Shubho Banerjee

Thermodynamic Limit for Dipolar Media

We prove existence of a shape- and boundary-condition-independent thermodynamic limit for fluids and solids of identical particles with electric or magnetic dipole moments. Our result applies to fluids of hard-core particles, to dipolar soft spheres and Stockmayer fluids, to disordered solid composites, and to regular crystal lattices. In addition to their permanent dipole moments, particles may further polarize each other. Classical and quantum models are treated. Shape independence depends on the reduction in free energy accomplished by domain formation, so our proof applies only in the case of zero applied field. Existence of a thermodynamic limit implies texture formation in spontaneously magnetized liquids and disordered solids analogous to domain formation in crystalline solids.

Elongation of confined ferrofluid droplets under applied fields

Ferrofluids are strongly paramagnetic liquids. We study the behavior of ferrofluid droplets confined between two parallel plates with a weak applied field parallel to the plates. The droplets elongate under the applied field to reduce their demagnetizing energy and reach an equilibrium shape where the magnetic forces balance against the surface tension. This elongation varies logarithmically with aspect ratio of droplet thickness to its original radius, in contrast to the behavior of unconfined droplets. Experimental studies of a ferrofluid-water-surfactant emulsion confirm this prediction.

How Multivalency Controls Ionic Criticality

To understand how multivalency affects criticality in z:1 ionic fluids, we report an ion-cluster association theory embodying ionic solvation and excluded volume for equisized hard-sphere models with z=1-3. In accord with simulation but contradicting integral equation and field theories, the reduced critical temperature falls when z increases while the density rho(c) rises steeply. These trends can be explained semiquantitatively by noting that 80%-90% of the ions near T(c) are bound in neutral or charged clusters, depleting the ionic strength. For z not equal 1, predicted interphase Galvani potentials vanish at T(c).

Orbital Motion of Electrically Charged Spheres in Microgravity.

The similar mathematical forms of Coulombs law and Newtons law of gravitation suggest that two uniformly charged spheres should be able to orbit each other just as two uniform spheres of mass are known to do. In this paper we describe an experiment that we performed to demonstrate such an orbit. This is the first published account of a successful orbit using electrostatic forces.

Exact closed-form solution for the electrostatic interaction of two equal-sized charged conducting spheres

We provide an exact closed-form solution for the electrostatic interaction of two equal-sized conducting spheres. We calculate the capacitance coefficients for the spheres in terms of the q-analogue of the digamma function. In the near limit, when the two spheres are about to touch, the closed-form exact solutions allow for much faster numerical calculations than the well-known infinite series solutions. By analyzing the exact solution in the near limit, we provide Taylor series expressions for the capacitance coefficients in terms of the surface-to- surface separation of the two spheres.

Asymptotic expansions of Lambert series and related q-series

Abstract. We study the Lambert series Lq(s, x) = ∑ ∞ k=1 k sqkx/(1 − qk), for all s ∈ C. We obtain the complete asymptotic expansion of Lq(s, x) near q = 1. Our analysis of the Lambert series yields the asymptotic forms for several related q-functions: the q-gamma and q-polygamma functions, the qPochhammer symbol, and, in closed form, the Jacobi theta functions. Some typical results include Γ2( 1 4 )Γ2( 3 4 ) ≃ 2 π log 2 and θ4(0, e−1/π) ≃ 2πe−π 3/4, with relative errors of order 10 and 10 respectively.We study the Lambert series ℒq(s,x) =∑k=1∞ksqkx/(1 − qk), for all s ∈ ℂ. We obtain the complete asymptotic expansion of ℒq(s,x) near q = 1. Our analysis of the Lambert series yields the asymptotic forms for several related q-series: the q-gamma and q-polygamma functions, the q-Pochhammer symbol and the Jacobi theta functions. Some typical results include Γ2(1 4)Γ2(3 4) ≈ 213/32π log 2 and 𝜗4(0,e−1/π) ≈ 2πe−π3/4, with relative errors of order 10−25 and 10−27 respectively.

On the stability of electrostatic orbits

We analyze the stability of two charged conducting spheres orbiting each other. Due to charge polarization, the electrostatic force between the two spheres deviates significantly from 1∕r2 as they come close to each other. As a consequence, there exists a critical angular momentum, Lc, with a corresponding critical radius rc. For L>Lc two circular orbits are possible: one at r>rc that is stable and the other at r<rc that is unstable. This critical behavior is analyzed as a function of the charge and the size ratios of the two spheres.

Thermodynamic Limit for Polydisperse Fluids

We examine the thermodynamic limit of fluids of hard core particles that are polydisperse in size and shape. In addition, particles may interact magnetically. Free energy of such systems is a random variable because it depends on the choice of particles. We prove that the thermodynamic limit exists with probability 1, and is independent of the choice of particles. Our proof applies to polydisperse hard-sphere fluids, colloids and ferrofluids. The existence of a thermodynamic limit implies system shape and size independence of thermodynamic properties of a system.

Shapes and textures of ferromagnetic liquid droplets

Theoretical calculations, computer simulations and experiments indicate the possible existence of a ferromagnetic liquid state. Should such a state exist, demagnetization effects would force a nontrivial magnetization texture governed by the shape of the liquid droplet. Since liquid droplets are deformable, the droplet shape couples to the magnetization texture. This paper solves the joint shape/texture problem subject to the assumption of cylindrical droplet symmetry. The shape undergoes a change in topology from spherical to toroidal as exchange energy grows or surface tension decreases.

Exact and Approximate Capacitance and Force Expressions for the Electrostatic Interaction Between Two Equal-Sized Charged Conducting Spheres

We analyze the electrostatic interaction between two equal-sized charged conducting spheres. We obtain exact closed-form expressions for the capacitance coefficients and the electrostatic force in terms of the special q -digamma function. Additionally, we provide simpler-to-use approximate expressions for the capacitance coefficients and the force between the spheres, and then compare the approximations with the exact results.