3D calcite heterostructures for dynamic and deformable mineralized matrices

Scales are rooted in soft tissues, and are regenerated by specialized cells. The realization of dynamic synthetic analogues with inorganic materials has been a significant challenge, because the abiological regeneration sites that could yield deterministic growth behavior are hard to form. Here we overcome this fundamental hurdle by constructing a mutable and deformable array of three-dimensional calcite heterostructures that are partially locked in silicone. Individual calcite crystals exhibit asymmetrical dumbbell shapes and are prepared by a parallel tectonic approach under ambient conditions. The silicone matrix immobilizes the epitaxial nucleation sites through self-templated cavities, which enables symmetry breaking in reaction dynamics and scalable manipulation of the mineral ensembles. With this platform, we devise several mineral-enabled dynamic surfaces and interfaces. For example, we show that the induced growth of minerals yields localized inorganic adhesion for biological tissue and reversible focal encapsulation for sensitive components in flexible electronics.


General growth process for calcite crystals
The growth of calcite crystals follows: where Ca 2+ ions come from CaCl 2 salt, and CO 3 2-ions were supplied by the following reactions: The precursors can yield various polymorphs of calcium carbonate, i.e. hydrated amorphous calcium carbonate (ACC), anhydrous ACC, vaterite, aragonite, and calcite, etc. In this discussion, hydrated ACC and anhydrous ACC are treated the same since they have similar characteristics, and aragonite is excluded from consideration given its relatively high crystallization temperature 1 . Considering enthalpies of formation (∆H ACC to calcite = -21.5 ~ -17.6 kJ/mol, ∆H vaterite to calcite = -3.4 kJ/mol) 2 and solubilities of calcium carbonate allotropes (K sp,ACC = 10 -6.30 , K sp,vaterite = 10 -7.91 , K sp,calcite = 10 -8.49 ) 3,4 , it is generally accepted that the crystallization of calcite occurs via nanoparticulate ACC precursors as passing through three phases: ACC → vaterite → calcite (Specifically, (i) initial deposition of ACC nanoparticles; (ii) dehydration of ACC and crystallization to vaterite; and (iii) transformation of vaterite to calcite via a dissolution and reprecipitation mechanism) 5 . Depending on the conditions, a direct transformation from ACC to calcite may also be possible 2 .

Growth kinetics of calcite crystals on patterned substrates
In order to elucidate the growth behavior of calcite crystals as well as to improve the controllability, we performed experiments with patterned hole arrays in the resist layers.
Briefly, we covered a single crystalline calcite substrate with a resist layer that was fabricated and isolated from a Si/SiO 2 wafer. The resist contains 7×7 square arrays of circular holes with different diameters (d h = 1, 2, 3, 4, 5, 8, 10, 15, and 20 µm) and center-to-center distance ( ! = 10, 20, 30, and 50 µm) (Supplementary Fig. 21). While the resist layer physically suppresses the nucleation of calcite, the exposed calcite surfaces via the holes served as the nucleation sites for the epitaxial calcite growth, producing micro-patterned arrays of rhombohedral calcite crystals. The growths from these smaller and well-controlled arrays (vs. 2D arrays with infinite sizes) allowed the exploration of growth mechanism and kinetics that are equally applicable over the large area matrix. Specifically, we have studied the effect of d h and d c on the effective size ( !"" ) of calcite crystals ( !"" is defined in the Supplementary   Fig. 7).
We observed several unique features in the experiments: (i) the !"" of calcite crystals grown out from the holes (Fig. 1d, left) is dependent on their locations within the arrays (i.e., !"" Considering all these observations into account, we formulated a hypothesis that the crystal growth is mainly governed by a lateral diffusion-limited kinetics. To justify our hypothesis, (1) The total volume of calcite crystals after the growth is proportional to the total amount of initial precipitate (i.e., precursors for calcite) from the supersaturated solutions.
(2) The existence of preferential nucleation sites in the exposed areas suppresses the nucleation of free-standing calcite crystals on the resist surface unless the ! is large.
(3) We consider two main routes for precursor supply: one is the vertical mass transport from the bulk solution, and the other is the lateral diffusion near the sample surface.
At the early stage, nanoparticulate ACC and vaterite are generated and dispersed in the bulk solution, and some of those particles settle down over the sample surface. Once nucleation from the exposed calcite sites initiates, the pre-deposited ACC and vaterite particles are dissolved and the resultant precursor ions are diffused to nearest calcite crystals and consumed for calcite growth. The ion concentration in the bulk solution remains largely steady and the mass transport along vertical direction is limited. The overgrowth of calcite via the holes is therefore driven mainly by the lateral diffusion of precursors that are released from ACC/vaterite in the immediately adjacent areas (Supplementary Fig. 24). Considering mass conservation, the relationship between volume of calcite crystal ( ) and supply area ( ) can be followed as where reflects the volume density of precursor particles (e.g., ACC and vaterite) per area, which is dependent on the degree of supersaturation, container geometry (e.g., depth) and the orientation of the growth substrate. ! is the radius of the circular area in which precursor particles are fully dissolved at the time . In our experimental system, the volume expension of calcite was limited when the ! exceeds ! 2 due to sharing of precursor supply with neighbouring crystals. The maximum becomes, Since is proportional to !"" ! , we then have: where is the (height/effective width) ratio for individual calcite crystal, and is the angle between two edges (~ 74.55°). This relation predicts that the maximum !"" (i.e., the effective length one can achieve after a single growth cycle is complete.) is linearly proportional to ! ! ! and also, volume limitation in ideal case of our experiemntal system. Indeed, this result agrees well the experimental data ( Fig. 1g and Supplementary Fig. 23, when ! < 30 µm).
Additionally, this lateral diffusion-limited crystal growth model can also explain locationdependent crystal size differences. Crystals with smaller 'coordination' number (i.e. crystals at the edge and corner) have extra exclusive area to collect precursors, and consequently, crystal size is bigger than that from the center area (Supplementary Figs. 22 and 26).
To evaluate the calcite growth dynamics, we first calculated the 2D diffusion of precursors at each time frame based on Fick's second law 6 , where is the ion concentration, is the distance. We considered two boundary conditions as follows: where ! is the apparent solubility of precursor precipitate, ! is the distance from center to the surface of calcite crystal and which is ~ !"" 2, and ! is the concentration at surface.
Thus, the concentration of precursor will be Under steady-state conditions, diffusion flux can be solved by Fick's first law, where is the diffusion constant. In ideal case, a total amount of diffusion fluxes at boundary of adjacent area ( ! should be same as growth rate of calcite crystal ( ), Plugging the diffusion flux (10) into this equation, we obtain From the definition of ! ( !~!"" 2) and equation (3) , growth rate can be derived as where and are the constant ( = 6 ! − ! , = 8 sin ! ). Given that ! is always bigger than ! (thus ! ! > 1), the growth rate of calcite crystal follows a 1 ln ! curve ( ! > 1). It is implying that the growth rate will be decreased as increase of ! (Supplementary Fig. 27). In addition, the formation of a neck (i.e. after filling of a hole in the mask) should precede the lateral overgrowth of rhombohedral calcite crystal (RE, Supplementary Fig. 3b) and as a result, ! is at least bigger than ! 2.
Although this model well explains growth behavior of calcite crystal in our system, there are some limitations. For example, we did not consider (i) the nucleation or growth of freestanding calcite crystal, which would occur with large ! (Fig. 1g and Supplementary Fig.   23, when ! > 30 µm), and (ii) the exact volume of neck.