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Air permeability in hierarchic porous media does not obey Fick's equation or its modification because fractal objects have well-defined geometric properties, which are discrete and discontinuous. We propose a theoretical model dealing with, for the first time, a seemingly complex air permeability process using fractal derivative method. The fractal derivative model has been successfully applied to explain the novel air permeability phenomenon of cocoon. The theoretical analysis was in agreement with experimental results.

Air permeation and moisture vapor diffusion in textile fabric have a close relation to the comfort ability of clothing [

Clothing exhibits a multiscale inner structure constructed by textile fiber. The micropores among the thin fibers provide a winding passage for air and moisture vapor exchange between the microclimate (between the skin and fabric) and atmosphere. For certain textile apparel, the arrangement style of the fiber in the fabric may result in different air permeation efficiency. Recently, Blossman-Myer and Burggren investigated the water loss and oxygen diffusion through the silk cocoon. And the experimental data indicated that the cocoon does not have any impact on the oxygen and water vapor transportation [

As a transport property, mechanism of gas permeation in different porous media has received continuous attention due to its significance in both science and engineering [

In the past two decades, fractal theory has been introduced to solve problems of diffusion in media with hierarchic configuration since Mandelbrot’s pioneering work [

A number of fractal geometry models have been developed by many researchers. However, most of these models include continuous parameters, which were against the basic property of fractal geometry that fractal objects are discontinuous. As alternative approaches, fractional and fractal derivatives have been found effective in modeling permeation problem in hierarchic porous media [

In this work, a fractal model for air permeation in hierarchic porous media was developed based on a new fractal derivative theory. And the excellent gas permeation of cocoon was investigated with assistant of the novel fractal derivative model.

Chen [

As an alternative modeling formalism of fractional derivative, fractal derivative presents the fractal space-time transforms to display explicitly how the fractal metric space-time influences physical behaviors in statistical description of the anomalous diffusion in diverse engineering fields.

Recently, He [

The new fractal derivative can convert fractional differential equations to ordinary differential equations, so that nonmathematicians can easily deal with fractional calculus.

In continuous media, gas permeation obeys Fick’s equation:

When the concentration of gas does not change with time, the gas diffusion can be regarded as in one-dimensional steady-state case. Equation (

Equation (

Gas permeation in the discontinuous hierarchic porous media can be expressed as

According to the definition of fractal derivative

Submit (

Suppose the gas concentration of the inner side of a porous medium equals to that of the outer side of the porous medium, that is,

For (

For the case the smallest measuring size

If

If

To investigate the novel gas diffusion phenomenon of cocoon, we need first to analyze the structure of cocoon. SEM observation shows that the cocoon was composed of mainly three layers. Silk fiber in the out layer of cocoon wall was relatively thick with a mean diameter of 26

Schematic diagram of hierarchic gas diffusion path in cocoon wall.

In Figure

Considering the construction of cocoon, the bifurcation number equals to 2. Since the thickness of the cocoon is about 500

The value of (

The result obtained in this presentation can explain the experimental phenomenon observed by Blossman-Myer and Burggren. The gas permeation property of cocoon is attributed to the inner hierarchic porous configuration constructed by hierarchic assemble of silk fiber. The structure feature of cocoon provides us with an optimized template, which could be duplicated in biomimic fabric design to improve the heat-moisture comfort ability of apparel.

A fractal derivative model for gas permeation has been proposed for the first time in hierarchic space. This model is able to describe a complex dynamic process completely from the theory, which is of critical importance for biomimic design in any fields, such as industry and biomaterial, functional textiles. The validity of the model has been proved in explaining the fascinating gas diffusion phenomenon of cocoon discovered by scientific experiments.

The work is supported by National Natural Science Foundation of China under Grant no. 51203114 and 10972053, China Postdoctoral Science Foundation Grant no. 2012M521122, Natural Science Funds of Tianjin under Grant No. 12JCQNJC01500, Subject of Science and Technology Development Fund at Universities in Tianjin under Grant no. 20110317, Natural Science Funds of Zhejiang Province under Grant no. 2012C21039, and PAPD (A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions).

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