Reversible, Selective, Ultrawide‐Range Variable Stiffness Control by Spatial Micro‐Water Molecule Manipulation

Abstract Evolution has decided to gift an articular structure to vertebrates, but not to invertebrates, owing to their distinct survival strategies. An articular structure permits kinematic motion in creatures. However, it is inappropriate for creatures whose survival strategy depends on the high deformability of their body. Accordingly, a material in which the presence of the articular structure can be altered, allowing the use of two contradictory strategies, will be advantageous in diverse dynamic applications. Herein, spatial micro‐water molecule manipulation, termed engineering on variable occupation of water (EVO), that is used to realize a material with dual mechanical modes that exhibit extreme differences in stiffness is introduced. A transparent and homogeneous soft material (110 kPa) reversibly converts to an opaque material embodying a mechanical gradient (ranging from 1 GPa to 1 MPa) by on‐demand switching. Intensive theoretical analysis of EVO yields the design of spatial transformation scheme. The EVO gel accomplishes kinematic motion planning and shows great promise for multimodal kinematics. This approach paves the way for the development and application of smart functional materials.


Supplementary Note1. Phase transition of EVO gel by heat and chemical perturbation.
Sodium Acetate collectively transits with 3 water molecules and has a great stability in supersaturated state appropriate. In case that a gel is swollen with the water manipulator, it effectively swells into the polymer strand retaining supersaturated state depicted in Supplementary Figure 1a. A chemical perturbation is a way to make supersaturated solution to undergo phase transition by giving energy to overcome the energy barrier ( . The phase transition may be evoked in many ways such as a touch of seed crystal 11 , cavitation effect 12 , laser radiation 13 and so on. For simplicity in our research, we touched a seed crystal (sodium acetate tri-hydrate) whose radius exceeds the critical radius.
Furthermore, in a mechanistic viewpoint as shown in Supplementary Figure 2, when phase transition to I-mode EVO gel, water manipulators start to crystallize, and polymer strands aggregate.
Reversely, heating the gel at 60⁰C will alter the gel from V-mode to I-mode. The crystallized water manipulators dissolves, and polymer strands get swelled and segregate. As a result, switching between two modes is reversible. Moreover, by tuning the amount of SBWM after the phase transition, rigidity in V-mode can be locally designated. This enabled a rod with homogeneous softness in I-mode and dramatically different rigidity in V-mode.

Supplementary Note2. Mechanical and selective behavior of EVO gel depending on the fabrication parameters.
EVO gel shows both elastic and plastic behavior. This can be identified in Supplementary Figure   10. For the curve drawn by c0v0.25, the curve after waiting for ten seconds keeps falling until going back to 0 compressed length. This behavior means that the sample exerts force till the original state representing elastic behavior. On the other hand, for the curve drawn by c1v0.  (S1) where is the mean of the observed data, is the total sum of squares, and is the residual sum of squares. The and in Figure 3d and 3e are the values that make the coefficient of determination maximum for each case. The coefficient is determined with the average experimental repetition of 3.84.

Supplementary note4. Scaling of strongly bounded water manipulator
In order to count the number of strongly bounded water manipulator, we starts applying the Poisson-Boltzmann equation to gel single cell as depicted in Supplementary Figure 7a.
where is electrical potential, is local charge density following the Boltzmannstatics, is bulk ion concentration, is Boltzmann constant, T is absolute temperature, e is elementary charge, and is electrical permittivity. Since 1D polymer strand closes the 2D surface, a unit cell of gel should be treated as 2D geometry. Also amorphorous nature of gel allows us to assume the the circumferential symmetry (Supplementary Figure 7b). It simplify the Equation

S5
with linearizing 1 an exponential of charge density, so calld Debye-Huckel approximation. (S6) Appropriate boundary condition for yields the solution of Equation S6. (S7) where is Debye length and is zeta potential of gel. (S8) For considering effectively bounded water manipulator, we should count excess amount of water manipulator. (S9) Strongly bounded water manipulator is evaluated by integration of excess amount from 0 to cell diameter and circumferential integration. (S10) Upper bar represents the averaged operation and the dimensionless number, potential, and Debye length are expressed as follows.
(S11) (S12) For scaling by polymer parameters, The bulk concentration should be given by polymer volume fraction . (S13) where is number density of water manipulator. Therefore, Debye length and diameter of cell 2 are scaled as follows.
(S14) (S15) (S16) In this scale, the exponential in Equation S11 is effectively vanished. (S17) Note that the asymptote of strongly bounded water goes to zero when and with since exponential in Equation S7 is no longer omitted.Finnally, approximation of last factor in is applied. (S18) Consequantly, strongly bounded water manipulator monotonically increses with polymer zeta potential and polymer volume fraction.

Supplementary note5.1. Free energy concerning the first order phase transition
The phase transition involving latent heat is refered as the first order transition. Landau 3 phenomenologically proposed that the free energy is able to be expanded in a power series of the order parameter m with the spirit of the mean field theory. The symmetric group generates the set of the order parameter which is finite on the ordered state and zero on the disordered state. In general, The 6th order even polynomial of the order parameter describes the first order transition. (S19) where is the crystallization free energy change to reference state, Greeks are some constant, , is temperature difference with the critical temperature. We chose the order parameter of the system as fraction of solid component. For the moment, we should utilize 3 conditions to evaluate the coefficients: 1) latent heat, 2) the discontinuous jump of an order parameter and 3) susceptibility (specific heat) during a transition.
Unfortunately, in order to take the discontinuous jump of susceptibility, we have to regard free energy above the critical temperature ( ), which is not defining the order parameter. Therefore, one needs to reduce the order of the free energy or seek another condition. Near the onset of solidification (positive limit of disordered state i.e. ), rigorous consideration of the free energy 4 had been made that a small bump exists. Since we are going to investigate the behavior of an ordered state, a small bump of disordered state can be properly ignored. While this approximation allows a reduction of the number of coefficients, we inevitably lost an opportunity to analyze the order parameter adjacent the critical temperature in a detailed manner. (Note that we cannot produce variables.) However, the equilibrium order parameter describing an ordered state is only factor we should consider. The remaining conditions, the latent heat and a jump of the order parameter constrains the free energy as follow.

Equation S21
seems to be more powerful constraints than original condition, but it is natural since the order parameter was defined the number fraction of liquid component which is not sensitive to temperature under single phase state. And is latent heat per SAT molecules. Thus, the free energy regarding the transition of supercooling of SAT can be read as follow. (S23)

Supplementary note5.2. Hydrogel induced molecular field consideration
Free energy contribution of a hydrogel in a SAT solution is analogous to Flory's work 5 . (S24) The first term of Equation S24 is considering the mixing phenomenon and the second stands for the polymer-solvent interaction where is the effective interaction energy between polymer strand, analogous to Flory-Huggins parameter. Polymer volume fraction, should be carefully dealt with that Equation S25 can be only applied to solvent-polymer system, i.e. is function of n. (S25) where is number of polymer monomer, and x is the ratio of the molar volumes of the polymer and solvent. Note that the crystallization free energy, Equation S24 is applied to red region in Figure 3b and molecular field is working in the blue, thus, the control volume for Equation S24 shrinks with development of the crystal, therefore, the residual SBWM is being n itself not ,

Supplementary note5.3. Evaluation of the proportion of SBWM
The overall free energy of the system can be constituted by addition of Equation S23 and S24 as shown in Equation 3 of the manuscript. Differentiating it with respect to n yields the chemical potential of the system. Let The proportion of residual bound water we wish to evaluate is such that the chemical potential is set to be equilibrium. was evaluated by iteration method using Equation S26 and transition enthalpy data in the Figure 3d since the is function of and transition enthalpy relates to both.

Supplementary note6. Diffusive polymerization and Analysis of the Gradient Structure into the EVO gel
Diffusive polymerization consists of three steps that are the diffusing step, the halfway polymerization step, and the complete polymerization step. In the diffusing step, the solution is swelled into the halfway polymerized hydrogel for 30 minutes in room temperature for monolithic integration between two different hydrogels. In the halfway polymerization step, the gel is polymerized in the 60⁰C oven for 60 minutes considering the complete polymerization step takes about 90 minutes. This step allows the gel to have a gradient structure which will be discussed in the next paragraph. In the complete polymerization step, the gel is polymerizaed in the 60⁰C oven for 90 minutes allowing the gel to be prepared thoroughly.
To verify that the gradient structure is successfully implemented to the EVO gel, we calculated the characteristic diffusion length. (S27) With diffusion coefficient, [6][7][8] and time for 1800 seconds. The calculated diffusion length is about 2mm. According to the fact that we poured the precursor solution before the gel was fully fabricated, this value guarantees that the gradient structure is succesfully implemented.

Supplementary note 7. Measurement of elastic modulus and latent heat
We set up the indentating profile with 3 steps. First, the sample is compressed with the speed of 50 m/s for 10 seconds. Then, waiting time is set to be 10 seconds to distinguish the sample's elastic and plastic modulus. Finally, the indentation tip returns to the starting position, measuring the elastic force of the sample. The elastic modulus for each sample is determined by the equation below 9 where, A is truncated area of the sphere-shape tip, is the slope of the graph when returning to the original position. All indentation curves for control parameters are available in Supplementary   Figure 9.

Supplementary note8. Local Transformation -Re-programming Mode, Amending State
EVO gel has elasticity in I-mode. This means that one can deform to any shape. After one deforms the shape of the gel, one can fix that shape by giving chemical perturbation to the gel. Taking advantage of this feature, one can intentionally make a certain part of the V-mode gel to I-mode, deform to any shape, and fix the shape which is so called re-programming mode. In this research, by heating only a certain part of the gel, one can change the V-mode gel to re-programming mode and proceed as explained.
We conducted the experiment to investigate transient behavior of EVO gel when local heating. Bone Converted area was exactly equal to heated region (mode interface is exactly same as surface of water bath in each temperature) because conductive heat transfer in gel is effectively smaller than latent heat. Scaling comparison between these two terms can be read as followed. (S29) We presented material and geometric properties inSupplementary Table 1

Supplementary note10. Advantage of volumetric dehydration in EVO
Although polymer aggregation during dehydration contributes to enhancement of mechanical modulus in both cases, the main differences between evaporation and EVO are the domain of dehydration and net outward water flux as depicted in Supplementary Figure 19. In evaporation, outward water flux develops in surface of gel while EVO can dehydrate a gel volumetrically and keep water component. We explored micro dehydration phenomenon near single polymer cell in the manuscript (Figure 3b). The result was extensively expanded to entire system and well matched to the experiment (Figure 3e), which suggests that phase transition of water manipulator dehydrates an entire volume of EVO gel. Furthermore, reversibility of EVO presented in Figure 2c of the manuscript shows absence of outward water flux.
Volumetric dehydration by phase transition of water manipulator has two advantageous properties in altering mechanical modulus compared to evaporation; speed (domain of dehydration) and reversibility (no outer water flux) of the transition.

Supplementary note11. Mechanistic analogy between EVO and Salting-out
Sodium acetate interacts more strongly with water molecules in a solid phase than in a liquid state, leading to its negative and large latent heat of solidification. Therefore, the component that mainly interacts with water molecules is different according to the mode of EVO gel as shown in Supplementary Figure 20a.
In I-mode, three components of EVO gel (polymer, salt, and water) almost equally interact similar to "Salting-in" state. The solubility of polymer should be high which leads to a stable gel state is achieved. However, when the EVO gel undergoes phase transition, a set of sodium acetate ion vigorously attracts three water molecules to build energetically stable tri-hydrate crystal. In this process, polymer strands become de-swollen and aggregate which is very similar to precipitation of polymer in "Salting-out". Mechanistic analogy between EVO and salting-out is depicted in The salting-out phenomenon would play a crucial role in discontinuous formation of salt hydrate crystal inducing the aggregation of polymer strand that enhances the mechanical property of EVO gel.
It is expected that this effect might contribute to exponent of Equation 4 in the manuscript by ~2.3.