Introduction

A previous Etomica module (Interfacial Tension) presented the molecular basis for surface tension of a planar interface between coexisting liquid and vapor phases. This module extends the discussion to curvature of the surface of a liquid drop, and also to liquid-liquid interfacial tension. Interfacial tension (force per unit length) can be interpreted equivalently in units of an energetic penalty per unit interfacial area. With reference to the spherical reference shape that minimizes the surface area for a given volume of liquid contained inside, we will explore two consequences of this contractile tendency of the interface.

: 1. Static: resistance of a drop to a steady squeezing force that would otherwise deform it indefinitely.

: 2. Dynamic: relaxation of an initially deformed drop back the spherical shape.

In considering the behavior of a liquid nanodrop, we can distinguish between two alternative perspectives: a “grainy” atomistic (or microscopic) view that corresponds most closely with the description of matter in chemistry courses ($10^{-10}$ – $10^{-8}$ meter) versus a “smooth” continuum (or macroscopic) view that is more familiar from courses in thermodynamics and transport phenomena ($10^{-6}$ – 1 meter). The indicated ranges for microscopic versus macroscopic scales are only rough guidelines. Between them it is common to identify an intermediate mesoscopic scale ($10^{-8}$ – $10^{-6}$ meter).

Although the mesoscale is also vague in its definition, we can think of it operationally as the regime in which (1) there are so many atoms that some collective (e.g., thermodynamic) properties begin to emerge, and (2) there are not enough atoms to statistically cancel out temporal and spatial fluctuations due to their collisions. For example, a colloidal particle 50 nm (i.e., 0.05 $\mu$m) in diameter is large enough to be regarded as a solid particle in a continuum liquid, yet it still “feels” impacts of water molecules that cause its erratic, random walk (Brownian motion) that underlies diffusion. For both the underlying physics and mathematical modeling, the mesoscale poses particular theoretical challenges. This module contains two simulations: Atomistic versus Continuum. The Atomistic simulation is based upon the molecular dynamics (MD) method. The particles that you see on the screen really represent individual atoms of Argon. They crowd and jostle each other in the liquid phase, collide with each other in the gas phase, and exchange between the phases (local evaporation and condensation) in a dynamic equilibrium. Because of the sheer numbers of atoms involved, you can see why reaching out toward the mesoscale is so computationally intensive. For tractable simulations in real time, we use 2607 atoms, which corresponds to a radius of about 1.5 nanometers – as you will soon see. Atomic Lennard-Jones atomic interaction potentials are of very short range, so each particle significantly affects and is affected by only the first two or three shells of neighboring particles. This is an example of local interactions.

For the Continuum simulation we use an interesting grid-free numerical method that solves the Stokes equations of continuum, viscous-dominated flow – also by tracking the trajectories of particles. But here a single particle represents a lumped collection of an enormous number of atoms. Aside from the difference in meaning of the particles, what makes this model macroscopic is that the particles do not interact directly in a local fashion (jostling or colliding via inter-atomic potentials) like atoms do. Rather, each particle disturbs the surrounding continuum liquid with a characteristic streamline pattern (well known in the theory of hydrodynamics) and thereby pushes around all the other particles that are floating around in the same liquid. In other words, the particles interact indirectly through the intervening continuum liquid, and these viscous interactions are of such a long range that every particle affects every other particle. Thus the interactions are global. In both theory and experiment, a suspension of particles has been shown to accurately model a continuum liquid drop inside another continuum liquid of the same viscosity. Because the liquid droplet is modeled with a swarm of discrete particles (whether in the Atomistic or Continuum simulation), there is no “interface” in the macroscopic sense of a smooth, well-defined, infinitesimally thin mathematical surface separating the droplet phase from the surrounding fluid. Rather, what we recognize visually is a jagged zone of transition between high number density of particles packed closely together inside the droplet and low (or zero) density of particles outside. Mathematical techniques are used to extract an equivalent droplet deformation parameter based upon the coordinates of all the particles that collectively comprise the “droplet.” The Atomistic simulation allows you to subject the droplet to a steady squeezing force and then see how much it deforms. The Continuum simulation can be run in the same way, but it also lets you start off with a deformed droplet and see how quickly that shape snaps back to a sphere. The interfacial tension can be inferred from both categories of numerical experiments. The control panels for both simulations enable you to monitor the size and shape of the droplet as a function of time (in units of picoseconds). In this module you will learn about and apply the following concepts.

: 1.) Axisymmetric drop shape analysis (ADSA) as an indirect mathematical way of deducing interfacial tension without measuring forces directly. The shape of a drop is characterized by a dimensionless deformation parameter whose quantitative value (in the presence of static forces) or time trace (under the action of viscous flow) allows to be calculated. You will distinguish between prolate (rod-like) and oblate (disc-like) deformations.

: 2.) Dimensional analysis whereby continuum capillarity theory and Stokes flow can be suitable rescaled in both space and time for comparison with atomistic simulations for drops on the nanometer scale. You will see that continuum theory works unexpectedly well even at molecular scales.