# On the critical nucleus and nucleation process of
$\beta$-cristobalite crystal

Shangcong Cheng

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Shangcong Cheng. On the critical nucleus and nucleation process of $\beta$-cristobalite crystal. Academia Materials Science, 2025, 2 (1), 10.20935/AcadMatSci7498. hal-05522266

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# On the critical nucleus and nucleation process of β-cristobalite crystal

Shangcong Cheng<fnref n="1" />,*

Academic Editor: Raul D.S.G. Campilho

## Abstract

Despite several decades of study, there has been much controversy regarding the structure of the glass and the nature of glass transition. Understanding the nucleation process of β-cristobalite is an important step in studying the formation of silica glass structure. The knowledge of structural change in the system prior to critical nucleus formation is crucial to follow the structural evolution of silica glass in the cooling process. The existing classical nucleation theory (CNT) has limitations in this aspect as it does not evaluate the critical role of the disorder-order transition in the nucleation process nor does it provide structural information before the critical nucleation. This work proposes a new hypothesis describing the critical nucleus's shape, size, and formation pathway. An analysis based on the newly proposed hypothesis shows that the most popular continuous random network (CRN) theory cannot completely describe the heterogeneous glass structure. A new theory that recognizes the small clusters in glass structures is needed.

**Keywords:** critical nucleus, disorder-order transition, nucleation pathway, high-speed transmission electron microscopy, silica glass, β-cristobalite

**Citation:** Cheng S. On the critical nucleus and nucleation process of β-cristobalite crystal. *Academia Materials Science* 2025;2. https://doi.org/10.20935/AcadMatSci7498

## 1. Introduction

Nucleation is a thermodynamic process in which structures transform from one phase to another based on free energy reduction consisting of a series of events at the molecular level, which is challenging to describe in detail. Among these steps, the most important is the one that creates nuclei with a critical size beyond which growth will continue. Owing to the critical size of a few nanometers, it is difficult to conduct experimental and computational studies of critical nuclei. However, crystal nucleation in supercooled liquids is a key phenomenon in glass science and engineering, and it plays an essential role in glass structures and production control of glass ceramics. The process has been extensively studied experimentally and theoretically in recent decades [1–9].

The nucleation process of β-cristobalite crystal is especially important due to its association with silica glass, the simplest silicate glass. Understanding the association between the β-cristobalite nucleation process and silica glass structure is fundamental to deciphering the structures of silicate glasses [10, 11]. Previously, the study of β-cristobalite nucleation was based on the classical nucleation theory (CNT), which displays many contradictions with experimental data [12–15]. In this work, the CNT and its weaknesses are briefly reviewed. A model different from various versions of the CNT is then introduced to better describe the nucleation process of β-cristobalite. Based on the new model, the β-cristobalite critical nucleus's shape, size, and formation pathway will be presented. The consequences of the

two models and their different impacts on the final glass structure will be discussed.

## 2. Classical nucleation theory

The CNT was originally derived by Gibbs in his study of the formations of liquid drops from vapor and gas bubbles in boiling liquids [16]. To extend the theory to the formation of crystals in supercooled liquids, additional uncertain assumptions are made about how the molecules in the original phase become part of the crystalline phase. In general, two factors change the system's free energy during the formation of clusters at temperatures lower than the crystal melting temperature $T_m$. One is crystallization, which lowers the volume of free energy, and the other is the creation of an interface between regions of different structures, which increases the system's free energy. In the CNT model, the clusters are assumed to be spherical with radius $r$; thus, the reduction in free energy from crystallization is $(4/3)\pi r^3 \Delta G_x$, where $\Delta G_x$ is the free energy difference per unit volume between crystal and liquid. The free energy increase caused by the interface is $4\pi r^2 \sigma$, where $\sigma$ is the interface energy per unit area. The total change in energy for the cluster, $\Delta W$, is given by the following equation [17]:

$$ \Delta W = (4/3)\pi r^3 \Delta G_x + 4\pi r^2 \sigma \quad (1) $$

<footnote n="1">¹Molecular Foundry of Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.</footnote>

<footnote>*email: cheng303@gmail.com; shangcongcheng@lbl.gov</footnote>
$\Delta W$ is a function of $r$; when $r$ is sufficiently small, $\Delta W$ tends to increase. This means that small clusters are unstable, and clusters may survive with a probability of 50% only when their size reaches a critical value. The critical radius, $r^*$, can be calculated by differentiating $\Delta W$ in Eq. (1) with respect to $r$ and setting its value equal to 0:

$$r^* = -2\sigma/\Delta G_x \qquad (2)$$

Equation (2) shows that at temperatures below the melting temperature, $T_m$, $\Delta G_x$ is negative, and $r^*$ has a temperature-dependent positive value.

Although the CNT has a history of over a hundred years and several attempts have been made to enhance the theory, it has gained only limited success in describing crystal nucleation. On some occasions, disagreements between CNT and experimental data can be as large as many orders [13, 14]. This may be because nucleation in the supercooled liquid differs greatly from the nucleation mechanisms of gas bubbles in boiling liquids and liquid droplets from vapor. Although successfully applicable for gas bubbles and liquid droplets, the CNT could not be readily applied to crystal nuclei.

Crystals have a three-dimensional ordering structure, and crystal nucleation is a self-organizing process that leads to the formation of a new thermodynamic phase. The crystal nuclei are usually polyhedrons, not spherical. The nonspherical shape of the interface indicates that the interface energy may not play an important role in this process. The correct CNT needs to consider this and emphasize the formation of an ordering structure rather than the spherical interface. Based on this insight, a new model is needed to better describe the β-cristobalite critical nucleus's shape, size, and formation pathway.

### 3. New hypothesis about the critical nucleus of β-cristobalite

As discussed in the previous section, the nucleation process of β-cristobalite crystal impacts the structural formation of silica glass and has attracted much research attention. Understanding the structure of silica glass is fundamental to deciphering the structure of all other silicate glasses. The existing CNT does not evaluate the critical role of the disordering-ordering transition in the nucleation process and may have overemphasized the contribution of interface energy in the critical nucleation.

This work proposes a hypothesis on the critical nucleus's shape, size, and formation pathway for the β-cristobalite crystal and explains why the nucleation rate is significantly lower than the crystal growth rate. If the critical nucleus of a β-cristobalite crystal is spherical, as suggested by the CNT or modified CNT, then the shape of the crystal after increasing its size must also be near-spherical. However, the experimental observations show that their shape is not spherical but octahedral. Figure 1a is a photograph of several β-cristobalite crystals that grow nearby after liquid silica nucleation. The edge of the crystal is close to 1 mm long [18]. Figure 1b is the microscope image of a cross-section of a small β-cristobalite crystal [19]. The shape of the cross-section is a polygon, not a circle, as the CNT expected. The bar in the image is 100 µm.

<figure><img src="image_3.png" /><figcaption>Figure 1 • Images of β-cristobalite crystals. (a) Image showing that their shape is not spherical but octahedral, or half octahedral. The field of view is 3 mm. Reproduced from Wolfsried [18]. (b) Cross-section microscope image. The bar is 100 µm. Reproduced from Wagstaff [19].</figcaption></figure>

The above-observed nonspherical shape of the interface contradicts with that proposed by the CNT that the interface energy plays an important role in the crystallization process. Otherwise, the new crystal phase from the liquid phase should be spherical, which would reduce the interface area.

Since the nucleus of β-cristobalite crystal is not spherical, the size of the nucleus should be described using a different perimeter from radius $r$. The critical size of the nucleus can be calculated based on the characteristics of the crystal.

β-Cristobalite crystal has the face-centered cubic (FCC) structure, shown in Figure 2a, and the (111) is the habit plane. Figure 2b shows that the eight facets of the octahedron crystal are formed by the (111) crystal planes. In Figure 2b, the triangle ABC represents one of the facets. Because of its inherent high degree of symmetry, the estimated thickness in the [111] direction can be used to calculate the size of the crystal in directions perpendicular to the [111] direction. In Figure 2b, the triangle ABC (ΔABC) represents a facet
of the crystal perpendicular to the [111] direction; AB is the length of the ΔABC's side and represents the crystal's size L. OE is perpendicular to ΔABC and represents the distance from the center of the crystal to the facet, i.e. thickness t in the [111] direction of the crystal. From symmetry, the ratio of AB to OE can be computed as follows [10]:

$$L/t = 2.45 \qquad (3)$$

This result means that in forming a β-cristobalite crystal, the tetrahedra SiO₄ units connect 2.45 times faster in the plane perpendicular to the [111] direction than in the [111] direction itself. This is consistent with the bonding structure of the basic SiO₄ tetrahedron building block. Among the four oxygen atoms of the SiO₄ tetrahedron that bridge the neighboring tetrahedra, only one has its Si-O bond in the [111] direction. In contrast, three oxygen atoms have Si-O bonds in the plane perpendicular to the [111] direction. This results in a structure with a layer dimension larger than its thickness. The structure of the facet is a two-dimensional character in the plane perpendicular to the [111] direction, in which all Si-O bonds would form six-membered rings. This is shown in **Figure 3a**. **Figure 3b** represents the side view of the structure from the [−110] direction of the crystals.

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 2** • Schematic images of the octahedral and isometric β-cristobalite crystal. (a) The structure of β-cristobalite. (b) The triangle ABC (ΔABC) represents a facet of the crystal perpendicular to the [111] direction; OE is perpendicular to ΔABC and represents the distance from the center of the crystal to the facet.</figcaption></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 3** • Two-layer structure of β-cristobalite crystal. (a) Top view along the [111] direction of the cristobalite crystal: the structure is two-dimensional, and all Si-O bonds form six-membered rings. (b) Side view of the structure from the [−110] direction of the crystals.</figcaption></figure>

As shown in **Figure 3b**, the orientations of SiO₄ tetrahedra change alternatively along both layers; the smallest stable β-cristobalite crystal particles consist of two layers of (111) planes. With a layer spacing of 0.41 nm in the [111] direction [20], the thickness of the two layers of planes is 0.82 nm. Using Eq. (3), the β-cristobalite nucleus's size L is calculated to be approximately 2.0 nm. Thus, the nucleus size depends only on the cristobalite structure and is unrelated to temperature. This is a significant difference between the proposed hypothesis and the CNT. In the new model, although the critical nucleus has the same size at various temperatures, the nucleation rate is a unimodal function of temperature between T_m and 1,470°C, the polymorphic inversion temperature between β-cristobalite and β-tridymite. As the temperature decreases from T_m, the driving force for crystal formation increases, thus increasing the nucleation rate. Lowering the temperature also reduces the mobility of molecules, so the nucleation rate reaches a maximum value at a particular temperature and then decreases when the temperature is further lowered. At the temperature of 1,470°C, the nucleation rate of β-cristobalite becomes 0 [10]. Thus, the temperature dependence of the β-cristobalite nucleation rate is properly asymmetrical with a maximum close to the middle in the range from T_m to 1,470°C.
Another major feature of the proposed hypothesis that differs from the CNT is how the critical nucleus is formed. In the new hypothesis, the critical nucleus could not spontaneously form, as suggested by the CNT. Instead, clusters with critical size or smaller are organized step by step according to the crystal structure. As soon as the cooling temperature decreases to below $T_m$, the groups of $SiO_4$ tetrahedra will start to self-organize. In the beginning, all tetrahedra do not know the exact positions they should be. Over time, from their initial randomized positions, all $SiO_4$ groups in the nucleation region can adjust to form six-membered rings on eight facets of the cluster.

In some cases, other structures, such as five- or seven-membered rings, are formed because the internal energies of these rings are close to that of the six-membered ring. According to thermodynamics, such structures can appear with particular probabilities. Non-six-membered rings mean different structures from $\beta$-cristobalite structures. We assume that a five- or seven-membered ring was formed on one or two facets, and all other facets were formed correctly according to the $\beta$-cristobalite structure. Since all facets share the octahedron's edges with their neighboring facets, the atoms cannot find stable positions on the octahedron's edge to satisfy structures on adjacent facets with six-membered and non-six-membered rings. Thus, although a group of $SiO_4$ tetrahedra on several facets may reach the correct arrangements initially, the facets cannot grow new layers if the $SiO_4$ tetrahedra on the neighboring facets are arranged incorrectly. It is possible that the correct cluster structure could not be obtained by adjusting the positions of several $SiO_4$ tetrahedra locally; the cluster has to break down completely and rebuild itself. Only when each facet of the cluster is organized in order of the (111) plane of $\beta$-cristobalite and the correct ordered structures on all facets of the particle are constructed, the critical nucleus is formed. The critical nucleus can be viewed as the structural foundation of the crystal. When the foundation is built, the $SiO_4$ tetrahedra can find proper positions relatively quickly and attach to existing facets, allowing the crystals to grow at a rate higher than the nucleation rate.

## 4. Discussion and conclusions

The nucleation process discussed in Section 3 is only the homogeneous kind, where nucleation occurs away from the impurities and the surface of the container. The process might also begin at nucleation points on the surface of the container or from foreign particles within. This would be heterogeneous nucleation. The size and shape of the critical nucleus of heterogeneous nucleation differ from those of homogeneous nucleation. One well-established characteristic of heterogeneous nucleation is that its nucleation rate is significantly higher than that of homogeneous nucleation. According to the newly proposed hypothesis, this is because the shape of a heterogeneous critical nucleus is less symmetrical than that of a homogeneous nucleation. For example, starting at a point on the container surface, small clusters have only half the space to develop, and the shape of the critical nucleus might be only half of an octahedron with four facets. The time needed and ease to arrange all disordered $SiO_4$ tetrahedra into crystal structures on the four facets is much less than that for eight. Thus, the nucleation rate of heterogeneous nucleation is significantly higher than that of homogeneous nucleation.

Observing the nucleation process of materials at the molecular level and at the appropriate time scale has been challenging for a long time. An article published in *Science* reported an investigation of the heterogeneous nucleation process of gold nanocrystals on graphene surfaces with atomic spatial resolution and millisecond temporal resolution [21]. High-speed *in situ* transmission electron microscopy (TEM) observations of a real atomic system are able to reveal the nucleation pathway in detail and confirm the nonclassical nature of the nucleation process during atomic crystallization. **Figure 4** is a part of the TEM experimental results describing the formation of a thermodynamically stable crystalline lattice structure. **Figure 4a-e** shows temporal snapshot images in a particular area at starting time $t=0$ and after starting 150, 160, 170, and 5,310 ms, respectively. It is found that the particle in **Figure 4e** has a stable crystalline lattice structure, continuously grows, and thus becomes large.

<figure><img src="image_8.png" /><figcaption>**Figure 4** • High-speed transmission electron microscopy images showing the formation of a thermodynamically stable crystalline lattice structure. (a–e) Snapshot images of structures in a particular area at starting time $t=0$ and after starting 150, 160, 170, and 5,310 ms, respectively. The length of the bar is 1 nm. Reproduced from Jeon et al. [21].</figcaption></figure>

In contrast, the structures of the particles in **Figure 4b** and **c** are unstable and can even become disassembled after a short time, as shown in **Figure 4d**. During such events, particles can appear and disappear several times before forming a stable particle, as shown in **Figure 4e**. Thus, high-speed *in situ* observations of a real atomic system confirm that the nucleation process during atomic crystallization is more complex than that described by the CNT. The observed phenomenon is more consistent with the new hypothesis proposed in Section 3, i.e., the atomic structures in the nucleation process evolve from disorder to order and exhibit

fluctuations between two states with size-dependent relative stability.

The proposed nucleation hypothesis is not limited to pure silica. It may also apply to binary and more complicated materials. Sodium silicate may serve as an example. According to the $Na_2O-SiO_2$ phase diagram [22], sodium silicate liquid with a 0–11.3 weight% $Na_2O$ concentration crystallizes to $\beta$-cristobalite at temperatures between 1,713°C and 1,470°C. This means that if the $Na_2O$ concentration is lower than 11.3 weight%, the sodium silicate crystallizes in the same way as the pure silica, and the
proposed nucleation hypothesis can be applied to these sodium
silicates without any modification. However, for sodium silicate
liquid with 11.3–24.5 weight% Na₂O concentration, the liquid
crystallizes to β-tridymite at temperatures between 1,470°C and
1,140°C [22]. β-Tridymite has a hexagonal structure with a
(0001) habit plane. The shape and size of the nucleus of these
sodium silicates are expected to differ from that of pure silica and
must be modified according to experimental data.

The proposed nucleation hypothesis indicates that the materials'
good glass-forming capability, allowing a relatively low cooling
rate, is closely related to the ease and time required for disorder-
to-order structural transition. A β-cristobalite crystal has an FCC
structure formed by SiO₄ tetrahedra. On the (111) planes, the
orientation of all SiO₄ tetrahedra has to be arranged alternately.
Such an ordered structure is not easily arranged from its initial
random constituents. It is much more complex than structures
found in metals and simple alloys that are without preferred
bonding directions. Thus, glass is significantly easier to form
from silica than all metals and alloys.

The proposed nucleation hypothesis fundamentally impacts the
theory of glass structure. It is the result of the hypothesis that
clusters exist before the formation of critical nuclei in the
supercooled liquids. As temperature continuously decreases,
although the clusters do not develop into critical nuclei, they do
not disappear. Glass scientists have recognized these clusters and
studied their structural rearrangements during the entire glass
cooling process [10, 23–28]. Currently, the most popular theory for
glass structure is the continuous random network (CRN) theory
developed by Zachariasen and Warren [29–31]. The proposed nu-
cleation hypothesis and the newly reported experimental data
indicate that the CRN theory cannot completely describe the
heterogeneous glass structure.

In summary, the proposed hypothesis describes the cristobalite
critical nucleus's shape, size, and formation pathway and explains
why the nucleation rate is lower than the crystal growth rate. It is
consistent with the recent high-speed TEM experimental data and
indicates that a valid glass structure theory must recognize small
clusters in the structure.

## Acknowledgments

This work at the Molecular Foundry of Lawrence Berkeley National
Laboratory was supported by the Office of Science, Office of Basic
Energy Sciences, of the US Department of Energy under Contract
No. DE-AC02-05CH11231.

## Funding

The author declares no financial support for the research,
authorship, or publication of this article.

## Author contributions

The author confirms sole responsibility for this work. The author
approves of this work and takes responsibility for its integrity.

## Conflict of interest

The author declares no conflict of interest.

## Data availability statement

Data supporting these findings are available within the article, at
https://doi.org/10.20935/AcadMatSci7498, or upon request.

## Institutional review board statement

Not applicable.

## Informed consent statement

Not applicable.

## Additional information

Received: 2024-09-29

Accepted: 2025-01-08

Published: 2025-01-31

*Academia Materials Science* papers should be cited as *Academia Materials Science 2025*, ISSN 2997-2027, https://doi.org/10. 20935/AcadMatSci7498. The journal's official abbreviation is *Acad. Mat. Sci*.

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