# Coefficient of Thermal Expansion in contrained films

Hello,

I am trying to determine the CTE of an polymer by coating a thin film (~1 micron) onto a thick substrate (1 mm) and observing the thin film thickness change as a function of temperature.  I have the ability to measure the film thickness and temperture very accurately, but I am unsure how to calculate the CTE from the resulting data. The complication arises because the thin film is constrained to the substrate surface due to adhesion.  The thin film and substrate can only move independently by changing thickness.  How do I go about calculting CTE from this "contrained thin film" data?

Thanks for the help,

Frank Palmieri

### a simple thermoelastic solution

If the polymer is incompressible, the rate of thickness change is identical to the rate of volume change, from which you should be able to determine the CTE (assumed to be isotropic).

If the polymer is compressible, you also need to know Poisson's ratio (nu): CTE = (dh/dT)/h * (1-nu)/(1+nu).

RH

### correction

Oops, I forgot the CTE of the substrate. The correct solution should be: CTE(film) = (dh/dT)/h * (1-nu)/(1+nu) + 2*nu/(1+nu)*CTE(substrate).

Take nu = 0.5 for incompressible cases.

RH 