Recognition of Spatial Finiteness in Meniscus Splitting Based on Evaporative Interface Fluctuations

The geometric deformation of viscous fingering is useful for understanding natural multiscale patterns and designing dissipative structures in materials. Although the spatio‐temporal patterns in soft materials are reported previously, there is a lack of research on the spatial finiteness and boundary effects. In this study, the recognition of spatial finiteness in “meniscus splitting phenomena” in aqueous polymer dispersions during water evaporation is demonstrated. By providing heat energy to polymer dispersions in a Hele‐Shaw cell, an interface fluctuation with concentration unevenness is induced to split the evaporative interface. The spatial finiteness of the interface causes asynchronous nucleation, which is demonstrated using polysaccharide dispersions. The results of the quasi‐natural experiments revealed that the nonequilibrium drying/wetting period for repositioning polymer clusters allows for considerable changes in Reynolds number in a low range (<10−6) to form multiple nuclei. This splitting method will be universally useful in various fields, including fluid dynamics, biology, and microfluidics, as well as non‐equilibrium, colloid, interface, polymer, and materials sciences.


Introduction
[8] The spontaneous pattern formation has been extensively studied at various scales not only in non-equilibrium systems driven by chemical reactions but also in systems dominated by physical conditions, such as symmetry breaking, [9,10] fluid dynamic DOI: 10.1002/admi.202300510[22] In the development of phenomena of viscous fingering driven by thermal gradients, [23,24] we successfully obtained millimeter-scale patterns by drying some aqueous polysaccharide dispersions, such as pectin, xanthan gum, and sacran. [25,26]Spatial patterns are formed by the splitting of the meniscus with ordered depositions of selfassembled polysaccharides (from one space into multiple spaces) (Figure 1).[30][31] There are optimum ranges for controllable parameters in meniscus splitting, such as the initial polymer concentration, drying temperature, cell geometries, and types of polymers with specific characteristics, such as viscosity. [32,33]s a result of interface fluctuations, the deposited polymer membranes have 3D-ordered microstructures. [34,35]The interface contact line induces the polymeric orientation and layered structures in the submicron scale using capillary forces. [26]he polymer membranes have huge potential for designing anisotropically swellable hydrogels that work as drug delivery systems or actuators responding to changes in aquatic environments. [36,37]Based on these flows, meniscus splitting is a method that has been used to prepare advanced materials, but the phenomena are yet to be discussed from a hydrodynamic viewpoint.During the period in which an initial interface transforms into multiple interfaces, there exist obvious fluctuations, and it is still uncertain how the second nucleus will form.
In this study, we report on interfacial fluctuations following the splitting of the meniscus into three menisci to expose the aspects of the dissipative structure.In contrast to the splitting of the meniscus into two parts, the interface becomes more unstable when three menisci are formed, and a larger degree of freedom is provided for the two nuclear positions.When evaporation conditions, such as the cell width, are fixed, the number of split menisci is determined with high probability, as shown in Figure 1.The first nucleus is deposited to split the distance of the cell width equally rather than on the center of the width.For example, the cell width is ≈25 mm, and one initial evaporative interface changes into a wavy shape and splits into three menisci.The interface never changes shape to form the first nucleus around the center of the width nor does it ever split into four.Before deposition nucleation, the interface with a precursor makes three concave menisci, that is, the interface recognizes the finite space.During water evaporation, the polymeric precursor accumulates on the interface in a nonequilibrium environment, that is, drying and wetting.This situation induces a wavy fluctuation and causes the recognition of spatial finiteness.Considering this process, meniscus splitting should include physical rules based on temporal evolution.
Aqueous pectin dispersion, which shows self-assembled particles in pure water with a hydrodynamic diameter of ≈1 μm at 40 °C in Figure S1 (Supporting Information) was used as the drying experimental model.The drying process was performed in a box and monitored in controlled temperature and humidity conditions.The phenomena were statistically examined and spatiotemporally analyzed by deriving the evaporation velocity at characteristic positions.Re was evaluated to understand the effects of physical parameters on the splitting.Life in low-Re cases (Re < ∼10 −4 ) exhibits attractive mass-transportation or swimming motions in microorganisms using flagella or cilia. [38]n the realm of material design, laminar flow at low Reynolds numbers enables molecules to exhibit their flexibility, leading to the creation of distinct microstructures tailored for specific materials. [39]These examples in actual, viscous underwater environments show that huge potentials exist at low Re, regardless of the size or scale.While surface tension plays a crucial role in the interfacial fluctuation generation during meniscus splitting, quantitative analysis from surface tension remains challenging due to the rheological properties of high-concentration polymer dispersion.Thus, the in vitro meniscus-splitting phenomena were examined from a hydrodynamic perspective instead of a mechanical perspective.

Results and Discussion
Figure 2A and Movie S1 (Supporting Information) respectively show the time course of the evaporative interface and the deposition nucleation from the aqueous pectin dispersion in cells with 1 mm gaps and 5, 15, and 25 mm widths at 40 °C.As the water evaporated, under the dominance of capillary forces in Ydirection, the local polymer concentration on the interface increased drastically to deposit pectin by bridging the 1 mm gap.During drying, the evaporative meniscus split into two parts from the 15-mm-width cell and into three parts from the 25mm-width cell.The split interfaces induced a continuous polymeric accumulation at specific positions by bridging the gap. [26]oreover, the number of nuclei and the interval were strongly affected by the cell width.Based on the results, the interval in the width direction was less than 10 mm with a high probability.This is likely because there exists a state of quasi-equilibrium between the air pressure and liquid pressure, like the stable semiellipse that formed when an air bubble was squashed in a 1mm gap. [40,41]The reproducibility of the number of nuclei was evaluated with 20 trials in each condition at 40, 50, and 60 °C (Figure 2B).At any drying temperature, the 15-mm-width cell induced the generation of one nucleus with a probability of 100%, and the 25-mm-width cell induced the generation of two nuclei with a probability > 80%.Thus, multiple nucleations occur because the split meniscus has the advantage of increasing the area of the evaporative interface with polymer depositions at specific positions.
To validate the specific depositions in detail, the positional information of the nucleus was collected.Figure 2C shows the distribution of the nuclear position (X N , Z N ) in cells with widths equal to 15 and 25 mm.The positions in the cell with a 15 mm width were estimated to be (7.5 ± 0.1, 9.9 ± 0.4), (7.3 ± 0.1, 10.5 ± 0.2), and (7.6 ± 0.2, 15.0 ± 0.7) at 40, 50, and 60 °C, respectively (see Figure S2, Supporting Information), whereas X N was mostly around the center at any drying temperature, and Z N became deeper as the temperature increased.This is because the viscosity of the dispersion became lower at higher temperatures.As in the case of the 25-mm-width cell, the distributions were normalized based on the following rule: the nucleus with a shallower Z N was placed in a smaller X-position as the first nucleus.Herein, two clear distributions were observed in the X-direction; for example, at 40 °C, (X N1, 40C , Z N1, 40C ) = (6.5 ± 0.3, 7.9 ± 0.5) and (X N2, 40C , Z N2, 40C ) = (15.9± 0.6, 11.2 ± 0.7) (Figure 2C; Figure S2, Supporting Information).The two distributions result from the interfacial instability of the accumulated polymer and the critical wavy meniscus.At the meniscus with the concentrated polymeric layer, the mechanical stress would be loaded from both sidewalls.Even such a situation, the polymer particles are provided and accumulated from the liquid phase.To release the mechanical stress and keep water evaporation, the interface is deformed by making a wavy shape with multiple specific points for polymer deposition.This is similar to skin layer generation and pattern formation in the volume changes of polymer gels, which is described as mechanical instabilities. [42,43]Furthermore, the difference between Z N1 and Z N2 is due to the insufficient widening of the area of the evaporative interface by the first nucleus, which induces the second deposition nucleus to widen the area further.As the drying temperature increased, both Z N1 and Z N2 increased.This result supports the consideration of the 15-mm-width cell, which was affected by viscosity.Thus, this statistical analysis revealed that the X-width of the cell determines the number of nuclei in meniscus splitting and nuclear positions.
By focusing on the interfacial fluctuation and two nucleations, spatiotemporal changes were analyzed using a 6 wt% aqueous pectin dispersion in a cell (25 mm, 1 mm, ≈23 mm) that was used for drying at 40 °C.The interfacial Z-depth was monitored at the characteristic X-positions for meniscus growth (Figure 3A: M1, M2, and M3) and nuclear growth (N1 and N2). Figure 3B,C shows the time course of the Z-depth (Z) the interfacial velocity as the interfacial position motion (dZ/dt), respectively.Figure 3B shows that the change in interfacial Z-depth of the pectin dispersion is slower than that of pure water.This is because the dispersed polymer inhibits water evaporation.This situation induces an unevenness in polymer concentration beneath the evaporative interface as a colloidal suspension. [44]In fact, the rates of the time courses at N1 and N2 gradually decrease and fix the Z-positions at time lags of 1200 and 1800 min, respectively.This is also confirmed by the time course changes of the velocity in Figure 3C.Comparing the snapshots visually helps us understand relative differences.To examine the time course changes before the visible nucleation (0-900 min), the differential in the Z-depth at each position was analyzed and plotted as dZ/dt in Figure 3C.In the time courses of 100-300 min, values of dZ/dt at N1, N2, and M2 (≈8.6 μm min −1 at 200 min) are smaller than those at M1 and M3 (≈10.4 μm min −1 at 200 min) near the side walls.This is because the two side walls provide larger evaporative interface near points of M1 and M3 than near point of M2.During the water evaporation, the M1 and M3 interface initially expand to counteract the inhibition of evaporation caused by the increase in the polymer concentration.These results for the induction time (0-300 min) mean that the interface fluctuation and nuclear X-positioning have already started.After ≈600 min, dZ/dt at M2 approached the same velocity as that at M1 and M2; dZ/dt did not synchronize and behaved as a monotonically decreasing function with respect to the induc-tion time.This interface fluctuation was due to the inhomogeneous rearrangement of the accumulated polymer clusters in the X-direction.
Considering our previous works, [26,35,37] the fluidic behaviors beneath the evaporative interface during the splitting process is critical for the formation of ordered polymeric micro-layered structures.Due to this precise arrangement of polymeric deposition, we focus on the correlation of fluid inertial force and the viscous force.By analyzing the temporal evolution of the interface as fluid flows downward, we verify the phenomena from a fluid dynamics perspective as a function of the dimensionless number Re, the ratio of the fluid inertial force to the viscous force was estimated as follows, where : density (kg m −3 ), u: interfacial velocity (m s −1 ), μ: dynamic viscosity (Pa s), and D H : hydraulic diameter (m).Because the accumulated state just before the polymer deposition is essential, the experimental parameters  and μ were evaluated as a function of polymer concentration C at constant temperatures (40, 50, and 60 °C); u was expressed as a function of time t Figure 4A shows the estimation of the time course of Re by introducing fixed values of the concentration (6, 9, 12, 15, 18, and 21 wt.%) into F(C, t) to provide approximate guides of ΔRe at characteristic X-positions (N1, M2, and N2) in Figure 3A.Considering that the experimental limitation for dispersing pectin homogeneously in pure water seemed less than ≈12 wt.%, the saturated polymer concentration (C s ) should be close to this value.In a virtual experiment using the estimated Re, the drying starts from C 0 of 6 wt.%, and C should increase constantly from Re ∼10 −6 .At ≈300 min, a critical inflection point was confirmed at all Xpositions, and a plateau period was ensured in cases in which the concentration was fixed.This critical point was also confirmed in the analysis of the evaporation velocity, dZ/dt (see Figure 3C).After this timepoint, all the Re curves for N1, N2, and M2 exhibited distinct slopes (see Figure S4, Supporting Information).These results indicate that the induction time (0-300 min) should include the well-ordered process of creating concentration unevenness on the interface to form two nuclei.Figure S4C (Supporting Information) shows the superposition of possible Re regions at given X-positions (N1, N2, and M2).When C approaches C s (>12 wt.%), the polymer deposits are attributed to the nuclei at positions N1 and N2 causing considerable ∆Re (10 −6 → less than 10 −8 ).ΔRe in the laminar flow region should allow the wellordered process from the submicrometer scale to form two nuclei in the millimeter scale.
Based on the spatiotemporal analysis and the validation of Re, the interface fluctuation is summarized as shown in Figure 4B.Considering that Re is the ratio of the fluid inertial and viscous forces, the temporal change in Re directly reflects a wavy fluctuation.When the water evaporation process begins, the polymer cluster accumulated on the interface and interfacial Zdepth simultaneously increases (state I).Initially, the interface near the two side walls increases as the boundary temporarily yields a larger change in the Z-depth to maintain the area for the water evaporation (state II).This process determines the shape of the interface fluctuation and the number of pre-nuclei.After the recognition of the space finiteness, definite splitting occurred to generate the first nucleus (N1), which was not generated from the center, and the interface predicted that the second one (N2) would be generated soon thereafter (state III).During state II, the concentration unevenness on the evaporative interface was induced, and a gradual deposition was allowed at a specific position.
To close upon the concentration unevenness before the characteristic deposition, a comparison with some Y-gaps that are directly related to parameter D H is discussed (Figure 5).When cells with 0.5 and 1.5 mm Y-gaps have the same parameter values, the polymer is deposited with different trends in each cell (Figure 5A).For the cell with a 0.5 mm Y-gap, the polymer is deposited horizontally at a smaller depth with lid-or T-shapes.Conversely, for the cell with a 1.5 mm Y-gap, the polymer is deposited at one or two specific points at a greater depth.These results can be expressed in the form of a correlation of D H and Z N1 (Figure 5B).When the X-width was fixed at 25 mm, Z N1 was strongly affected by D H and the shape.This result suggests that the Y-gap is an important factor in meniscus splitting.D H shows a slight difference at an X-width > 15 mm (Figure S5, Supporting Information).Consequently, as the X-width increases, the fluctuation can be developed into meniscus splitting with multiple nuclear formations.In fact, the evaporative meniscus is split into multiple menisci with similar Z N1 values in conditions of a fixed 1 mm Y-gap and X-widths >15 mm.In our previous studies, [26,45] the effect of the cell width was also confirmed using the widths of 50 or 100 mm for other polysaccharides.
From these validations, the deposition trends can be explained by the initial unevenness on the evaporative interface.Figure 5C shows a schematic illustration of the polymer clusters in the fluctuation to form pre-nuclei in the cases of the 25 mm width.From the cell with a 0.5 mm gap, the clusters easily bridge the gap at any X-position and make the evaporative meniscus larger near the sidewalls.By contrast, from the cell with a 1.5 mm gap, the clusters generate and retain multiple pre-nuclei because of the degree of freedom on the larger area of the interface.During water evaporation, these multiple pre-nuclei accumulate, and their numbers decrease to increase the area of the evaporative interface by forming multiple menisci.Thus, D H critically affects the unevenness on the evaporative interface and number of specific depositions; for example, three splittings occur in the cell with a 1.0 mm gap.

Conclusion
In summary, meniscus splitting with multiple nuclei was verified statistically and kinetically based on the parameters of viscous fluids using aqueous pectin dispersions.Multiple nucleations occurred asynchronously in stages following interface fluctuations of the polymer cluster.By controlling the cell width, a analysis specified the two nuclear generations with high probability and the asynchronous trends of the spatial distributions for the first and second nuclei.The spatiotemporal analysis revealed that there existed an induction period for the differentiation of the velocity dZ/dt of the interface, that is, an "interface fluctuation" as a non-equilibrium state between drying and wetting states.This period enabled the recognition of the cell sidewalls as the boundaries and the finiteness of the space for the nucleation in the horizontal X-direction.The evaporative interface formed a wavy shape before depositing a nucleus.This was due to polymer concentration unevenness, which increased the area of the evaporative interface.The system spontaneously achieved dynamic equilibrium and resulted in the dissipation process in the horizontal direction and meniscus splitting.After the interface fluctuation, asynchronous nucleation occurred to generate the first nucleus (not from the center but near the trisection of the cell width), followed by the generation of the second one.
To evaluate the splitting process, the time course of Re was estimated to cause a drastic change in the low range, less than 10 −6 using the experimental values of , μ, u, and D H .This methodology will help the application of meniscus splitting in other materials that have inherent physical values.Furthermore, the experimental evaluations conducted with datadriven fluid dynamics and long short-term memory [46][47][48][49][50][51] will be useful in predicting unique spatiotemporal developments.In the future, we will develop a dimensionless model in terms of the dominant parameters such as the capillary number [52] and the Marangoni number.We envision the universalization of this meniscus-splitting phenomenon not only on the millimeter scale but also on the micrometer scale, where geometrically precise structures and systems are generated as microorganisms.This understanding could contribute to designing adaptable microdevices using this technique for material applications.

Experimental Section
Materials: Citrus pectin with a molecular weight in the range of 20-400 kDa was purchased from Nacalai Tesque, Japan.As a low-methoxy pectin, it contains more than 50% galacturonic acid units and 11% methoxy groups.The pectin powder was dispersed in pure water at 25-40 °C, and the small number of impurities and air bubbles were removed by centrifugation.The supernatant was used to construct the samples.The supernatant dispersion was used for sample preparation.The aqueous pectin dispersion was kept in a refrigerator and used within a week to avoid changes in properties.The glass substrates were purchased from Matsunami Glass Ind. Ltd., Japan.
Drying Experiments for Statistical Analysis: The sample was poured into a top-open cell (X-width, 1 mm Y-gap, and ∼23 mm Z-depth), a Hele-Shaw cell (X-width = 5, 15, and 25 mm).The cell had a U-shape and was composed of two nonmodified glass slides and rubber spacers made of polyvinyl chloride with a thickness of 1 mm.The cell was placed in an oven (EYELA, VOS-210C) at a given constant temperature at atmospheric pressure maintained using an air circulator.Considering that the volume of the oven (192 mm × 270 mm × 192 mm, ≈10 L) with the air circulator (effective exhaust velocity ≈40 L min −1 ) was much larger than the volume of the samples (<10 mL), the relative humidity in the oven was controlled by the set temperature.
Drying Experiments for Spatiotemporal Analysis: The sample in the cell was placed in a ≈1 L observation box under controlled temperature and humidity through a temperature-controlled air circulation pump (Figure S1, Supporting Information).The internal space temperature of the box was controlled by using two ways.One was connecting a heat pen with a circulating tube and another one was rubber heaters attached on the inner walls of the box.The relative humidity in the box was controlled by keeping the temperature constant and flowing air circulation.The correlation of the temperature and the relative humidity was checked and it could be confirmed that it was possible to control the humidity constant during the drying experiments.During the drying process, the samples were monitored through glass windows, and images were captured at certain time intervals (10 min).The interfacial positions (X, Z) were determined by setting the cell's top corner as the coordinate origin (O).To obtain the Z-positions from the images in a time course, the image sequences at each X-position were prepared using the software package ImageJ (National Institutes of Health, Bethesda, MD, USA) and analyzed using the plugin Multi Kymograph.The Z-positions in the time course were plotted using GetData Graph Digitizer, a software package.The interfacial velocity (dZ/dt) was estimated using the plots (t, Z).
Characterization of Aqueous Polymer Dispersions: The absolute viscosities of aqueous polymer dispersions were measured using an electromagnetically spinning viscometer (EMS-1000S, Kyoto Electronics Manufacturing Co., Ltd.) in temperature-controlled and inhibition of water evaporation conditions.Measurement conditions were as follows: Probe: Ti-5.0 mm, probe rotation speed: 1000 revolutions per minute, maintenance time: 600 s, repeat times: 10, measurement interval: 10 s, measurement times corresponding to different viscosities in the range of 0.5-45 min.The density of the sample was measured using a benchtop density meter (DMA4501, Anton Paar).The characterization was performed at specific constant temperatures.

Figure 1 .
Figure 1.Schematics of meniscus splitting into three parts owing to interface fluctuation and recognition of spatial finiteness.When the conditions (including the cell width) are fixed, one meniscus splits into three parts (but rarely into two) with nucleation at the center of the cell width (following a split into three or four menisci).

Figure 2 .
Figure 2. Recognition of spatial finiteness and the effect of cell width on the number of nuclei.A) Images of the drying process with meniscus splitting in given cells (X-width, 1 mm gap, 23 mm depth, drying temperature: 40 °C, and initial concentration of pectin, 6 wt.%).B) Probability of the number of nuclei at a given temperature and cell width.Twenty trials were conducted at each condition.C) Statistical space distribution of the nuclear position (X N , Z N ) from cells with 15 and 25 mm widths at a given temperature.

Figure 3 .
Figure 3. Spatiotemporal analysis of the meniscus splitting into three.A) Images of the drying process with meniscus splitting in a cell (25 mm width, 1 mm gap, 23 mm depth, initial concentration of pectin: 6 wt.%, drying temperature: 40 °C).The lines represent characteristic X-positions: concave meniscus bottom parts (M1, M2, and M3), and nuclear top parts (N1 and N2).B) Time course curves of the interfacial Z-position at characteristic X-positions; that of pure water is also shown as a control.C) Evaporation velocity at characteristic X-positions and snapshots at different times.

Figure 4 .
Figure 4. Spatiotemporal estimation of the Reynolds number during interfacial fluctuation.A) Estimation of the Reynolds number (Re) in a time course, calculated with actual μ, , u, and D H at given polymer concentrations.Arrows indicate the ΔRe values at the X-positions (N1, N2, and M2).B) Hypothesis of the transition from accumulation to splitting to enlarge an evaporative interface based on three steps: I) induction period with polymeric accumulation on the evaporative interface; II) fluctuation period with a temporary acceleration on concave menisci sides; III) meniscus splitting into three parts with two asynchronous nucleations based on concentration unevenness.

Figure 5 .
Figure 5. Trends of characteristic depositions subject to the control of the cell gap.A) Images of a typical deposition in cells with a given Y-gap and probability.Cell (25 mm width, Y-gap, 23 mm depth, initial concentration of pectin: 6.0 wt.%, drying temperature: 40 °C).Twenty trials were conducted.B) Effect of the Y-gap in 25-mm-wide cells on the first nucleating Z-position, Z N1 .Effect of the X-width in 1-mm-gap cells on Z N1 .D H is the hydraulic diameter calculated using the X-width and Y-gap.Z N1 is determined by the statistical average of experimental results from 20 trials.C) Schematic of the evaporative interface and nucleation for meniscus