Fractal diffusion of silver ions in hollow cylinders with unsmooth inner surface

Abstract

This article studies the ion release from an unsmooth inner surface of a hollow fiber. A fractal diffusion model is established using the fractal derivative, and the effect of the fractal dimension on the ion release is elucidated. The present theory provides a theoretical basis for the optimization of a hollow fiber contained silver ions for practical applications.

Introduction

Hollow fibers with silver ions have wide antibacterial applications in various fields from biomaterials to tissue engineering. Silver ions are distributed on the inner surface of a hollow fiber and will be released when met with water. Hollow fibers also possess environment-friendly property, giving its considerable potential as a medical material; however, a safe and controllable release process has become a pressing issue in various applications. In practical applications, a fast release rate is much needed at the initial stage to kill fast and effectively viruses and bacteria in its environment, but afterward the ions should be used to prevent viruses and bacteria from reproduction for a long and reliable durability.

The ion release includes diffusion at the initial stage and the oscillation afterward.^1–4 Figure 1 shows a hollow fiber, and Ag ions are uniformly distributed on the inner surface of the fiber.

Figure 1.

Hollow fiber with unsmooth inner surface containing silver particles. The solid circles represent the silver ions. All ions are accumulated on the inner face of the fiber.

The release process takes up three steps when the fiber is immersed into water: (1) fast diffusion of ions from the inner surface to the center of the fiber, the ions are still kept inside of the fiber; (2) diffusion of ions near two ends to its environment to kill viruses and bacteria, this process is also fast and release rate is independent of fiber length; (3) oscillation release from the middle fiber gradually to its environment, this release rate of this process depends upon the frequency of the oscillation of the vibrating fluid inside the fiber.^4,5

The initial release rate mainly depends upon the fiber’s inner diameter and ion’s concentration on the inner surface; a smaller inner diameter or a higher concentration results in a faster release rate. This property is widely used to have an immediate effect on killing viruses and bacteria. However, an accurate prediction of the release rate is the main factor in practical applications. Liu et al.^5,6 suggested a fractional model for release oscillation and a fractal rate model for adsorption kinetics, the former was verified experimentally. ⁷

In this article, we will establish a fractal diffusion model for the ion diffusion at the initial release process.

Fractal diffusion

The diffusion equation due to concentration difference can be generally written as

$$\frac{\partial C}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r}\left(rD\frac{\partial C}{\partial r}\right)$$

(1)

where C is ion concentration, r is the circular cylindrical coordinate, t is time, and D is the diffusion coefficient.

Equation (1) with given initial and boundary conditions can be easily solved by Taylor method,^8–11 the variational method,^12–17 and the variational iteration method.^18,19 In this article, we assume that D is constant during all process of transport, that means that there are no molecular interactions between the diffusing molecule and the support.

For the steady case, equation (1) becomes

$$\frac{\partial }{\partial r}\left(rD\frac{\partial C}{\partial r}\right)=0$$

(2)

with the boundary conditions

$$C(r=0)=0\mathrm{and}C(r=R)=C_0$$

(3)

Equation (2) is the Fick’s law in a circular cylindrical coordinate system. The ions are distributed on the inner surface at r = R, where R is the inner radius of the hollow fiber, and we assume its initial concentration is $C(r=R)=C_0$ , while at the center of the fiber (r = 0), the initial condition for ion concentration is zero, $C(r=0)=0$ .

The general solution of equation (2) is

$$C=\frac{1}{D}\left(p\ln r+q\right)$$

(4)

where p and q are integral constants.

We cannot identify p and q by the boundary conditions given in equation (3) due to the singularity of the logarithmic function at r = 0. This paradox can be solved using the two-scale thermodynamics.^20,21 Actually the inner surface is not smooth enough (see Figure 1), so a continuum model with smooth boundary on a large scale leads to a wrong result; however, if we observe the problem on a smaller scale so that the unsmooth inner surface can be measured, the paradox can be completely solved by the fractal calculus,^22,23 which is to study various phenomena in discontinuous space, and has widely applied in electrochemistry, ²⁴ biomechanics,^25,26 Tsunami model, ²⁷ wool fiber, ²⁸ thermal insulation, ²⁹ fractal solitary wave, ³⁰ and fractal convection-diffusion model. ³¹

As shown in Figure 1, the inner surface of the hollow fiber is not smooth, and the inner boundary of the cross section is a coastline-like curve with fractal dimensions larger than 1, so the fractal calculus has to be adopted in our study. Equation (1) has to be modified as

$$\frac{\partial C}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r^{\alpha }}\left(rD\frac{\partial C}{\partial r^{\alpha }}\right)$$

(5)

where $\frac{\partial C}{\partial r^{\alpha }}$ is the fractal derivative. The fractal derivative at r = $r_0$ is defined as ²²

$$\frac{\partial C}{\partial r^{\alpha }}(r_0)=\mathrm{\Gamma }(1+\alpha )\underset{\begin{array}{l}r-r_0\to \mathrm{\Delta }r\\ \mathrm{\Delta }r\ne 0\end{array}}{\lim }\frac{C-C_0}{{(r-r_0)}^{\alpha }}$$

(6)

We consider the steady diffusion

$$\frac{1}{r}\frac{\partial }{\partial r^{\alpha }}\left(rD\frac{\partial C}{\partial r^{\alpha }}\right)=0,C(r=0)=0\mathrm{and}C(r=R)=C_0$$

(7)

Using the He-Li transform^32–34

$$r^{\alpha }=s$$

(8)

Equation (7) can be approximately converted into an ordinary differential equation. Hereby α is the fractal dimension of the perimeter of the fiber’s section. As the perimeter of a circle is minimal, α must be larger than 1.

Explanation of He-Li transform is given in the study by Ain and He ²⁰ , He and Ji, ²¹ and it is also called as the two-scale transform,^20,21 and α is the two-scale fractal dimensions.^20,21 By equation (8), we have

$$\frac{d}{ds}\left(s^{1/\alpha }D\frac{dC}{ds}\right)=0,0<s<R^{\alpha }$$

(9)

and the boundary conditions become

$$C(0)=0,C(R^{\alpha })=C_0$$

(10)

Integrating equation (9) with respect to s twice results in

$$DC=\frac{\alpha }{\alpha -1}P{s}^{(\alpha -1)/\alpha }+Q$$

(11)

where P and Q are integral constants, which can be identified by the boundary conditions given in equation (10). After the identification of P and Q, we have

$$C(r)=C_0{\left(\frac{r}{R}\right)}^{(\alpha -1)},1<\alpha <2$$

(12)

Release kinetics

The traditional release kinetics predicts that the released ion concentration scales with time at the initial stage ³⁵

$$C\propto \mathrm{\Delta }t$$

(13)

But in a fractal space, all variables scale with ${(\mathrm{\Delta }t)}^{\alpha }$ instead of Dt $\mathrm{\Delta }t$

$$C\propto {(\mathrm{\Delta }t)}^{\alpha }$$

(14)

where Dt $\mathrm{\Delta }t$ is the time scale.

The fractal release kinetics can be expressed as

$$\frac{\partial C}{\partial t^{\alpha }}=kC$$

(15)

The solution of equation (15) reads

$$C=C_0\exp (k{t}^{\alpha })$$

(16)

This is the fractal release kinetics. For $k{t}^{\alpha }<<1$ , equation (16) can be approximated as

$$C=k{t}^{\alpha }$$

(17)

While the traditional release kinetics is

$$C=kt$$

(18)

In order to verify the fractal property of the release process, an experimental was carried out, and the results are given in Figure 2. In the experiment, Ag/PET hollow fibers were used, which were soaked in a glass vial containing 2 L deionized water, the ion concentrations were measured at 0, 8, 16, 24, 72, and 168 h, respectively.

Figure 2.

Release kinetics. Dots are experimental data, the red line is the traditional release kinetics, and the green one is the fractal release kinetics.

By the experimental data, the release kinetics at initial stage can be expressed as

$$C=0.0535t^{0.817}$$

(19)

and

$$C=0.0535t$$

(20)

Comparison between the fractal release and traditional release is given in Figure 2; it is obvious the fractal release kinetics gives an accurate prediction at the initial stage, showing our fractal model is reliable. Please note equation (19) is only valid for small time, and only for comparison with the first-order release kinetics. For large time, equation (16) has to be used.

Discussion

The release process takes up three steps when the fiber is immersed into water: (1) fast diffusion of ions from the inner surface to the center of the fiber, the ions are still kept inside of the fiber; (2) diffusion of ions near two ends to its environment to kill viruses and bacteria, this process is also fast and release rate is independent of fiber length; (3) oscillation release from the middle fiber gradually to its environment, this release rate of this process depends upon the frequency of the oscillation of the vibrating fluid inside the fiber.^4,5

Differentiating equation (12) with respect to r, we have

$$\frac{dC(r)}{dr}=\frac{C_0(\alpha -1)}{R^{(\alpha -1)}}{r}^{(\alpha -2)},1<\alpha <2$$

(21)

When r tends to zero, we have

$$\underset{r\to 0}{\lim }\frac{dC(r)}{dr}=\underset{r\to 0}{\lim }\frac{C_0(\alpha -1)}{R^{(\alpha -1)}}{r}^{(\alpha -2)}\to \infty$$

(22)

That means that the release rate is extremely high at initial stage, while the diffusion rate (J) can be calculated as

$$J={D\frac{dC(r)}{dr}|}_{r=R}=\frac{D{C}_0(\alpha -1)}{R}$$

(23)

Equation (23) implies that the diffusion rate depends upon only the fiber radius; a smaller fiber predicts a faster diffusion process, so nanoscale hollow fibers by the electrospinning process^36–44 are much needed for a fast ion release at the initial stage.

Conclusion

To be concluded, this article, for the first time ever, establishes a fractal derivative model for ion release from a hollow fiber, which can overcome the inconsistency of the traditional release model, and the theoretical model can be used to optimize the fiber structure for practical applications. The initial release rate mainly depends upon the fiber’s inner diameter and ion’s concentration on the inner surface; a smaller inner diameter or a higher concentration results in a faster release rate. This property is extremely helpful for an immediate killing viruses and bacteria in its environment, but afterward the ions should be used to prevent viruses and bacteria from reproduction for a long and reliable durability, and the low frequency oscillation of the fluid inside the fiber plays an import role in the extremely slow release.^4,5