Introduction

Crystal nucleation is a ubiquitous phenomenon and highly relevant to many industrial and technological applications (e.g., bone formation, pharmaceuticals, meteorology, and metallurgy)1,2,3. In glass science, it is the initial stage of glass devitrification (uncontrolled crystallization) and glass–ceramics formation. Devitrification, mostly observed on the glass surface, is an undesired phenomenon in glass manufacturing because the formation of crystals in the glass products often deteriorates their transparency and strength. In the case of nuclear waste vitrification, radiogenic heat induces the formation of soluble crystalline phases4,5, resulting in radioactive pollution due to the leakage of radioactive waste from the glassy matrix. On the other hand, technologies for controlling crystallization have contributed to society since they enable us to develop glass–ceramics products with exceptional optical and mechanical properties6. Knowledge on the mechanism of the process of crystal nucleation and crystallization in oxide glass is thus fundamental to design a glass–ceramic system that exactly fulfils the requirements for any applications.

Crystal nucleation occurs with appreciable rate when the glass is heated and held near the glass-transition temperature (Tg). The kinetics and thermodynamics of crystal nucleation are usually explained by the Classical Nucleation Theory7,8,9,10 (CNT), which simplifies the initial germ of nucleation as a spherical shape of radius r since it may minimize the surface energy between a crystal and the supercooled liquid. The other assumption of CNT is that an initial germ of nucleation has the same macroscopic properties, such as density, structure, composition and thermodynamic properties, with those of a stable crystalline phase to be formed.

In this framework, the nucleation rate10 is given by Eq. 1:

$$ I = I_{O} exp\left( { - \frac{{W^{*} + \Delta G_{D} }}{{k_{B} T}}} \right), $$
(1)

where \(I_{O}\) is the pre-exponential term, \(k_{B}\) and T are the Boltzmann constant and temperature. \(\Delta G_{D}\) is the kinetic contribution to the nucleation rate, that is, the activation energy for the transfer of species through the melt/nucleus interface, and \(W^{*}\) is the free energy change due to the formation of a new nucleus of critical radius r*. Assuming that, the atomic clusters or embryos below the critical size are not stable and can be dissolved into the original melt without producing a nucleus. Once an embryo overcomes the critical size, it can grow to form a crystalline phase.

Even though the CNT is widely accepted as a fundamental theory, unraveling the origin of the crystal nucleation mechanism in glass forming liquids is still a challenging issue in glass science community. The nucleation mechanism is categorized into two main types: one is homogeneous nucleation and the other one is heterogeneous one11. The heterogeneous nucleation at surfaces and interface boundary between different materials is the most common mechanism since it requires lower activation energies to occur. Indeed, a variety of nucleating agents (i.e. metal oxides such as ZrO2, TiO2 and CeO212, non-metal oxides13,14 like a P2O5 and colloidal metal nanoparticles15) are used to accelerate bulk crystallization.

Contrarily, the homogeneous nucleation rarely occurs because of the higher activation energy since the nucleus forms from the bulk without the help of any phase inhomogeneities or boundaries. However, in the case of homogeneous nucleation, the nuclei form throughout the entire glass matrix, allowing a better control of the processing conditions and of the final glass–ceramic microstructure and properties. The homogeneous crystallization is usually associated with the similarity of microstructures between glass and crystals. Nevertheless, the mechanism is not well explained only by the simple assumption16,17,18 because intermediate metastable phases might also play an important role. It is thus worth investigating the origin of the homogeneous crystallization in glass forming liquids19, using theoretical simulations20,21.

A typical homogeneous crystallization is observed in the lithium silicate glass system close to the Li2O·2SiO2 (lithium disilicate, LS2) composition7, and many experimental investigations using a large variety of techniques22,23,24,25,26,27,28,29 have been investigated. Accordingly, two possible phase evolution mechanisms have been proposed7,28,29: (1) both LS2 and metastable Li2O·SiO2 (lithium metasilicate, LS) phases simultaneously nucleate homogeneously but the latter phase disappears during the heat treatment or (2) the LS crystal is firstly nucleated and subsequently promotes the heterogeneous LS2 crystal on it. The latter hypothesis was suggested by the fact that the CNT underestimates the steady-state nucleation rate in several orders of magnitude. To fill the gap between CNT and experimental observation, it is assumed that the metastable crystalline phase in the early stage would decrease the activation energy. In addition, some experimental groups have observed the formation of metastable phases with stoichiometry close to the LS2 composition (called α’-LS2 and β’-LS2 in ref.23). Meanwhile, others have obtained the LS crystalline phase27,28,29 although the conversion of the LS crystal (if formed before) to the LS2 crystal would require a high energetic cost because the two phases have quite different stoichiometry and structures, as shown in Fig. 1 (LS2 has a double-layered silica structure with Li ions in the interlayers, whereas LS has a chain-like silica structure surrounded by Li ions).

Figure 1
figure 1

Structure of the LS and LS2 crystals along different orientations and unit cell parameters. Blue tetrahedral represents silicon, red and green spheres represent oxygen and lithium ions, respectively.

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In spite of the many efforts made in the past, there is still no consensus neither on the precipitation of metastable phases prior to the LS2 one, nor on their composition, and thus the detailed process of the homogeneous crystallization is still open to question. In this work, we employ Molecular Dynamics (MD) simulations coupled with an original algorithm to search crystal-like structures in amorphous models and Free-Energy calculations on a crystal-embedded in a glass model in order to answer the following queries: (i) Is homogeneous nucleation in oxide glasses associated with the existence of pre-formed nuclei (embryos) whose stoichiometry and structure are similar to those of the crystals precipitating? (ii) Does the LS crystal form before the LS2 one inside the glass or on the glass surface? (iii) How do temperature and P2O5 nucleating agent affect the nucleation? (iv) Does the nucleation of the LS2 and LS phases obey the Classical Nucleation Theory?

It is worth noting that although, as stated before, nucleation and crystallization occur (with appreciable velocity) from the melt rather than the glass (defined as the undercooled liquid under Tg) the latter has the same structure of the undercooled melt (this is especially true for MD-derived glass models). Therefore, in this work we have sought the fingerprints of nucleation analyzing computational glass bulk and surface models with compositions 33.3Li2O·66.7SiO2 (LS2 Glass) and 33Li2O·66SiO2·1P2O5 (LS2P1 Glass) instead of melts. Moreover, it is also worth to note that the high viscosity of the oxide glasses investigated here does not allow to follow the dynamic evolution of nucleation and crystal growth with unbiased MD simulations within a reasonable simulation time, contrary to model systems30,31.

Computational details

The LS2 and LS2P1 glasses were generated using MD simulations by employing the modified PMMCS force-field published in ref.32 and described in the ESI. The LS2P1 composition was investigated to understand the effect of P2O5 on the structure and nucleation of Li2Si2O5 and Li2SiO3 crystals in the glass since it is well known that addition of P2O5 favors the precipitation of the LS2 crystal and the formation of glass–ceramics13,33.

Glass structural models containing 13,500 and 12,160 atoms were generated for LS2 and LS2P1 glasses, respectively, through the melt and quench approach by MD simulations34. Four replicas of each glass model have been examined to confirm the reproducibility of the results and to estimate the variability in the glass properties. The leap-frog algorithm encoded in the DL_POLY2.14 package35 was used to integrate the equation of motions with a time step of 2 fs. The initial configurations were generated by randomly placing the atoms in a cubic box, whose size corresponds to the experimental density. Table S1 lists the number of the atomic species and the experimental density used to determine the unit cell size (5.4 nm for LS2 and 5.2 nm for LS2P1).

The systems were heated and held at 3500 K for 100 ps, which is sufficient to melt the samples and remove the memory of the initial configurations. The liquids were then monotonically cooled to 300 K with a cooling rate of approximately 5 K/ps. The cooled glass structures were subjected to a final equilibration run of 200 ps. In these cases, the canonical ensemble (NVT) was employed, and Berendsen thermostat36 was used to control the temperature (frictional constants set to 0.2 ps). The coulomb interactions were calculated by the Ewald summation method with a cutoff distance of 12 Å. The short-range interactions were evaluated using cutoff values of 5.5 Å.

In addition to the bulk glass models with periodic conditions, slab models with two surfaces were created by eliminating periodic boundary conditions in the z-axis. The slab models were replicated in x-axis by one time in order to have larger surface areas and thus more statistics, then the systems were heated from 300 to 1800 K with a heating rate of 5 K/ps and cooled down at 300 K with the same cooling rate to relax the surface structure. It is worth to note that the melting temperature of the LS2 crystal computed by heating the system in the isobaric-isothermal ensemble (NPT) with the same rate (5 K/ps) is around 1750–1800 K, whereas the glass transition temperature is around 1250–1300 K. Both values overestimate the experimental data (Tm: 1306 K, Tg: 727 K17) because of the application of periodic boundary conditions, which eliminate surfaces where melting is initiated in real samples, and the huge cooling rate used in computer simulations with respect to the one used in experiments, as discussed in ref.37,38.

Cluster analysis: the algorithm

The aim of this analysis is to explore atomic aggregates whose stoichiometry and structure are similar to a particular crystalline phase that is thought to nucleate and crystallize from the glass forming liquid. The cluster analysis has been performed using a new FORTRAN90 code based on the one developed in the past by Pedone18 (Cluster code). The program analyzes the glass structure generated through MD simulations and systematically samples the stoichiometry and local structure around each atomic species (for example lithium, oxygen or silicon in lithium silicate glasses). Then, it compares the extracted clusters with a reference crystal phase (Li2Si2O5 for example), which is expected to form in the glass.

A similar approach, known as the Adaptive Template Method, has been applied to identify crystal lattice types (FCC, BCC and HPC) in hard sphere systems39. In our algorithm, a similar idea with the Adaptive Template Method was combined with our original Cluster code18, which enables us to investigate more complicated crystals with multiple species in multicomponent glasses. Here, we introduce the detailed procedure of the extended algorithm.

The program performs the following steps:

  1. 1.

    Read a trajectory file of MD simulations (in this work, DL_POLY REVCON file) and a crystal structure with a unit cell (P1 symmetry) to be explored. Then, specify a reference atom of the crystal structure.

  2. 2.

    Count the number of each atomic species within a spherical region with radius rk around the reference atom of the crystal structure. The numbers of species are stored in a reference vector RA, which is noted as (NLi, NSi, NO) for the LS2 glass, for example. In the case of the LS2P1 glass, the reference vector RA has one more dimension as (NLi, NSi, NO, NP).

  3. 3.

    Subsequently, put a center of a spherical probe on an atom whose atomic type is the same with the reference atom in the MD trajectory. Then, count the number of each species in the spherical probe with a radius rk, which defines a vector χA,j for atom j.

  4. 4.

    Evaluate the Hamming distance \(\delta_{j} \left( {r_{K} } \right) = \left| {R_{A} \left( i \right) - \chi_{A,j} \left( i \right)} \right|\) to measure the similarity of the local structure in the MD trajectory to the reference crystal.

  5. 5.

    Repeat the steps from #2 to #4 with varying the proof radius rk from a minimum (rmin) to a maximum (rmax) value every small increment (dr). Then, evaluate the cumulative distance for atom j at the radius rk is computed, as \(\Delta_{j} \left( {r_{k} } \right) = \sum\nolimits_{{r_{min} }}^{{r_{k} }} {\delta_{j} } \left( {r_{k} } \right)\). The atom with smaller cumulative displacement is judged to possess more similar microstructure to that of the crystal explored.

At the end the code outputs: i) PDB files with atomic structure of the best cluster found for each radius; ii) a statistic on the best clusters found at each radius and a distribution of the cumulative distances of all the target species in the glass.

It is worth to note that when the increment dr is small (i.e. 0.1 Å), the cumulative distance of atom j \(\Delta_{j} \left( {r_{k} } \right)\) gives an idea of the degree of matching between the total distribution function of the reference atom in the crystal and the total distribution function of each atom in the glass. In this work, the analysis has been carried out using rmin = 2.0 Å, rmax = 6.0 Å, dr = 0.2 Å without considering oxygen atoms.

Free-energy seeding method (FESM)

In addition to the morphological analyses, we have also evaluated the thermodynamics on the nucleation of both the LS and LS2 crystals inside the LS2 glass. To determine the work necessary to create a critical nucleus of such crystals in the LS2 glass, the Free-Energy Seeding Method (FESM) has been used. This method inserts a cluster of a given shape into the supercooled liquid, as the seeding method proposed by Espinosa et al.30 but determines the thermodynamic and kinetic parameters that characterize the nucleation process differently. In the original seeding method, the critical nucleus is determined by following the dissolution or growth of the inserted clusters in the fluid; the thermodynamic and kinetic parameters, such as the interfacial free energy, the chemical potential difference between crystal and liquid phases, and the attachments rates of particles to the critical cluster, are determined independently. This interesting method is well-examined and a reasonable choice to low viscous systems or model systems, such as Lennard–Jones, Hard Sphere and spherical coarse-grained water models, whereas it is impractical for high viscous and complex systems like multicomponent silica-based glass forming liquids. Indeed, we conducted several test simulations on the LS2 system but could not observe crystal growth for any nucleus sizes in the timescale accessible to classical MD simulations (up to 1 µs), even though it is possible to observe dissolution of the crystal nuclei at temperature above the computational glass transition temperature (> 1300 K in this case).

Therefore, to apply the idea analogous to the seeding method to viscous glass melt, here we simplified the approach. In the case of FESM, we do not simulate dissolution and nucleation, explicitly, even though our approach also embeds a spherical nucleus of the LS or LS2 crystal with different radii into a fully explicit lithium disilicate (LS2) glass melt. Instead, we evaluated the energy profile as a function of the cluster radius. From this energy profile, it is possible to extract the nucleation activation energy, the critical size of the cluster, and the interfacial free energy, which can avoid the possible timescale problem to observe dissolution and growth in our highly viscous systems.

The embedded systems have been built as follows. First, we built a crystal model. For the modeling of LS2 nucleus in the LS2 glass, we replicated the unit cell of the LS2 crystal 10 × 10 × 4 times to generate a box containing 14,400 atoms, and the side lengths of the simulation box varied accordingly to the experimental density of the LS2 glass. On the one hand, to model the LS nucleus in the LS2 glass matrix, after replicating the lithium metasilicate unit cell by 5 × 10 × 8 times to create a supercell with 9600 atoms, subsequently, the simulation box was enlarged to have the same density of the LS2 glass. Then, 1600 SiO2 formula units were randomly added in the vacuum spaces in the simulation box to maintain the overall compositions and density same as those of LS2 glass. It is worth to mention that system size effect is presumable in these calculations, and thus, we restricted the volume of the embedded crystal nucleus less than 10% of the total simulation box to avoid the artifact.

Finally, in both cases, an atom was randomly chosen and the positions of the atoms locating within a distance rsphere from the central atom were fixed as an embedded crystal cluster (seed). Then, the cluster embedded systems were melted and quenched using the same procedure employed to generate the bulk structural models with fixing the embedded crystal, generating a model with a crystal embedded in a glassy matrix. Finally, the model was equilibrated for 2.4 ns at 800 K and 1000 K to investigate the temperature effect. During this equilibration, the positions of the atoms of the embedded crystals were also relaxed without constraints. Nuclei with rsphere from 6 to 14 Å were created and 7 replicas for each system were examined to have a sufficient statistic. An example of the starting and final structures for both the LS2 and LS crystal nuclei into the LS2 glass is shown in figure S1 of the ESI.

To evaluate the energy differences between crystal, glass, and the cluster-embedded models, MD simulations on the LS2 glass and LS2 crystal were also performed (four replicas for each). For all the aforementioned systems, the internal energy (\(E_{S}\)) of the system was accumulated and the Helmholtz free energy (\(A_{S}\)) of the state S (S refer to the crystal, glass and embedded system states) computed using the thermodynamic formulas36

$$ A_{S} = - k_{B} Tln\left\langle {{\text{e}}^{{ - \frac{{E_{S} }}{{k_{B} T}}}} } \right\rangle $$
(2)

The free energy of nucleation (W) with cluster radius r is then computed as

$$ W\left( r \right) = A_{CryGlass} \left( r \right) - A_{Glass} $$
(3)

where ACryGlass and AGlass are Helmholtz free energy of the crystal embedded system and the glass, respectively. In our calculations, we used the Helmholtz free energy of the more stable glass as a reference. In CNT, the free energy of nucleation of a spherical nucleus with negligible strain energy is assumed to be composed of surface free energy and volume free energy as follows:

$$ W = 4\pi r^{2} \gamma_{sl} - \frac{4}{3}\pi r^{3} \Delta A_{V} , $$
(4)

where \(\Delta A_{V}\) is the free energy difference between the melt and the crystal per unit volume of the crystal, whereas \(\gamma_{sl}\) is the interfacial free energy per unit area required to create a liquid/crystal nucleus interface. The interfacial free energy, which governs largely the crystal nucleation rate, is assumed to be equal to that of a planar interface and thus to be independent on the nucleus size. This is known as the capillary approximation and is one of the most serious shortcomings of CNT. A benefit of our approach is that the interfacial free energy is not an input parameter independent of the nucleus size and temperature but is a property that varies inherently with the nucleus dimension and the temperature. The interfacial free energy can be computed from the MD simulations as:

$$ \gamma_{sl} \left( r \right) = \frac{{W\left( r \right) + \frac{4}{3}\pi r^{3} \Delta A_{V} }}{{4\pi r^{2} }}. $$
(5)

where \(\Delta A_{V} = A_{Crystal} - A_{Glass}\) at a defined temperature. We used T = 800 K in this work to make our results comparable to that reported by McKenzie et al.40 (that used a hybrid MD/MC/implicit approach to compute the free energy of nucleation of the LS and LS2 crystals inside the LS2 glass) and to the experimental data41.

It should be noted that since the temperature, 800 and 1000 K, are below the computational glass transition and melting temperatures, the effect of the temperature on the free energy profiles is investigated with assuming that the topology of the crystal/glass interface is similar. In other words, we do not investigate the dynamic evolution of the crystal growth with temperature but generate equilibrated model systems containing a crystal nuclei in an amorphous LS2 matrix, as previous Monte Carlo simulations40.

Results and Discussions

Bulk and surface structures

Homogeneous nucleation is usually associated with the similarity of the short and intermediate-range structures between the glass forming liquid and the crystal that crystallizes from it16,17,42,43. Since the short-range structures measured by the cation-oxygen distances, coordination numbers and the O-Si–O, O-Li–O bond angle distributions are not distinguishable between LS2 and LS crystals as well as between the crystals and the frozen melts, we only give our attention to the structural differences in intermediate-range order. The intermediate-range structures are described by Li–Li distributions and the Qn distributions, which are the populations of the quaternary species (Q) of the network former cations connected to n bridging oxygens (BO), representing degree of polymerization of the network.

Figure 2 shows the bulk structures of the LS2 and LS2P1 glasses at 300 K. A visual inspection of the atomistic configurations reveals that phosphorous atoms form orthophosphate units and tend to segregate each other. The units attract Li ions, forming Li-richer regions around PO4 units. Indeed, the Qn distributions of the P species (not reported) reveals that 98.8% of PO4 units are Q0 species, in excellent agreement with NMR experiments on similar glasses27. Table S4 compares the Qn distribution of silicon and the network connectivity (NC) representing the average number of BO per silicon for the two glasses. More Q4 silicon and larger NC of the LS2P1 glass reveal that the silica network is slightly more polymerized than the LS2 glass. This is a consequence of the existence of the orthophosphate units, which attract Li ions to balance their negative charge, depleting Li ions around the silica network and thus Q4 silicon increases. In both glasses, silicon is predominantly present as Q3 species as in the case of LS2 crystal, which is formed by two tetrahedral layers of silicon (100% Q3 species). The amount of the Q2 silicon species, which are the majority in the LS crystal, are below 20% in both glasses. The Li-rich regions can be seen in Fig. 2 as a formation of percolation channels of Li ions; these are highlighted by the worm-like Li distribution sandwiched between chains and sheets of silica. An extended ordering over four alkali coordination shells is also revealed by the Li–Li pair distribution function (PDF) drawn in Fig. 3. In this figure, the Li–Li PDFs of LS and LS2 crystals at 300 K are also compared, and we can find that the peak positions of both crystals are close to the four peaks in the PDF of the glasses. However, the intensities of the first two peaks of the glasses are more similar to those of the LS2 crystals.

Figure 2
figure 2

Snapshot of the MD-derived bulk structural model of the LS2 glass showing all the ions in the box (a) and Li ions only (b). Snapshot of the LS2P1 glass showing all the ions in the box (c) and Li (cyan spheres) and PO4 tetrahedra only (d).

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Figure 3
figure 3

Li–Li Pair Distribution Functions of the LS2 and LS2P1 Glasses compared to that of the LS2 and LS crystals.

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The other useful parameter for probing structural resemblance or dissimilarity between crystalline and glassy phases is the Li–Li homonuclear M2 dipolar moment16. The M2 dipolar moment is related to the Li–Li distance correlation and can be an ability index of a glass to be transferred to a crystal with the same stoichiometry. It is computed using the van Vleck equation:44.

$$ M_{2, Li - Li} = 0.9562\frac{1}{{N_{Li} }}\left( {\frac{{\mu_{0} }}{4\pi }} \right)^{2} \gamma_{Li}^{4} \hbar^{2} \mathop \sum \limits_{i = 1}^{{N_{Li} - 1}} \mathop \sum \limits_{j = i + 1}^{{N_{Li} }} r_{ij}^{ - 6} , $$
(6)

where \( N_{Li}\) is the number of the Li atoms in the simulation box, \(\mu_{0}\) is the vacuum permittivity, \(\gamma_{Li}\) is the gyromagnetic ratio and \(r_{ij}\) is the distance between atoms i and j. The calculated \(M_{2,LiLi}\) for the LS2 and LS2P1 glasses are of 49.7 and 49.6 × 106 rad2 s−2 and closer to that of the LS2 crystal (\(M_{2,LiLi} = 57.8\) × 106 rad2 s−2) than the LS crystal (\(M_{2,LiLi} = 75.7\) × 106 rad2 s−2). Accordingly, it seems unlikely that the LS crystal can nucleate homogeneously from the bulk of the melt, as suggested in some of the previous experimental28 and computational works40.

Nucleus of crystals in bulk and on surface

Another hypothesis to explain and predict the crystallization of a specific phase from a glass forming liquid involves the clustering of specific atoms forming embryos of the crystal with subcritical dimensions. These embryos, formed during the fast quenching of the melt, can trigger the nucleation during heat treatment and be considered as structural markers for the prediction of the crystallization of a particular phase. MD simulation is an exceptional method to detect such extremely small embryos, which is indeed difficult to be found experimentally. To test this hypothesis we performed a cluster analysis18 on the bulk structure of the two simulated glasses using the modified Cluster code described before.

Figure 4 reports the distribution of the cumulative displacements of atomic clusters similar to the Li2SiO3, Li2Si2O5 and Li3PO4 crystals in the LS2P1 glass within a sphere of radius of 3.6, 4.0 and 4.6 Å centered on Li, Si and P, respectively. The figure shows that silicon and lithium environments in the glass structure are similar to the Li2Si2O5 crystal rather than the Li2SiO3 one at any radii. However, the discrepancy is more evident for silicon since the two distributions are more separated than the ones for the clusters centered on lithium. It is interesting to note that the distribution of the clusters centered on Li shifts to higher cumulative distances by P2O5 addition, revealing that the formation of embryos similar to the metasilicate phase is disfavored in the LS2P1 glass. Figure 5 shows the most Li2Si2O5 –like embryo obtained by means of a sphere of 4.6 Å radius centered on Li in the LS2P1 glass. The embryo contains 7 Li and 6 Si atoms as in the reference cluster of the crystal and shows a layered structure similar to that of the Li2Si2O5 crystal. Moreover, an embryo whose stoichiometry and structure is similar to the Li3PO4 crystals (see Fig. 5), which is also found in the LS2P1 glass when the radius of sphere is 4.0 Å, agrees with the experimental fact that the lithium orthophosphate crystal phase appears as a secondary phase27.

Figure 4
figure 4

Distribution of cumulative distances (with respect to Li2SiO3, Li2Si2O5 and Li3PO4 crystals) of clusters of radii 3.6, 4.0 and 4.6 Å centered on Silicon (left) and Lithium (middle) and Phosphorous (right) for the LS2 and LS2P1 glasses.

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Figure 5
figure 5

Embryos of Li2Si2O5 (LS2), Li2SiO3 (LS) and Li3PO4 (LP) found in the LS2P1 glass and structures of the three crystals.

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Interestingly, the situation drastically changes at the glass surfaces. Figure 6 shows the top and lateral views of the slab models for the LS2 and LS2P1 glasses. The figure demonstrates that the surface is much richer in Li ions than the bulk. During the formation of the glass surface the structure undergoes a drastic rearrangement with Li ions (violet spheres) aggregating in layers and forming percolation channels that flow to a surface to the other side. The abundance of Li ions at the glass surface is also confirmed by analyzing the fraction of each element along the z-direction as shown in Fig. 7. Indeed, the first atomic layer at the surface is exclusively composed by lithium ions, the second by oxygen ions and the third by silicon ions. The concentration profiles also show that the P ions, albeit present in small quantities, tend to gather at the subsurface rather than in the bulk.

Figure 6
figure 6

Top and lateral view of the surface of the LS2 and LS2P1 glasses. Li, Si and O ions are, respectively, represented by violet, yellow and red spheres. Orthophosphate units are represented as green tetrahedral.

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Figure 7
figure 7

Z-depth profiles of the atomic fraction of Li, Si, P and O ions and Qn distributions of Si ions in the LS2 and LS2P1 glass surface models.

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In addition, the Qn distributions of silicon change considerably during the surface formation since the migration of Li ions to the surface leads to breaking of Si–O-Si bonds and reducing the connectivity of silicon. In fact, Fig. 7 reveals that silicon ions at the glass surface are almost exclusively Q2 species (~ 80–90% in the LS2 and 100% in the LS2P1 glass) as in the case of the LS crystal. This finding is extremely interesting and demonstrates that the surface provides favorable stoichiometric conditions for the nucleation and crystallization of the LS crystal rather than LS2 one. Indeed, in Fig. 5, an atomic cluster possessing structure and stoichiometry analogous to the LS crystal is found on the LS2 glass surface. The cluster is constituted by a chain of three SiO4 units surrounded by 10 Li ions, similarly to the chain present in the LS crystal. On the contrary, LS-like embryos with the same cumulative displacements were not found in the bulk of the glass model. It seems thus unlikely that the LS crystal phase nucleates homogeneously from the bulk of the glass in these glasses.

Free energy of crystal nucleation

To obtain further insights on the possible mechanisms of nucleation from the LS2 glass forming liquid, we computed the Helmholtz free energies of the LS and LS2 crystals formation using the FESM method (see Method section). In this free energy calculation, we modeled a spherical crystal nucleus embedded into a glass model as seen in Fig. 8 and examined size dependency of the nucleation free energy for both LS2 and LS crystals at 800 K. The Helmholtz free energy profiles are reported in Fig. 9a. The error bar in this figure represents standard deviation of free energies evaluated by seven independent replicas. As expected from CNT, the free energy first increases and then decreases after reaching the maximum energy (barrier for nucleation) at the critical radius. It is worth highlighting that LS2 clusters with radius less than 5 Å were not considered to build the free energy profile in Fig. 8 since the free energy of such embryos formed during the glass formation is expected to be almost zero. Indeed, LS clusters of such small sizes lost the crystalline structure during relaxation, which indicates the instability. According to the figure, at 800 K the critical radii for both LS and LS2 crystals are around 7 Å in excellent agreement with previous experimental estimations45.

Figure 8
figure 8

Structure of boxes with critical size nuclei. On the left, glass with LS2 composition with the critical nucleus of Li2Si2O5 crystal. On the right, glass with LS2 composition with the critical nucleus of Li2SiO3 crystal.

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Figure 9
figure 9

Free energy of nucleation for (a) the LS and LS2 crystals in the LS2 glass at 800 K and (b) the LS2 crystals in the LS2 glass at 800 K and 1000 K with relative standard deviation.

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The computed Helmholtz free energy barrier for nucleation (W*) of the LS2 crystal at 800 K is about 64 kT, agrees fairly well with the experimentally estimated ones, which are ranging from 35 and 50 kT at the temperature from 700 to 800 K41. The large variability associated with the free energy is due to the variability of the energy of frozen liquid structures generated through the standard MD quenching protocols. In fact, the fast quenching of the melts prevents the good sampling of the free energy surface and thus glass structures with different energies are easily generated46. Nevertheless, considering the unfavorable conditions in stoichiometry and structural similarities for the LS crystal in the bulk glass as discussed above, it is expected that the LS crystal can crystallize preferentially on the glass surface through the heterogeneous nucleation mechanism. Indeed, the energy barrier of heterogeneous nucleation \((W_{het}^{*} )\) is lower than that of the homogeneous one \((W_{hom}^{*} )\) according to a geometric factor \(S\left( \theta \right)\), which depends on the wetting angle between the solid and liquid at the interface47. Further developments are required to extend the FESM method to investigate heterogeneous nucleation.

McKenzie et al.40 used a hybrid MD/MC approach to compute the free energy of nucleation of the LS and LS2 crystals inside the LS2 glass considered as an implicit medium. In their method, the nucleation free energy (NFE) was evaluated by considering three contributions: the cluster formation, the cluster to crystal transition and the cluster solvation energy. The thermodynamic barrier was estimated to be 40 kT and 34.5 kT for the LS2 and LS crystals, respectively.

They showed that the NFE is higher for the LS2 than for the LS at smaller cluster dimension, but it crossovers at around 4–5 formula units, and thus it was stated that the LS crystal first nucleates, and then it transforms to the LS2 with a thermodynamic barrier of 28.3 kT. A smaller free energy of nucleation at small cluster radius for the LS phase is also observed in our simulations, but since none of the two methods investigates the reactive pathway and the kinetics associated to an eventual crystalline phase transition from the LS to LS2, explicitly, it is difficult to make speculation on the phase transformation. Both our results and those by McKenzie et al.40 are in good agreement with the facts experimentally measured by Differential Thermal Analysis (DTA) and SEM observations by Soares et al.29. In fact, the DTA data has shown that two exothermic peaks appear at 615–680 °C and 750–800 °C, and these peaks are associated with the crystallization of the LS and LS2 phases, respectively, by XRD measurement. Morphological observations by SEM have also confirmed the occurrence of these two crystallization mechanisms since needle-like and granular crystals were detected in the microstructure of samples including LS and LS2, respectively. The authors have also found that, in the early stage of crystallization in the LS2 glass, both the LS and LS2 crystals nucleate simultaneously and independently28. Further, the analysis of the crystallization kinetics through the Johnson–Mehl–Avrami equation suggested that the surface crystallization (Avrami exponent n = 1) associated to the first peak rather than volume crystallization (Avrami exponent n = 3) associated to the second peak is dominant in the crystallization process. This is because the activation energy (225–275 kJ/mol) of the former is substantially lower than that of the latter (425–500 kJ/mol). Therefore, the LS crystal nucleates at the glass surface because favorable stoichiometric and thermodynamic conditions for its precipitation are present, whereas the LS2 crystal appears in the bulk of the glass.

The effect of the temperature on the nucleation free energy profile was investigated only for the nucleation of the LS2 crystal, as shown in Fig. 9b. Although error bars are relatively large, it is clear that both the critical size and the activation energy barrier increases with temperature as expected.

In conclusion of this section, it is worth summarizing advantages of the FESM method, despite the not negligible error associated with the calculation of the free energy of the glass state. Firstly, it should be emphasized that our method inherently includes the surface energy penalty during the creation of the melt/crystal interface and the strain energy at the interface. To the best of our knowledge, the surface energy contribution was implicitly subsumed in the solvation term (parameterized for a particular system) of the hybrid MC method used by McKenzie et al.40,48 The MC simulation highlighted that the strain energy term does not play an important role in nucleation above the glass transition temperature, thereby the contribution is usually neglected49.

The interfacial free energy associated to the formation of the melt/crystal interface for the LS2 nucleus with critical size using Eq. (5) calculated at 800 K and 1000 K is respectively 0.3 ± 0.1 and 0.4 ± 0.1 J/m2. These values are comparable to the value estimated by Fokin et al.50 when accounting both the temperature and size dependence of the crystal/liquid surface energy.

Moreover, as shown above, our approach can be used to compute the thermodynamic barrier for homogeneous crystal nucleation at different temperatures. This would be useful because the thermodynamic barrier has been shown to exhibit an unusual increase with a decrease in temperature below the maximum nucleation rate for a variety of oxide glass-forming liquids49. Although the internal elastic stresses arising from the density misfits between the crystal and liquid phases might play a role, the phenomenon is still not completely understood and should be object of future investigations49.

The approach is applicable to any system of interest since the parameters needed are only the interatomic force field for MD simulations. An advantage with respect to the original seeding method30 is that it allows extracting thermodynamic parameters of the nucleation process in high viscous liquids. Finally, all-atomistic MD simulations can provide information on the structure of the melt/crystal interface, which are hardly accessible from experiments. For instance, Fig. 10a shows the computed local order Q6 Steinhardt parameter (see ESI for definition) of silicon atoms from the center of the LS and LS2 crystal nuclei (of radius 10 Å shown in Fig. 10c) to the glass matrix at 800 K. The gradual change of Q6 Steinhardt parameter demonstrates gradual structural transition from the crystal to the glass (melt) structure and the profile allows us to measure the interface thickness in both cases. The LS2 crystal/LS2 glass interface thickness is of about 4 Å and less than that of the LS crystal/LS2 glass, which is about 8 Å. This may be due to the greater flexibility of the Q2 chains in the LS crystal that easily lose the perfect order at the interface with the glass. In Fig. 10b, the Q6 parameter was compared at 800 K, 1000 K and 1200 K to understand the effect of the temperature on the LS2 crystal/glass interface. The figure shows that the interface slightly enlarges in about 1–2 Å at 1000 K, while the Q6 value in the cluster approaches to that in the glass matrix and the crystalline order is almost lost at 1200 K. However, a visual inspection according to Fig. 10d reveals that the layered structure of the crystal is still maintained as well as the original topology of the silicate network and no Si–O bonds break. Only disordered crystalline structure is observed.

Figure 10
figure 10

(a) Q6 Steinhardt parameter of silicon atoms from the center of the LS or LS2 crystal nuclei to the glass matrix at 800 K. (b) Computed local order Q6 parameter for silicon atoms from the center of the LS2 crystal nuclei (of 10 Å) to the glass matrix at 800 K, 1000 K and 1200 K. (c) Glass with LS2 composition with a nucleus of Li2Si2O5 crystal with radius of 10 Å equilibrated at 800 K. (d) Glass with LS2 composition with a nucleus of Li2Si2O5 crystal with radius of 10 Å equilibrated at 1200 K.

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Conclusions

Classical MD simulations were used to shed light on the crystal nucleation process inside and at the surface of the stoichiometric lithium disilicate glasses. In this work, we employed a modified cluster-exploring algorithm to detect subnano-scale nuclei in the initial crystallization stage, and the free energy calculation using a modified seeding method for the first time in multicomponent oxide glasses. Our simulations suggest that lithium disilicate and lithium metasilicate crystals nucleate independently. The former would appear homogeneously from the bulk, whereas the latter can heterogeneously nucleate only on the glass surface where favorable exogenous conditions are present. Therefore, the failure of CNT in predicting the magnitude of the steady state nucleation rate is not due to the formation of metastable phases as suggested in some of earlier investigations27,28,29.

As observed in previous works, homogeneous nucleation is associated with the similarity in intermediate range-order morphology of modifier cations between the crystal and the glass. In fact, the Li–Li second dipolar moments computed on the bulk glass model is very close to that of the LS2 crystal. The structural analysis using the improved cluster algorithm reveals that embryos of about 4–4.5 Å of radius with stoichiometry and structure very similar to those of the LS2 and LS crystals nucleate during glass formation both in the bulk and on the surface. The presence of such embryos in the MD-derived structural models is considered an indication of which crystal phase can crystallize from the glass forming liquid. The method thus hopefully serves the possibility to predict crystallization in the other multicomponent glasses in future investigations. Another interesting finding is that the addition of the nucleating agent P2O5 leads isolated orthophosphates units formation that attracts Li ions and promotes Li3PO4 embryos in the glass. This phenomenon facilitates crystallization of lithium metasilicate crystals on the surface where chains of Q2 silica species surrounded by Li ions are dominant rather than inside the bulk.

The FESM method allowed us to compute the free energy of nucleation for the LS2 and LS crystals in the stoichiometric LS2 glass and to investigate the effect of the temperature. The critical radii obtained are in excellent agreement with experimental estimations, and the activation energy barriers are comparable to the experimental data. This seems to confirm that both crystal phases form obeying the CNT if the capillarity approximation is overcome. That is, those thermodynamic properties such as the interfacial free energy assume size dependent values as inherently assumed in the FESM approach. Therefore, we confirm that the FESM is a powerful approach to study nucleation in glass, for instance to predict which crystals would nucleate more easily or extract key thermodynamic parameters as a function of nucleus size, temperature and pressure.