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DOI: 10.1039/D5CE00467E (Communication) CrystEngComm, 2025, 27, 5265-5278

Nucleation rate and Gibbs free energy of nucleation of APIs, large molecule, amino acid and inorganic materials in solution at different cooling rates†

Mayank Vashishtha a and K. Vasanth Kumar *b
aDepartment of Chemical Sciences, Synthesis and Solid State Pharmaceutical Centre, Bernal Research Institute, University of Limerick, Limerick V94 T9PX, Ireland
bDepartment of Chemical and Process Engineering, Faculty of Engineering and Physical Sciences, University of Surrey, Guildford GU2 7XH, UK. E-mail: v.kannuchamy@surrey.ac.uk

Received 5th May 2025 , Accepted 16th July 2025

First published on 16th July 2025

Abstract

We propose a new mathematical model based on the classical nucleation theory to predict the nucleation rate, kinetic constant, and Gibbs free energy of nucleation using metastable zone width (MSZW) data as a function of solubility temperature. Unlike widely used models by Nývlt, Kubota, and Sangwal, which are limited in capturing the impact of varying cooling rates, the proposed model allows direct estimation of nucleation rates from MSZW data obtained under different cooling conditions. This is particularly advantageous for continuous or semi-batch crystallisation design, where cooling rate is a critical variable. The model has been successfully validated using experimental data from 22 solute–solvent systems, including 10 APIs, one API intermediate, lysozyme, and glycine, as well as 8 inorganic compounds. Predicted nucleation rates span from 1020 to 1024 molecules per m3 s for APIs, and up to 1034 molecules per m3 s for lysozyme, the largest molecule studied. Gibbs free energy of nucleation varies from 4 to 49 kJ mol−1 for most compounds, reaching 87 kJ mol−1 for lysozyme. The model also enables accurate prediction of induction time and key thermodynamic parameters such as surface free energy, critical nucleus size, and number of unit cells—based solely on MSZW data obtained at different cooling rates.

In this communication, we propose a useful mathematical model to predict the nucleation rate and Gibbs free energy of nucleation from metastable zone width (MSZW) values measured at different cooling rates. The model is validated using a broad dataset comprising 11 different active pharmaceutical ingredients (APIs)/solvent combinations involving 10 active pharmaceutical ingredients (APIs), 8 inorganic compound–solvent systems, one API intermediate, one amino acid, and one large molecule. The MSZW is crucial in crystallization processes because it defines the range of supersaturation where no spontaneous nucleation occurs, but crystal growth is possible.1–4 MSZW helps to determine the precise supersaturation levels to avoid spontaneous nucleation, which can lead to undesirable crystal sizes or polymorphs.5–7 Operating within the MSZW allows controlled crystal growth, ensuring consistent size and quality of the crystals. On the theoretical side, MSZW allows to calculate the nucleation rate which is a key factor in understanding and optimising the crystallisation.5 Theoretically the nucleation rate is obtained using the MSZW measured at different cooling rates using theoretical models that includes the one developed by Nývlt,8 Sangwal,9 and Kubota.1 These models allow to theoretically calculate the nucleation rate kinetic constant and the order of nucleation.1,8–21 Moreover, these models relate the MSZW obtained at fixed saturation temperature with the rate of cooling. It will be useful to develop a theoretical model that can theoretically determine the nucleation rate at different cooling rates. Thus, to complement the established model of Nývlt, Sangwal and Kubota, we present a new mathematical model based on the classical nucleation theory that allows to calculate nucleation rate at different cooling rates using the MSZW data collected from different solubility onset temperature. Unlike existing models, which do not explicitly account for cooling rate, our approach directly incorporates its influence, enabling accurate prediction of nucleation rates across varying cooling conditions. Furthermore, as the model rely on the classical nucleation theory, it immediately allows to theoretically calculate the Gibbs free energy of nucleation as a function of temperature and supersaturation at the point of nucleation. From the Gibbs free energy, it is possible to calculate the surface free energy or the interfacial tension and the radius of the critical nucleus as a function of the operating variables that mainly include the supersaturation and the temperature at which the nucleation is observed. To demonstrate the usefulness and universality of the proposed model, it was applied to experimental MSZW data from 22 different solute–solvent combinations. This dataset includes 10 different APIs across 11 API–solvent systems, eight inorganic compound–solvent combinations, as well as one API intermediate (L-arabinose), one amino acid (glycine), and one large biomolecule (lysozyme). The MSZW data, reported at different cooling rates, were used to extract key nucleation parameters, including the nucleation rate kinetic constant, Gibbs free energy of nucleation (ΔG), surface energy, and the critical nucleus radius, all evaluated as functions of supersaturation and nucleation temperature.

The metastable zone widths (MSZW) of solutions containing these single-component compounds, experimentally measured using the polythermal method, were obtained from the literature.22–32 This method involves changing the temperature of the solution from a reference solubility temperature image file: d5ce00467e-t1.tif at a predefined cooling rate and detecting the onset of nucleation at Tnuc.9 The relationship between the reference solution temperature, temperature at which nucleation can be observed or detected is shown in Fig. 1. In Fig. 1, we also showed the line of supersolubility concentration, solubility concentration and their relationship with the MSZW, ΔTmax and the supersaturation at the MSZW, ΔCmax.


image file: d5ce00467e-f1.tif
Fig. 1 The relationship between solubility concentration, c* and solubility temperature, T*. Also shown in the figures are the different parameters that can be experimentally fixed or obtained and their relationship between the MSZW (or ΔTmax = image file: d5ce00467e-t2.tif − Tnuc) and the supersaturation (or image file: d5ce00467e-t3.tif) at the MSZW.

According to the classical nucleation theory, the nucleation rate is defined as

 
J = kn[thin space (1/6-em)]exp(−ΔG/RT)(1)
During the cooling the crystallization, the undersaturated solution will be cooled from approximately image file: d5ce00467e-t4.tif + 5 °C at a fixed cooling rate of dT*/dt until the nucleation or the phase change is observed or can be experimentally detected measured in the solution. The temperature where the nucleation is observed is Tnuc. So, at the point of nucleation, T = Tnuc and thus eqn (1) becomes:
 
J = kn[thin space (1/6-em)]exp(−ΔG/RTnuc)(2)
The nucleation rate can be defined in terms of cooling rate, image file: d5ce00467e-t5.tif and the slope of solubility curve image file: d5ce00467e-t6.tif as follows
 
image file: d5ce00467e-t7.tif(3)
From eqn (2) and (3), it is possible to write that
 
image file: d5ce00467e-t8.tif(4)
At the point of nucleation,
 
image file: d5ce00467e-t9.tif(5)
where, image file: d5ce00467e-t10.tif is the MSZW and Δcmax is the supersaturation (or the driving force required for crystallization) achieved at Tnuc, where primary nucleation is most likely to occur. The relationship between the solubility concentration c* and the parameters that include T*, image file: d5ce00467e-t11.tif, Tnuc, and MSZW (i.e., ΔTmax) and the supersaturation concentration at ΔTmax (i.e., Δcmax) are given in Fig. 1.

Substituting eqn (5) in (4), we get

 
image file: d5ce00467e-t12.tif(6)
Eqn (6) can be linearized as follows:
 
image file: d5ce00467e-t13.tif(7)
From eqn (7), it is immediately evident that the model explicitly incorporates the cooling rate, along with the metastable zone width (ΔTmax), supersaturation at ΔTmax, and the nucleation temperature, to predict the nucleation rate constant (kn) and related parameters under varying experimental conditions. Importantly, eqn (7) establishes a direct link between key nucleation parameters such as the nucleation rate constant (kn), Gibbs free energy of nucleation (ΔG), and induction time (as later shown in eqn (12)) and experimentally measurable quantities. Theoretically, a plot of ln(ΔCmax/ΔTmax) versus 1/Tnuc should be linear with a negative slope whose value will be equal to ΔG/R with an intercept equal to ln(kn). Once the nucleation kinetic constant, kn and the ΔG is determined, then mathematically it is possible to estimate at the nucleation rate J using eqn (2) at a fixed cooling rate R′ and the MSZW measured at different image file: d5ce00467e-t14.tif (and the corresponding Tnuc). Furthermore, from the classical nucleation theory, it is also possible to calculate the surface energy, or the interfacial tension associated with the formation of stable nucleus using the expression:9
 
image file: d5ce00467e-t15.tif(8)
Once the surface energy is calculated using eqn (8), we can mathematically calculate the radius of the critical nucleus using the expression:9
 
image file: d5ce00467e-t16.tif(9)
In Fig. 2a and b, we showed the plot of ln(ΔCmax/ΔTmax) versus 1/Tnuc for 11 different combinations of APIs and solvents. In Fig. 2c and d we showed the plot of ln(ΔCmax/ΔTmax) versus 1/Tnuc for 8 combinations of inorganic compounds and solvents, as well as for one API intermediate (L-arabinose), one amino acid (glycine) and a large biomolecule (lysozyme). The ΔCmax, ΔTmax, and the corresponding nucleation temperature during the crystallization of these different compounds from their solution and the solvents involved are given in Table 1. All the experimental values were obtained from literature (see Table 1). From Fig. 2, it is evident that the proposed model provides an excellent fit to the experimental data across all systems studied, regardless of the nature of the solutes (APIs, inorganic compounds, or biomolecules), the solvents used, or the specific experimental conditions, including the studied cooling rates. From Fig. 2 and using eqn (7) we calculated the nucleation rate kinetic constant, kn and the Gibbs free energy of nucleation, ΔG. The calculated kn, ΔG and the corresponding r2 values are given in Table 1. In most cases the coefficient of determination was always greater than 0.97, indicating the proposed model provides an excellent representation of the experimental data across systems studied including API/solvent combinations, inorganic compounds, the API intermediate, the amino acid (glycine) and lysozyme. While some systems, such as lysozyme/NaCl, demonstrate a very high coefficient of determination (r2 = 0.9952), others, like NaNO3/NaCl–H2O, show relatively lower but still acceptable values (e.g., r2 = 0.8911 – see Table 1). In crystallisation studies, particularly those relying on thermodynamic or kinetic parameters derived from indirect measurements such as MSZW, an r2 ≥ 0.9 is generally considered indicative of a strong model fit. The observed variation in r2 across different systems does not point to a fundamental limitation of the proposed model but rather reflects statistical constraints inherent to linear regression when applied over narrow MSZW or solubility ranges. Specifically, systems with limited experimental variability, where MSZW data span only a small solubility temperature range are more susceptible to experimental noise, leading to slightly lower r2 values. This is a well-recognised limitation of linear regression, which benefits from broader data coverage to enhance fit robustness. For example, in the CoSO4/water and NaNO3/NaCl–H2O systems, ΔTmax showed minimal change across the studied solubility concentrations. Moreover, in these systems, ΔTmax does not increase consistently with solubility temperature, which disrupts the expected linear trend and leads to greater deviation when the data are transformed and fitted using eqn (7). This inconsistency contributes to higher residuals and reduces the overall r2 value. Theoretically, nucleation depends on multiple interconnected factors, including the degree of supercooling, nucleation temperature (Tnuc), reference solubility temperature image file: d5ce00467e-t17.tif, solubility concentration at image file: d5ce00467e-t18.tif (i.e., image file: d5ce00467e-t19.tif), supersaturation at the Tnuc (i.e., Δcmax), and cooling rate. The proposed model performs particularly well in systems where ΔTmax increases with solubility temperature, regardless of the applied cooling rate supporting its general applicability across a wide range of crystallisation scenarios.


image file: d5ce00467e-f2.tif
Fig. 2 Plot of ln(ΔCmax/ΔTmax) + ln(R′) versus 1/Tnuc for (a) 6 different combinations of API and solvents involving 4 APIs, (b) 11 different combinations of API and solvents involving 6 APIs, (c) combination of inorganics/solvents involving 8 different compounds and (d) API intermediate, amino acid and a large molecule. Note: these values are valid only at the cooling rates studied given in Table 1. Fig. 2a: image file: d5ce00467e-u1.tif: paracetamol/water (R′: 0.5 K s−1); image file: d5ce00467e-u2.tif: dextrose/water (R′: 0.0278 K s−1); image file: d5ce00467e-u3.tif: L-asparagine monohydrate/water (R′: 0.005 K s−1); image file: d5ce00467e-u4.tif: L-asparagine monohydrate/water (R′: 0.0033 K s−1). Fig. 2b: image file: d5ce00467e-u5.tif: ibuprofen/ethanol (R′: 0.0167 K s−1); image file: d5ce00467e-u6.tif: paracetamol/ethanol (R′: 0.0167 K s−1); image file: d5ce00467e-u7.tif: pyrazinamide/acetone (R′: 0.0167 K s−1); image file: d5ce00467e-u8.tif: fenofibrate/ethyl acetate (R′: 0.05 K s−1); image file: d5ce00467e-u9.tif: gestodene/ethanol (R′: 0.0083 K s−1); image file: d5ce00467e-u10.tif: gestodene/ethanol (R′: 0.005 K s−1); image file: d5ce00467e-u11.tif: carbamazepine_saccharine/ethanol (R′: 0.0167 K s−1); image file: d5ce00467e-u12.tif: pyrazinamide/acetone (R′: 0.0033 K s−1); image file: d5ce00467e-u13.tif: gestodene/ethanol (R′: 0.00167 K s−1). Fig. 2c: image file: d5ce00467e-u14.tif: NaNO3/NaCl + NaNO3 + H2O solution (R′: 0.000833 K s−1); image file: d5ce00467e-u15.tif: NaNO3/NaCl + NaNO3 + H2O solution (R′: 0.001667 K s−1); image file: d5ce00467e-u16.tif: CoSO4/water (R′: 0.000833 K s−1); image file: d5ce00467e-u17.tif: CoSO4/water (R′: 0.001667 K s−1); image file: d5ce00467e-u18.tif: CoSO4/water (R′: 0.0025 K s−1); image file: d5ce00467e-u19.tif: (NH4)2B4O7·4H2O/water (R′: 0.00833 K s−1); image file: d5ce00467e-u20.tif: (NH4)2B4O7·4H2O/water (R′: 0.0025 K s−1); image file: d5ce00467e-u21.tif: Na3VO4/NaOH solution (R′: 0.005 K s−1); image file: d5ce00467e-u22.tif: Sr(OH)2·8H2O/water (R′: 0.001388 K s−1); image file: d5ce00467e-u23.tif: Sr(OH)2·8H2O/water (R′: 0.00277 K s−1); image file: d5ce00467e-u24.tif: ZnL2/water (R′: 0.00138 K s−1); image file: d5ce00467e-u25.tif: ZnL2/water (R′: 0.00275 K s−1); image file: d5ce00467e-u26.tif: borax/water (R′: 0.003472 K s−1); image file: d5ce00467e-u27.tif: borax/water (R′: 0.01066 K s−1); image file: d5ce00467e-u28.tif: ammonium oxalate/water (R′: 0.000833 K s−1 – plotted in secondary axis). Fig. 2d: image file: d5ce00467e-u29.tif: L-arabinose/water (R′: 0.0083 K s−1); image file: d5ce00467e-u30.tif: glycine/water (R′: 0.001667 K s−1); image file: d5ce00467e-u31.tif: glycine/water (R′: 0.005 K s−1); image file: d5ce00467e-u32.tif: lysozyme/NaCl solution (R′: 0.002778 K s−1 – we assumed this cooling rate – shown in secondary axis).
Table 1 Nucleation parameters as a function of cooling rate obtained using the proposed model as in eqn (7) for 11 different combinations of APIs/solvents that involve 10 APIs, 8 inorganic compounds in their solution, an API intermediate, amino acid and lysozyme (a large molecule)
API/solvent R′, K s−1 k n, m−3 s−1 T nuc, K

image file: d5ce00467e-t20.tif

 Δcmax, molecules per m3 ΔTmax, K ΔG, kJ mol−1 J, m−3 s−1 γ , mJ m−2 r c, m V 1/3cell, m λ = 2rc/V1/3cell N m/V t ind, s r 2 Ref.
Note: the unit cell volumes of all compounds studied were obtained from the crystallographic information files (CIFs) retrieved from the International Centre for Diffraction Data (ICDD) powder diffraction file (PDF-5+, 2025), using the database tools provided by the ICDD.
Psilocybin/water 0.0167 6.4 × 1032 326.02 3.86 6.6 × 1025 14.87 62.1 7.12 × 1022 28.35 9.32 × 10−10 1.38 × 10−9 1.35 6.35 × 1025 892 0.9789 23
330.35 3.13 7.2 × 1025 12.38 9.63 × 1022 25.55 9.82 × 10−10 1.42 7.15 × 1025 743
332.70 2.83 7.63 × 1025 11.38 1.13 × 1023 24.13 1.01 × 10−9 1.46 7.71 × 1025 683
334.91 2.61 8.17 × 1025 11.01 1.3 × 1023 22.98 1.04 × 10−9 1.50 8.65 × 1025 661
338.30 2.21 8.31 × 1025 9.01 1.64 × 1023 20.39 1.10 × 10−9 1.59 8.85 × 1025 541
341.90 1.87 8.2 × 1025 6.09 2.06 × 1023 17.52 1.19 × 10−9 1.72 7.55 × 1025 366
Paracetamol/ethanol 0.0167 2.3 × 1026 285.21 1.59 5.14 × 1026 22.99 15.22 3.75 × 1023 7.97 8.70 × 10−10 1.14 × 10−9 1.51 5.17 × 1026 1379 0.9823 23
288.86 1.54 5.13 × 1026 21.05 4.06 × 1023 7.63 8.90 × 10−10 1.54 5.14 × 1026 1263
296.63 1.38 4.72 × 1026 15.61 4.80 × 1023 6.42 9.70 × 10−10 1.68 4.50 × 1026 937
299.74 1.41 5.2 × 1026 16.85 5.11 × 1023 6.75 9.46 × 10−10 1.64 5.18 × 1026 1011
303.93 1.41 5.72 × 1026 16.70 5.56 × 1023 6.81 9.41 × 10−10 1.63 5.58 × 1026 1002
304.63 1.49 6.56 × 1026 19.73 5.64 × 1023 7.55 8.94 × 10−10 1.55 6.68 × 1026 1184
310.22 1.54 7.76 × 1026 21.05 6.29 × 1023 8.00 8.69 × 10−10 1.50 7.95 × 1026 1263
Pyrazinamide/acetone 0.0016 1.7 × 1025 294.45 1.10 3.5 × 1024 3.68 22.63 1.64 × 1021 3.31 1.65 × 10−9 8.30 × 10−10 3.97 3.62 × 1024 2210 0.9007 30
299.72 1.11 4.4 × 1024 3.60 1.92 × 1021 3.42 1.62 × 10−9 3.9 4.17 × 1024 2162
305.51 1.07 3.85 × 1024 2.89 2.29 × 1021 2.75 1.81 × 10−9 4.36 3.97 × 1024 1734
311.14 1.05 2.97 × 1024 2.26 2.69 × 1021 2.05 2.09 × 10−9 5.05 3.65 × 1024 1354
Pyrazinamide/acetone 0.0033 2.4 × 1025 292.70 1.18 5.67 × 1024 5.43 21.42 3.62 × 1021 4.56 1.37 × 10−9 8.30 × 10−10 3.29 5.90 × 1024 1628 0.9602 30
298.10 1.18 6.93 × 1024 5.23 4.25 × 1021 4.59 1.36 × 10−9 3.28 6.67 × 1024 1568
303.73 1.15 7.24 × 1024 4.67 4.99 × 1021 4.19 1.43 × 10−9 3.43 7.00 × 1024 1402
309.84 1.10 6.08 × 1024 3.56 5.89 × 1021 3.28 1.61 × 10−9 3.88 6.31 × 1024 1069
L-Asparagine monohydrate/water 0.0016 1.3 × 1028 290.65 1.27 2.16 × 1025 4.40 34.69 7.81 × 1021 6.79 1.42 × 10−9 8.67 × 10−10 3.28 2.06 × 1025 2639 0.9723 24
297.87 1.26 2.92 × 1025 5.11 1.11 × 1022 6.75 1.43 × 10−9 3.29 3.39 × 1025 3068
303.94 1.22 3.22 × 1025 4.05 1.46 × 1022 6.13 1.50 × 10−9 3.46 3.56 × 1025 2433
308.46 1.20 3.68 × 1025 3.50 1.78 × 1022 5.90 1.53 × 10−9 3.52 3.76 × 1025 2098
315.24 1.17 4.24 × 1025 2.76 2.39 × 1022 5.38 1.60 × 10−9 3.69 3.96 × 1025 1654
L-Asparagine monohydrate/water 0.0033 2.5 × 1028 288.69 1.39 2.88 × 1025 6.36 34.55 1.42 × 1022 8.38 1.28 × 10−9 8.67 × 10−10 2.95 2.71 × 1025 1907 0.9887 24
294.24 1.49 4.67 × 1025 8.74 1.86 × 1022 9.66 1.19 × 10−9 2.75 4.88 × 1025 2622
301.39 1.37 4.90 × 1025 6.60 2.6 × 1022 8.36 1.28 × 10−9 2.95 5.15 × 1025 1980
305.81 1.36 5.83 × 1025 6.15 3.17 × 1022 8.28 1.29 × 10−9 2.97 5.86 × 1025 1844
313.68 1.26 6.02 × 1025 4.32 4.46 × 1022 6.92 1.41 × 10−9 3.25 5.79 × 1025 1295
L-Asparagine monohydrate/water 0.005 1.6 × 1028 284.41 1.70 4.19 × 1025 10.64 32.35 1.87 × 1022 11.13 1.07 × 10−9 8.67 × 10−10 2.48 3.99 × 1025 2127 0.9891 24
292.21 1.64 5.53 × 1025 10.77 2.69 × 1022 10.84 1.09 × 10−9 2.51 5.82 × 1025 2154
299.61 1.49 5.95 × 1025 8.38 3.75 × 1022 9.52 1.16 × 10−9 2.68 6.28 × 1025 1676
302.44 1.59 8.21 × 1025 9.52 4.23 × 1022 10.61 1.10 × 10−9 2.54 8.06 × 1025 1904
312.16 1.35 7.63 × 1025 5.84 6.32 × 1022 8.09 1.26 × 10−9 2.91 7.38 × 1025 1168
Dextrose/water 0.0027 1.9 × 1024 286.42 1.25 3.28 × 1026 17.01 8.55 5.41 × 1022 3.99 9.22 × 10−10 7.57 × 10−10 2.44 3.31 × 1025 6124 0.9824 29
290.45 1.26 3.64 × 1026 17.03 5.68 × 1022 4.16 9.03 × 10−10 2.39 3.49 × 1026 6131
294.52 1.25 3.70 × 1026 17.07 5.97 × 1022 4.12 9.07 × 10−10 2.40 3.67 × 1026 6144
298.60 1.24 3.76 × 1026 16.55 6.26 × 1022 4.08 9.12 × 10−10 2.41 3.73 × 1026 5958
302.46 1.23 3.80 × 1026 16.36 6.54 × 1022 4.03 9.18 × 10−10 2.42 3.86 × 1026 5889
Paracetamol/water 0.5 7.2 × 1027 292.08 1.66 3.16 × 1025 14.85 21.34 1.09 × 1024 9.58 9.40 × 10−10 1.14 × 10−9 1.64 3.24 × 1026 30 0.9951 29
296.03 1.78 4.30 × 1025 16.99 1.22 × 1024 10.57 8.95 × 10−10 1.56 4.16 × 1026 34
304.37 1.49 3.56 × 1025 11.65 1.55 × 1024 8.40 1.00 × 10−9 1.75 3.62 × 1025 23
303.19 1.61 4.29 × 1025 14.21 1.50 × 1024 9.47 9.46 × 10−10 1.65 4.28 × 1025 28
313.66 1.32 3.22 × 1025 8.12 1.99 × 1024 6.77 1.12 × 10−9 1.95 3.24 × 1025 16
318.36 1.28 3.32 × 1025 7.37 2.25 × 1024 6.34 1.16 × 10−9 2.02 3.32 × 1025 15
321.35 1.22 2.80 × 1025 4.77 2.42 × 1024 5.48 1.24 × 10−9 2.17 2.32 × 1025 10
Gestodene form1/ethanol 0.0017 1.6 × 1025 301.58 1.41 2.67 × 1025 11.43 20.90 3.81 × 1021 7.49 1.05 × 10−9 1.20 × 10−9 1.75 2.62 × 1025 6857 0.9814 25
308.32 1.33 2.62 × 1025 9.71 4.57 × 1021 6.70 1.11 × 10−9 1.85 2.67 × 1025 5829
315.44 1.25 2.43 × 1025 7.62 5.51 × 1021 5.75 1.20 × 10−9 2 2.52 × 1025 4571
321.62 1.22 2.59 × 1025 6.54 6.42 × 1021 5.43 1.24 × 10−9 2.06 2.52 × 1025 3924
Gestodene form1/ethanol 0.0050 3.4 × 1025 298.15 1.56 3.31 × 1025 14.86 19.95 1.09 × 1022 8.70 9.54 × 10−10 1.20 × 10−9 1.59 3.25 × 1025 2971 0.99 25
305.02 1.46 3.36 × 1025 13.02 1.31 × 1022 7.98 9.96 × 10−10 1.66 3.41 × 1025 2603
312.40 1.36 3.28 × 1025 10.67 1.57 × 1022 7.07 1.06 × 10−9 1.76 3.37 × 1025 2133
319.46 1.30 3.33 × 1025 8.70 1.87 × 1022 6.41 1.11 × 10−9 1.85 3.26 × 1025 1740
Gestodene form1/ethanol 0.0083 2.5 × 1025 296.18 1.65 3.64 × 1025 16.83 19.15 1.06 × 1022 9.27 9.05 × 10−10 1.20 × 10−9 1.51 2.15 × 1025 2019 0.9896 25
303.62 1.52 3.65 × 1025 14.41 1.28 × 1022 8.40 9.51 × 10−10 1.59 2.23 × 1025 1730
311.76 1.39 3.45 × 1025 11.30 1.56 × 1022 7.25 1.02 × 10−9 1.71 2.13 × 1025 1356
318.83 1.32 3.53 × 1025 9.33 1.84 × 1022 6.61 1.07 × 10−9 1.79 2.07 × 1025 1120
Fenofibrate/ethyl acetate 0.0050 3.1 × 1027 279.80 2.59 1.07 × 1027 27.37 22.51 1.94 × 1023 14.43 7.87 × 10−10 9.92 × 10−10 1.586 1.07 × 1027 5474 0.999 31
284.53 2.54 1.23 × 1027 27.00 2.28 × 1023 14.43 7.87 × 10−10 1.586 1.24 × 1027 5400
287.48 2.52 1.33 × 1027 26.63 2.52 × 1023 14.42 7.87 × 10−10 1.587 1.34 × 1027 5327
290.16 2.49 1.44 × 1027 26.53 2.75 × 1023 14.40 7.88 × 10−10 1.588 1.46 × 1027 5306
292.79 2.47 1.55 × 1027 26.11 2.99 × 1023 14.38 7.88 × 10−10 1.588 1.56 × 1027 5222
295.26 2.45 1.67 × 1027 25.79 3.24 × 1023 14.38 7.88 × 10−10 1.589 1.67 × 1027 5159
296.89 2.43 1.75 × 1027 25.64 3.40 × 1023 14.37 7.88 × 10−10 1.589 1.74 × 1027 5127
Vismodegib/MIBK 0.0167 1.5 × 1030 345.96 2.71 2.74 × 1025 19.04 51.54 2.48 × 1022 22.62 9.51 × 10−10 1.28 × 10−9 1.49 2.84 × 1025 1142 0.9975 28
349.89 2.32 3.92 × 1025 21.12 3.07 × 1022 20.35 1.00 × 10−9 1.57 3.85 × 1025 1267
354.00 2.13 5.08 × 1025 22.10 3.73 × 1022 19.10 1.03 × 10−9 1.62 4.95 × 1025 1326
372.98 3.86 3.24 × 1025 6.02 9.09 × 1022 29.14 8.38 × 10−10 1.31 3.28 × 1025 361
379.98 5.82 2.30 × 1025 7.98 1.23 × 1023 35.20 7.62 × 10−10 1.19 5.92 × 1025 479
Carbamazepine–saccharin cocrystal/water 0.0167 9.1 × 1025 297.27 1.84 7.27 × 1024 19.17 23.70 6.24 × 1021 11.39 9.08 × 10−10 1.57 × 10−9 1.16 8.34 × 1026 415 0.9865 32
298.89 1.80 7.32 × 1024 18.66 6.57 × 1021 11.15 9.18 × 10−10 1.17 9.74 × 1026 419
300.00 1.78 7.42 × 1024 18.33 6.81 × 1021 11.03 9.23 × 10−10 1.18 1.08 × 1027 411
301.23 1.77 7.69 × 1024 18.27 7.08 × 1021 11.01 9.24 × 10−10 1.18 1.15 × 1027 384
302.46 1.76 7.93 × 1024 18.07 7.36 × 1021 10.97 9.26 × 10−10 1.18 1.31 × 1027 380
303.37 1.77 8.26 × 1024 18.10 7.57 × 1021 11.03 9.23 × 10−10 1.18 1.53 × 1027 376
303.85 1.77 8.41 × 1024 18.10 7.68 × 1021 11.05 9.22 × 10−10 1.18 1.73 × 1027 372
Ibuprofen/ethanol 0.0167 1.7 × 1030 287.99 1.58 8.37 × 1026 6.91 32.75 2.09 × 1024 10.27 1.12 × 10−9 1.36 × 10−9 1.65 7.18 × 1024 1150 0.9994 26
291.09 1.59 9.66 × 1026 6.98 2.32 × 1024 10.40 1.12 × 10−9 1.64 7.36 × 1024 1120
293.67 1.58 1.07 × 1027 6.85 2.62 × 1024 10.39 1.12 × 10−9 1.64 7.49 × 1024 1100
296.64 1.56 1.16 × 1027 6.40 2.99 × 1024 10.24 1.13 × 10−9 1.66 7.77 × 1024 1096
299.74 1.55 1.30 × 1027 6.33 3.43 × 1024 10.21 1.13 × 10−9 1.66 7.98 × 1024 1084
303.68 1.54 1.53 × 1027 6.27 4.07 × 1024 10.25 1.13 × 10−9 1.66 8.22 × 1024 1086
306.79 1.53 1.72 × 1027 6.20 4.65 × 1024 10.21 1.13 × 10−9 1.66 8.34 × 1024 1086
NaNO3/NaCl–NaNO3–H2O solution 0.00083 1.7 × 1024 282.1 1.06 2.24 × 1026 5.99 9.23 3.29 × 1022 1.69 1.47 × 10−9 7.6 × 10−10 3.89 2.37 × 1026 7188 0.9226 54
287.7 1.06 2.39 × 1026 5.65 3.55 × 1022 1.71 1.46 × 10−9 3.87 2.41 × 1026 6780
298.8 1.04 1.96 × 1026 4.73 4.10 × 1022 1.40 1.61 × 10−9 4.27 2.33 × 1026 5676
303.0 1.03 1.55 × 1026 3.86 4.32 × 1022 1.17 1.77 × 10−9 4.67 2.00 × 1026 4632
305.7 1.03 1.32 × 1026 3.6 4.45 × 1022 1.04 1.87 × 10−9 4.96 1.92 × 1026 4320
309.2 1.02 1.24 × 1026 2.66 4.64 × 1022 0.97 1.94 × 10−9 5.13 1.48 × 1026 3192
NaNO3/NaCl–NaNO3–H2O solution 0.001667 2.2 × 1023 281.06 1.08 2.74 × 1026 7.06 4.45 3.30 × 1022 1.52 1.08 × 10−9 7.6 × 10−10 2.8 1.40 × 1026 4236 0.8911 54
286.9 1.07 2.76 × 1026 6.39 3.44 × 1022 1.48 1.09 × 10−9 2.9 1.32 × 1026 3834
298 1.05 2.42 × 1026 5.52 3.68 × 1022 1.27 1.18 × 10−9 3.1 1.22 × 1026 3312
302.3 1.04 1.96 × 1026 4.52 3.78 × 1022 1.08 1.28 × 10−9 3.4 1.03 × 1026 2712
308.9 1.03 1.45 × 1026 2.99 3.92 × 1022 0.85 1.44 × 10−9 3.8 7.04 × 1025 1794
Ammonium oxalate monohydrate/water 0.00083 1.0 × 1029 299.36 1.40 7.34 × 1025 1.16 30.07 5.64 × 1023 8.28 1.20 × 10−9 6.8 × 10−10 3.53 7.88 × 1025 139.7 0.9335 55
300.45 1.40 7.62 × 1025 1.00 5.89 × 1023 8.33 1.20 × 10−9 3.52 7.09 × 1025 120.4
302.41 1.41 8.23 × 1025 1.08 6.37 × 1023 8.47 1.19 × 10−9 3.49 8.29 × 1025 130.1
306.15 1.40 8.98 × 1025 0.96 7.37 × 1023 8.46 1.19 × 10−9 3.49 8.52 × 1025 115.6
308.75 1.39 9.40 × 1025 1.00 8.15 × 1023 8.38 1.19 × 10−9 3.51 9.82 × 1025 120.5
L-Arabinose/water 0.0022 8.7 × 1025 306.84 1.60 2.15 × 1027 26.04 15.84 1.76 × 1023 8.53 8.58 × 10−10 8.5 × 10−10 2.01 2.06 × 1027 11[thin space (1/6-em)]718 0.9615 56
311.19 1.61 2.38 × 1027 28.12 1.92 × 1023 8.72 8.48 × 10−10 1.99 2.42 × 1027 12[thin space (1/6-em)]654
315.54 1.63 2.62 × 1027 28.68 2.08 × 1023 8.89 8.40 × 10−10 1.97 2.69 × 1027 12[thin space (1/6-em)]906
320.08 1.64 2.89 × 1027 29.06 2.27 × 1023 9.06 8.32 × 10−10 1.96 2.97 × 1027 13[thin space (1/6-em)]077
325 1.64 3.19 × 1027 28.68 2.49 × 1023 9.20 8.26 × 10−10 1.94 3.21 × 1027 12[thin space (1/6-em)]906
329.73 1.66 3.55 × 1027 28.12 2.70 × 1023 9.41 8.17 × 10−10 1.92 3.42 × 1027 12[thin space (1/6-em)]654
Cobalt sulfate/water 0.00083 2.3 × 1024 284.13 1.09 1.32 × 1026 9.14 12.28 1.30 × 1022 2.39 1.43 × 10−9 6.5 × 10−10 4.39 1.43 × 1026 10[thin space (1/6-em)]965 0.9407 15
290.27 1.10 1.50 × 1026 7.99 1.45 × 1022 2.54 1.38 × 10−9 4.26 1.39 × 1026 9583
296.80 1.08 1.28 × 1026 6.30 1.63 × 1022 2.25 1.47 × 10−9 4.53 1.23 × 1026 7556
310.01 1.04 7.46 × 1025 3.22 2.01 × 1022 1.51 1.80 × 10−9 5.53 7.78 × 1025 3870
Cobalt sulfate/water 0.00167 1.5 × 1024 282.75 1.10 1.50 × 1026 10.52 9.59 2.56 × 1022 2.40 1.26 × 10−9 6.5 × 10−10 3.87 1.61 × 1026 6313 0.9399 15
288.12 1.12 1.80 × 1026 10.14 2.76 × 1022 2.64 1.20 × 10−9 3.69 1.67 × 1026 6083
293.27 1.11 1.80 × 1026 9.83 2.96 × 1022 2.60 1.21 × 10−9 3.73 1.74 × 1026 5897
308.40 1.06 1.01 × 1026 4.84 3.59 × 1022 1.70 1.49 × 10−9 4.60 1.04 × 1026 2902
Cobalt sulfate/water 0.0025 9.8 × 1023 279.83 1.13 1.88 × 1026 13.44 7.63 3.70 × 1022 2.58 1.08 × 10−9 6.5 × 10−10 3.33 1.99 × 1026 5375 0.9453 57
285.05 1.15 2.22 × 1026 13.21 3.93 × 1022 2.82 1.04 × 10−9 3.19 2.08 × 1026 5283
289.20 1.15 2.37 × 1026 13.90 4.12 × 1022 2.90 1.02 × 10−9 3.14 2.29 × 1026 5559
296.72 1.12 1.97 × 1026 11.52 4.46 × 1022 2.52 1.10 × 10−9 3.37 2.06 × 1026 4607
306.78 1.07 1.27 × 1026 6.45 4.94 × 1022 1.84 1.28 × 10−9 3.94 1.27 × 1026 2580
Glycine/water 0.00167 2.4 × 1025 293.86 1.19 3.82 × 1026 8.94 14.29 7.00 × 1022 4.11 1.17 × 10−9 6.9 × 10−10 3.40 3.76 × 1026 5365 0.9905 57
303.32 1.16 3.96 × 1026 8.06 8.40 × 1022 3.85 1.21 × 10−9 3.51 4.06 × 1026 4833
311.52 1.14 4.02 × 1026 6.87 9.76 × 1022 3.60 1.25 × 10−9 3.63 4.02 × 1026 4123
318.61 1.13 4.06 × 1026 6.06 1.10 × 1023 3.39 1.29 × 10−9 3.74 4.01 × 1026 3636
Glycine/water 0.005 1.8 × 1025 290.75 1.26 4.95 × 1026 12.05 13.54 6.76 × 1022 4.86 1.05 × 10−9 6.9 × 10−10 3.04 1.63 × 1026 2409 0.9927 57
300.95 1.22 4.99 × 1026 10.42 8.17 × 1022 4.46 1.10 × 10−9 3.18 1.70 × 1026 2084
309.75 1.18 4.92 × 1026 8.65 9.53 × 1022 4.08 1.15 × 10−9 3.32 1.65 × 1026 1729
316.62 1.17 5.22 × 1026 8.06 1.07 × 1023 3.97 1.16 × 10−9 3.37 1.72 × 1026 1611
Na3VO4/NaOH solution 0.005 4.6 × 1030 306.55 1.16 1.23 × 1025 6.39 49.94 1.44 × 1022 5.76 1.85 × 10−9 7.6 × 10−10 4.88 1.84 × 1025 1279 0.98 58
314.08 1.63 6.84 × 1025 8.98 2.30 × 1022 13.06 1.23 × 10−9 3.24 4.13 × 1025 1795
330.52 1.70 1.58 × 1026 12.42 5.95 × 1022 14.31 1.18 × 10−9 3.10 1.48 × 1026 2484
340.55 1.63 2.19 × 1026 12.79 1.02 × 1023 13.73 1.20 × 10−9 3.16 2.60 × 1026 2557
Ammonium biborate tetrahydrate/water 0.00083 4.2 × 1025 283.53 1.21 3.03 × 1025 9.50 22.72 2.76 × 1021 5.01 1.34 × 10−9 3.6 × 10−10 7.27 3.14 × 1025 11[thin space (1/6-em)]396 0.9981 59
287.85 1.24 4.03 × 1025 10.14 3.19 × 1021 5.49 1.28 × 10−9 6.95 3.88 × 1025 12[thin space (1/6-em)]173
293.03 1.23 4.56 × 1025 9.93 3.77 × 1021 5.36 1.30 × 10−9 7.03 4.49 × 1025 11[thin space (1/6-em)]914
306.07 1.15 4.64 × 1025 6.98 5.61 × 102 4.21 1.46 × 10−9 7.93 4.70 × 1025 8375
Ammonium biborate tetrahydrate/water 0.0025 3.4 × 1026 281.37 1.31 4.08 × 1025 11.66 24.71 8.86 × 1021 6.40 1.24 × 10−9 3.6 × 10−10 6.71 4.13 × 1025 4662 0.9988 59
285.90 1.33 5.14 × 1025 12.09 1.05 × 1022 6.75 1.21 × 10−9 6.53 5.06 × 1025 4834
290.80 1.33 6.08 × 1025 12.16 1.25 × 1022 6.81 1.20 × 10−9 6.50 6.07 × 1025 4863
303.69 1.25 7.18 × 1025 9.35 1.93 × 1022 5.92 1.29 × 10−9 6.98 7.20 × 1025 3741
Lysozyme/NaCl solution 0.00028 1.1 × 1034 300.86 1.82 2.80 × 1023 9.52 87.15 8.15 × 1018 17.5 1.41 × 10−9 7.1 × 10−9 0.39 2.79 × 1023 26 0.9952 60
303.92 2.80 4.12 × 1023 10.45 1.16 × 1019 19.4 1.34 × 10−9 0.37 4.35 × 1023 29
306.43 2.54 7.47 × 1023 14.15 1.54 × 1019 23.8 1.21 × 10−9 0.34 7.82 × 1023 35
307.74 3.13 1.12 × 1024 17.66 1.78 × 1019 27.3 1.13 × 10−9 0.32 1.13 × 1024 39
308.96 3.62 1.15 × 1024 20.43 2.03 × 1019 29.6 1.08 × 10−9 0.30 1.49 × 1024 42
Sr(OH)2·8H2O/water 0.00139 1.4 × 1024 279.65 1.31 2.57 × 1024 7.37 18.62 4.76 × 1020 5.81 1.13 × 10−9 9.8 × 10−10 2.30 2.52 × 1024 5306 0.9981 61
292.96 1.15 2.13 × 1024 4.39 6.85 × 1020 3.96 1.37 × 10−9 2.79 2.16 × 1024 3161
305.02 1.08 1.79 × 1024 2.76 9.26 × 1020 2.75 1.64 × 10−9 3.35 1.84 × 1024 1987
316.88 1.02 7.03 × 1023 1.06 1.22 × 1021 1.14 2.54 × 10−9 5.18 9.30 × 1023 763
327.63 1.01 6.52 × 1023 0.58 1.54 × 1021 0.85 2.94 × 10−9 6.00 6.42 × 1023 418
Sr(OH)2·8H2O/water 0.00278 3.5 × 1022 279.11 1.34 2.81 × 1024 8.09 8.51 9.90 × 1022 4.77 8.42 × 10−10 9.8 × 10−10 1.72 2.88 × 1026 2912 0.9802 61
291.81 1.21 2.77 × 1024 5.64 1.16 × 1023 3.67 9.59 × 10−10 1.96 2.36 × 1026 2030
303.57 1.14 2.87 × 1024 4.13 1.33 × 1023 2.94 1.07 × 10−9 2.19 1.98 × 1026 1487
316.08 1.05 1.71 × 1024 1.87 1.52 × 1023 1.61 1.45 × 10−9 2.96 1.02 × 1026 673
327.28 1.02 1.13 × 1024 0.84 1.70 × 1023 0.95 1.88 × 10−9 3.84 5.14 × 1025 302
Zinc lactate/water 0.002775 8.9 × 1024 300.78 1.30 1.14 × 1025 11.32 20.16 2.80 × 1021 6.25 2.27 × 10−9 1.1 × 10−9 2.06 1.14 × 1025 4080.4 0.99832 62
319.68 1.34 1.97 × 1025 12.32 4.52 × 1021 6.88 2.15 × 10−9 1.96 2.01 × 1025 4438.7
308.81 1.37 1.67 × 1025 13.29 3.46 × 1021 7.08 2.12 × 10−9 1.93 1.66 × 1025 4789.9
329.97 1.40 2.97 × 1025 14.23 5.72 × 1021 7.74 2.03 × 10−9 1.85 2.93 × 1025 5127.7
Zinc lactate/water 0.003472 1.1 × 1025 299.18 1.35 1.29 × 1025 12.92 19.99 3.46 × 1021 6.78 2.16 × 10−9 1.1 × 10−9 1.97 1.29 × 1025 3722.2 0.9984 62
318.00 1.39 2.20 × 1025 14.00 5.56 × 1021 7.45 2.06 × 10−9 1.88 2.24 × 1025 4032.9
307.11 1.42 1.84 × 1025 14.99 4.25 × 1021 7.62 2.04 × 10−9 1.85 1.84 × 1025 4317.2
328.25 1.45 3.26 × 1025 15.95 7.04 × 1021 8.30 1.95 × 10−9 1.78 3.23 × 1025 4594.0
Borax/water 0.0033 1.4 × 1027 278.32 1.65 2.28 × 1025 10.78 28.07 7.69 × 1021 9.97 1.06 × 10−9 1.2 × 10−9 1.83 2.50 × 1025 3235 0.9982 63
282.02 1.62 2.66 × 1025 10.66 9.03 × 1021 9.99 1.06 × 10−9 1.83 2.89 × 1025 3198
288.28 1.48 2.62 × 1025 8.35 1.17 × 1022 8.61 1.14 × 10−9 1.97 2.93 × 1025 2504
292.35 1.43 2.85 × 1025 7.68 1.38 × 1022 8.22 1.16 × 10−9 2.01 3.17 × 1025 2303
295.62 1.41 3.13 × 1025 7.31 1.56 × 1022 8.01 1.18 × 10−9 2.04 3.43 × 1025 2193
Borax/water 0.01067 2.3 × 1027 273.93 2.01 2.93 × 1025 15.17 26.28 2.22 × 1022 12.12 9.27 × 10−10 1.2 × 10−9 1.60 3.16 × 1024 1422 0.9967 63
279.98 1.86 3.21 × 1025 13.71 2.74 × 1022 11.47 9.53 × 10−10 1.65 3.52 × 1025 1285
285.23 1.69 3.35 × 1025 11.39 3.51 × 1022 10.29 1.01 × 10−9 1.74 3.75 × 1025 1068
288.94 1.67 3.83 × 1025 11.09 4.04 × 1022 10.19 1.01 × 10−9 1.75 4.20 × 1025 1039
292.76 1.60 4.10 × 1025 10.17 4.67 × 1022 9.71 1.04 × 10−9 1.79 4.45 × 1025 954

The ΔG values typically range from 8 to 62 kJ mol−1 for the APIs, 13.5 for amino acid, 15.84 kJ mol−1 for the API intermediate. The ΔG values for the inorganic compounds range from 4.45 to ∼50 kJ mol−1. Most of these values agree with the ΔG values reported for several organic/inorganic compounds.7,33–51 Furthermore, we also noticed that there doesn't exist a perfect correlation between the molecular weight and the Gibbs free energy of nucleation (see Table 1). However, the ΔG values are higher for compounds with higher molecular weight. Noticeably for the case of the largest molecule studied (lysozyme), eqn (7) predicted a ΔG value of 87 kJ mol−1. This agrees with the classical nucleation theory which states that the higher the molecular volume is tougher in the nucleation. The surface energy, calculated using our model in conjunction with classical nucleation theory, ranges from 0.8517 mJ m−2 to 35.2 mJ m−2 across all systems studied. The radius of the critical nucleus of all the studied APIs, API intermediate, amino acid and lysozyme was found to be in the range of ∼10−10 m to ∼10−9 m. Similar magnitudes were also observed for the inorganic compounds, indicating that the critical nucleus size remains within the nanometre scale across both organic and inorganic solutes. Using the size of the critical radius, we mathematically estimated the number of unit cells constituting the critical nucleus. This was done by calculating the normalized critical nucleus size (λ), defined as the ratio of the diameter of the stable nucleus (2rc) to the cube root of the unit cell volume, Vcell. The unit cell volumes for the API crystals were obtained from their respective crystallographic information files (CIF file) (see Table 1). Finally, using this normalized size, the approximate number of unit cells in the critical nucleus was calculated as equal to λ.3 This estimation provides a simple theoretical approximation of the structural size of the critical nucleus in terms of unit cells. Note that, this approach assumes that the spherical critical nucleus has a diameter approximately equal to that of a cube-shaped nucleus composed of unit cells arranged in a simple cubic packing, i.e., the linear dimension of the nucleus is equivalent in both cases. This simplification allows for a geometric approximation of the number of unit cells. While this is a simplification and does not account for molecular packing or anisotropic crystal structures, it provides a first-order estimate of the number of unit cells that compose the critical radius. As shown in Table 1, the estimated diameter of the critical nucleus corresponds to approximately 1–5 times the average linear dimension of a unit cell for all the API–solvent systems studied. For the API–solvent combinations considered, the nucleus is estimated to consist of approximately 1 to 125 unit cells, calculated as λ.3 For inorganic compounds, where λ was found to range from 2 to 8, the critical nucleus is estimated to consist of approximately 8 to 512 unit cells, which is notably larger than the corresponding values estimated for the organic compounds. In the case of small organic molecules such as L-arabinose and glycine (excluding the large biomolecule lysozyme), the critical nucleus was estimated to contain approximately 8 to 64 unit cells, again based on the cube of the corresponding λ values reported in Table 1. In the case of lysozyme, the estimated value of λ suggests that the critical nucleus diameter is approximately equal to the linear dimension of a single unit cell, indicating that the nucleus may be composed of only one unit cell. As previously noted, these estimates are approximate and derived from theoretical calculations, as no direct experimental evidence is currently available to validate these values for the systems studied. Nevertheless, the results show good alignment with prior literature in which similar nucleus sizes (that falls within the range of λ3 values calculated here) have been reported using both computational simulations and advanced experimental techniques such as in situ transmission electron microscopy. For example, Prado et al., used molecular dynamics simulations to estimate the size of the critical nucleus for a model inorganic compound, BaS, and reported similar values.52 In specific their studies shows that the critical nucleus size typically ranges from 5.9 × 10−10 to 6.5 to 10−10 m depending on the degree of supercooling. These values fairly agree with the ones that we obtained for most of the inorganic and organic compounds. Additionally, Nakamura et al., employed in situ transmission electron microscopy (TEM) to experimentally observe nucleation from amorphous phases and identified clusters composed of several unit cells forming during early nucleation stages.53 In specific the TEM results showed experimental evidence of the growth of unit cells containing 24 to 86 unit cells. While their work focused on amorphous-to-crystalline transitions, it provides conceptual support for the cluster sizes estimated in solution-phase crystallisation. In the specific case of lysozyme, which has the largest molecular volume among the systems studied, the calculated ΔG indicates a critical nucleus composed of approximately one unit cell. This observation is consistent with classical nucleation theory, which predicts that systems with large molecular volumes and high interfacial energy require larger energy barriers for nucleation but may only form stable nuclei at very small sizes due to packing constraints and limited supersaturation.

The higher number of unit cells estimated within the critical nucleus of inorganic compounds, compared to organic systems, can be explained by several key factors. First, inorganic crystals typically possess smaller and more symmetrical unit cells, owing to their simpler atomic or ionic structures. In contrast, organic compounds, particularly active pharmaceutical ingredients (APIs), often feature larger and more complex unit cells due to their flexible molecular frameworks and lower packing densities. As a result, for a given critical radius, more inorganic unit cells can be accommodated simply because each unit cell occupies less volume. Second, inorganic compounds tend to exhibit higher interfacial energies (γ), due to the stronger ionic or covalent bonding at the interface between the crystal and the surrounding phase. According to classical nucleation theory, a higher interfacial energy leads to an increased Gibbs free energy barrier and, consequently, a larger critical nucleus radius. When this larger critical radius is compared to the smaller unit cell dimension of an inorganic crystal, the number of unit cells required to form a stable nucleus is further amplified. Finally, the high packing efficiency and structural rigidity of inorganic lattices—featuring uniform, tightly arranged ions or atoms—contrasts with the relative disorder in organic systems, where flexible side chains, hydrogen bonding, and solvent inclusions are more common. This inherent structural order in inorganic systems may favour the formation of larger, more compact nuclei, again increasing the number of unit cells involved in the initial nucleation event.

Finally, we used our model as in eqn (7) to theoretically estimate two more parameters that are important to homogeneous nucleation, this includes in the number density or the number of molecules formed per unit volume, Nm/V when the time is equal to the induction time tind. According to Kubota, MSZW is assumed to be a point where the accumulated crystals that are grown up to detectable size had reached a fixed value, (Nm/V) when the time t = tind that can be related to the nucleation rate J by the relation:

 
image file: d5ce00467e-t21.tif(10)
Eqn (2) and eqn (10) can be combined to form:
 
image file: d5ce00467e-t22.tif(11)
Eqn (10) and (11) can be combined to theoretically calculate the tind as follows:
 
image file: d5ce00467e-t23.tif(12)
where, Nm/V can be obtained from eqn (10) as follows:
 
image file: d5ce00467e-t24.tif(13)
Eqn (11) to (13) allows us to theoretically calculate, tind, and Nm/V for each of the cooling rate provided we have the experimentally obtained information on the MSZW (i.e., ΔTmax) and theoretically obtained nucleation rate J (which can be obtained eqn (2) and (7)). From eqn (12) and (13), it can be understood that the induction time is defined as the ratio of the metastable zone width to the cooling rate. In this context, the induction time represents the duration required to observe nucleation at image file: d5ce00467e-t25.tif starting from image file: d5ce00467e-t26.tif. The calculated tind and Nm/V calculated using eqn (12) and (13) are given in Table 1. Clearly for all the APIs studied, at the MSZW, the Nm/V was found to be equal to 1024 to 1027 molecules per m3.

For demonstration purposes, in Fig. 3, we plotted the nucleation rate obtained using the classical nucleation theory as a function of temperature based on the ΔG values calculated using the model proposed as in eqn (7) for all the combinations of solute and solutions involved. It is worth mentioning here that, in this work we considered up to 10 different APIs whose molecular weight ranges from 73.09 to 421.3 g mol−1. For these APIs, according to the model proposed in this work and the nucleation temperature studied, the nucleation rate ranges from 1021 to 1024 m−3 s−1. In Table 1, we showed the theoretically obtained tind and ΔTmax for all the 11 different combinations of API/solvent systems. The Nm/V typically ranges from 1024 to 1027 molecules per m3 when time, t = tind. In the case of inorganic compounds, Fig. 3c shows that the nucleation rate (J) ranges from 1020 to 1023 m−3 s−1, with corresponding number densities (Nm/V) between 1023 and 1026 molecules per m3. Comparable values were observed for the API intermediate and glycine, where the predicted nucleation rates ranged from 1022 to 1023 m−3 s−1, and the corresponding Nm/V values were between 1026 and 1027 molecules per m3. Among all the systems studied, lysozyme, which has the largest molecular volume, exhibited the lowest nucleation rate, on the order of 1018 to 1019 m−3 s−1, with a corresponding number density of 1023 to 1024 molecules per m3. According to classical nucleation theory, such low nucleation rates are expected for large molecules, as the Gibbs free energy barrier (ΔG) for nucleation increases significantly with both molecular volume and interfacial energy. For lysozyme, the calculated ΔG was approximately 87 kJ mol−1, which is substantially higher than the ΔG values estimated for smaller organic molecules in this study. This outcome is consistent with the theoretical prediction that systems with larger molecular volumes or lower supersaturation ratios exhibit higher nucleation barriers and therefore lower nucleation rates.


image file: d5ce00467e-f3.tif
Fig. 3 Plot of nucleation rate versus the temperature for (a) 9 combinations of solvents/APIs involving 5 APIs, (b) 6 combinations of solvents/APIs involving 5 APIs, (c) combination of inorganics/solvents involving 8 different compounds and (d) API intermediate, amino acid and a large molecule. Fig. 3a: image file: d5ce00467e-u33.tif: dextrose/water (R′: 0.00278 K s−1); image file: d5ce00467e-u34.tif: L-asparagine monohydrate/water (R′: 0.005 K s−1); image file: d5ce00467e-u35.tif: L-asparagine monohydrate/water (R′: 0.0033 K s−1); image file: d5ce00467e-u36.tif: gestodene form I/ethanol (R′: 0.0083 K s−1); image file: d5ce00467e-u37.tif: gestodene form I/ethanol (R′: 0.005 K s−1); image file: d5ce00467e-u38.tif: carbamazepine_saccharine/ethanol (R′: 0.0167 K s−1); image file: d5ce00467e-u39.tif: Gestodene form I/ethanol (R′: 0.00167 K s−1); image file: d5ce00467e-u40.tif: pyrazinamide/acetone (R′: 0.00167 K s−1); image file: d5ce00467e-u41.tif: pyrazinamide/acetone (R′: 0.0033 K s−1). Fig. 3b: image file: d5ce00467e-u42.tif: ibuprofen/water (R′: 0.0167 K s−1); image file: d5ce00467e-u43.tif: paracetamol/water (R′: 0.5 K s−1); image file: d5ce00467e-u44.tif: paracetamol/ethanol (R′: 0.5 K s−1); image file: d5ce00467e-u45.tif: fenofibrate/ethyl acetate (R′: 0.005 K s−1); image file: d5ce00467e-u46.tif: psilocybin/water (R′: 0.0167 K s−1); image file: d5ce00467e-u47.tif: vismodegib/MIBK (R′: 0.0167 K s−1). Fig. 3c: image file: d5ce00467e-u48.tif: NaNO3/NaCl + NaNO3 + H2O solution (R′: 0.000833 K s−1); image file: d5ce00467e-u49.tif: NaNO3/NaCl + NaNO3 + H2O solution (R′: 0.001667 K s−1); image file: d5ce00467e-u50.tif: CoSO4/water (R′: 0.000833 K s−1); image file: d5ce00467e-u51.tif: CoSO4/water (R′: 0.001667 K s−1); image file: d5ce00467e-u52.tif: CoSO4/water (R′: 0.0025 K s−1); image file: d5ce00467e-u53.tif: (NH4)2B4O7·4H2O/water (R′: 0.00833 K s−1); image file: d5ce00467e-u54.tif: (NH4)2B4O7·4H2O/water (R′: 0.0025 K s−1); image file: d5ce00467e-u55.tif: Na3VO4/NaOH solution (R′: 0.005 K s−1); image file: d5ce00467e-u56.tif: Sr(OH)2·8H2O/water (R′: 0.001388 K s−1); image file: d5ce00467e-u57.tif: Sr(OH)2·8H2O/water (R′: 0.00277 K s−1); image file: d5ce00467e-u58.tif: ZnL2/water (R′: 0.00138 K s−1); image file: d5ce00467e-u59.tif: ZnL2/water (R′: 0.00275 K s−1); image file: d5ce00467e-u60.tif: borax/water (R′: 0.003472 K s−1); image file: d5ce00467e-u61.tif: borax/water (R′: 0.01066 K s−1); image file: d5ce00467e-u62.tif: ammonium oxalate/water (R′: 0.000833 K s−1 – plotted in secondary axis). Fig. 3d: image file: d5ce00467e-u63.tif: L-arabinose/water (R′: 0.0083 K s−1); image file: d5ce00467e-u64.tif: glycine/water (R′: 0.001667 K s−1); image file: d5ce00467e-u65.tif: glycine/water (R′: 0.005 K s−1); image file: d5ce00467e-u66.tif: lysozyme/NaCl solution (R′: 0.002778 K s−1 – we assumed this cooling rate – shown in secondary axis).

The theoretical model developed in this work offers a practical and physically grounded framework for extracting key nucleation parameters from metastable zone width (MSZW) data obtained under different cooling rates. Using only three experimentally measurable quantities, MSZW, nucleation temperature, and cooling rate, the model enables the prediction of critical nucleation characteristics, including the nucleation rate kinetic constant, Gibbs free energy of nucleation, induction time, and the number density of nuclei at the onset of crystallisation. A key strength of the model lies in its ability to capture the dependence of nucleation rate on cooling rate, in contrast to existing models (e.g., those by Nývlt, Kubota, and Sangwal), which assume a single, cooling-rate-independent nucleation rate derived from MSZW data. By directly incorporating classical nucleation theory, the proposed model provides a more realistic and adaptable tool for interpreting experimental results under varying process conditions. In addition to kinetic parameters, the model facilitates the estimation of important thermodynamic quantities such as surface free energy, critical nucleus radius, and the approximate number of unit cells constituting the stable nucleus. These estimates are derived by comparing the critical nucleus size to the cube root of unit cell volume, offering valuable molecular-level insight into nucleation behaviour. The model was validated using MSZW data from a diverse range of 22 solute–solvent systems, including 11 API–solvent combinations, 8 inorganic compounds, one API intermediate (L-arabinose), one amino acid (glycine), and one biomolecule (lysozyme). In most cases, the model fit the experimental data with coefficients of determination (r2) exceeding 0.97, confirming its robustness, generality, and predictive accuracy across a wide spectrum of chemical systems.

In terms of theoretical limitations, the model, by design, does not predict the order of nucleation, which is commonly estimated using alternative models such as those developed by Nývlt, Kubota, and Sangwal. As the proposed approach is based on classical nucleation theory, it does not assign a fixed reaction order but rather an Arrhenius-like temperature dependence. While classical nucleation theory predicts a first-order dependence on supersaturation under low-supersaturation conditions, at higher supersaturation, the theory implies a nonlinear and more complex relationship. If needed, an empirical nucleation order can still be inferred using a phenomenological expression such as J = k′(Δcmax)n; where k′ is a kinetic constant. In this case, based on the J obtained using eqn (7), the order of nucleation and the kinetic constant, k′ according to the power law type expression can be obtained from the linear expression: ln[thin space (1/6-em)]J = ln(k′) + n[thin space (1/6-em)]ln(Δcmax). It is worth to mention here that, the primary aim of this work was to develop a model capable of predicting the nucleation rate from MSZW data obtained at different cooling rates. Nucleation fundamentally depends on both supersaturation and induction time, both of which are directly influenced by the cooling rate. This is important because cooling rate is a key process variable that governs the properties of the final crystals. At higher cooling rates, the solution cools rapidly into the labile zone, leading to faster attainment of the critical supersaturation required for nucleation. In such cases, by the time nucleation occurs—after the induction period involving the formation of prenucleation clusters and their structural rearrangement into a stable nucleus—the solution achieves a higher supersaturation. This promotes a higher nucleation rate and results in a larger number of smaller crystals. Conversely, at lower cooling rates, the solution cools more slowly, reaching the induction point at a lower temperature and thus at a lower supersaturation. Here, the longer time allows molecular clusters to form and reorganise at lower driving forces, typically leading to fewer nuclei and larger crystals. This relationship between cooling rate, supersaturation, and nucleation rate is well established, and industries often exploit this by adjusting cooling rates to tailor crystal size and size distribution. In this context, the proposed model is particularly valuable as it can predict nucleation rates as a function of cooling rate. If experimental correlations between MSZW, cooling rate, and solubility are available, the nucleation rate predictions can be linked to the resulting crystal population characteristics. This makes the model useful for designing batch crystallisation processes to achieve targeted product attributes. In continuous crystallisation, knowledge of nucleation rate at different cooling rates is even more critical. The nucleation rate directly determines the production rate (JV) and thus the required residence time to reach a target suspension density or productivity. Higher J values at a given cooling rate reduce the necessary residence time, improving process efficiency. Furthermore, understanding how J varies with cooling rate is essential for designing washout-safe operations, as insufficient nucleation rates relative to dilution rates can lead to washout of nuclei from the crystalliser. Overall, the proposed model is a simple yet useful for analysing nucleation phenomena using MSZW data observed at different cooling rates. Its broad applicability and minimal data requirements make it particularly valuable for both research and industrial crystallisation processes, where understanding and controlling nucleation is essential.

Notation

c*Solubility concentration (molecules per m3) (see eqn (3))
image file: d5ce00467e-t27.tif Solubility concentration at nucleation temperature (molecules per m3)
image file: d5ce00467e-t28.tif Solubility concentration at reference temperature (molecules per m3)
ΔcmaxSupersaturation at metastable zone width (molecules per m3) image file: d5ce00467e-t29.tif (see eqn (5))
ΔGCritical Gibbs free energy of nucleation (kJ mol−1) (see eqn (1))
J Nucleation rate (molecules per m3) (see eqn (2))
k n Nucleation rate constant (molecules per m3 s) (see eqn (1))
image file: d5ce00467e-t30.tif Number density (molecules per m3) (see eqn (10))
r c Critical radius (m) (see eqn (9))
R Gas constant, (J mol−1 K−1) (see eqn (1))
R′Cooling rate (K s−1) image file: d5ce00467e-t31.tif (see eqn (3))
S Supersaturation image file: d5ce00467e-t32.tif (dimensionless) (see eqn (8))
t time (s)
t ind Induction time (s) (see eqn (12))
T Solubility temperature (K) (see eqn (3))
T nuc Nucleation temperature (K) (see eqn (6))
image file: d5ce00467e-t33.tif Reference temperature (K)
ΔTmaxMetastable zone width (K) image file: d5ce00467e-t34.tif (see eqn (5))
V cell Unit cell volume (m3)
γ Interfacial surface energy (mJ m−2) (see eqn (8))
ϑ Molecular volume (m3) (see eqn (8))

Data availability

The ESI† file includes the experimental data, calculation procedures, model equations, and graphical outputs for four representative systems: pyrazinamide, glycine, borax, and lysozyme. Data and calculations for the remaining systems discussed in the manuscript, along with the corresponding theoretical parameters, are available from the authors upon reasonable request.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We acknowledge the financial support of the Science Foundation Ireland (Grant 12/RC/2275, 12/RI/2345/SOF and 18/SIRG/5479). M. V. would like to acknowledge the Bernal Institute, Boston Scientific, Department of Chemical Sciences, and the University of Limerick Foundation for the funding support through the mULtiply program.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5ce00467e
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