A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal

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Fratzl, P., Fischer, F.D., Svoboda, J., Aizenberg, J. Acta Biomaterialia, 2010, 6, 1001-1005

Author: Sofia Magkiriadou, Fall 2011

Keywords: single-domain crystal, polycrystalline, Gibbs free energy, diffusion, Onsager's principle, biomimetics

Introduction

The authors study the self-assembly of large single crystals with complex shapes. Based on the experimental observation that such materials often come from an amorphous precursor, they provide a simple thermodynamic model for this process which shows how the geometry of the precursor may affect the number of domains in the final product. Many organisms have been observed to form organs, such as teeth or exosceletons, from large single-domain crystals. This process has recently been replicated in the lab using amorphous calcium carbonate to make calcite crystals. Remarkably, it involves the transformation of one substance into a much denser one. At first glance, one would think that the resulting stresses would easily break the crystallization fronts and disrupt the formation of a single domain. Intriguingly, this process has been observed to succeed when the precursor is perforated by holes with a specific range of size and spacing. It is thus believed that the initial geometry plays a crucial role in the successful formation of large single crystals.

Single-crystal-figure1.jpg

Theoretical Model

The authors believe that an important initial condition is that the precursor be a soft material which can deform without maintaining high stresses as the crystal forms. This is indeed the case with amorphous calcium carbonate, which can contain a lot of water and is relatively flexible. As a model system they use a quasi-two-dimensional layer of amorphous calcium carbonate, of thickness d, containing holes of initial radius ρ_wo and final radius ρ_w on a square lattice with constant a (see Figure 2).

Single-crystal-figure2.jpg

Initially, it is asserted that in the middle of a square occurs a small circular crystallization nucleus with radius ρ_c. As the crystal grows, water is released, which results in the enlargement of the holes ρ_w. The dynamics of what follows must obey two conservation laws: calcium ions must be conserved, as must the number of water molecules. These two facts, in combination with the universal condition that the change in Gibbs free energy must decrease during a self-assembly process, leads to a condition on the maximum hole size, ρ, in the precursor for the formation of a single crystalline domain. If the holes are smaller than ρ they prevent the growth of the crystallization center; if they are larger than ρ their existence does not affect the growth of the crystallization center. Thus, if the precursor is perforated with holes smaller than ρ, the number of nucleation sites allowed to evolve into crystalline domains will be very small and potentially only one.

In order to gain further understanding into the dynamics of this process, the authors consider the diffusion of water out of the amorphous calcium carbonate (the calcium crystals contain no water). In particular, they consider the case where the holes are spaced closely enough that water preferentially leaves the system through them and not through the upper and lower surfaces, i.e. the case where a/d is sufficiently small.

Single-crystal-figure3.jpg

To understand the dynamics, the relevant principles are the energy dissipated during the diffusion of water, combined with Onsager's principle which requires that this energy dissipation balance the change in Gibbs free energy of the system (Ref[1]). Combining these considerations, the authors arrive at a relation between the diameter of the crystallization nucleus to the time it takes for it to grow until the next nearest hole. Interestingly, they observe that this time scales with the square of the lattice size, as opposed to linearly. The implication of this observation to the kinematics is that, if the holes in the precursor are too scarce, there is enough time for many crystalline domains to form before each crystal front reaches the nearest hole, thus preventing the formation of a single crystal.

Single-crystal-figure4.jpg

In summary

Based on these mathematical observations, the authors conclude that the existence of a perforated precursor is key to the formation of a large single crystalline domain, for the following three reasons:

1. the holes act as a sink for the excess water that must be removed as the hydrated amorphous calcium carbonate transforms into anhydrous calcium crystal, provided that their spacing is smaller than the material thickness

2. if the holes are sufficiently small, crystallization is suppressed, making the formation of a single crystal more likely

3. if the holes are sufficiently close together, crystallization of numerous sites is again suppressed, since the velocity of the crystal fronts scales as the inverse square of the distance between the holes; if the holes are too far, many crystalline fronts have time to form independently of each other.

These observations agree with the results of experimental attempts to recreate large crystals of complex shapes in the lab. Modeling the system might promote better understanding of this process and encourage its use for the creation of novel structures. The authors warn against the over-generalization of these principles, since not all occurrences of this phenomenon in nature require a hydrated precursor; however, they don't exclude the possibility that diffusion of other molecules or ions might play a similar role in those cases.

The author of this summary believes that, if this knowledge can be applied accurately to controlled experiments in the lab, it opens up a wide range of possibilities for the creation of interesting materials - potentially materials with high strength, since polycrystallinity is often associated with mechanical instability.