E. Kinetic stabilization of thread breakup and glasslike phenomenology

Hammond [25] explained the slow growth rate of the "lobe bulging" instability of the wall fluid in the highly confined fluid regime (no thermodynamic fluid-surface interactions incorporated into the modeling) as being due to the slow rate at which the lubricating fluid surrounding the thread can "drain out" to allow the lobes to grow. A lubrication theory approximation and linearized hydrodynamic theory predicts that the rate of instability growth occurs on a time scale scaling as (R_tube η_m / σ)[(R_tube−R_thread) / R_tube]⁻³ where (R_tube−R_thread) / R_tube is small [see Eq. (7)]. In our units and notation, this corresponds to an instability growth rate of the confined system q(Λ) relative to the bulk rate q_∞ which scales as q(Λ) / q_∞ ~ [(Λ−1) / Λ]³, Λ ≈ 1. We see that the relative value of the instability growth rate vanishes as the thickness of annular liquid layer vanishes. This qualitatively explains the strong effect of confinement on thread breakup kinetics under high confinement, but this result of the linearized lubrication theory is untrustworthy from a quantitative standpoint. Hammond further suggested that plug formation should be inhibited for Λ ≈  1 since at some point there should be insufficient fluid for the lobes to grow large enough to pinch off to form a plug by simple volume conservation. Instead, he suggested the formation of periodic lobe structures along the tube axis or "unduloid surfaces" that ultimately become separated from each other due to the pinch-off of the wall fluid at points along the tube where the layer becomes critically thin. Measurements [25, 47] have not indicated the growth of such isolated fluid "collars" distributed regularly along stable fluid threads, raising questions about even the qualitative conclusions of the lubrication theory of thread breakup beyond early times. Subsequent numerical work by Gauglitz and Radke [47], based on a lubrication approximation, but with inclusion of nonlinear hydrodynamic effects relevant to describing the surface evolution at later times, indicated that thread rupture occurs beyond a "critical value of confinement," Λ* (Λ* ≈ 1.12 for "inviscid" threads, i.e., bubbles). For Λ less than this "critical" value, the fluid thread continuity is preserved, while for greater values rupture ensues in this treatment. Measurements of the breakup of (air bubble) threads by Gauglitz and Radke [47] indicated that plug formation is largely "suppressed" for Λ < Λ*(expt.) = 1.09. Moreover, thread breakup was found to be a "statistical phenomenon" in the accompanying measurements, becoming more infrequent and occurring at random along the tube for Λ, < Λ*(expt.). While thread breakup may occur after sufficiently long times, as suggested by the work Preziosi et al. [48], "effective stabilization" is obtained from a practical standpoint. Numerical studies of Newhouse and Pozrikidis [24] for viscosity-matched fluids (p = 1) indicate that an arrays of lobes form below Λ = 1.2. (This estimate of the "critical confinement parameter" Λ* is the most relevant to the present study since our fluids are likewise viscosity matched.) However, these lobes structures were found to be unstable to plug formation, which is the first stage of the end-pinch instability in our simulations. Thus, the stabilization of the threads is not an equilibrium phenomenon.

To gain insight into this regime of high thread confinement, we considered an additional simulation for the series shown in Fig. 5, where the tube radius was 12 and Λ = 1.4. In this case, we observe that the fluid thread remains stable up to a long time, t_red = 800, consistent with the confinement-induced "stabilization" effect indicated theoretically by Hammond [26] and experimentally by Gauglitz and Radke [47]. Because of limited resolution in the lattice fluid description and the finite interfacial width arising from the relatively weak fluid immiscibilty, we performed several other simulations to see if lattice effects were influencing the apparent "stabilization" phenomenon. For example, we considered Λ ≈ 1.28 for R_tube=25, 50, and 100. It was found that for R_tube=25 and 50, the system appeared very "stable" and we simply stopped the simulations at about t_red = 100 because it did not appear to be any significant evolution of these systems, but for R_tube = 100 there was a clear indication of the onset of thread breakup after a similar time period. While it is likely that the apparent stability of these highly confined systems is affected by the lattice discretization, it should also be noted that the interfacial width becomes increasingly small relative to the thread radius in this series of simulations and this could also be a factor in the pinning phenomenon. The influence of interfacial width on thread stability will be the subject of future research.

It seems relevant at this point to note that a critical value of Λ for thread stabilization near 1.2 has been observed for fluid threads subjected to flow in a tube [7]. The existence of such stabilization is supported theoretically where the stabilization conditions (range of Λ for which thread stability exists depends on flow rate, viscosity ratio, etc.) [23, 48], suggesting that the kinetic stabilization observed in the absence of flow can be converted into absolute stability under suitable flow conditions (see Sec. IV). The kinetic stabilization effect observed in our simulations apparently occurs over a comparable confinement range where genuine stabilization under flow occurs.

Simulation of thread stabilization by the LB method is difficult for highly confined threads (Λ 1.5) using the current method because the interfacial width between the coexisting phases starts to become comparable to the scale of confinement (distance between tube wall and thread surface). The LB method becomes strictly inapplicable when the confinement scale becomes comparable to one lattice spacing, corresponding to the physical coarse graining scale of the model. (This scale is approximately the interfacial width in an infinitely deep temperature quench and can be appreciable in polymer blends and other complex fluids where the particle dimensions and the associated coarse graining scale correlation length amplitude, ξ_o) are large [28].) To obtain further insight into the thread stabilization effect, we next consider how the time scale of thread breakup τ_B depends on Λ in a confinement regime where we are more confident in the method.

Two timescales are evident in our simulations of thread breakup, regardless of the changes in the character of the thread breakup process caused by finite size effects: (1) the time at which the fluid thread breaks τ_B, and (2) the "induction time" τ_I at which the boundary deformation α(t) first grows to 2% of R_thread so that the surface undulations are first appreciable". This latter time is defined somewhat arbitrarily, but it does capture the notion of the onset of the thread breakup process, while τ_B characterizes its end. Figure 10 shows τ_B (solid line) and τ_I (dashed line) in reduced time units as a function of Λ for the simulations shown in Fig. 4(a) ("tapped" thread; ε = 0.1). The data show a sharp increase of τ_B and τ_I with decreasing Λ, but it is not clear if the divergence of these times occurs for a critical value, Λ* > 1. The functional dependence of the increase in τ_B and τ_I is strong and, indeed, a power law in (Λ-Λ*) with a negative exponent does not seem to fit the data at all. Recognizing that this dramatic slowing down of the dynamics is due to the restricted "free volume" accessible for the displacement of the thread surface due to the confining tube, we then tried a function with an essential singularity to describe τ_B, as often employed to phenomenologically describe the strong T dependence of relaxation time data in glass-forming liquids,

where (Λ−Λ*)² corresponds to the mean-square particle displacement in the glass-forming liquid analogy of Eq. (9) [48]. The resulting fit led to a Λ* value close to the prediction of Newhouse and Pozrikidis [24], so we simply fix Λ* in Eq. (9) to this value, i.e., Λ* = 1.2. Similarly, the "dimensionless induction time" τ_I data fit this same function reasonably well,

where Λ* = 1.2 and the maximum residual is 2.1. The solid line and dashed lines in Fig. 10 represent the fits to Eqs. (9) and (10), respectively. The simulation data accord with the fits to within a maximum residual of 3.2 in reduced time units for the Λ range indicated. These empirical expressions provide only a convenient parametric description that gives some sense of the rapid rate at which the dynamics of the thread breakup slows down under confinement. Note that τ_B and τ_I become fairly constant over the confinement range Λ between 2 and 2.5, despite the fact that this is still in a Λ range where finite size effects have an appreciable effect on q(Λ) (see Fig. 4). The minimum in the data is probably a real effect associated with entering the weak confinement regime. This transition of regimes is also apparent in Fig. 4 where a substantial "kink" in the q(Λ) data occurs near Λ = 2.5. These expressions for τ_I and τ_B apply to only highly confined threads (Λ < 2.5).

FIG. 10. Induction time τ_I and thread breakup time τ_I vs Λ. Lines are fit to Eq. (9) and (10). The solid line and the dotted line are fits to the breakup and the induction times given by Eqs. (9) and (10), respectively. The relatively small difference between these times reflects the long induction times where the thread is "thinking" about breaking up.

Although there is some quantitative uncertainty in the data regarding the lattice discretization effect described above, there is no doubt that the breakup time τ_B becomes prohibitively long to observe in both simulation and measurement with high confinement. We refer to this nonequilibrium condition as "kinetic stabilization".

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