Thermal density fluctuations in amorphous polymers as revealed by small angle X-ray diffraction

Abstract

In the case of equilibrium the mean square relative fluctuations of the thermodynamic parameters vanish asymptotically as the number of degrees of freedom approach infinity. There are various observable effects, however, which are related to local fluctuations of the thermodynamic quantities within small parts of the macroscopic system. In particular the scattering of electromagnetic waves by a one-component, one-phase system is due to thermal density fluctuations within small volumes V of the sample. Considering a grand canonical ensemble the phenomenological theory of local fluctuations (1) for a one component system shows that the fluctuation of the number of particles N in the volume V is given by

$$\overline{{{N}^{2}}}-{{\bar{N}}^{2}}=kT{{\left( \partial \bar{N}/\partial \mu \right)}_{V,T}}$$

(1)

$$\overline{{{\left( N-\bar{N} \right)}^{2}}}/\bar{N}=\left( -kT\bar{N}{{\left( \partial V/\partial p \right)}_{N,T}} \right)/V=kTx\varrho $$

(2)

where ϰ is the isothermal compressibility, µ the chemical potential and ϱ the average number density.