Cylindrical Droplets Research Articles

Fluctuations and instabilities of curved interfaces

We study the stability of cylindrical and spherical interfaces with respect to density fluctuations within the square-gradient approximation. That is, we determine the stability matrix (of the second derivatives of the free energy functional with respect to the density) when the stationary state is a cylindrical or spherical droplet of a stable phase embedded in the metastable phase. For these geometries the stationary states are unstable, some of the eigenvalues are negative and their eigenfunctions represent those fluctuations that are amplified in a process where the equilibrium state is reached. At early times, in a simple model A kinetics, the eigenfunctions represent the fluctuations that grow or decay with a simple exponential law and with a characteristic time that is proportional to the inverse of the eigenvalues. Our results agree with the stability criteria obtained from the Laplace equation, that is, the nucleation of critical droplets and in the case of cylinders also the Rayleigh instability. In the limit of infinite radius we recover the known results for the planar interface between two stable phases. The modes with lowest energy correspond to the customary capillary waves, while other modes with higher energy associated to changes in the interfacial width are shown to be related to a novel interfacial coefficient.

ＳＯＬＡ‐ＶＯＦ法による板上液滴形状の数値解析

The purpose of this report is to predict the shape of the droplet on a solid surface through numerical analysis. The shape of the droplet is determined by the equilibrium of the forces of gravity and surface tension. The two-dimensional cylindrical droplet is analyzed numerically by SOLA (solution algorithm)-VOF (volume of fluid) method using the contact angle as the initial condition. The curvature of the droplet surface was calculated from the position of marker particles put on the droplet surface, subsequently, the droplet shape is obtained with high accuracy. The droplet shape on a solid surface predicted by this method is in good agreement with the photograph of the droplet as well as the contact angle. As the inclination angle of solid surface gets larger, both the front side angle and the back side angle of the droplet gradually changed, until finally, the droplet slipped along the slope. The droplet shape could be satisfactorily simulated by use of the SOLA-VOF method.

Extensional flow-induced crystallization of a polyethylene melt

Measurements of flow-induced orientation and crystallization have been made on a high-density polyethylene melt (HDPE) undergoing a planar extensional flow in a four-roll mill. The HDPE was suspended as a cylindrical droplet at the flow stagnation point in a linear low density polyethylene (LLDPE) carrier phase. Deformation and crystallization of the HDPE droplet phase were monitored using video imaging in conjunction with measurement of the birefringence and dichroism to quantify the in-situ transformation kinetics. Planar deformation rates along the symmetry axis of the molten HDPE phase were on the order of 0.03 s−1. Measurements of the initial transformation rate following flow cessation at 131.5°C and 133.2°C show a dependence on initial amorphous phase orientation and the total Hencky strain achieved during flow. The flow-induced crystallization rate is enhanced over the quiescent transformation rate by orders of magnitude, however, the dependence on temperature is less dramatic than expected for a nucleation-controlled growth mechanism. Analysis demonstrates that the melting point elevation model cannot account either qualitatively or quantitatively for the phenomena observed, suggesting that alternative explanations for the strong orientation dependence of the transformation rate are needed.