F. Fluid thread breakup under confinement: Single and multiple threads between parallel plates

We now briefly consider the influence of boundary confinement on fluid threads positioned between two parallel plates, where the confining effect can be expected to be weaker than in the tube geometry. This geometry is interesting in its own right and arises in experimental studies on confined polymer blends in phase separating films [49] and immiscible polymer blends subjected to flow in a Couette apparatus where the initial droplet size is comparable to the gap spacing of the instrument [20]. Preliminary LB calculations were performed of thread breakup under parallel plate confinement to contrast this type of confinement with the tube case. These computations are briefly compared to measurements for this geometry in a separate paper where the focus was on aspects of thread breakup that are difficult to observe experimentally [21]. This subsection considers simulations for this geometry from the separate perspective of how tube confinement compares to confinement by parallel plates.

The confinement parameter Λ_plate for the parallel plate confinement geometry is naturally defined as the ratio of the plate gap distance H to the thread diameter. For Λ_plate ≡ H / (2R_thread) = 1.2, we find that the thread remains stable for a long time (t_red =200), while for Λ_plate 2 the capillary breakup is not qualitatively changed from the bulk case. We show a stage of thread breakup in Fig. 11 for an intermediate value of confinement, Λ_plate = 1.43( = 5 x 10⁻⁴). The side view in Fig. 11 gives a perspective of the capillary breakup process, corresponding to looking into the gap between the plates and in a direction normal to the thread orientation. The presence of the boundary attenuates the largest amplitude deformations at this stage of thread breakup as lubrication forces resist the approach of the perturbed thread surface towards the tube wall [21]. This leads to a "blunting" of the undulations on the thread into a more square-shaped waveform, an effect discussed in some detail by Son et al. [21]. We also observe that the thread has a nearly elliptical shape near the maximum amplitude undulations, while the thread cross-section is nearly circular in the thinned portions of the thread which are far from the tube walls [21]. A view of the thread breakup process from above the plates is shown in the top view. This is the usual experimental perspective [20, 21]. Large amplitude growth regions in this projection have a more circular shape and the amplitude of these deformations is largest in regions where the lubrication forces slow the growth [21]. Finally, a profile view of the distortion in the undulating thread is shown in Fig. 11.

FIG. 11. Thread breakup between parallel plates under intermediate confinement conditions. LB simulation of a fluid thread having a diameter D = 106 (in units of lattice spacing), confined between two parallel plates having a gap width H = 232. The thread length is 600 and the width of the plates in the orthogonal direction is 480, where the boundary conditions are taken to be periodic along both these directions. The extent of confinement H/D is equal to 1.45 and t_red = 78.5. Side view and end-on views at points of maximum and minimum deformation from initial cylindrical shape are shown.

Comparison between the tube and parallel plate simulations indicates similarities in the thread breakup process, although evidently a greater confinement is required to achieve comparable finite size effects in the parallel plate geometry. For Λ_plate 2, the breakup geometry is bulklike, while for Λ_plate < 1.2 the process is kinetically too slow to be observable on the time scales of our simulations. For intermediate confinement (e.g., Λ_plate = 1.5), an unstable distorted droplet ("capsule") morphology forms. These findings accord well with the measurements of Son et al. for polymer threads confined between parallel plates [21] and further results are briefly summarized below after discussing the LB simulations. Here we do not investigate whether a transition to nucleation thread breakup occurs for high confinement (1.2 < Λ_plate), as in the threads in highly confined tube, because of the time consuming nature of these simulations. However, we do anticipate a "kinetic stabilization" of the capillary instability thread breakup process and the emergence of a new mode of thread breakup at high confinement, as in the thread confined to a cylinder. Notably, the parallel plate geometry allows for the novel situation in which Λ_plate < 1, where the cylinder must be distorted by the boundary at the outset (more of a "ribbon" than a thread). Such structures can also be considered as extended fluid plugs. Recent measurements have shown that these structures are stable under both quiescent and shear flow conditions [20]. This stability is natural given that ribbons do not break up by capillary instability in two dimensions [20, 49, 50]. Next, we highlight some recent measurements on thread breakup in a parallel plate geometry that are relevant to our simulations and discussion. Son et al. [21] find "stabilization" against thread breakup for strongly confined threads (Λ_plate < 1.3) in the case of nylon-6 fluid threads breaking up within a confining polystyrene matrix. This finding is reminiscent of the thread breakup measurements of Gauglitz and Radke [47] for confining tubes and the simulations of threads confined to tubes. Notably, the highly confined threads of Son et al. [21] appeared to be stable over a time scales on the order of a day. However, these measurements indicate that finite size effects on the rate of capillary instability growth q(Λ_plate) are weaker than for the case of tube confinement, for the same extent of confinement. Specifically, the finite size effects apparently saturate for Λ_plate ≈ 4−5 in the plate geometry, which a factor of 2 less than for the tube geometry. Based on these striking observations of thread "stabilization" under confinement, Son et al. [21] present a simple geometrical model of thread deformation between plates indicating that this kind of confinement leads to a thermodynamically stable state when Λ_plate is sufficiently small (i.e., Λ_plate 1.3). (Notably, these arguments neglect consideration of fluid-surface interactions, which are likely relevant to determining thread stability under general circumstances.) Regardless of the exactness of this geometrical argument, there is no question that confinement leads to effective thread stabilization over very long time scales and that the thread distortion on which their argument is based actually occurs.

In summary, the "extent of confinement" (1/Λ) must be larger in the parallel plate geometry than for the tube to achieve the same relative effect on the slowing down of the thread breakup kinetics. This trend is natural given that the tube involves confinement along two (orthogonal) directions, while confinement occurs only along one direction in the parallel plate geometry.

We also briefly consider some aspects of thread breakup in confined geometries that arise when the confining boundary is flexible. Our simulations are partly motivated by our previous LB calculations [28] that showed a tendency of adjacent threads, formed under phase separation and steady flow, to undulate out of phase. (This remarkable "string phase" has been observed experimentally [51−55].) It was our impression that these collective inter-thread interactions could be crucial for understanding thread ("string") stability since isolated threads generally disintegrate under steady shear flow. We were also influenced by observations of strong inter-thread interactions in measurements modeling thread breakup in extruded polymer blends [57−60]. These fragmentary observations suggested that a "flexible" confining boundary can substantially influence thread breakup under confinement, and we thus considered a simple model to explore this effect.

Following the experiments of Elemans et al. [57], we consider a parallel array of threads confined between two parallel plates under weak confinement (Λ_plate = 2.49) so that the interthread interactions are the primary source of confinement. The initial spacings between the threads was chosen to be equal to 1.7 times the thread diameter so that the interthread interactions are moderate. We use five threads in our simulation system and extend this computational cell periodically into the plane of confinement. Figures 12(a) and 12(b) show early and late stages of interacting thread breakup starting from random initial perturbations of the thread, where the method of applying the random perturbations and their magnitude ( =5 x 10⁻⁴) is the same as described for the tube case described in Sec. III C. First, we find that there is a long period of time occurs over which the growth of perturbations is suppressed by interstring interactions. (Elemans et al. [57] have observed a thread interaction induction time of this kind experimentally.) This long-lived, transient regime is followed by a relatively rapid thread breakup process during which the threads undulate out of phase with each other [Fig. 12(a)]. Once the instability starts, it develops rapidly and collectively, leading to the formation of a fairly regular droplet array [Fig. 12(b)]. Measurements of this multiple thread instability for polymer blend threads are in progress [60], but for the present we note the similarity of Fig. 12 to previous experimental observations [56]. An interesting and as yet unexplained problem is the process by which the thread undulations phase lock before breakup. The long induction time before thread breakup is apparently related to this phase locking phenomenon which has also been reported in the motion of microorganisms [61].

For an inter-thread spacing less than or equal to 1.5 thread diameters, the threads tended to fuse at apparently random points and subsequently formed droplets that were substantially larger than those in Fig. 12(b). This phenomenon is reminiscent of the transition to nucleated-rupture for threads subjected to high tubular confinement. Preliminary results indicate that once rupture occurs, it tends to propagate outward from its source in a wavelike fashion. This is apparently a two-dimensional generalization of the end-pinch instability occurring for tubular confinement. (We note that morphologically similar two-dimensional, wavelike instabilities are seen in the dewetting of thin films [62−67], suggesting another way to think about these instabilities.)

FIG. 12. Thread breakup for a thread confined between parallel plates and a periodic array of surrounding threads. The ratio of the inter-thread center spacing to the thread diameter at initial time is 1.7. The image shows a top view of plates at early stages of the thread breakup process. Simulation values of t_red are indicated in the figure.