The role of the pre-exponential factor in determining the kinetic selection of polymorphs during solution crystallization of organic compounds

Abstract

Generally, pairs of polymorphs can be characterized by their ratios of equilibrium solubilities and interfacial energies (γ_st/γ_me) for a given temperature and solvent. We refer to this point as the solubility-interfacial energy characteristic point (characteristic point for short) of a polymorphic pair. The equations of the classical nucleation theory have been used to determine the influence of supersaturation, the absolute size of the interfacial energies and the ratio of the pre-exponential factors for pairs of polymorphs to predict the experimental conditions in which metastable or stable polymorphs crystallize first. Domain diagrams for polymorph pairs based on the equilibrium solubility ratios and the ratio of interfacial energies (γ_st/γ_me) have been developed. Separate zones are identified where the metastable and stable polymorphs are favoured kinetically; generally higher supersaturation kinetically favour the metastable form. This contribution investigates the circumstances where large values for the pre-exponential factor, particularly for the metastable polymorph, in the classical nucleation theory description of nucleation can expand the zone where the metastable zone is kinetically favoured. The results indicate that the pre-exponential factor has a strong influence in expanding the kinetically metastable zone when the interfacial energies of the metastable and stable polymorphic are low (less than 3.5 mJ m⁻²) but has little or no effect when these values are high (greater than 5.5 mJ m⁻²). This work also identifies the circumstances where a metastable polymorph with a higher interfacial energy than the stable polymorph will crystallize first.

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Introduction

This paper follows on from ref. 1 which looked at the 95% probability that the thermodynamic solubility ratio between pairs of polymorphs is less than 2 fold.^2–4 There are exceptions to this observation, for example in the case of ritonavir, where a 4–5 fold difference in solubilities has been reported for selected solvents.⁵ In parallel with this work Nyman and Day⁶ and, independently, Cruz-Cabeza, Reutzel-Edens and Bernstein⁷ demonstrated that the free energy and lattice energy differences between pairs of polymorph rarely exceed 5 kJ mol⁻¹ with differences extending to 10 kJ mol⁻¹ for pairs of conformational polymorphs.

Ref. 1 took as its starting point the calculation of critical free energies of nucleation for pairs of polymorphs using the classical nucleation theory:⁸

Where

N_a is Avogadro's number, γ is the interfacial energy, v_m is the molecular volume, k is the Boltzmann constant, T is the temperature in Kelvin and S is the supersaturation ratio. Inclusion of N_a in eqn (1) allows the critical free energy of nucleation to be expressed as J mol⁻¹. A consistent outcome was that as the ratio of equilibrium solubilities approaches 2 then the value of the supersaturation with respect to the metastable form (S_me) is typically half the supersaturation with respect to the stable form (S_st).¹ This has profound effects on the value of for each of the pairs of polymorphs. Generally, for values above 2 it is easy to find circumstances where is less than . This work also illustrated that can is less than at low supersaturations (but still supersaturated with respect to both polymorphs) and when the ratio of γ_st/γ_me is low. This approach explains the numerous literature reports which specify that high supersaturation favour the formation of the metastable polymorph and low supersaturations favour the stable form, where in each case the system was supersaturated with respect to both polymorphs.^9–22 These results were interpreted in the light of the single nucleation event hypothesis whereby one particle of a particular polymorph forms and is propagated throughout the solution via a secondary nucleation mechanism.^23–28 Ultimately, that work led to the development of a series of domain diagrams by identifying for any selected value of supersaturation the combinations of and γ_st/γ_me at which equals .¹

A limitation of the work in ref. 1 was that it was based on the determination of critical free energies of nucleation, rather than the full classical nucleation equation which determines nucleation rates namely:

This paper addresses this problem and in addition it explores a larger range of interfacial energies than was possible in ref. 1.

Eqn (2) comprises two parts, namely the exponential term, which basically ranges from 0 to 1 and a pre-exponential term which can take on any value. There is a good deal of literature regarding the nature of the pre-exponential factor.^29–35 According to Li et al.³¹ reducing the interfacial energy or enhancing supersaturation does not increase the nucleation rate as effectively as reducing the kinetic barrier. Dimensional analysis of eqn (2) would indicate that A and J should have the same units. For J this is usually expressed as the number of nuclei of a size greater than the critical nucleus size generated per unit volume and per unit time (no. of stable nuclei per m³ s⁻¹). On this basis, A may be defined as the total number of nuclei (or clusters) of any size generated per unit volume and per unit time (no. of pre-critical and stable nuclei per m³ s⁻¹). The exponential factor is the fraction of these pre-critical nuclei which can advance to the critical size.

There is a limited amount of literature data which records interfacial energies and pre-exponential factors for various crystallizations of organic compounds. Table 1 presents the influence of solvent on these parameters for tolbutamide,³⁶ salicylic acid,³⁷ and risperidone³⁸ which clearly show a solvent effect in determining the values of the pre-exponential factors and interfacial energies. Two studies on curcumin³⁹ and 3-nitrophenol⁴⁰ demonstrate the effects of impurities on the pre-exponential factors and interfacial energies and finally there are 4 studies which show how these factors change for polymorph pairs, namely eflucimibe,¹¹D-mannitol,¹² mefenamic acid,⁴¹ and famotidine.¹³

Table 1 A selection of literature values for pre-exponential factors and interfacial energies for a range of organic compounds

Compound	Polymorphic form	Solvent	Pre-exponential factor A (m^−3 s^−1)	Interfacial energy γ (mJ m^−2)	Ref.
a DMA: dimethyl acetamide; DMC: demethoxy curcumin; BDMC: bisdemethoxy curcumin. b These are B values.
Tolbutamide	Form I^L	Acetonitrile	15.6	1.25	36
		Ethylacetate	23.3	1.90
		n-Propanol	11 210	3.99
		Toluene	220	3.46
Salicylic acid		Chloroform	57	0.71	37
		Ethyl acetate	148	1.82
		Acetonitrile	289	2.40
		Acetone	8645	3.81
		Methanol	586	4.13
		Acetic acid	175	5.50
Risperidone		Cumene	348	1.72	38
		Toluene	181	1.70
		Acetone	161	1.77
		Ethyl acetate	71	1.58
		Methanol	134	2.18
		1-Propanol	129	2.25
		1-Butanol	61	2.04
Curcumin		2-Propanol	659	4.45	39
		2-Propanol with 0.1 mM DMC^a	113	4.70
		2-Propanol with 0.1 nM BDMC^a	165	5.01
3-Nitrophenol		Toluene	4.8 × 10^8	5.1 ± 1.3^b	40
		Toluene with 0.25 mol% 3-aminobenzoic acid	1.3 × 10^5	3.7 ± 1.0^b
Eflucimibe	Form B metastable	Ethanol : n-heptane (7 : 3)	118	4.23	11
	Form A stable		14	5.17
D-Mannitol	Form δ metastable	Water	610	1.78	12
	Form β stable		3000	3.23
Mefenamic acid	Form II metastable	40% DMA^a–60% water	1324	2.92	41
	Form I stable	70% DMA^a–30% water	160	2.86
Famotidine	Form B metastable	Water	10^9	14.36	13
	Form A stable		1.25 × 10^4	9.16

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Perusals of the data in Table 1 indicated that ratio of pre-exponential factors for the various crystallizations encountered rarely exceed 10³. This range of ratios cover the effect of solvent (rows 1–3, Table 1), the effects of impurities (rows 4 and 5) and the effect of polymorph selection (rows 6–8). The single exception is in the case of famotidine (row 9) where the ratio A_me/A_st is just less than 10⁵. This work reports that the interfacial energy is 14.36 mJ m⁻² for the metastable form of famotidine and 9.16 mJ m⁻² for the stable polymorph. This case will be discussed further below but as a working hypothesis we will proceed on the assumption that the 10³ ratio is attainable and should be considered as a reasonable upper ratio limit for this study.

Methods

The present study seeks to explore further the role of the pre-exponential factor in eqn (2) in determining the kinetic outcome (namely the first polymorph in a pair which will crystallise). The work does not explore the role of polymorphic transformations neither in the solid state nor through a solution mediated process. Clearly, there is a temporal aspect to polymorphism and if the metastable polymorph crystallizes first it will transform into the stable form on a timescale which can vary from milliseconds to years. The approach taken here has been to seek the conditions whereby the nucleation rate for the metastable polymorph (J_me) is equal to the nucleation rate of the stable polymorph (J_st) eqn (3). Throughout these calculations the pre-exponential term of the stable polymorph (A_st) was set at 1 and the value of A_me required to achieve J_me = J_st was calculated.

J_me = J_st (3)

whenand where A_st = 1An upper value of 10¹⁰ was set the A_me; where larger values were encountered these are reported in the tables below as >10¹⁰.

For this work a range of scenarios was explored principally for ratios between 1.2 and 4.0; the range of interfacial energies (γ_st and γ_me) was 2–11 mJ m⁻² and the range of supersaturations with respect to the stable form (S_st) was 3–8. Generally, the results are presented below as values of A_me needed to satisfy eqn (5), namely the conditions in which the nucleation rate of the metastable (J_me) and stable polymorphs (J_st) are equal.

Results

Tables 2–4 present the values of A_me needed to satisfy eqn (5) in circumstances where the interfacial energies for each pair of polymorphs is set at low values (2–2.7 mJ m⁻²) for Table 2, set at intermediate values (4–5.5 mJ m⁻²) for Table 3 and set at high values (8–11 mJ m⁻²) for Table 4. The range explored as 1.2 to 4.0 with set values of supersaturation with respect to the stable polymorph (S_st) of 3.5 and 5.5. In these tables' circumstances where γ_st = γ_me, γ_st > γ_me and γ_st < γ_me were explored.

Table 2 For interfacial energies in the range 2.0–2.7 mJ m⁻² the calculated minimum value of A_me at which J_me = J_st (eqn (5))

S st	γ st	γ me	Min. Ame at = 1.2	Min. Ame at = 1.5	Min. Ame at = 2.2	Min. Ame at = 3	Min. Ame at = 4
			S me = 2.91	S me = 2.33	S me = 1.59	S me = 1.17	S me < 1.0
3.5	2.7	2.0	0.84	0.96	2.2	2.7 × 10^4	n/a
3.5	2.7	2.7	1.15	1.60	13	>10^10	n/a
3.5	2.0	2.7	1.46	2.00	16.5	>10^10	n/a

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S st	γ st	γ me	Min. Ame at = 1.2	Min. Ame at = 1.5	Min. Ame at = 2.2	Min. Ame at = 3	Min. Ame at = 4
			S me = 4.58	S me = 3.67	S me = 2.5	S me = 1.83	S me = 1.38
n/a not applicable because Sme < 1.
5.5	2.7	2.0	0.90	0.94	1.1	1.6	9.4
5.5	2.7	2.7	1.05	1.16	1.7	4.3	3.4 × 10^2
5.5	2.0	2.7	1.20	1.32	1.9	4.9	3.8 × 10^2

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Table 3 For interfacial energies in the range 4.0–5.5 mJ m⁻² the calculated minimum value of A_me at which J_me = J_st (eqn (5))

S st	γ st	γ me	Min. Ame at = 1.2	Min. Ame at = 1.5	Min. Ame at = 2.2	Min. Ame at = 3	Min. Ame at = 4
			S me = 2.91	S me = 2.33	S me = 1.59	S me = 1.17	S me < 1
3.5	5.5	4.0	0.21	0.59	5.7 × 10^2	>10^10	n/a
3.5	5.5	5.5	3.4	49	>10^10	>10^10	n/a
3.5	4.0	5.5	25.6	372	>10^10	>10^10	n/a

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S st	γ st	γ me	Min. Ame at = 1.2	Min. Ame at = 1.5	Min. Ame at = 2.2	Min. Ame at = 3	Min. Ame at = 4
			S me = 4.58	S me = 3.67	S me = 2.5	S me = 1.83	S me = 1.38
n/a not applicable because Sme < 1.
5.5	5.5	4.0	0.40	0.55	1.9	45	7.0 × 10^5
5.5	5.5	5.5	1.6	3.6	1.0 × 10^2	2.2 × 10^5	>10^10
5.5	4.0	5.5	4.7	10.7	3.0 × 10^2	7.0 × 10^5	>10^10

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Table 4 For interfacial energies in the range 8.0–11.0 mJ m⁻² on the calculated minimum value of A_me at which J_me = J_st (eqn (5))

S st	γ st	γ me	Min. Ame at = 1.2	Min. Ame at = 1.5	Min. Ame at = 2.2	Min. Ame at = 3	Min. Ame at = 4
			S me = 2.91	S me = 2.33	S me = 1.59	S me = 1.17	S me < 1
3.5	11.0	8.0	3.9 × 10^−6	1.6 × 10^−2	>10^10	>10^10	n/a
3.5	11.0	11.0	1.7 × 10^4	>10^10	>10^10	>10^10	n/a
3.5	8.0	11.0	>10^10	>10^10	>10^10	>10^10	n/a

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S st	γ st	γ me	Min. Ame at = 1.2	Min. Ame at = 1.5	Min. Ame at = 2.2	Min. Ame at = 3	Min. Ame at = 4
			S me = 4.58	S me = 3.67	S me = 2.5	S me = 1.83	S me = 1.38
n/a not applicable because Sme < 1.
5.5	11.0	8.0	6.4 × 10^−4	8 × 10^−3	1.6 × 10^2	>10^10	>10^10
5.5	11.0	11.0	3.7 × 10^1	3 × 10^4	>10^10	>10^10	>10^10
5.5	8.0	11.0	2.3 × 10^5	>10^10	>10^10	>10^10	>10^10

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When the interfacial energies for the metastable and stable form are both low the metastable form seems to be kinetically attainable at S_st = 5.5 for all values explored. No values of A_me/A_st greater than 10³ are ever required to arrive at equal nucleation rates for the metastable and stable polymorphs, even in circumstances where γ_st < γ_me. A very similar results pertains when S_st = 3.5, except we can identify very high values of A_me/A_st required to attain equal nucleation rates when the value of is 3 and above. In this table the value of S_me calculated as is presented for each entry to illustrate the extent to which the supersaturation with respect to the metastable polymorph reduced dramatically as the ratio increases.

When the interfacial energies for the pair of polymorphs take on intermediate values (4–5.5 mJ m⁻²) very large values of A_me/A_st are required when S_st = 3.5 to satisfy eqn (5) for all values of at and above 2.2. However, at S_st = 5.5 more reasonably achievable values of A_me/A_st (<10³) are required at = 2.2 and 3. This trend is again consistent with our earlier observation that the metastable form becomes more kinetically favoured as the applied supersaturation increases.¹

When the highest interfacial energies (8.0–11.0 mJ m⁻²) very large values of A_me/A_st are required when S_st = 3.5 and 5.5 to satisfy eqn (5) for all values of above 2.2.

As γ increases through the ranges 2.0–2.7, 4.0–5.5 and 8.0–11.0 mJ m⁻² the minimum value of A_me needed to generate identical nucleation rates of the metastable and stable polymorphs, i.e. J_me = J_st with A_st = 1 increases very significantly. For Tables 2–4 the reader should be aware that the values are those calculate to J_me = J_st, namely the value at which both polymorphs would be expected to appear concomitantly. To attain “clean” metastable polymorphs which would be expected if a characteristic point were placed firmly within the metastable zone would require significantly higher values of A_me/A_st.

As reported previously¹ higher supersaturations favour the metastable polymorph so that smaller values of A_me are generally required at S_st = 5.5 than the corresponding values at S_st = 3.5.

The results presented in Tables 2–4 are further confirmed when the range of interfacial energies is allowed to vary in the range γ_me = γ_st ± 3 mJ m⁻² (Table 5). Unreasonably high values of A_me (greater than 10³) are required for J_me to become equal to J_st at S_st equal to 3.5 and 5.5 and where γ_st is set at 11.0 kJ m⁻².

Table 5 The influence of γ_me = (γ_st ± 3) mJ m⁻² on the calculated minimum value of A_me at which J_me = J_st (eqn (5))

γ st	γ me	S st	Min. Ame at = 2.2	S st	Min. Ameat = 2.2
2.7	0.35	3.5	0.7	5.5	0.8
	0.7		0.7		0.8
	1.7		1.4		1.0
	2.7		13		1.7
	3.7		1.4 × 10^3		5.5
	4.7		4.1 × 10^6		41
	5.7		>10^10		8.9 × 10^2
5.5	2.5	3.5	0.4	5.5	0.3
	3.5		23.5		0.95
	4.5		3.3 × 10^4		5.3
	5.5		>10^10		1.0 × 10^2
	6.5		>10^10		5.4 × 10^3
	7.5		>10^10		1.4 × 10^6
	8.5		>10^10		>10^10
11.0	6.0	3.5	4.8 × 10^2	5.5	2 × 10^−3
	7.0		>10^10		0.3
	8.0		>10^10		1.7 × 10^2
	9.0		>10^10		6.2 × 10^5
	10.0		>10^10		>10^10
	11.0		>10^10		>10^10
	12.0		>10^10		>10^10

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When γ_st is set at 5.5 kJ m⁻² and S_st is equal to 3.5, reasonable values of A_me (10³ or less) according to Table 1 are only attainable in circumstances where γ_me < γ_st. For the same set of circumstances where S_st = 5.5 a reasonably attainable value of A_me can only be achieved when γ_me = γ_st.

When γ_st is set at 2.7 kJ m⁻² and S_st is equal to 3.5 and 5.5 reasonable values of A_me (10³ or less) according to Table 5 are attainable in most circumstances except where γ_me ≥ (γ_st + 2) mJ m⁻².

Taken together Tables 2–5 demonstrate that as γ_st increases we need higher and higher values of A_me to make J_me = J_st, i.e., A_me is less effective at expanding the metastable zone at high values of γ_me, and accordingly γ_st. In addition, Table 5 demonstrates that when γ_st is less than γ_me, J_me becomes equal to J_st only in circumstances where very high ratios of A_me/A_st are possible.

Of particular interest to this paper is to add the influence of A_me/A_st to the domain diagram presented as Fig. 7 of ref. 1 where ratios of equilibrium solubilities were plotted against the ratio of interfacial energies (γ_st/γ_me) for a range of S_st values. The outcome was a series of supersaturation lines in the range S_st = 2–8 representing the combinations of and γ_st/γ_me for which the critical free energies of nucleation are equal for pairs of polymorphs. For the purposes of the domain diagrams, each pair of polymorphs is characterized by the ratio of equilibrium solubilities and the ratio of interfacial energies (γ_st/γ_me). Together these ratios are referred to as the characteristic point for a pair of polymorphs and will vary depending on temperature and solvent choice. Where the characteristic point lies below the domain line for a particular supersaturation, the metastable polymorph is favoured kinetically, namely it will be the first polymorph to appear. If the characteristic point lies above the domain line the stable form will be favoured kinetically.

The assumption in ref. 1 was that A_me = A_st = 1 that the critical free energy of nucleation alone determined the nucleation rates, i.e. if then J_st = J_me. In Fig. 1–3 below domain diagrams are constructed which include ratios of A_me/A_st in the range 1–1000. Briefly when the nucleation rate of each of the pair of polymorphs are equal as expressed in eqn (4) and (5) the value of A_st is set at 1 and the value of A_me varied from 1–1000.

Fig. 1 The influence of the size of the pre-exponential factor A_me/A_st on the domain diagram for pairs of polymorphs with fixed values of γ_me set at 3.5 (A), 5.5 (B) and 8 (C) mJ m⁻² at a fixed value of S_st = 5.0.

Fig. 2 The influence of the size of the interfacial energy (γ_me) set at 3.5, 5.5 and 8 mJ m⁻² at a fixed ratio of A_me/A_st = 1 and 1000, on the expansion of the domain diagram for pairs of polymorphs at a fixed value of S_st = 5.0.

Fig. 3 The influence the ratio of A_me/A_st for supersaturations in the range S_st = 3–8 on the expansion of the metastable zone when γ_me = 5.5 mJ m⁻².

Expanding eqn (5) leads to eqn (6):

A_me exp(−16πN_aγ_me³v_m²/3k²T²ln²S_meRT) = exp(−16πN_aγ_st³v_m²/3k²T²ln²S_stRT) (6)

where, and

Substituting the values of N_a = 6.023 × 10²³ mol⁻¹, v_m,me = v_m,st = 4 × 10⁻²⁸ m³ per molecule, K = 1.38 × 10⁻²³ m² kg s⁻² K⁻¹, T = 293 K, R = 8.3142 J mol⁻¹ K⁻¹, S_st = 3, 5, and 8, γ_me = 3.5, 5.5, and 8 mJ m⁻², A_me = 1, 10, 100, and 1000, and S_me as defined earlier will help to determine the value of γ_st. These calculations allow us to plot against the ratio of interfacial energies (γ_st/γ_me) for a range of selected supersaturation (S_st) and to assess the influence of A_me on the location of the domain line provided that the term S_me does not fall below 1 at which point the solution would be undersaturated with respect to the metastable polymorph.

An assumption of this work is that is always greater than (essentially the working definition of the metastable and stable phases). The extent of the domain along the y-axis is determined by the applied supersaturation (S_st) and where = S_st. So then S_st is set at 5 the maximum value of is also 5 at which point the value of S_me is equal to 1. The purpose of the domain diagram is to determine the zones where the metastable and stable forms are favoured kinetically, i.e. the forms which appear initially when both are supersaturated. Hence, the absolute maximum value of interest to this work is where = S_sti.e. where the value of S_me is 1 and so cannot crystallize. Hence, the extent of the combined domains along the y-axis is 1 < = S_st. Along the x-axis the ratio of γ_st/γ_me cannot be less than zero. Equally, it would be unrealistic to set the upper limit of γ_st/γ_me at greater than 4.

Fig. 1A–C illustrates the situation where the domain diagram is presented for the range in the range 1–4. With the value of the interfacial energy for the metastable polymorph (γ_me) set at 3.5, 5.5 and 8.0 mJ m⁻², the required value of the interfacial energy of the stable polymorph (γ_st) required to satisfy eqn (5) is calculated and expressed as the ratio of interfacial energies (γ_st/γ_me). One domain line in each of Fig. 1A–C represent the situation where A_me = A_st = 1. The other lines represent the situations where A_me = 10, 100 or 1000. The examples shown in Fig. 1A–C set the supersaturation with respect to the stable polymorph (S_st) at 5.

Fig. 1A illustrates that for low values of interfacial energies (γ_me = 3.5 mJ m⁻²) the domain within which the metastable form is kinetically favoured increases very significantly as the value of A_me increased over the range 1–1000. In fact at the highest A_me value applied (A_me = 1000) the kinetically favoured metastable zone nearly covers the entire domain. In such circumstances, the combination of low interfacial energies with high A_me/A_st ratios does allow the for ratios in excess of 2. According to ref. 1 just 5% of polymorph pairs fall into this category.

Fig. 1B and C illustrate the situation where the value of the interfacial energy of the metastable form increases to 5.5 and 8 mJ m⁻², respectively while allowing A_me/A_st to vary from 1 to 1000. For the higher values of interfacial energies the value of A_me needed to expand the kinetically favoured metastable zone increases significantly. At γ_me equal to 5.5 mJ m⁻² there is a modest expansion of the kinetically favoured zone for the highest value of A_me modelled whereas for γ_me equal to 8.0 mJ m⁻² the expansion of the kinetically favoured zone for the metastable polymorph is very small. The expansion of the kinetically favoured metastable zone in these diagrams is entirely consistent with data in Tables 2–5 which demonstrate the need for higher values of A_me/A_st as the interfacial energies applied increase.

Overall, the pre-exponential factor plays a dominant role in determining the relative sizes of the kinetically favoured zones in the domain diagrams then the interfacial energies are low but have a diminishing to negligible effect as the interfacial energies increase.

Fig. 2 summarises this situation further: here the original domain line is reproduced. This line is almost coincident when γ_me is in the range 3.5–8 mJ m⁻², S_st = 5 and A_me = A_st = 1. The corresponding domain lines for γ_me equal to 3.5, 5.5 and 8 mJ m⁻² with A_me = 1000 are combined in this figure and clearly show that the expansion of the metastable favoured kinetic zone depends strongly on the absolute value of γ_me applied and by association with eqn (6), the absolute value of γ_st. The expansion of the metastable zone is most pronounced at low values of γ_me.

In all circumstances where γ_st < γ_me the only possible circumstances where the metastable form is favoured kinetically is if A_me ≫ A_st. In Fig. 2 this situation is illustrated for γ_st/γ_me in the range 0.5–1. Above γ_st/γ_me = 1 a zone emerges where the metastable form is favoured kinetically. When γ_st/γ_me < 1 the stable phase only is favoured kinetically when A_me = A_st = 1. The metastable form starts to be favoured kinetically when A_me becomes greater than A_st by a significant amount. The metastable zone is hardly attainable according to Fig. 2 when γ_me is equal to 11 mJ m⁻², but becomes readily attainable for γ_me values of 3.5 and 5.5 mJ m⁻². For example, the characteristic point for a pair of polymorphs where γ_st/γ_me = 0.8 and is equal to 2 falls in the stable zone when A_me = 1 but is in the metastable zone when A_me = 10³ where γ_me is equal to 3.5 or 5.5 mJ m⁻². This is the situation outline in Table 1 for famotidine.¹³ In that case the reported value of γ_st = 9.16 mJ m⁻² and γ_me = 14.36 mJ m⁻², i.e. γ_st/γ_me = 0.64. The reported ratio of is less than 1.3. In these circumstances we would predict that the metastable form becomes kinetically attainable provided that the ratio of A_me/A_st is high. The reported ratio is 10⁹/1.25 × 10⁴ = 10⁵. Both polymorphs can be attained; the kinetically favoured metastable form and the stable form which can be formed through a solution mediated transformation into the stable form.

Fig. 3 presented the influence of applied supersaturation. Three values of S_st are considered, namely, 3, 5 and 8 and the value of γ_me is set at 5.5 mJ m⁻². The original domain lines where A_me = A_st = 1 are presented for each supersaturation. Superimposed are the corresponding domain lines when A_me = 10³ is applied at each supersaturation. At all supersaturations applied there was a significant expansion of the zone where the metastable form of the polymorph pair was kinetically favoured. The extend of the expansion was greater at interfacial energies γ_me less than 5.5 mJ m⁻² and less for higher values of γ_me.

Conclusions

Looking then at the overall conclusions of the influence of the pre-exponential factor in determining the relative sizes of the kinetically favoured metastable and stable zones for a polymorph pair, we can conclude that low values of interfacial energies lead to very significant expansion of the kinetically favoured zone for the metastable polymorph; the extent of the expansion depends on the ratio of A_me/A_st. The higher interfacial energies explored in this work lead to small and sometimes insignificant expansion of the kinetic metastable zone. The general conclusion of ref. 1 still stands, namely that for a given polymorph pair, higher supersaturations kinetically favour the metastable form although the extent of the expansion is similar for all supersaturation explored in the range S_st = 3–8. The observation that above 2 are rarely encountered for polymorph pairs still stands although the kinetically favoured metastable zone does expand when the ratio of A_me/A_st increases. The attainment of above 2 in a limited number of polymorph pairs may well be possible in circumstances where A_me/A_st is large. This work has also expanded the application of the domain diagrams and explains how metastable polymorphs are kinetically favoured in circumstances where γ_me > γ_st, namely through the occurrence of very high A_me/A_st values.

Conflicts of interest

There is no conflict to declare.

Acknowledgements

Results incorporated in this standard have received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 101026339. KH thank the Science Foundation Ireland (SFI) for support (award Grant Numbers 12/RC/2275_P2).

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