# Calculating the Diffusion Constant for Polymer Films using Dynamic Vapor Sorption

This article shows how to determine moisture diffusion coefficients in polymer thin films by measuring adsorption isotherms on the thin films in a DVS instrument.

Moisture diffusion in thin polymer films is of special interest to a wide range of industrial sectors, including membrane technologies and packaging materials.

## Method

Diffusion equations first utilized by Crank and Park are used to compute the diffusion constants for the thin films [1]. A thin film sample is suspended in the DVS instrument, and the sorption kinetics for a series of steps in humidity are recorded as usual. For a single step in humidity and a double-sided thin film of thickness d, the initial kinetics of sorption into the bulk may be described by the following equation.

 (1)

Where Mt = amount adsorbed at time t
M∞ = amount adsorbed at thermodynamic equilibrium
D = diffusion constant

Generally, the equation is valid for values of Mt/M∞ <0.4, where a plot of Mt/M∞ against t1/2/d should be linear. The diffusion constant D may then be calculated from the slope of this line.

In the present study, the diffusion of moisture into a 15 mg sample of 7.5 microns thick polyimide film was studied in a DVS instrument. The sample was gradually exposed to increasing steps in humidity from 0% RH to the desired humidity and back to 0% RH, so that both sorption and desorption steps could be measured for each discrete humidity. The data acquisition interval was set to 2 seconds as the kinetics were expected to be fast.

## Results

The sorption and desorption kinetics for the polymer film for steps from 0% RH to 20% RH, 40% RH and 60% RH are shown in Figure 1. The blue line indicates the humidity profile and the red line shows the change in mass referenced to the mass after drying.

Figure 1. Sorption and Desorption kinetics on a 7.5 µm polyimide film.

As shown in Figure 2, each discrete step in humidity Mt/M∞ is plotted against t1/2/d, and a least squares line of best fit is fitted to the initial slope of this plot for Mt/M∞ <0.4, indicated by the blue line. From this plot, values of the diffusion coefficient are calculated, and shown in Table 1.

Table 1. . Diffusion coefficients from initial slopes.

Previous RH (%) Target RH (%) Diffusion Coeff. (cm2/s) R-squared (%)
0.0 20.0 7.63E-10 99.55
20.0 0.0 4.38E-10 99.58
0.0 40.0 9.04E-10 99.52
40.0 0.0 6.05E-10 99.59
0.0 60.0 9.30E-10 99.54
60.0 0.0 6.55E-10 99.57

Figure 2. Diffusion plot for 0% RH to 20% RH step in humidity on a 7.5 µm polyimide film.

The tabulated data demonstrates that for steps from 0% RH to the specified RH, the diffusion constant increases slightly with the increasing partial vapour pressures of water. However, this increase is relatively small and it would seem that the diffusion constant does not strongly depend on the concentration of water vapor present.

## Conclusion

The above DVS methodology can be used rapidly to assess moisture diffusion in thin polymer films, such as membrane and packaging materials. Using the Advanced Data Analysis ad-in suite, the analysis of the experimental data for these experiments can be carried out to achieve fast and reliable evaluation of experimental data.

## Acknowledgement

SMS acknowledges the contributions of Mr. C. L. Levoguer and Mr. J. Booth, for the article developed.

## References

[1] Crank J. and G. S. Park. Diffusion in Polymers, Academic Press New York (1968).

This information has been sourced, reviewed and adapted from materials provided by Surface Measurement Systems Ltd.

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