Divergent Deborah number-dependent transition from homogeneity to heterogeneity

Heterogeneous structures are ubiquitous in natural organisms. Native heterogeneous structures inspire many artificial structures that are playing important roles in modern society, while it is challenging to identify the relevant factors in forming these structures due to the complexity of living systems. Here, hybrid hydrogels consisting of flexible polymer networks with embedded stiff cellulose nanocrystals (CNCs) are considered an open system to simulate the generalized formation of heterogeneous core-sheath structures. As the result of the modified air drying process of hybrid hydrogels, the formation of heterogeneous core-sheath structure is found to be correlated to the relative evaporation speed. Specifically, the formation of such heterogeneity in xerogel fibers is found to be correlated with the divergence of Deborah number (De). During the transition of De from large to small values with accompanying morphologies, the turning point is around De = 1. The mechanism can be considered a relative humidity-dependent glass transition behavior. These unique heterogeneous structures play a key role in tuning water permeation and water sorption capacity. Insights into these aspects can prospectively contribute to a better understanding of the native heterogeneous structures for bionics design.

The liquid phase velocity is small compared to the moist air velocity, which means advective transport rate (in the inner part of hydrogel) is lower than diffusive transport rate (on the interface of hydrogel/air).Therefore, the water velocity in hydrogel is determined by gas phase pressure gradient (based on Dancy' law).Under the assumption that all hydrogels are of similar porous structures, the whole evaporation process is mainly dominated by the volatility of media.
Here, three media were applied in the system, including the original sodium hydroxide, borax-NaOH buffer (pH = 10.00)(buffer), a faster evaporation system containing 10% ethanol and a slower evaporation system containing 5% glucose.To quantitatively describe the volatility difference between these media, an approximate calculation was applied based on Henry's law and Raoult's low, and the saturated vapor pressure was set as the evaluation criterion.Under constant ambient pressure and temperature, the volatility and the saturated vapor pressure is positive correlation, which means that the liquid of high saturated vapor pressure allows for easier transition from liquid to gas.Here, the temperature was set as 293 K and the ambient pressure was 1 atm.
To simplify things, we would use the saturated vapor pressure of water to refer to that of buffer solution.Thus, the saturated vapor pressure of the system is around 2.34 kPa.
As for the faster evaporation system containing 10% ethanol, the Henry's law was applied.
Here, x refers to the molar percentage of chemicals, and the P 0 means the saturated vapor pressure of chemicals.Herein, the saturated vapor pressure of the medium is 2.69 kPa.
As for the slower evaporation system containing 5% glucose, the Raoult's law was applied.
Here, ΔP is the decreased vapor pressure of medium due to colligative properties of glucose solution.Thus, the saturated vapor pressure of the medium is 2.22 kPa.
In brief, the order of evaporated speed can be regarded as: The experimental data are also consistent with the above conclusion.Typically, the free liquid drops were placed on the glass slide.The evaporation area is kept constant (Constant contact radius mode).The degree of air drying completion was defined as evaporated mass/original mass of the drops.When the value reaches 1, it means that the evaporation process is complete.
Supplementary Fig. 1.The evaporation speeds of different media in air drying process.
From the above experimental data, it is clear that the difference in evaporation rate between these three media follows the theoretical expectation.
Supplementary Fig. 2. The energy dissipation mechanism of dynamic hydrogel in the process of uniaxial stretching via the reconstruction of non-covalent bonding.Supplementary Fig. 3. Typical TEM images of TEMPO-CNCs.The length and width were analyzed, and the number of samples for statistical analysis was 300.The diffusion of water in samples was simplified and described using Fick's Low.Thus, the evolution of water concentration depends on the position (x) and diffusion time (t).
The concentration of water with respect to time was described at: Where  is the concentration and D is the diffusion coefficient.Thus, for the spherical sample, the diffusion process was the same along the evaporation direction.When the sample changes to cylindrical specimens, there are two possible diffusion pathways, one along the radial direction and the other one along the longitudinal direction.Taking the central zone of cylinder samples, we can define the speed of concentration change in radial direction (vr) as Eq.(4) and in longitudinal direction (vl) as Eq. ( 5) (5) We can see that the ratio between these two independent directions is proportional to the square of aspect ratio (cylinder).For example, for the cylinder sample with a high aspect ratio, such as more than 50, the vr is 2500 times vl.Therefore, for high aspect ratio cylindrical samples, the diffusion along the radial direction dominates the overall water diffusion process.

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Schema of plants inspired spontaneously formed core-sheath structural hybrid fibers.Based on numerical analysis, it has been demonstrated that the air drying process is isotropic for cylindrical samples.The initial strain rate was defined as dλ / dt to eliminate the scale effect.Scale bar: 5 mm

Supplementary Fig. 12 .
The optical microscopy images, polarized microscopy images and SEM images of various obtained xerogel fibers.The colored arrows indicate the regions with different structural features.