WLH) for the Wang-Li-Hu (WLH) model [6] are given by

$$\tau _{{\rm 1,cylinder}}^{{\rm (WLH)}} = \frac{2 + B_1 }{2}\frac{a^2}{D\nu _{{\rm 1,cylinder}}^{{\rm (LT)2}}}$$

(\rm A9)

$$\tau _{{\rm 1,disk}}^{{\rm (WLH)}} = \left( {1 + 2B_1^\prime} \right)\frac{a^2 }{D\nu _{{\rm 1,disk}}^{{\rm (LT) 2}}}$$

(\rm A10)

where B ₁ and B ₁′ are the quantities related to R as

$$B_1 = \frac{{2\left( {3 - 4R} \right)}}{{\alpha _1^2 - \left( {4R - 1} \right)\left( {3 - 4R} \right)}}$$

(\rm A11)

$$B_1^{\prime} = \frac{4}{{\alpha _1 }}\frac{{\left( {{{\rm sin}}\alpha _1 - \alpha _1 \cos \alpha _1 } \right)}}{{\left[ {2\alpha _1 - {{\rm sin}}(2\alpha _1 )} \right]}}{{\rm sin}}\alpha _{{\rm 1}} $$

(\rm A12)