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Introduction

It is well-known that a long liquid thread surrounded by an immiscible liquid matrix exhibits sinusoidal distortions, which grow and cause the thread to break up into a row of smaller droplets.^1,2 For example, in polymer blending, threads are formed by flow, and their subsequent disintegration occurs in either the presence or absence of the flow.³ Thus, a fundamental understanding of this phenomenon is important to predict the size and morphology of the final polymer blends.

The original studies by Rayleigh and Tomotika concern the case where a fluid thread is surrounded by another fluid of infinite extent. As discussed below, a linear stability analysis shows that the thread is unstable because fluctuations for which > 2R₀ grow exponentially, where is the wavelength of a given fluctuation and R₀ is the initial thread radius. Recent progress in this area stems from the construction of linear stability analyses for more complex cases, such as when the interface between the immiscible fluids is covered with an insoluble surfactant,^4,5 or to consider the effects of elasticity at the interface.⁶ These cases are also relevant to bioengineering applications.

The effects of experimental geometry on the Rayleigh-Tomotika instability can be profound. In the two-dimensional case where the stability of a flat ribbon is investigated, it was theoretically shown by Miguel⁷ that thermal fluctuations decay; intuitively, this can be understood by the observation that the fluctuations cause an increase in surface area. The case of a liquid annular coating on the inside or outside of a cylinder was investigated by Goren.⁸ Newhouse and Pozrikidis⁹ studied numerically the case where the thread of radius R₀ encompasses a matrix fluid which is confined to a tube of radius R_t. In the limit R₀/R_t 1 they recover the results of Rayleigh-Tomotika; as R₀/R_t increases, the thread still breaks into an array of alternating large and small droplets and the dominant wavelength of instability does not change significantly, but the amplitude growth slows down significantly and the shape of the axisymmetric fluctuation changes. When R₀/R_t > 0.82 the thin outer layer evolves into an array of lobes or collars. Their case represents confinement in an axisymmetric geometry.

In a recent paper by Migler,¹⁰ a combination of simple shear and confinement between parallel plates was utilized to generate threads (called "strings" in that work). Upon cessation of flow, threads of sufficiently large diameter were found to be stable with respect to the capillary instability. That experiment then considers a different geometry from those considered previously: a thread confined between parallel plates, which involves both confinement and nonaxisymmetry. This regime is important because there is great current interest in micro- and nanolength scale technologies in which polymer blends can play an important role,^11,12 but the understanding of the processing of polymer blends when the size of the minor phase is comparable to a sample dimension is poor.

A second application of thread breakup between parallel plates occurs during the measurement of interfacial tension of polymers.^13-16 To ensure that the elastic effect of polymer is negligible in the interfacial tension measurement, the growth rate of the distortion must be sufficiently low because Tomotika's theory assumes that both liquid phases are Newtonian fluids. As the kinetics are inversely proportional to a characteristic length scale, bigger threads are experimentally desirable. However, as the size of the thread increases and becomes comparable to the gap width, the hydrodynamic interaction between the drop and the wall may not be negligible. A study of the thread instability on the confined regime can provide a guide for exact interfacial tension measurement. Finally, our work is relevant to two-phase microfluidics, as it demonstrates a strategy whereby one may be able to stabilize liquid threads.

In the present study, we examine in detail the effects that occur when the thread diameter is comparable to the gap between two confining parallel plates via a combination of experimental, lattice-Boltzmann (LB) numerical simulations, and surface area calculations. The experimental work demonstrates the basic phenomena of how the confinement acts to inhibit the instability. The lattice-Boltzmann simulations agree qualitatively with the experiments and allow visualization in directions that are experimentally inaccessible. The surface area calculations show that if the confinement causes the thread to grow nonaxisymmetrically (somewhat flattened as depicted in Figure 1), then the critical wavelength increases and the driving force for the instability (the reduction of surface area) decreases. The lattice-Boltzmann simulations validate key assumptions of the surface area calculations.

Figure 1 Schematic view of a thread undergoing a nonaxisymmetric capillary instability. The fluid is unconfined in the experimentally observable top view (x-z plane). The side view shows how the matrix fluid is confined in the y direction by parallel plates that are separated by a distance H.

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