Nicholas Goble Department of Physics, Case Western Reserve University, Cleveland, Ohio Equilibrium shape of a dielectric droplet in an electric field Abstract Strong electric fields beyond a threshold cause droplets of dielectric fluid to form conical ends. The equilibrium shape of these droplets has eluded an exact solution for over a century. However, we know that the shape is dictated by the competition between the surface tension and the dielectric contribution to the free energy. Our efforts embody a numerical approach in solving theses equations resulting in a series approximation for the shape of a droplet with a given dielectric constant in a uniform electric field. We are able to find the nature of the conical angle, verify the presence of hysteresis, and confirm that the development of pointed ends is rapid and abrupt. Introduction It is well established that droplets of dielectric will form conical, pointed ends when exposed to in a uniform electric field. However, a solution for the equilibrium shape has proved to be elusive. Rosenkilde [1] investigated the equilibrium shape by assuming the droplet took the shape of a prolate spheroid. Bacri et al. [2] found an approximation by using experimental data fitted to spheroids. Most recently, Li et al. [3] took a semi-analytical approach to the problem by using a local-force balance and global-energy argument. Although these approaches yielded respectable results, their use of an assumed shape, by nature, do not consider all possible shapes. We consider the problem by using a polynomial to approximate the shape of the droplet, which is used to calculate the opposing electrostatic and surface energies. The equilibrium shape is then determined by minimizing the total energy. The motivation for this research stems from the fact that experimentation in microgravity environments has recently become available. In addition, these calculations can be applied to more complex systems, such as suspensions of droplets of nematic liquid crystal in a polymerizing solution. Our results should have significance for the design of ink-jet printers, the electrospinning of polymers, and electrically switchable privacy windows. Weak Field Strong Field Problem: At what field does the droplet become pointed? or Results We find that the coefficient is zero when the droplet becomes pointed. As the field increases, suddenly jumps to 0, which confirms that the formation of conical ends is abrupt. The term n refers to the number of intervals over which is integrated. By changing the location at which the minimization of begins, we can detect local minima in the function. When multiple minima exist, the droplet becomes unstable and hysteresis occurs. We also studied how the angle of a pointed droplet droplet changes with respect to the applied field. Surprisingly, this relationship is nonlinear. Our data suggests that there is a particular angle to which the droplet approaches. It is worth noting that as we increase n, the precision of our calculations increases. We have found that a bin count of 5, 10, and 20 will calculate with relative uncertainties of 6.70%, 1.13%, and 0.19%, respectively. Acknowledgements This project was supported by the NSF REU grant DMR , and was done in collaboration with Professor Philip Taylor at Case Western Reserve University. References [1] C. E. Rosenkilde, Proc. Roy. Soc. A 280, 211 (1964). [2] J. C. Bacri and D. Salin, J. Physique Lett. 43, 649 (1982). [3] H. Li, T. C. Halsey, and A. Lobkovsky, Europhys. Lett. 27, 575 (1994). For any given, a total energy can be calculated using: The equilibrium shape of the droplet is that at which the energy is lowest. Therefore, minimizing the two-variable function, gives us the equilibrium shape of the droplet. We minimize with a gradient search method. The coefficients and converge to their minimum by obeying the algorithm: Once the values for and are calculated, the equilibrium shape has been found. Methods The shape of the droplet is defined by. However, it is convenient to use. The electrostatic energy is defined as: where is the uniform electric field,, is the dielectric constant of the fluid, and is the electrostatic potential on the surface of the droplet. The potential of the dielectrics surface is expressed as: where is the observation point and is the integration variable. The surface energy of the droplet is defined as: Conclusion It is well established that droplets of dielectric will form conical points in an electric field. For the first time, the equilibrium shape of a dielectric droplet in such a system has been calculated. The development of pointed ends, hysteresis, and endpoint angle have been analyzed. Future work includes using a fourth-order polynomial for the shape of the droplet and finding a critical dielectric constant below which no pointed ends form.