Origins of the failure of classical nucleation theory for nanocellular polymer foams

Abstract

Relative nucleation rates for fluid bubbles of nanometre dimensions in polymer matrices are calculated using both classical nucleation theory and self-consistent field theory. An identical model is used for both calculations showing that classical nucleation theory predictions are off by many orders of magnitude. The main cause of the failure of classical nucleation theory can be traced to its representation of a bubble surface as a flat interface. For nanoscopic bubbles, the curvature of the bubble surface is comparable to the size of the polymer molecules. Polymers on the outside of a curved bubble surface can explore more conformations than can polymers next to a flat interface. This reduces the free energy of the curved interface which leads to a significantly smaller barrier energy to nucleation and thus a much higher nucleation rate. Also, there is a reduction of unfavorable energetic contacts between polymer and fluid molecules in the vicinity of a curved interface. Polymers on the outside of a curved interface are less likely to find a portion of themselves in the interior of the unfavorable fluid bubble. A secondary cause of the failure of classical nucleation theory is due to the collapse of the bulk region inside the bubble. As the radius of a bubble is reduced, eventually the diffuse walls collide causing increased mixing of polymer and fluid molecules everywhere. This causes a reduction of internal energy associated with the interface, leading to smaller nucleation barrier energies and, again, a reduced barrier energy to nucleation.

Bubble nucleation in polymer–CO2 mixtures

“…Some of the structural features also cannot be captured using crude theories based on incompressible, pseudo-one-component treatment of the polymer-CO 2 mixture. 32 At the very early stage of bubble formation ( Fig. 6(b)), the total density proles exhibit a slight oscillation.…”

“…The key features reported in this work are incapable of being captured by either the classical nucleation theory or previous self-consistent eld theory based on an incompressible, pseudo-single-component model for the mixture. 32 The combination of the string method and accurate DFT opens the way for studying nucleation in complex polymeric systems, such as polymer-CO 2 mixtures in the presence of nanoparticles 36 and polymer-polymer-CO 2 mixtures. 37 …”

Maximal cell density predictions for compressible polymer foams

“…The hole-based SCFT that we use in this work was first introduced by Hong and Noolandi [8]. As there are many very good reviews of SCFT, such as references [9] and [10], we will not present the details of SCFT here but rather refer the reader to our previous works for incompressible [2,3] and compressible [6,11] polymer foams. The SCFT model for a compressible polymer solvent system can however be summarized by the free energy functional…”

“…The SCFT equations are solved numerically as described in previous work [3,6,7,11]. For any given hole volume fraction and box size, the sample pressure can be calculated using the formula…”

“…Polymer foaming is after all a non-equilibrium process and the SCFT method is an equilibrium model. Nonetheless, one can use SCFT for polymer foaming as has been argued in references [3] and [4]. Our method was not quan titative however but rather provided only qualitative guidance such as finding that foaming should take place at as low a practical temperature as possible, that the maximum amount of blowing agent that doesn't cause spinodal de composition should be used, and that polymer surface tension is not always an important factor in predicting cell densities [2].…”