Experiment.

Figure 2 is a typical set of optical micrographs showing the growth of the capillary instability of a PA-6 thread in PS matrix at 230 C. We present two sets of micrographs. Figure 2A is for the case of a PA-6 thread immersed in a PS matrix where the gap width is much greater than the initial thread diameter (H D₀). In Figure 2B, the other case, the thread diameter and gap are of comparable size (H 1.5 D₀). Hereafter, we designate the former case as unconfined and the latter case as moderately confined. In both cases, initially cylindrical threads transform gradually into a row of droplets via an increase in the amplitude of a sinusoidal fluctuation. However, the sequence of micrographs immediately reveals significant differences between the two cases. The radius of the final spheres is approximately 2 times larger than that of the original thread (R₀) in the unconfined regime. This is consistent with the predicted radius by volume conservation,¹⁴ i.e., R₀(1.5/X_m)^1/3. However, that in the moderately confined regime is larger than 2R₀, and the distance between the droplets is therefore greater than in the unconfined case. Thus, the wavelength of the initial fluctuation in the moderately confined regime is larger than that in the unconfined case, and the final drop shape is a "squashed sphere" because the gap width is smaller than the circle which is observed from the x-z plane. Later, we show that the dimensionless wavenumber of the distortion, 2R₀/, is a function of (H/D₀).

Figure 2. Sinusoidal distortions on a PA-6 thread embedded in a polystyrene matrix. The measurements were performed at 230 C. Viscosity ratio (p) is 0.25. (a) H/D0 = 10.6; initial thread diameter = 127 µm; the times for subsequent photographs are 0, 471, 542, 615, and 754 s. (b) H/D0 = 1.45; initial thread diameter = 138.4 µm; the times for subsequent photographs are 0, 344, 894, 1073, 1358, and 2315 s.

To determine whether the confinement causes the thread to grow in a nonaxisymmetric manner, we calculate the dimensionless apparent volume of the sinusoidal thread as a function of time. The apparent volume of the sinusoidal thread, V_app, is calculated from the fluctuation in the x direction (parallel to the plates) with the assumption that the fluctuation is axisymmetric. This is necessary since we observe the fluctuations in the x direction (refer to Figure 1), but not in the y direction. The dimensionless apparent volume V'_app is then obtained by dividing by the volume of the initial thread V'_app = V_app/R₀². If a fluctuation is axisymmetric, then V'_app = 1, whereas if the fluctuation is greater in the x direction than y, then V'_app > 1.

Figure 3 is a plot of V'_appas a function of amplitude (), showing that in the unconfined regime V'_app is constant within experimental error. The solid lines are the results of a numerical simulation and will be discussed later. This confirms the expected result that when H D₀, the fluctuation is axisymmetric. However, in the moderately confined case V'_app increases as the instability grows. Note that the deviation from V'_app = 1 first becomes noticeable when /R₀ 0.1. This occurs well before an axisymmetric surface would impinge on the wall (which occurs at /R₀ 0.45). This result implies that the growth toward the glass wall (perpendicular to the observation plane) in the moderately confined regime is smaller than that in the direction parallel to the observation plane. As the PS matrix between the glass wall and a PA-6 thread is squeezed out as the amplitude grows, the hydrodynamic interaction between the thread and the wall increases. As the distance between the wall and the largest perimeter of the sinusoidal thread decreases by the growth, the hydrodynamic resistance for flow of the matrix phase increases. Therefore, the growth toward the wall is hindered, resulting in the nonaxisymmetric (flattened) sinusoidal thread.

Figure 3. Discrete points represent the normalized apparent volume of the thread, 4Vapp/(D02), as a function of the relative amplitude of distortion (2/D0). Apparent volumes of the threads were calculated from the 2D photographs assuming distorted threads are axisymmetrical. Viscosity ratio (p) is 0.25. The standard uncertainty (one standard deviation) is less than 2%. The solid lines are the simulation results (upper: H/D0 = 1.4; lower: H/D0 = 2.1; viscosity ratio, p = 1).

Figure 4 is a plot of relative amplitude (2/D₀) vs time for the same experiment as in Figure 3. For a given run, the straight-line fit to the semilogarithmic plot for the unconfined regime demonstrates the well-known early-stage exponential growth of the fluctuation as a function of time. The situation for the moderately confined regime is different. A straight line for lower H/D₀ (the confined regime) is drawn by a linear regression with the initial five data points. The early-stage growth rate in the confined regime is much slower, even though the initial thread diameters in both cases are similar. At later times (t > 700 s), the growth rate increases and becomes comparable to the unconfined case. As indicated in Figure 3, the growth rate toward the glass wall is hindered by the hydrodynamic interaction with the wall. The data points begin to deviate upward from the straight line at about 720 s, which corresponds to the largest diameter of the sinusoidal thread/initial thread diameter ratio of 1.25, which is still smaller than H/D₀ = 1.45. As the amplitude grows, the thread becomes more flattened by the hydrodynamic interaction.

Figure 4. Relative thread distortion amplitude, 2/D0, as a function of time for two different H/D0 ratios (gap width/initial thread diameter). Viscosity ratio (p) is 0.25. The standard uncertainty (one standard deviation) is less than 3%.

Figure 5 is a plot of the scaled growth rate vs H/D₀. The growth rate is the slope of 2R₀ ln(/R₀) as a function of time, as suggested by Tomotika. This slope is directly related to the interfacial tension by eq 3. The slope is constant within the experimental uncertainty for H/D₀ > 3; the solid line is drawn to aid the eye. The interfacial tension obtained above for H/D₀ > 3 is (5.7 ± 0.49 mN/m). This value is reasonable compared to the values reported elsewhere.^13,16,23,24 However, the slope decreases rapidly for H/D₀ < 3. It is reasonable to expect that the growth rate is hindered by the hydrodynamic interaction between the wall and the thread as the thread size increases. One experimental run shows that at H/D₀ = 1.3 ± 0.05 the thread does not show the distortion growth. It maintains its original shape for several hours. We refer to this case as strongly confined. This result is consistent with Migler's¹⁰ previously mentioned experimental observation, whereby sufficiently large diameter PDMS threads were stable upon cessation of shear. In that case, the viscosities were matched and H/D₀ = 0.83 ± 0.15. So in the present experiment, where p = 0.25, the confinement effect is even stronger, in that thread stabilization occurs for a lesser degree of confinement. In both the present work and the prior PDMS experiment¹⁰ the strongly confined thread is stable over a time scale at least 40 times longer than the unconfined breakup time. Figure 6 plots the dimensionless wavenumber, 2R₀/, vs H/D₀. The dimensionless wavenumber is constant within experimental uncertainty for H/D₀ > 3. At H/D₀ < 3, it decreases.

Figure 5 Slope for the plot ln(2/D0) vs time as a function of H/D0. Viscosity ratio (p) is 0.25. The standard uncertainty (one standard deviation) is less than 10%.

Figure 6. Dimensionless wavenumber vs H/D0. Viscosity ratio (p) is 0.25. Horizontal line represents theoretical dimensionless wavenumber calculated from Tomotika's theory. The standard uncertainty (one standard deviation) is less than 10%.