Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Dana
Zöllner
and
Igor
Zlotnikov
*

B CUBE – Center for Molecular Bioengineering, Technische Universität Dresden, Dresden, 01307, Germany. E-mail: igor.zlotnikov@tu-dresden.de

Received
17th September 2018
, Accepted 15th January 2019

First published on 15th January 2019

Grain size distribution is a fundamental property of polycrystalline materials and the phenomenon of grain boundary motion has major repercussions for materials processing and in a variety of technological applications. In classical models for microstructural evolution during grain boundary motion, special attention is given to the case of “ideal” grain growth defined by the seemingly unrealistic assumption of homogeneity of the physical properties of the boundaries. Whereas different models were rarely shown to demonstrate ideal microstructural evolution of selected synthetic and geological granular networks, here we present a biogenic polycrystalline material the formation of which is consistently described by all principal predictions of an ideal grain growth behaviour. We show that biomineralization of the prismatic architecture in the shell of the mollusc Atrina vexillum is fully quantified by conventional thermodynamic, kinetic and topological considerations, thus making a material formed by a living organism archetypical of ideal grain growth.

## Conceptual insightsWe present an intricate 3D biomineralized architecture the formation of which is fully predicted by key thermodynamic, kinetic and topological models that were classically developed to describe an analytically ideal behavior during grain growth of man-made and geological materials. Whereas ideal grain growth was rarely shown to occur even in generic materials systems using some of the theoretical models, the morphogenesis of the prismatic assembly in the mollusc shell of Atrina vexillum is the first material that adheres perfectly to all, both qualitatively and quantitatively. Surprisingly, the polycrystalline assembly, that evaded physicists and materials scientists since the early 1950s, is found to be formed by a living organism. |

Since the early 1950s, many attempts have been made to understand and to predict polycrystalline microstructure evolution during grain growth and coarsening analytically. From a general point of view, this process is rather simple. A polycrystalline microstructure consists of many mono-crystalline grains, where the grain boundaries between two neighbouring crystals are usually slightly curved. This curvature is the result of the system striving to achieve an angle of 120 degrees between the three arms at a triple junctions, where three grains meet, necessary to maintain an energetic equilibrium.^{7,8} This configuration results in a curvature-driven boundary motion during which the free energy is reduced by a motion of the boundaries to the centre of their curvature. Since small grains with less than the average amount of edges tend to have convex boundaries, they shrink, whereas large grains with more than the average amount of edges tend to increase their size. As a result, the number of grains decreases and the total grain boundary area and the coupled free energy decreases, while the average grain size increases. Of course, such a transformation during production and application can be both wanted or unwanted, but in any case leads to a change of the properties of the material.

A special situation that has been drawing a majority of research efforts is the case of curvature-driven ideal grain growth, where it is assumed—mostly for reasons of simplification—that all grain boundaries in a polycrystalline network are characterized by the same physical properties.^{7} Specifically, by the same value of surface tension, γ, as well as by the same boundary mobility, m. Hence, any possible dependence of the boundary properties m and γ on misorientation between the associated neighbouring crystals is neglected in agreement with the uniform boundary model.^{5} As a result, over the years, many analytical laws have been derived to describe ideal grain growth of polycrystalline materials in two and three dimensions. Whereas large experimental evidence exists showing that different analytical theories are successful in describing microstructural evolution of a variety of structures, so far not a single materials system has been shown to obey all predictions of ideal grain growth qualitatively nor quantitatively.

In this work, we present the first microstructure whose morphogenesis consistently fulfils a number of classical thermodynamic and kinetic relationships for ideal grain growth, both qualitatively and quantitatively. Interestingly, the different theories are shown here to converge and describe a biogenic mineralized material formed by a living organism. Namely, the growth of the prismatic architecture in the shell of the bivalve Atrina vexillum (Fig. 1). Similar to other bivalves, which are molluscs that have a shell made of two hinged valves that enclose the soft body of the animal, the shell of A. vexillum consists of two mineralized assemblies, which are arranged in layers parallel to the outer surface of the shell. One layer is called the prismatic ultrastructure composed of elongated columns made of calcite (Fig. 1a), and the other layer is a nacreous ultrastructure composed of flat platelets made of aragonite. Earlier studies were successful in demonstrating that physical models have the capacity to describe the morphogenesis of the prismatic architecture in a variety of bivalves^{9–11} and ideal behaviour in A. vexillum was recently suggested.^{12} However, ideal growth that perfectly adheres to all principal theoretical predictions has not yet been shown.

To obtain a complete history of microstructural evolution of the prismatic layer in A. vexillum, we performed a microtomography experiment at beamline ID19 of the European Synchrotron Radiation Facility (ESRF). The organic interfaces, having the thickness of approximately 1 μm, were easily resolved using an effective voxel size of 0.649 μm. The measurement covered a total thickness of approximately 1 mm from the main body of the prismatic layer. A representative 3D segment reconstructed from these data clearly shows the coarsening behaviour of the prismatic ultrastructure along the direction of growth, z (Fig. 1d). In addition, when comparing 2D slices, taken from the microtomography data perpendicular to the direction of growth, one during the initial stage of the prismatic layer formation (closer to the periostracum) (Fig. 1e) and one at a later stage (Fig. 1f), the reduction of the amount of the boundary and the number of prisms is evident. To follow this behaviour as a function of the growth direction quantitatively, similar 2D sections throughout the entire thickness of the prismatic layer were analysed. It is important to note that in this study, the time parameter, t, that is commonly used to follow the coarsening process, was substituted with the direction of growth, z. Here we assume a linear correlation between the two and, essentially, reduce the three-dimensional growth in space into a two-dimensional temporal process.^{9–11}

Assuming that the migration of the boundaries within a grain network is driven by transport of matter under pressure because of curved interfaces between the grains, the direction of migration of a boundary is always directed towards the centre of its curvature. This leads to a reduction of the total inner interface area of the microstructure and, therefore, of the total interface energy. The associated velocity of a segment of any grain boundary is given by v = mγκ, thus, introducing the curvature κ. The velocity is always directed to the centre of curvature. In ideal growth, assuming that all boundaries are characterized by a unique value of surface tension, γ, and mobility, m, and that a grain can be described in average by a spherical shape, the curvature relates directly to a corresponding average radius, 〈R〉, calculated as a grain-volume-equivalent sphere resulting in κ ∝ 〈R〉^{−1}.

Recently, the subsequent relation for the change of the average grain size with annealing time, t, was derived with the same result but using a completely different ansatz:^{19}

(1) |

(2) |

In the prismatic ultrastructure of A. vexillum, the average grain size indeed follows the well-known growth law, eqn (2), in terms of 〈R〉 = 〈R〉(z) (Fig. 2a) and the growth exponent is n = 0.4976 ± 0.0027. Here, the calculations were performed every 20 sections, corresponding to a step size of 12.98 μm and the area of each section is 973.5 μm × 973.5 μm (1500 pixels × 1500 pixels). The radii were calculated as prism-area-equivalent circles using R = √(A/π), where A is the measured prism area. The growth factor is b = 2mγ/Γ = 0.2369 ± 0.0011 μm, from which the reduced mobility, mγ, can be calculated under the assumption that the growth constant is the same as in the above mentioned case for ideal grain growth with Γ = 6.8639 resulting in mγ = 0.8130 ± 0.0038 μm.

Fig. 2 Average growth law and self-similarity. (a) The relationship between average prism radii, 〈R〉, and the growth direction, z. The curve represents a fit to eqn (2) – average growth law (the regression coefficient R^{2} is 0.9985). (b) Scaled prism size distributions showing self-similarity at three different sections, z_{1} < z_{2} < z_{3}, along the growth direction of the prismatic layer. The curve represents a fit to Weibull distribution function (the regression coefficient R^{2} is 0.9345). (c) and (d) Prism size distribution and a corresponding natural double logarithmic plot of different sections along the growth direction, respectively. Solid curves represent a fit to the envelope of the numerical data (the regression coefficient R^{2} is 0.8822). Dashed curves outline the first and the last data set along the direction of growth. The data have been put into 13 bins. |

F(R,t) = g(t)f(x). | (3) |

The time-dependent function that follows from volume conservation yields g(t) = N/〈R〉 ∝ 〈R〉^{−(d+1)},^{23} where the dimension of the structure is d ∈ {2,3} and N = N(t) is the number of grains, which decreases with time. In contrast, the function f(x) is time-independent function that scales the grain radii of all the grains in the microstructure by the average radius such that x = R/〈R〉 and .

Time-independence of f(x) and therefore, self-similarity of the prismatic ultrastructure in A. vexillum is shown in Fig. 2b. Identical scaled size distributions are obtained at three randomly selected sections along the direction of growth, in z_{1} = 25.96 μm, z_{2} = 376.42 μm and z_{3} = 726.88 μm. The evolution of the distribution function F(R,z) is presented in Fig. 2c and also in Fig. 2d in a double-logarithmic plot showing the changes of the distribution with the direction of growth every 60 sections (corresponding to a step size of 38.94 μm). In both figures, the data points of the first and the last size distribution are connected by dashed lines. The linear least squares fit in Fig. 2d represents an envelope to the numerical data.^{24} In Fig. 2c, this unique curve, F_{e}(R), touches every single size distribution in exactly one point. It is very interesting to note that theoretically the envelope F_{e}(R) is proportional to R^{−(d+1)}. In the present case, the exponent is −2.9924 ± 0.0838, which represents a two-dimensional microstructure (d = 2). Moreover, the Weibull distribution function, which was previously proposed to describe two-dimensional grain growth, perfectly describes the scaled size distribution (Fig. 2b).^{25} Although the Weibull distribution is not physically motivated, it fits the numerical data of 2D grain growth.^{20,25}

The law states that the number of edges, n, of a grain is related to the average number of edges of all neighbouring grains, , by the approximation ·n = 5n + 8. Based on Euler's theorem for planar graphs,^{32} Weaire^{27} extended this consideration taking the central second moment of the neighbour distribution, μ_{2} = 〈n^{2}〉 − 〈n〉^{2}, into account, thus yielding ·n = 5n + (6 + μ_{2}). This extension takes into account that every two-dimensional cellular network can be characterized by its own distribution of the number of edges. Further studies formed what is known today as the Aboav–Weaire-law:^{28}

·n = (〈n〉 − α)n + (〈n〉α + μ_{2}). | (4) |

In the present case, the predicted linear relation between the product of the average number of edges of all neighbouring prisms, , and the number of edges of the corresponding central prism, n, versus n is indeed fulfilled in the three different sections (z_{1},z_{2},z_{3}) in the prismatic layer (Fig. 3a). A linear least squares fit of the data from the section at z_{1} to eqn (4) yields α = 1.2028 ± 0.0217 and α = 1.2039 ± 0.0264 extracted from the first and the second term, respectively. This result is perfectly consistent considering that the two terms were fitted independently. Furthermore, the obtained average number of edges is 〈n〉 = 5.9930 ± 0.0217, which is again fully consistent with ideal growth.

Fig. 3 Aboav–Weaire-law, Lewis-law and von Neumann–Mullins-law. (a) The relationship between the average number of edges in all neighbouring prisms, , the number of edges, n, of the central prism at three different sections, z_{1} < z_{2} < z_{3}, along the growth direction of the prismatic layer. The curve represents a fit of the data from the section at z_{1} to eqn (4) – Aboav–Weaire-law (the regression coefficient R^{2} is 0.9210). (b) The relationship between the scaled grain area, ρ = A/〈A〉, and the number of edges, n, at three different sections, z_{1} < z_{2} < z_{3}, along the growth direction of the prismatic layer. The curve represents a fit of the data from the section at z_{1} to eqn (5) – Lewis-law (the regression coefficient R^{2} is 0.6996). (c) The relationship between prism area change rate and the number of edges, n, at three different sections, z_{1} < z_{2} < z_{3}, along the growth direction of the prismatic layer. The curve represents a fit of the data from the section at z_{1} to eqn (6) – von Neumann–Mullins-law (the regression coefficient R^{2} is 0.3832). It is important to note, that the deviation from the predicted behaviour observed for n = 3, 9 and 10 is due to the limitation in data size and that the small values of the obtained regression coefficients are the result of naturally occurring vertical data spread. |

(5) |

(6) |

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