Perhaps no mystery of the mineral kingdom has remained shrouded in the details of its chemistry and physics longer than that of their origin. Scholars have been fascinated by the formation of natural crystals since ancient times. Plato, who was so intrigued by the origin of crystal shapes that the five regular polyhedra now bear his name, speculated in ~360 BCE that “when earth is compressed by air into a mass that will not dissolve in water, it forms stone, of which the transparent sort made of uniform particles is fairer, whereas the opposite kind is coarser” (Caley and Richards 1956). In later ages, luminary figures such as Theophrastus (Caley and Richards 1956), Pliny the Elder (Bostock and Riley 1855), and Georgius Agricola (Hoover and Hoover 1950) all made their own attempts at explaining the formation of mineral particles and crystals.In the modern scientific era, the advent of atomic theory made it clear that through some combination of processes, atoms could align themselves into regular arrays whose bonding would serve to minimize the energy of the system. The groundwork for studying rates of crystallization was arguably first laid out by Johnson, Mehl, Avrami, and Kolmogorov (Avrami 1939, 1940, 1941) with the derivation of the widely used “JMAK” kinetic equation. We now have many other theoretical tools with which to piece together the formation of minerals. Classical nucleation theory describes the thermodynamics of the incipient stages of crystal formation (Volmer and Weber 1925; Becker and Döring 1935). The electrical double layer model describes the chemical interactions that take place at the surface of a growing particle (Helmholtz 1853), and the order in which minerals will crystallize from a complex, silicate-bearing melt was famously worked out by Bowen (1928). Processes such as homogeneous and heterogeneous nucleation, Ostwald ripening, step and edge formation, oriented attachment, and surface-induced phase stability reversal have been studied in detail, each one contributing a new piece to a complex but ever-growing puzzle.However, despite the many great advances that have been made in the past century, the complex array of interacting effects and processes has made it seemingly impossible to unite crystallization into a single understandable theory, and many fundamental mysteries remain. In this issue of American Mineralogist, Eberl (2024) argues that the so-called “law of proportionate effect” may be a guiding principle that describes a majority of systems in which a large number of crystalline particles form and grow within a homogeneous medium. This law (as used by Eberl) holds that whenever crystals grow via advective transport of nutrients, each crystal grows by an amount that is, on average, proportional to its original size during each time step (muting the absolute growth rate of very small crystals while exaggerating the growth rate of large ones). Eberl (2024) outlines how this law, when applied to an initially uniform distribution of crystal nuclei, quickly transforms the population into a lognormal distribution of sizes. This distribution then increases its mean and variance as it evolves but retains its fundamental lognormal shape so long as transport-limited growth dominates the system.This mathematical description of crystallization is obviously not completely universal, and Eberl himself discusses several other processes that can result in alternative distribution styles. He does, however, argue (with abundant examples from lab syntheses and natural systems) that a great majority of homogeneous nucleation and growth processes will result in lognormal size distributions via the law of proportionate effect. In this framework, the mean and variance of a mineral’s lognormal size distribution can help us retrace its crystallization history, whereas non-lognormal distributions can serve as telltale indicators of additional processes that took place at an early stage of crystallization (even when the physical remnants of that process are too heavily overprinted to directly observe).Regardless of whether the law of proportionate effect is truly universal, this framework offers a chance to systematize crystallization processes and identify them in natural systems using measurable predictions. It also offers hope that even when an array of molecular mechanisms is at work, an element of statistical randomness can (ironically) average them into an understandable and more deterministic net result. Though he would surely have been vexed by descriptions of the underlying electrostatic and quantum interactions, Plato would no doubt have been pleased to have a working classification scheme not only for the beautiful natural crystals that he so admired, but also for the processes that create them.

中文翻译：

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寻找晶体生长的普遍规律：比例效应定律？

也许矿物王国的神秘面纱在其化学和物理细节中的存在时间比其起源还要久。自古以来，学者们就对天然晶体的形成着迷。柏拉图对晶体形状的起源非常感兴趣，以至于现在五个正多面体都以他的名字命名，他在公元前约 360 年推测：“当地球被空气压缩成不溶于水的物质时，它会形成石头，由均匀颗粒组成的透明种类更公平，而相反的种类则更粗糙”（Caley and Richards 1956）。在后来的时代，泰奥弗拉斯托斯（Caley and Richards 1956）、老普林尼（Bostock and Riley 1855）和乔治斯·阿格里科拉（Georgius Agricola）（Hoover and Hoover 1950）等杰出人物都尝试解释矿物颗粒和晶体的形成。在现代科学时代，原子理论的出现清楚地表明，通过某种过程的组合，原子可以将自己排列成规则的阵列，这些阵列的结合可以最大限度地减少系统的能量。研究结晶速率的基础可以说是由 Johnson、Mehl、Avrami 和 Kolmogorov（Avrami 1939、1940、1941）首先通过推导广泛使用的“JMAK”动力学方程奠定的。我们现在拥有许多其他理论工具来拼凑矿物的形成过程。经典成核理论描述了晶体形成初期阶段的热力学（Volmer 和 Weber 1925；Becker 和 Döring 1935）。双电层模型描述了生长颗粒表面发生的化学相互作用（Helmholtz 1853），而矿物质从复杂的含硅酸盐熔体中结晶的顺序是由 Bowen（1928）提出的。均质和异质成核、奥斯特瓦尔德熟化、阶梯和边缘形成、定向附着和表面诱导的相稳定性反转等过程均已得到详细研究，每一个过程都为复杂但不断增长的难题贡献了新的一块。过去一个世纪取得的许多巨大进步，一系列复杂的相互作用的影响和过程，使得将结晶统一成一个单一的可理解的理论似乎是不可能的，并且许多基本的谜团仍然存在。在本期《美国矿物学家》中，Eberl（2024）认为，所谓的“比例效应定律”可能是描述大多数系统的指导原则，在这些系统中，大量晶体颗粒在均匀介质中形成和生长。该定律（由埃伯尔使用）认为，每当晶体通过养分的平流输送生长时，每个晶体在每个时间步长中平均生长量与其原始尺寸成正比（减弱非常小的晶体的绝对生长速率）同时夸大了大型企业的增长率）。Eberl (2024) 概述了该定律如何 当应用于最初均匀分布的晶核时，很快会将总体转变为尺寸的对数正态分布。然后，这种分布随着其演化而增加其均值和方差，但只要传输限制增长在系统中占主导地位，就保留其基本对数正态形状。这种结晶的数学描述显然不完全通用，埃伯尔本人讨论了其他几个可能导致替代的分发方式。然而，他确实认为（通过实验室合成和自然系统的大量例子）绝大多数均匀成核和生长过程将通过比例效应定律产生对数正态尺寸分布。在这个框架中，矿物对数正态尺寸分布的均值和方差可以帮助我们追溯其结晶历史，而非对数正态分布可以作为结晶早期阶段发生的其他过程的指示指标（即使当物理残留物存在时）无论比例效应定律是否真正普遍，该框架都提供了系统化结晶过程并使用可测量的预测在自然系统中识别它们的机会。它还提供了希望，即即使一系列分子机制在发挥作用，统计随机性元素也可以（讽刺地）将它们平均化为可理解且更具确定性的最终结果。尽管他肯定会对潜在的静电和量子相互作用的描述感到烦恼，但柏拉图无疑会很高兴有一个有效的分类方案，不仅适用于他如此欣赏的美丽的天然晶体，而且还适用于创造它们的过程。无论比例效应定律是否真正普遍，该框架都提供了一个将结晶过程系统化并使用可测量的预测在自然系统中识别它们的机会。它还提供了希望，即即使一系列分子机制在发挥作用，统计随机性元素也可以（讽刺地）将它们平均化为可理解且更具确定性的最终结果。尽管他肯定会对潜在的静电和量子相互作用的描述感到烦恼，但柏拉图无疑会很高兴有一个有效的分类方案，不仅适用于他如此欣赏的美丽的天然晶体，而且还适用于创造它们的过程。无论比例效应定律是否真正普遍，该框架都提供了一个将结晶过程系统化并使用可测量的预测在自然系统中识别它们的机会。它还提供了希望，即即使一系列分子机制在发挥作用，统计随机性元素也可以（讽刺地）将它们平均化为可理解且更具确定性的最终结果。尽管他肯定会对潜在的静电和量子相互作用的描述感到烦恼，但柏拉图无疑会很高兴有一个有效的分类方案，不仅适用于他如此欣赏的美丽的天然晶体，而且还适用于创造它们的过程。