Next: Confined fluid thread breakup: Localized Up: Simulation Results Previous: Simulation Results
As a reference case and a further check of our LB mixture model of Newtonian fluid mixtures [28], we briefly consider capillary breakup of a fluid thread where finite size effects have a weak effect on the morphology of thread breakup. In Fig. 2(a), we show a progression of images of thread breakup as a function of tred. The tube radius is 24 (lattice spacings) and the length of the tube (and thread) in the simulation is 600. Periodic boundary conditions are applied along the axial direction. The initial thread radius equals Rthread=9.9, so that the tube length-thread aspect ratio (the length of tube L divided by Rthread) is approximately 60. This is about an order of magnitude larger than the Rayleigh-Plateau length (2 πRthread) for thread breakup in bulk so that we can consider the threads as "long." In Fig. 2(a) we show only a section of the simulation (about two wavelengths) that is comparable to the measurement image for a polymer thread in a polymeric matrix shown in Fig. 2(b) (experimental conditions are summarized below). The thread was perturbed by introducing random impulsive perturbation throughout the thread, as described in Sec. III C.
FIG. 2. Simulated and experimental fluid thread breakup under weak confinement. (a) LB simulation of a thread of radius 9.79 (in units of lattice spacing) confined to a tube of radius 24(Λ = 2.45) and a length of 600. This length corresponds to about ten Rayleigh-Plateau lengths, but the image only shows a section corresponding to a couple of instability wavelengths. The time units of the simulation are given in the reduced units of thread breakup in the bulk, tred (see the text). The walls of the tubes are omitted for visual clarity and to facilitate a comparison with the measurements in the accompanying figure. (b) Representative experiment of the thread breakup of a polymer thread in a polymer matrix (see the text). The time t is given in units of seconds. This measurement is for a thread confined between parallel plates and having gap to thread diameter ratio of 10.6 [21]. Confinement effects are weak in this measurement, so that these can be considered "bulk limit" observations.
The rupture of the fluid thread in Fig. 2(a) occurs through the growth of collective sinusoidal undulations about the original circular cylindrical thread, as in the schematic image shown in Fig. 1(b). At a late stage of this instability, the large amplitude regions of positive deformation (i.e., "bulges") are separated by thin, nearly circular filaments that break up by a secondary capillary instability, leading to the formation of satellite droplets [1, 33–36]. A whole hierarchy of droplet sizes can be created by thread breakup through a recursive occurrence of capillary instabilities to ever-finer scales [33, 35]. Treatment of these higher generations of the droplet breakup and the fine structure of the singular thread breakup morphology requires a finer discretization of the lattice model calculations. In our simulations, we observe only the leading order satellite droplets shown in Fig. 2(a). This is also often the case in measurements where various physical effects (surface tension, viscosity, non-Newtonian fluid characteristics, impurities) cut off this hierarchical instability. For example, the breakup of a polymer thread under weak confinement conditions shown in Fig. 2(b) exhibits only one well-formed "generation" of satellite droplets. (These measurements are actually performed for fluid threads under confinement between parallel plates, but the scale of confinement is so large that confinement effects are small.) We discuss these measurements further below, after further summarizing our results for the LB simulations under weak confinement.
We next quantify the growth rate of the capillary instability. In Fig. 3, we show a semi-log plot of the growth rate of the thread undulation log α(tred). The nonlinear increase of log αtred) at long times is associated with the thread rupture process and the data terminates at the time of rupture. The solid curve is a fitted "steady state" growth rate of the capillary instability and the dotted curve represents the simulation data for all times, including early and late stages where the instability growth is non-exponential. Although the size of the confining tube is sufficiently large that confinement effects do not have an appreciable influence on the geometry of the thread breakup process, the confinement is sufficient to influence the rate of thread breakup (see below). This situation is evidently similar to measuring fluid viscosity by studying the sedimentation of a sphere in a capillary [37] or the Brownian motion of a sphere in a capillary where finite size effects act over appreciable distances to affect particle mobility [38]. A typical rule of thumb is that the tube diameter should be at least an order of magnitude larger than that of the sphere diameter in order to avoid significant finite size effects [39].
FIG. 3. Growth of capillary undulations under weak confinement conditions. α is the amplitude of the thread surface undulations and is defined as ½ the difference between the maximum and minimum distances from the original cylindrical thread surface. Data are taken from the run shown in Fig. 2(a) and terminate at the point of thread rupture. The solid curve indicates the data range where we have fitted to the exponential growth law predicted by linearized stability theory [18], and dashed lines indicate the early "induction regime" and late stage "rupture regime" where the growth dynamics exhibits a more complicated behavior. Note the acceleration of the breakup process near the point of thread rupture in this example of confined thread breakup.
The simulations described below indicate that finite size effects can substantially influence the rate of thread breakup over a large range of tube confinement and that the rule of thumb, just mentioned, describes these confinement effects reasonably well. Notably, these corrections are relevant to an accurate estimate of surface tension by observations of the dynamics of capillary breakup. Despite the potential importance of these corrections, there has been little discussion of them in the experimental literature, apart from recent work of Son et al. [21]. Indeed, the thread confinement scales are not normally reported [40]. Such corrections are quantified be low for the tube geometry in the case where the matrix and thread fluids have the same viscosities and where the tube boundary does not have an energetic preference for either fluid component (see Sec. III D). Our results should be suitable for comparison with the breakup of real fluids, provided the viscosity mismatch is not too large. Equation (8) provides an estimate of the uncertainties caused by this approximation in the bulk case and we expect this expression to provide a rough estimate for the viscosity mismatch effect for the breakup of confined threads. This approximation remains to be tested, however, and should not be adopted uncritically.
In our next test of the LB model, we consider the wavelength of the most rapidly growing undulatory instability in our simulations in comparison with the analytic theory of Tomotika [18]. The dimensionless wave number δ for the data in Fig. 2(a) is equal to δ ≈ 0.58 ±0.03, where the confidence interval reflects the uncertainty in determining the λmax and α due to the lattice discretization of the thread boundary. This value accords within experimental uncertainty with the theoretical value 0.56 for thread breakup in bulk (see Sec. II).
In order to quantify the role of finite size effects on the kinetics of thread breakup in the limit of "weakly confined" threads, we simulated the thread breakup for a range of Λ values between 2 and 8. The resulting growth rate data are shown in Fig. 4(a). The data from Fig. 3 (Λ = 2.45) are included for comparison in this figure. Growth rates q(Λ) for confined threads, obtained from the linear (solid curve) portions of these curves are indicated in Fig. 4(b). (Curves have been shifted horizontally so that they do not overlap one another.) Dotted portions of curves correspond to early and late-stage regimes. Each curve in Fig. 4(a) is shifted relative to the previous curve by an amount tred = 5, from left to right. For Λ > 2.45, the growth rate q(Λ) increases monotonically, becoming relatively constant for Λ ~ O(5−10). This finding accords well with the usual intuition about the scale where finite size effects tend to "saturate."
It is interesting to compare the kinetic data for thread breakup to the generalization of the Tomitika theory to con- fined threads derived by Mikami and Mason [41]. This theory is rather algebraically complex and does not lend itself to a closed analytic description of the rate of thread breakup, but we can accurately fit the results of this theory to analytic approximants that are useful in comparisons to our simulation data and experiment. Rigorous application of the "linearized" hydrodynamic theory is limited to "short" times, but experience has shown that this type of approximation can be a remarkably good at longer times approaching the thread rupture time and we next compare our calculations to these predictions. Figure 4(b) shows estimates of the reduced breakup rate q(Λ) / q(Λ → ∞ ) determined numerically from the analytic results of Mikami and Mason [41] for the cases where the viscosity of the thread ηthr is ten times that of the matrix fluid (dotted line), equal to that of the matrix fluid ηm (solid line) and a factor of 0.1 times ηm (dot-dashed line). By utilizing the equation discovery algorithm of Judith Devaney a NIST [41], we find that the exact numerical values of q (Λ) / q∞ determined from the Mikami-Mason theory can be described by a simple exponential function of Λ over a broad confinement range (2.3 < Λ < ∞), i.e., q(Λ, p = 1) / q∞ ≈ 1 − Q exp[− νΛ] where q∞≡ q(Λ → ∞), ν = 0.637, and Q = 2.67. The magnitude of the deviation between this approximant and the exact numerical data is generally less than 0.005 so that we do not discriminate between the approximant and the exact numerical data in Fig. 4(b). We also observe that this simple analytic expression agrees well with our LB data for q(Λ, p=1) / q∞, indicated by the filled circles in this figure. Notably, the value of the "bulk" capillary growth rate (q∞) derived from this fit is used to define the time scales of our simulations below.
We also observe from Fig. 4(b) that order of magnitude changes in the ratio of the thread to matrix fluid viscosity p have a relatively small effect on the calculated q(Λ) / q∞ when p is small. The deviation becomes substantial, however, for large p and the upper (dotted) curve shows this effect for the representative case, p=10. This relative insensitivity of q(Λ) / q∞ to p does not extend to other properties, such as the "wavelength of the instability," λmax(Λ). This point is illustrated in Fig. 4(c) which shows the analytic prediction of Mikami and Mason [41] for λmaxΛ), relative to its bulk value λ∞ and for representative p values between 0.1 and 10. It is apparent that λmax(Λ) / (Λ∞ depends strongly on p and the finite size dependence of this ratio becomes increasingly large as p becomes larger.
The apparent increase in q(Λ) for Λ ≈ 2.1 and the minimum near Λ ≈ 2.54 in Fig. 4(b) deserves comment. Apparently, the onset of strong confinement can actually lead to an enhancement of the early-stage rate of capillary breakup. It must be noted, however, that the geometrical character of the thread breakup process becomes substantially modified in this confinement regime (see the detailed discussion below) so that the thread breakup process is not directly comparable to the weak confinement data (Λ > 2.54). For more confined systems (Λ < 2), we find below that thread breakup no longer occurs by a capillary instability process like that of the bulk fluid. Thus, it is not generally sensible to speak of the Rayleigh-Plateau instability under high confinement conditions. Nonetheless, we use q(Λ) to define the dimensionless time of our simulations since the bulk measurements still provide a natural reference point for describing the relative rate thread breakup in confined threads, regardless of the mode of thread breakup. Direct comparison to measurement can be made in the same reduced time units.
We now return to the representative experimental data [21] in Fig. 2(b) for the breakup of a polymer fluid thread and compare these results to the LB simulations above. The scales of the images in Fig. 2 have been adjusted so that the initial thread sizes are comparable (Rthread = 127 µm). The fluid thread is a polyamide-6 (nylon) polymer and the matrix is polystyrene where the molecular masses of the polymers are relatively low to avoid significant "entanglement" effects and the temperature is rather high (T = 503 K) to avoid non- Newtonian effects arising from the glass transition. We also note the viscosities of the nylon and polystyrene are 300 and 1200 Pa s, respectively, so that the viscosities are not exactly matched, as they are in the simulations. Confinement effects are weak in these measurements since the thread is confined between two parallel plates where the ratio of the gap width between the plates and the thread diameter equals 10.6. Measurement details are given by Son et al. [21] and similar observations for non-polymeric fluids are described by Mason and co-workers [30].) It is apparent from Fig. 2 that the LB simulation captures the geometrical form of the "bulk" thread breakup process rather well, including the process of satellite formation.
FIG. 4. Growth of capillary undulations as function of confinement, Λ. (a) The growth rate data shown in Fig. 3 is extended to the range of Λ indicated in the figure. The solid curve indicates the data range where we have fitted to the exponential growth law predicted by linearized stability theory [18], and dashed lines indicate the early "induction regime" and late stage "rupture regime" where the growth dynamics exhibits a more complicated behavior. (b) Capillary instability growth rates q(Λ) obtained from a fit to linear portions of curves shown in (a). Growth rate data have been reduced by the bulk growth rate, q(Λ → ∞) = q∞. (c) Theoretical prediction of the influence of confinement (Λ) on the wavelength λmax(Λ) of the maximum growth rate. Wavelength data have been reduced by the bulk value, λmax(Λ → ∞ = λ ∞. The curves are calculated from the linearized stability theory of Mikami and Mason [41].