Multimaterial fiber as a physical simulator of a capillary instability

Capillary breakup of cores is an exclusive approach to fabricating fiber-integrated optoelectronics and photonics. A physical understanding of this fluid-dynamic process is necessary for yielding the desired solid-state fiber-embedded multimaterial architectures by design rather than by exploratory search. We discover that the nonlinearly complex and, at times, even chaotic capillary breakup of multimaterial fiber cores becomes predictable when the fiber is exposed to the spatiotemporal temperature profile, imposing a viscosity modulation comparable to the breakup wavelength. The profile acts as a notch filter, allowing only a single wavelength out of the continuous spectrum to develop predictably, following Euler-Lagrange dynamics. We argue that this understanding not only enables designing the outcomes of the breakup necessary for turning it into a technology for materializing fiber-embedded functional systems but also positions a multimaterial fiber as a universal physical simulator of capillary instability in viscous threads.

and the silica cladding softens.While the core plus cladding system continues to move through the hot zone in the axial direction x, surface tension minimization drives the pinching-off of the core into droplets at an approximately constant pinching location for each feed speed (Figure 2), as is observed experimentally.
At this location, the core thread reshapes into a droplet and pinches off while a critical amount of material is fed into the droplet.This results in uniformly spaced spheres with a wavelength .
To understand the process shown above, we consider that the amount of core material that is fed into the droplet before it breaks off the core at the time = (pinch time, defined in the main text) is equivalent to that stored in a section of a core of length being fed for a time (growth time).This relationship is mathematically expressed as: (S1) , where may be a function of interface energy, viscosity and , such that .Moreover, we state that the derivative of the neck thickness around the breakup point is smooth and equal to 0 at .
Supplementary Fig. 1.Illustration of pinch-off process of the core thread at position  .The light red capped cylinder in the background represents the intact core thread shape, in comparison to the reshaped core in solid red shown on the foreground, representing the core during the pinch-off process at a time .The location of the pinch is highlighted by the yellow arrows.
In Supplementary Fig. 1, we show an illustration of the pinch-off process: Once the length is fed through the pinching front at , a breakup occurs, and at this time, the neck thickness is zero.We can also define the pinching speed as the ratio between and the pinch-off time where is the difference between the thread radius , and the neck half-thickness at the front location.
Let us now consider what happens in a position , where is an infinitesimally small quantity: at the same moment as the pinching is occurring at , it hasn't yet occurred at .Moreover, the material has begun to be fed through later than at by a factor of .In this way, we can define an "excessive pinching time" as the pinching time at minus : Similarly, for the pinching front , we have: As stated before, we propose that the derivative of the neck thickness at is smooth and equal to 0: Since , we write the partial derivative on as (S8) So we can rewrite Eq. (S7) as: (S9) Substituting from Eq. (S1), and dividing Eq. (S9) by , we obtain: Thus, from Eq. ( S1) and (S10), we obtain the following set of two equations in two variables from which we can obtain our solution: (S11) Solving this system, however, requires information about the breakup time .Considering that , we have , where is a system-dependent function of viscosity contrast between the core and the cladding materials.
In some specific cases, for large viscosity contrast, converges asymptotically to a constant derived by We can easily see that since , it follows that .Also, , and thus we have: We can verify that (S18) is true by performing the derivatives on : From Parikh (1985) 5 , we get that the surface tension of silica is .In Li et al. (1992) 6 , the wettability of silica by molten silicon is investigated through the sessile drop method, and the contact angle of silicon on silica was determined to be 92° at 1703 K.The work of adhesion W is then given by: (S24) With .

Thus, we can calculate as (S25)
which is consistent with the values found in the literature 7,8 .
Supplementary Fig. 4. Snapshot of a breakup experiment.It illustrates how the breakup location is measured with respect to the melting position of silicon, which can be determined as a dip in the intensity profile (y-scale on the bottom graph is intensity in arbitrary units, x-scale is the number of pixels starting from right along the red line 580 pixels long on the image, along which the intensity profile was collected).
To capture the dip in emissivity associated with the Si melting point, the dynamic range is compromised: it can be noted that the signal saturates around pixel 550, thus emissivity corresponding for T max is not captured.At the same time, in our experience, if the sensitivity of the camera is reduced, such that there is no saturation of the signal in the frame, the dip in the emissivity corresponding to the melting of silicon blends into the noise and can't be captured reliably.Hence the challenge of reconstructing the temperature profile from the emissivity signal, mentioned in the main text.
. Using AVG-IM interpolation (Supplementary Fig. 11d), we have reconstructed the temperature profile of the hot zone created by the flame.Using the same flame settings, we have performed a breakup of the fibers scaled down, as is shown in Supplementary Fig. 11c, from the etalon fiber such that the cladding dimensions of the scaled-down sections and are significantly smaller than , yet the core dimensions and are comparable to Fig. 11b).
The fiber in Supplementary Fig. 11b resulted from scaling the etalon fiber (the one used for the experiments in Figure 6, with a 280 silica cladding and a 4 Si in two steps according to the procedure schematically depicted in Supplementary Fig. 11c. Next, we verified that AVG-IM extrapolates well to the results of the breakup in fibers with thinned-out claddings (Supplementary Fig. 11e), thus supporting experimentally that AVG-IM is independent of the cladding dimensions, i.e., MS1 holds.
Even if MS1 holds, the cores of the tapered fibers are tapered too, i.e., , thus needs to be verified for varying core dimensions.Figures 6 and Supplementary µm, thus we know that AVG-IM will hold for adiabatically tapered cores in this range of core dimensions.
In the general tapered-fiber case, must be considered to be changing continuously and significantly within a single breakup period.We hypothesize that the central equation of AVG-IM, Eq. (2), will hold for such tapered cores if in it is replaced with a modified accordingly, i.e., , although the validity of the model and its limitations, in that case, need to be reexamined experimentally and numerically as part of the future follow-up study.
Model Simplification MS2: The fiber cores are solid crystalline materials that abruptly change the aggregation state upon melting, becoming an inviscid liquid.This simplification results in the core-cladding viscosity contrast in the temperature range relevant for the development of capillary instability and allows us to consider only the cladding viscosity past the melting point of the core to the AVG-IM.It holds true for a wide variety of core materials relevant for silica-fiber embedded optoelectronics and photonics, including but not limited to semiconducting, metallic, magnetic, optically non-linear, superconducting, thermoelectric, ferroelectric, and piezoelectric 2,12-20 .
The situation is different for the polymer-fiber embedded devices and systems, where amorphous materials, such as chalcogenides 21,22 or polymers different from this comprising the cladding 23 , are used for the cores functionalizing the fiber optically or electronically.For such fibers is Thus, we hypothesize that for the AVG-IM to hold true, in Eq. ( 2) needs to be replaced with a modified .
The educated guess that we suggest for for scenarios considering a finite is , where is the Tomotika instability rate.This choice is guided by the fact that converges towards for , as is demonstrated in Methods.Although, once again, the validity of the model and its for such extrapolation need to be reexamined experimentally and numerically.
pinch-off time by , we can obtain the half-thickness of the neck at the locations , at the moment of pinch-off, which are expressed in Eq. (S2) and (S3), and illustrated in

Supplementary Note 3 .
Silicon-Silica Surface TensionThe value of the surface tension between liquid silicon and silica was obtained as follows:In Fujii et al. (2006) 3 , the surface tension of molten silicon is measured by the microgravity oscillating drop method and the following relation between (in mN/m) and the temperature T, is presented: (S23) for a temperature range of 733 K to 1890K, in agreement with Lucas et al. (1984) 4 .
Fig. 11 present an experimental verification that AVG-IM describes well the breakup for four discrete core diameters -4, 1.8, 1.5, and 0.8 the Marginal Stability Criterion for propagating Plateau-Rayleigh instability case, as is done in Powers et al. (1998) 1 , and to a different constant assuming the classical Tomotika's growth rate.