The polymer surface at a microscopic level

We now turn to the question of what we can say about the structure of the polymer surface at the microscopic level. As we have seen, in addition to predicting the surface tension, square gradient theories also predict the density profile at the surface of the polymer melt. Typically, the density goes smoothly from the melt density to zero (the density of the vapour phase being vanishingly small for high polymers) over a few angstroni imits, in very much the same way as it does at the surface of a small-molecule liquid. 

There have not been many systematic studies of the degree to which the surface of a polymer is diffuse on this scale, although neutron and x-ray reflectivity measurements are sensitive enough to provide this information. What observations exist are consistent with an interface whose width is about 5 

We can now find the total contribution of the capillary waves to the roughness of the interface by integrating the mean squared displacement due to all the possible capillary waves. The munber of possible waves is controlled by a quantisation condition if we consider standing waves in a square of sides I, we know that the jc and y components of the wave-vector must be integral multiples of JtlL. Thus the density of possible standing wave states in /t-space is Ajn, and the total number of states whose wave-vector has a magnitude between k and A + d is AkAk/(2ji). Thus we find 

The integral does not converge, so we need to do the integration between maximum and minimum values of k, kmax and k , the choice of which we discuss below. The result, then, is 

In addition to affecting the observed interfacial width, these capillary waves also have an associated free energy. The magnitude of the correction that this implies for the surface tension is given by the expression 

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