Ostwald Ratio and the Appearance of Metastable Phases · Web view(31)Kitamura, M., Polymorphism in the crystallization of L-glutamic acid. Journal of Crystal Growth 1989, 96, (3),

The Ostwald Ratio, Kinetic Phase Diagrams and Polymorph Maps.

Peter T Cardewa and Roger J Daveya*

aSchool of Chemical Engineering and Analytical Sciences, University of Manchester,

M13PL, UK.

* Email: [email protected]

Abstract

From a general formulation of batch crystallisation in a polymorphic system it is shown

that the concept of an Ostwald Ratio provides a fundamental parameter with which to

understand and compute the variation in polymorph composition with time,

temperature and supersaturation. In particular, it defines the conditions under which a

system will exhibit sequential development of polymorphs, as per the widely known

Ostwald’s Rule of Stages and conversely when either only the stable polymorph will

result or when forms appear simultaneously resulting in concomitant polymorphism.

The Ostwald Ratio is not a constant but varies with temperature, the initial

supersaturation, crystal shape and density. It is shown that its evaluation can lead to a

kinetically based phase diagram enabling the construction of polymorph maps in

compositional and temperature space. A review of relevant existing kinetic data reveals

a lack of information concerning both the relative nucleation and growth rates of

polymorphic phases. The specific case of gestodene is used as a means of demonstrating

the significance and potential of this approach to the robust design of batch

crystallisation processes in polymorphic systems.

Keywords: Polymorphism, concomitant, crystallisation, nucleation, growth, Ostwald’s

Rule, gestodene.

Introduction

The occurrence of crystal polymorphism influences many fields of materials chemistry

e.g. pharmaceuticals, dyestuffs, pigments, electro-optic materials, explosives1, 2. In 19853

the first paper concerning the kinetics of solvent-mediated phase-transformations

1

between polymorphs was published. The utility of the kinetic approach was illustrated

using published data for the pigment copper phthalocyanide3. At that time such a

dataset was unique since the importance of polymorphism within the pharmaceutical

sector had hardly been recognised. Indeed it was arguably the Zantac patent cases of the

early 1990s that focussed the industry on the real importance of polymorphism in its

solid form products. The active component of this remedy, ranitidine hydrochloride,

was found to be polymorphic and from 1991 onwards became the subject of a number

of high profile patent infringement litigations2. Since then there has been an increasing

and consistent interest in this field with its impact in the pharmaceuticals sector having

been particularly figural: regulatory approval is required not only for a drug as an active

ingredient but also for the specific polymorph to be isolated and used in formulated

products4. This arises not least from the fact that at any given temperature and pressure

the various polymorphs available to a system will have different relative

thermodynamic stabilities, a factor which can influence bioavailabilities and lead to

phase transitions both during processing and upon storage. Over recent years this has

resulted in an increase in interest in the relationship between experimental

crystallisation conditions and the observed polymorphic outcome. The literature

suggests that two particular issues are of relevance. Firstly, from an experimental and

process development perspective, it is important to appreciate the operating factors

(including purity and solvent choice) that control the appearance of different forms.

This underpins our ability to design consistent and robust processes5 and in the future

will enhance the utility of crystal structure prediction as a product design tool.

Secondly, in our scientific quest to understand the molecular scale processes

surrounding nucleation, polymorphic systems have often been used as potential tools to

link solution chemistry to the state of aggregation in solution and hence to the

nucleation pathway itself6.

Historically, this area of endeavour began with the work of Ostwald and his elucidation,

from significant experimental data, of his Rule of Stages7. This simply states that, if a

metastable solution is allowed to crystallise, the solid phases will appear in order of

their increasing stability starting with the most metastable and transforming until the

most stable state is reached. This Rule remains of importance today because it is simply

the only heuristic that exists to guide experimental crystallisation scientists in their

exploration of polymorphic systems8-11. Attempts to give theoretical substance to the

2

Rule began in 1933 with the work of Stranski and Totomanov12, who used the (then)

recently developed nucleation theory (now called ‘Classical Nucleation Theory, CNT) to

discuss the problem of polymorph appearance in a system of two polymorphs. Volmer

reconsidered this in his 1939 book ‘Kinetik der Phasenbildung’13. More recently, Keller

et al14, Davey15, Bernstein et al16 and Verma and Hodnett17 returned to this issue, again in

the context of CNT. In this previous work, Ostwald’s Rule of Stages was seen as a

sequential process of polymorphic crystallisations determined solely by the nucleation

kinetics of the potential phases. However, it is readily apparent that such a narrow view

ignores the fact that the original Rule was based on macroscopic observations and

hence also reflects the potential role played by the relative growth rates of the available

polymorphs. It appears that only the very recent work of Black et al has recognised this

and given an interpretation of the outcome of crystallisation in polymorphic systems by

including not just relative nucleation rates but also the crystal growth kinetics of each

form18. This work, which built on an earlier study19 , utilised the simplest assumption of

first order crystal growth kinetics for both phases with their solubility ratio at or close

to 1 to show that a ratio of kinetic factors involving both nucleation rates and growth

rate constants plays an important role in determining the macroscopic development of

the metastable phase during a batch crystallisation. By referring to this ratio (Equation

1) as the Ostwald Ratio, ΦOs, B, they made a fitting tribute to Ostwald’s pioneering work

on crystallisation:

ΦOs, B=( N2∗ρ2∗ϕ2∗k23

N1∗ρ1∗ϕ1∗k13 ) (1)

where for each phase, j:  N j is the number of nuclei initially formed, ρ j the crystal

density,  ϕ j the shape factor (it is implicitly assumed that the growth shape is conserved

during this process) and  k j the growth rate constant. This model predicts that the

larger the value of the Ostwald Ratio (hereafter OR) the greater the proportion of

metastable phase that develops. There is no doubt that the implementation of a growth

rate model which is first order in supersaturation, while mathematically convenient,

does not reflect the many real situations where higher order or more complex rate laws

are observed20. One reason for this is that the growth of small crystals (< 10 m) is notμ

limited by volume diffusion to the surface but by surface kinetics which may be

3

controlled by surface nucleation or utilise structural defects such as screw dislocations.

In either case the relationship between growth rate and supersaturation can be

described by a non-linear function 21. Additionally, it is noted that the model was limited

to polymorphs with roughly equal solubilities. By definition, the solubility ratio,

metastable/stable phase, has to be a number greater or equal to one and a review of

polymorphic systems in drugs found that such ratios ranged from ~1 to over 23 (most

commonly < 2) so that there is good reason to explore the impact of this in more

detail22.

This paper is concerned with developing a theoretical framework for the batch

crystallisation of polymorphs which generalizes Equation 1 to enable the inclusion of

non-linear growth kinetics, polymorph solubility ratios, initial supersaturation and

nuclei size. One of its key objectives is to identify conditions under which Ostwald’s Rule

may provide a useful guide and when it is likely to fail.

The Ostwald RatioGeneral Theory

The following analysis is built around the limiting case that nuclei are formed on time-

scales that are very much faster than the time-scales for growth. In essence, the crystal

nuclei are all spontaneously generated att=0 and will be assumed monodisperse with

their initial size small relative to their final potential size. For simplicity only two phases

will be considered, a metastable polymorph 2 and a stable form 1. Over time these

phases will grow and compete for supersaturation. This competition period comes to an

end when the supersaturation with respect to the metastable phase is zero (i.e. the

solution composition reaches the solubility of the metastable phase). Further growth of

the stable polymorph will cause dissolution of the metastable phase in a solution

mediated process leading to a product of the pure stable form 3. A transition point (the

subscript p will be used to denote values at this point) may thus be defined as the time

at which the period of competitive crystallisation ends and the subsequent period of

solution mediated transformation starts. The current analysis is solely concerned with

the initial competitive period of crystallisation, leading up to this transition point, when

the solution is supersaturated with respect to both the metastable and stable phases

which nucleate and grow.

4

The crystal growth rate equations for this two phase system (1 is the stable and 2 the

metastable phase) with crystal sizes r1 and r2, growth rates R1 and R2 are

dr 1

dt=R1(σ1)

dr 2

dt=R2(σ2)

(2)

where σ j is the supersaturation of phase j (defined as σ j=(c−c j , e )/c j , e, with c and c j ,e the

actual and saturation concentrations respectively). The supersaturation for phase 1 and

phase 2 are not unique but are linked and it is readily shown that

σ 2=σ1−σ x

1+σ x(3)

In supersaturation terms, σ x is the driving force for transforming phase 2 into 1 ((

σ x=(c2 , e−c1 ,e )/c1 ,e ) and is related to the solubility ratio, Sx, of the forms by

Sx=1+σ x (4)

Where appropriate the subscript 1 will be dropped (i .e . σ1becomes σ) in the following

development. Completion of the growth equations requires the mass balance linking the

supersaturation to the crystal sizes:

ce σ=ce σ i−N 1∗ρ1∗ϕ1∗r13−N2∗ρ2∗ϕ2∗r2

3 (5)

Here N j is the number of nuclei per unit volume of phase j, ϕ j is the shape factor for

phase j, ρ j is the density of phase j, σ i is the initial supersaturation with respect to the

stable phase and ce is the solubility of the stable phase, 1.

It is useful to normalise the crystal sizes:

5

u j=r j

r j , max(6)

where r j , x is defined by

r j ,ma x≡( σ j ,i c j , eN j ρ jϕ j

)1 /3 (7)

r j ,max is the maximum size that phase j crystals will attain in the absence of the other

phase given an initial supersaturation σ i. With these definitions the supersaturation,

equation (5), can be expressed in the simpler form using the normalized crystal sizes, ui

σσ i

=1−u13−βu2

3(8)

where

β≡σ2 ,i c2 ,e

σ1 ,i c1 ,e=σ i−σ x

σ i(9)

From which it follows that β ranges from 0 to 1, becoming unity both when Sx equals

one and in the high supersaturation limit.

Using the scaled normalized crystal sizes (equation (6)) in equation (2) and normalizing

the growth rates with respect to their initial values gives

du1

dt= 1τ1

~R1(σ1)

du2

dt= 1τ2

~R2(σ2)

(10)

where two natural time constants τ1 and τ 2 are defined for each phase by the equation

6

τ j≡r j , maxR j ,i

=( σ j ,i c j , eN j ρ jϕ jR j , i

3 )1/3

(11)

where R j ,i is the initial growth rate, R j (σ j , i )and

~R j(σ j)≡R j (σ j )R j , i

(12)

is the normalised growth rate. These time constants reflect the natural characteristic

time-scales for the independent development of each phase. Their ratio gives the

dimensionless constantΦOs:

ΦOs=( τ1

τ 2 )=( r1 , maxR2 ,i

r2 , maxR1, i )=( 1βN2∗ρ2∗ϕ2∗R2 ,i

3

N1∗ρ1∗ϕ1∗R1 ,i3 )

1 /3

(13)

When τ1 << τ2 (i.e. ΦOs is small) the stable phase, 1, will dominate the crystallisation

outcome. When the converse is true (i.e. ΦOs is large) it would be expected that the

metastable phase 2 will dominate and Ostwald’s Rule of Stages will hold. It is noted that

ΦOs depends on both the relative number of nuclei (N2/N1) and the relative growth rates

(R2,i/R1,i) of the two phases. As will be shown later ΦOs , as defined by equation (13), is

the generalisation of the Ostwald Ratio (OR) originally introduced by Black et al18 for

the special case of linear growth kinetics and without the 1/3 exponent. This new

derivation shows that the ratio has a much wider applicability to polymorphic

crystallisations and, through the concept of the time constants, provides a physically

intuitive reason as to why this is the case.

While there is no analytical solution to the growth equations (10) further analysis

provides additional insights and offers a basis for obtaining useful approximations.

Taking the ratio of the pair of growth equations (10) gives the first order non-linear

differential equation:

du2

du1=ΦOsG (σ ) (14)

7

where the normalized relative growth rate function, G, is defined

G (σ )≡~R2 (σ2 )~R1 (σ1 )

=R2 (σ2 )R1 ,i

R2 ,iR1 (σ1 )(15)

Equation (14) shows explicitly that the sizes of metastable and stable crystals evolve

together in the normalised size space of u1 and u2 . Important conclusions can be drawn

from equation (14) when it is combined with the mass balance equation (9). In

particular the relationship between u1 and u2 is determined solely by the OR, the initial

normalised crystal sizes, (u1*, u2

*), the initial supersaturation, σ i, the solubility ratio

between the two phases,   Sx=1+σ x and the functional form of the crystal growth

kinetics.

In an unseeded system the relative number of nuclei of polymorphs 1 and 2 is

proportional to their initial nucleation rates, J1 ,i and J2 ,i , and hence the OR may be

written:

ΦOs=( 1βJ 2 ,i∗ρ2∗ϕ2∗R2, i

3

J 1 ,i∗ρ1∗ϕ1∗R1 , i3 )

1/3

(16)

Since both nucleation and growth rates vary with the initial supersaturation so does the

OR. For primary nucleation the rate can be described by the CNT equation

J=A exp( − Bln2 (1+σ ) ) (17)

where A and B are kinetic and thermodynamic constants respectively. ( Alternative

forms of this nucleation rate equation are in use6 which include an additional

supersaturation dependency in the pre-exponential constant. This has little impact on

the analysis presented here and can easily be accommodated if required.)

Polymorph Composition

In the context of existing and potential experimental data a particularly useful

parameter to consider is the polymorph composition of isolated product (e.g. the mass

8

fraction of crystalline material in the metastable form), χ2, which is related to the

normalized crystal sizes:

χ2=N2 ρ2ϕ2 r2

3

N 1 ρ1ϕ1 r13+N 2 ρ2ϕ2 r2

3 =βu2

3

u13+ βu2

3 =1− 11+βu2

3/u13

(18)

Of particular interest is the value that χ2 achieves at the end of the competitive period

i.e. its value, χ2 , p, at the transition point.

In the limit σ x /σ i=0 the relative normalized growth rate function tends to unity if the

two phases have the same growth law. In this limiting case equation (14) is readily

solved to give

u2−u2¿=(u1−u1

¿)ΦOs (19)

If the initial nuclei sizes (u1¿, u2

¿ ) are small compared to their sizes at the end of the

competition period (u1(t p), u2(t p)) it follows that

u2(t p)∼u1(t p)ΦOs (20)

Hence the composition at the transition point is given by

χ2 , p∼1− 11+βΦOs

3 =βΦOs

3

1+βΦOs3 (21)

In the absence of any general analytical solution the coupled growth equations

(equation (10)) have been solved numerically. To this end the R package deSolve was

used for all calculations23. The sensitivity of the crystal composition, χ2 , p , to the OR was

explored with particular consideration given to the influence of the initial nuclei sizes,

the initial supersaturation, the solubility ratio of the two phases and the functional form

of the growth kinetics.

The relationship between the composition of crystalline product, χ2 , p and OR is shown in

Figure 1 where β=1. The model predicts that for a system with ΦOs < 0.1 the product

9

will consist mainly of the stable phase while for a value > 10 it will follow Ostwald’s

Rule and comprise essentially the metastable phase. If the OR has a value around 1 then

the system will crystallise concomitantly16 as a mixture of both polymorphs. It is clear

from this that Ostwald’s Rule, rather than being generally true, is a special case which

applies only at large values of OR. In general, the outcome is a feature of both the

nucleation and growth rates of the polymorphic phases. As a corollary it is also true that

because a system obeys the Rule this does not imply anything, per se, about the relative

nucleation or growth rates of the individual polymorphs.

Figure 1. Variation in polymorph composition, χ2 , p with the Ostwald Ratio.

Equation (21) provides a generalization of equation (7) in Black et al18. It predicts that

the condition for concomitancy is Φ 1/ β1/3. Given that for most practical situations β

will be close to unity, an OR of 1 is a good indicator of concomitant behavior. The

sensitivity of this equation to the initial nuclei size, the initial supersaturation, the

solubility ratio and the functional form of the growth kinetics are explored in the

following section.

Crystal Growth Kinetics.

In seeking to generalise this analysis it is noted that, as discussed above, measured

crystal growth kinetics are often characterised by a power law expression20 i.e.

R j (σ )=k j σn j (22)

10

In such cases the relative normalized growth rate function and OR are

G(σ)=( σ−σxσ i−σ x

)m

( σ iσ )n

(23)

ΦOs=1β1 /3

1S x

m ( J 2 ,i∗ρ2∗ϕ2∗k23

J 1 ,i∗ρ1∗ϕ1∗k13 )

1 /3 (σ i−σ x )m

σ in

(24)

where for clarity, the exponents for polymorphs 1 and 2 are n=n1 and m=n2 . The

power law model will be referred to by (n, m). Inspection of the relative growth rate

function (eq 23) shows that for power laws it is only a function of the two ratios σ /σ i

and σ x /σ i. (For further details see S1).

More complex growth rate-supersaturation relationships can be considered such as

those relating to spiral growth (Burton et al24) or to surface nucleation (Gilmer and

Bennema25) but, as seen in S2, they create additional complexity which for a general

understanding is not required.

The formula for OR given by Black et al18 was restricted to first order kinetics (R=kσ)

with a solubility ratio of 1(i.e.σ x=0and hence β=1). Under these conditions equation

(24) becomes

limσ x→0

ΦOs=¿ ( J 2∗ρ2∗ϕ2∗k23

J 1∗ρ1∗ϕ1∗k13 )

1 /3

=[ΦOs , B ]1 /3 ¿ (25)

This equation shows that as expected the limit defaults to that given by Black et al 18

with the exception of the exponent. The exponent is an issue of convention. Defining the

Ostwald Ratio in terms of the time constants seems more intuitive and will be the

convention adopted in this paper.

In a further simplification it is often implicitly assumed26, 27 that, together with densities

and shape factors, the ratio of growth rate constants are all unity. In such cases the

crystal phase composition is determined by nucleation kinetics alone. With these

assumptions the OR (equations (25)) simplifies to

11

ΦOs3=J 2/J 1 (26)

from which the metastable mass fraction at the transition point becomes

χ2 , p 1− 11+βΦOs

3 =1− 1

1+J2

J1

=J 2

J1+J2 (27)

Supersaturation Effects.

At low supersaturations up to the equilibrium solubility of the metastable phase, the

Ostwald Ratio is zero since both the nucleation rate and growth rate of the metastable

phase are zero. As the supersaturation increases the OR will increase. Close to its

solubility the nucleation rate of the metastable phase will generally be the fastest

changing function with supersaturation so that the OR is largely dictated by the

nucleation kinetics of the metastable polymorph.

At high supersaturations the nucleation rate achieves its maximum at the nucleation

rate constant A (equation (17)). In the case that the crystal growth kinetics are of the

same order then OR must approach a constant

limσ i→∞

ΦOs=ΦOs∞ ≡ 1

Sxn ( A2∗ρ2∗ϕ2∗k2

3

A1∗ρ1∗ϕ1∗k13 )

1 /3

(28)

It follows that for high initial supersaturations the composition is solely determined by

the limiting OR, ΦOs∞ . If the growth kinetics of the phases have unequal powers then at

high supersaturation the situation is more complex. If the metastable phase is of higher

order than the stable phase (m>n) then at sufficiently high supersaturations the OR will

become large and favour the development of the metastable phase. If the stable phase is

of higher order (n>m) then the converse is true and the stable phase will dominate.

12

Exploring the impact of the Ostwald Ratio on polymorph appearanceFirst order growth kinetics

In this section the numerical solution of equation (10) is examined for the simple case of

first order growth kinetics with Table 1 providing conditions and numerical results for

selected cases. Figures 2 and 3 show how the various parameters of interest evolve for

two different cases in which the time constants have been chosen to give systems with

relatively low (Case 1 ; 5.0) and high values (Case 6; 0.2) of ΦOs respectively. Each figure

is made up of four charts. Thus for Case 6, Figure 3a and 3b show the temporal

evolution of the normalised crystal size of each phase and the resulting proportion of

crystalline material in the metastable phase, χ2, respectively. As expected for the low OR

case the stable phase 1 is dominant with both its crystal size and phase proportion

outstripping those of the metastable form. For the system with a high OR, Case 1,

Figures 2a and 2b show, as expected, the reverse with the metastable phase dominating.

In both cases the phase composition changes extremely rapidly from its initial value but

thereafter changes very slowly. Figures 2c and 3c show, not surprisingly, how the rate

of change in supersaturation is very small at first before gathering pace as the crystals

grow. A marked difference between the two cases can be seen at their relative transition

points which, as expected, are reached much more rapidly when the stable phase

dominates. Finally, Figures 2d and 3d show how the system evolves in normalised

crystal size space from the initial values (open circle) to the transition point (filled

circle). As can be seen in both cases the trajectory is virtually a straight line to the

transition point.

13

Figure 2. Four charts showing the evolution of size, composition and supersaturation for Case 1, ΦOs = 5.0. The dashed red line marks the boundary between the period of competitive growth of the two phases and the start of the region where solution mediated transformation between the metastable to the stable phase occurs i.e. the transition point.

14

Figure 3. Four charts showing the evolution of size, composition and supersaturation for Case 6, ΦOs = 0.2. The dashed red line marks the boundary between the period of competitive growth of the two phases and the start of the region where solution mediated transformation between the metastable to the stable phase occurs i.e. the transition point.

15

Consideration of Table 1, for the case of the linear growth law, illustrates the sensitivity

of the product crystal composition to the OR, initial supersaturation, initial crystal size

and the solubility ratio. Inspection of the data suggests that the initial nuclei size has

little bearing on the competition period or the final crystalline composition. A special

case occurs at the point of equilibrium between two polymorphs where σ x=0 and here

the composition is given exactly by equation (21). In this case there is only parallel

development of the two polymorphs. A limiting case occurs when σ x /σ i→0. In this limit

the competition period tends to infinity and as with the special case the composition

tends to that given by equation (21).

16

Table 1. Conditions and numerical results for selected cases with first order kinetics.

Cas

e

Normalised Size Initial

Supersaturation

Solubility

Ratio

Kinetic

Time

Constants

Ostwald

Ratio

Competition

Period

Mass

fraction

polymorph

2 at

transition

point

Fig.

Ref

u1 u2 σ i Sx τ1 τ 2 ΦOs t p χ2 , p

1 1e-03 1e-03 1 1.20 1.

0

0.2 5.0 0.54 0.977 Fig 2

2 1e-04 1e-02 1 1.20 1.

0

0.2 5.0 0.54 0.977 -

3 1e-02 1e-04 1 1.20 1.

0

0.2 5.0 0.54 0.974 -

4 1e-03 1e-03 1 1.01 1.

0

0.2 5.0 0.79 0.991 -

5 1e-03 1e-03 10 1.20 1.

0

0.2 5.0 0.74 0.990 -

6 1e-03 1e-03 1 1.20 0.

2

1.0 0.2 0.26 0.005 Fig 3

7 1e-04 1e-02 1 1.20 0.

2

1.0 0.2 0.26 0.006 -

8 1e-02 1e-04 1 1.20 0.

2

1.0 0.2 0.26 0.005 -

9 1e-03 1e-03 1 1.01 0.

2

1.0 0.2 0.47 0.008 -

10 1e-03 1e-03 10 1.20 0.

2

1.0 0.2 0.42 0.007 -

Power law growth kinetics.

For the case that both phases exhibit the same power law then as the ratio σ x /σ i tends

to zero the relative growth rate function approaches unity for all supersaturations

greater than σ x. This is not the case when the growth rates of different polymorphs

follow different power laws. For example, (see S1 for details) if the metastable phase

follows a linear law and the stable phase a quadratic law then as supersaturation falls

during growth the dominance of the metastable form will increase until the

supersaturation reaches twiceσ x. In the opposite case, where the metastable form

17

follows a quadratic growth law and the stable phase a linear growth law, the relative

growth rate function will show a uniform rate of descent between the initial and final

supersaturation as σ x /σ i tends to zero.

For the cases when the power law exponents are greater than 1 and different, more

complex limiting behaviour is observed. If the metastable phase has a higher growth

rate exponent than the stable phase (m>n) then its growth will win over the stable

phase (Ostwald’s Rule). Equally, in the reverse case (n>m) the stable phase will

dominate. If, however, the exponents of two phases are of the same order then as the

supersaturation increases a mix of the two phases will form (concomitancy). If m > n

then the relative growth rate function increases with increasing initial supersaturation

increasing the maximum yield of metastable phase, while when m < n the opposite

effect will be observed. Figure 4 summarises this behaviour, showing the relationship

between the yield of polymorph 2 and the OR for various power law combinations in

the limit (σ x /σ i=0). It is apparent that in moving away from the linear law situation

there is a slight shift and increasing width to the yield curve as the order difference

increases. (See S2 for results with other growth kinetic models).

Figure 4. The metastable yield, χ2 , p as a function of the Ostwald Ratio for selected growth power law relationships in a system with a solubility ratio of 1. The respective power laws of forms 1 and 2 are written as (n, m).

18

The impact of the Solubility Ratio and the Initial Supersaturation

As already pointed out the growth rate function depends on two ratiosσ /σ i and σ x /σ i.

This can be readily seen by rearranging the growth rate expression in equation (23) to

give

G(σ)=(σσ i

−σ x

σ i

1−σ x

σ i)m

( σ iσ )n

(29)

Here the solubility ratio, Sx appears only in the magnitude of (σ x /σ i)since

(Sx−1 )/σ i=(σx /σ i). The ratio σ x /σ i , can only vary between 0 and 1. Thus, the effect of

both the solubility ratio and the initial supersaturation can be determined by selecting

values in this range. To this end the differential equations (10) were solved for three

selected values of the ratio (0, 0.5 and 0.9). The results, Figure 5, show that as (σ x /σ i)

increases the metastable polymorph yield falls slightly for any given value of the OR.

However, even when (σ x /σ i) reaches 0.9 the impact on the metastable yield curve is

modest. (see S3 and S4 for further comment).

Figure 5. The metastable yield, χ2 , p , as a function of the Ostwald Ratio for selected values of σ x/σ i in a system with first order kinetics.

19

The impact of the initial size of the nuclei

Examination of cases 1, 2, 3 and 6, 7 and 8 in Table 1 indicates that there might only be a

slight dependence on nuclei size. This is confirmed in Figure 6 which shows that

increasing the normalized nuclei size from 10-6 to 10-1 produces only minor changes in

the metastable yield, χ2 , p.Hence for a given value of OR, unless the final size of nuclei is

a large fraction of the initial nuclei size it has minimal impact on χ2 , p. (See S4 for further

details.)

Figure 6. The metastable yield χ2 , p as a function of the Ostwald Ratio for selected nuclei sizes.

Experimental dataThe foregoing analysis has shown that the OR is determined by the product of four

ratios of polymorphic properties of the two phases, namely their nucleation kinetics,

growth kinetics, crystal shapes and crystal densities. Thus, in principle, values for all

four ratios are needed to calculate the OR. Table 2 summarises literature reports

(restricted to molecular materials) in which data on competitive crystallisation in

polymorphic systems is available. Fortunately, the density ratio of the different

polymorphs is invariably close to 1 and hence its influence is negligible. Information on

crystal shapes is restricted to qualitative descriptors such as needles, plates, rhombs

etc.

20

As seen in Table 2 it is frequently the case that Ostwald’s Rule is followed only at high

supersaturations while at intermediate supersaturation both polymorphs crystallise

and low supersaturations yielding the stable polymorph. To date, there appear to be no

complete studies in which both the nucleation and growth kinetics for two polymorphs

have been determined in full. In the majority of these examples only the identification of

the polymorphs appearing at different supersaturations and temperatures has been

reported. In cases where kinetic data are available the focus has nearly always been on

nucleation. Of the systems in Table 2 only the study of gestodene27 includes data for

both nucleation and growth kinetics of both forms. Even here the growth rate data are

limited to one concentration at two temperatures. This enantiotropic system, which has

a transition temperature at about 18.5°C, exhibits nucleation rates (measured via

induction times) which for the high temperature stable phase I decrease with increasing

temperature and for the low temperature phase II increases with increasing

temperature. Also the growth rate (measured on single crystals) of phase II is higher

than that of phase I at high temperatures while the inverse is true at low temperatures.

21

Table 2. Crystallisation studies of polymorphic systems in molecular crystals.Compound Polymorphs Shape/

density/relative density

Nucleation vs growth

Result

Eflucimibe26

No structures recorded

A and B. EnantiotropicA stable above -4C.

A needleB needleCrystal structures unknown

Only nucleation considered

Relative nucleation rate JA/JB decreases with and fraction σof B and increases with .σ

Famotidine28

FOGVIG01FOGVIG02

A and B. Monotropic A stable formconformational

A rod 1.599B rod 1.5810.989

Only nucleation

Low , AσHigh , Bσ

BPT PropylEster29

YENZUN01

A and BB stable

A needle 1.215B prism 1.2280.989

No kinetic data

Low , BσHigh , Aσ

Tolfenamic acid30

KAXXAIKAXXAI01

I and II EnantiotropicI stable above 0C

I needle 1.400II needle 1.3970.998

No kinetic data

Low , IσHigh , IIσSolvent dependentConcomitant at intermediate .σ

Gestodene27

FUXMOA01FUXMOA02

I and IIEnantiotropicI stable above 18.5C

I plate 1.177II needle 1.1890.990

Nucleation and growth

T>19 low , IσT<17 low , IIσConcomitant at higher . σRelative J as f(T) estimated.15C high , Gσ I>GII

At low , Gσ II > GI

Reverse at higher T.L-Glutamic Acid31

LGLUAC03LGLUAC11

and .α βMonotropic.

stable.β

prism 1.532α needle 1.576β

0.972

No kinetic data

Concomitant at 45C.Only at 25.α

L-Histidine32

LHISTD02LHISTD10

A and B.Monotropic.A stable.

A plate 1.441B plate 1.4350.996

Growth rates as a function of solvent composition.

Concomitant at low and σethanol volume fractions.GA > GB

D-Mannitol33

DMANTL07DMANTL08DMANTL10

, and α δ βMonotropic

stable.β

needle 1.496α needle 1.505δ needle 1.516β

0.987; 0.993

Nucleation rates

High , σ αIntermediate , σ δLow , σ βOccasional concomitancy

p-Aminobenzoic Acid18 34, 35

AMBNAC07AMBNAC08

and .α βEnantiotropic.

stable above α14C

needle 1.420α rhomb 1.417β

0.998

Nucleation and growth of αNo nucleation rates for .β

All favour except in water.σ α Gα >> Gβ.

o-Aminobenzoic Acid36

AMBACO01AMBACO05

I and IIEnantiotropicI stable below 81 C

I prisms 1.409II plate/needle 1.372O.974

Growth rate data

High , IσLow , IIσConcomitant at intermediate .σ

In the next section this example of gestodene is used as the basis for exploring the

importance of the relative kinetic data in determining both the OR and the related

polymorphic outcome.

22

Gestodene

In the earlier sections it was shown that the polymorph composition is highly

dependent on the value of the Ostwald Ratio which in turns varies with the initial

supersaturation and temperature (and pressure) (eq 16). In this section the example of

gestodene (Table 2) crystallising from ethanolic solutions is used to provide a real

context for the application of the OR and to highlight what is required to understand

crystallisation in polymorphic systems. It is clearly shown that growth kinetics can

significantly influence the outcome. With this in mind the subsequent calculations are

based on the solubility, nucleation and growth rate data for gestodene extracted from

the work of Zhu et al27. The density ratio of the two phases is 0.99 (Table 2) and has

negligible influence when compared to the magnitude of the nucleation and growth rate

terms. The crystal shapes of the two forms are described as plates (I) and needles (II)

and in the absence of any quantitative data the shape factor ratio will be taken as unity.

Calculations were made for three cases:

1. When the growth rate constants of the two phases are assumed to be identical.

2. When the growth rate constants of the two phases are as reported (Figure 7)

3. The growth rate constants of the two phases are interchanged.

The first of these cases equates to the assumption, often made6, 17, 18, that the

polymorphic outcome mirrors solely the relative nucleation rates of the two forms.

Indeed, in their original paper Zhu et al. used this assumption to attempt to predict the

outcomes based solely on their measured induction time data which they modelled with

the CNT equation (17). Within the analytical framework developed here they implicitly

assumed that the ratios of the growth kinetics, the solubility, the densities and the shape

factors for the two phases are all unity so that the OR simplifies to that given by

equation (26) from which the metastable mass fraction at the transition point becomes

the ratio of the nucleation rate of the metastable phase to the total nucleation rate

(equation (27)).

Cases 2 and 3 explore the role of relative growth kinetics in the final product

composition

23

The first stage in the calculations is to evaluate OR as a function of solute concentration

and temperature. To this end the nucleation kinetics were abstracted from Figure 9 in

reference27 (nucleation rate curves for both forms at 7 different temperatures 5, 10, 15,

20, 25, 30, 35°C) and these were used to estimate values for the temperature

dependence of the coefficients A and B (in equation (17)) for both forms (see S5). In

order to calculate the supersaturation as a function of the initial solute concentration

the solubility of each of the two phases was reconstructed from the solubility diagram

for gestodene in ethanol given in Figure 5a of reference27. This was done by using the

solubility of the two polymorphs at three temperature points, 2.5, 35 and 18.5°C (the

polymorphic transition temperature of the two phases) and fitting these to the van’t

Hoff equation:

c=coexp [ ∆HR ( 1T

− 1T o )] (30)

with c the solubility, T the temperature, To the enantiotropic transition temperature

(0K), co the solubility of both phases at the transition temperature (18.5°C), ∆ H the

enthalpy of dissolution and R the ideal gas constant. The available crystal growth rate

data for gestodene constitute kinetic measurements for the two phases at one

concentration and two temperatures (15 and 20°C). These data are inadequate to create

a realistic model for the growth kinetics of gestodene. However, they are perfectly

adequate for the purposes of demonstrating the potential impact that growth kinetics

can have on the crystal phase composition. By making the simple assumption of linear

kinetics for both phases and temperature dependencies that follow an Arrhenius

expression, a relationship for the growth kinetic constant as a function of temperature

is obtained as shown in Figure 7. As can be seen, the growth rate constants of phases I

and II span respectively 5 and 3 orders of magnitude in the temperature range 5 to

35°C, being equal at about 19.5°C which is surprisingly close to the transition

temperature (18.5°C). (See S5 for more details). These data allow the calculation of the

Ostwald Ratio ΦOs using equation (31) which is derived from equation (24) for first

order growth kinetics viz.

24

ΦOs=1β1 /3

1S x ( J 2∗ρ2∗ϕ2∗k 2

3

J 1∗ρ1∗ϕ1∗k13 )

1/3

(31)

Figure 7. The temperature dependence of the crystal growth rate constants of gestodene

polymorphs.

In the second stage the crystal phase composition at the transition point, χ2 , p is

calculated from OR by using equation (21) which in this instance provides an excellent

approximation (see S5 for further details). This enables the creation of a kinetic phase

diagram or polymorph map of the crystal phase composition, χ2 , p, as a function of both

initial concentration and temperature.

Figure 8 records the results of both sets of calculations for the three cases showing how

both the OR and the polymorph outcome vary with initial solute concentration and

temperature. In this case, because gestodene is enantiotropic the OR has been

calculated throughout with respect to the low temperature stable phase, II. It should be

noted that when the supersaturation is low (i.e. at solution compositions close to the

solubility curves) the OR is always close to zero for temperatures less than the

transition temperature and very large for temperatures above the transition

temperature. One general point worth noting from Figure 8 is that as the

supersaturation increases there comes a point when the outcome switches from stable

to the metastable form via a region where the product is concomitant. How the system

25

behaves and how wide such a concomitant region is seen to depend on the relative

nucleation and growth kinetics.

Figures 8(a1,a2) show the result of case 1 where the growth kinetics of the two phases

are assumed equal and the outcome is determined only by the relative nucleation rates,

as per equation (27). As expected the polymorph appearing at low supersaturations is

the stable phase while at high supersaturations the situation is completely reversed

with the metastable phase appearing at all temperatures. A very narrow concomitant

region is predicted between the phases and where these narrow lines cross close to the

transition temperature a slightly larger region is predicted where a mix of the two

polymorphs is observed.

The impact of including growth kinetics, cases 2 and 3, is substantial. Figure 8(b2)

shows that for case 2, while it remains true that the stable form always appears at low

supersaturations, its region of appearance is much more restricted below 18.5oC (Form

II stable) with the concomitant region around the transition temperature also much

reduced. On the other hand for case 3 the concomitant region extends to high

supersaturations and covers a wide temperature range (Figure 8(c2)) and it becomes

impossible to crystallise the pure metastable form. This difference can be understood

on the basis that in case 2 the growth kinetics of the two phases align with the

nucleation kinetics while in case 3 they work against each other. These three examples

show how the compositional outcome is critically dependent on the individual growth

kinetic constants. Since these constants are independent of each other and of the

nucleation data it is clear that they all have to be known in order to explain and predict

systems where more than one phase forms.

The computed polymorph maps can be compared to the experimental results of Zhu et

al27 as reflected in their Figure 5. Essentially they found that at low supersaturations the

stable form always appeared at all temperatures. This is indeed predicted by all three of

the computed cases. At high temperatures (27-35oC) they found that this region of pure

Form I extended to all compositions tested. At lower temperatures, however they found

concomitancy at all higher supersaturations (x > 0.004) with no region where the

respective metastable form could be isolated. This experimental outcome is only

consistent with the Case 3 prediction, Fig 8(c2), in which the estimated growth rate

constants have been reversed. The reason for this outcome is unclear but may be

26

underpinned by the limited growth rate data available and to the assumption of primary

nucleation. If, for example one form nucleated heterogeneously on the surface of the

other37 then the large observed concomitant region could result. Nevertheless it shows

very definitely that considering nucleation alone does not generate the experimental

situation and reinforces the conclusion that growth kinetics cannot be ignored.

27

Figure 8. OR (a1,b1,c1) and polymorph maps (a2,b2,c2) for (a) Case 1, nucleation only (eq

29), (b) Case 2 nucleation and growth, (c) Case 3, nucleation and growth.

28

Finally it is worth pointing out that the outcomes revealed through these computed

polymorph maps (Figures 8 a2, b2, c2) are in stark contrast to the trivial Ostwald’s Rule

outcome, shown in Figure 9, for which, incorrectly, the metastable phase is the only

predicted product for all conditions.

Figure 9. The polymorph map for gestodene according to Ostwald’s Rule of Stages.

Discussion and conclusionsA new analysis has been presented here, based on the kinetic processes of nucleation

and crystal growth, offering the simplest complete description for polymorph evolution

during the period of competitive crystallisation in a batch experiment. This has yielded

a new parameter, the Ostwald Ratio, with which to understand and predict

crystallisation outcomes in polymorphic systems. For simplicity, this has been

developed for a system of two polymorphs despite the fact that the appearance of more

polymorphs during crystallisation is not uncommon e.g. isonicotinamide38, d-mannitol39.

It is readily seen, however, that the approach developed here is extendable though it

would add further complexity.

It is important to appreciate that the OR is not a constant but varies with the initial

supersaturation and temperature (and pressure). At low initial supersaturations its

29

magnitude is dominated by nucleation while at high initial supersaturations a limiting

value is determined by the relative limiting nucleation and growth rates of the two

phases. A general finding for monotropic systems is that a concomitant region16 occurs

when the OR is about one. When the ratio is much greater than one the system will

become dominated by the metastable phase and sequential development of phases

occurs as per the Ostwald Rule of Stages. For small values of the Ostwald Ratio the

system will be dominated by the stable phase and Ostwald’s Rule will not apply. (S5.1

includes an equivalent discussion for enantiotropic systems where OR is defined with

respect to the low temperature stable phase. As a consequence in the high temperature

region the metastable phase dominates if 0 < OR<<1.)

One important outcome of this new kinetic analysis has been to show that Ostwald’s

Rule has no general validity. From an experimental perspective, according to the

literature data shown in Table 2, the same conclusion may also be drawn. In many

systems the polymorph to appear first will depend on supersaturation and will not

necessarily be the metastable form. Indeed it is apparent that among these recently

reported data Ostwald’s Rule only applies at high supersaturation provided OR >> 1. It

is also the case that most reports of crystallisation in polymorphic systems appear to

focus on nucleation kinetics26, 28 with scant attention paid to crystal growth. Very few

studies have taken into account both nucleation and growth and indeed only one study

has been found where kinetic measurements of both nucleation and growth have been

measured for both phases27.

In the derivation of the OR presented here the polymorph nucleation and growth

processes have been modelled using CNT together with a general power law

formulation for growth kinetics. While the approach can be readily applied (see S5) to

specific growth mechanisms, such as spiral growth24 or birth and spread surface

nucleation40, much of the simplicity of the analysis is lost. At the same time this analysis

removes the restriction18, 19 that the solubility ratio of the two phases is close to 1 and

that the growth kinetics are linear.

The application of this Ostwald Ratio approach to scoping out the domains of

polymorph appearance (polymorph maps) has significant ramifications for

experimental and process design. It allows not only the identification of operating

envelopes for the isolation of desired forms but also enables the experimentalist to test

30

the sensitivity of the product form to fluctuating conditions. This application has been

demonstrated through analysis of the data reported by Zhu et al for the crystallisation

of the enantiomorphic system of gestodene27. Using their thermodynamic and kinetic

data the pure phase and concomitant regions of the phase diagram have been computed

and of particular note is the dependence of the predicted extent of the concomitant

region on the kinetic data used. Thus, moving from the sole use of nucleation kinetics to

include the best estimate of growth rate constants leads to a reduction in the

concomitant zone. Switching the relative growth rate constants yields a very large

expansion of the concomitant zone and suggests that isolation of pure metastable forms

will not be possible. This example serves to demonstrate the predictive potential of this

overall concept in aiding process or experimental design and also highlights the need

for kinetic data for both nucleation and growth.

NomenclatureA j -nucleation kinetic coefficient for phase j (see eq. 17)

B j - nucleation kinetic coefficient for phase j (see eq. 17)

G – normalised relative growth rate function

J j – nucleation rate of phase j

J j ,i –initial nucleation rate of phase j

N j - number of nuclei per unit volume of phase j

R j – growth rate of phase j

R j ,i–initial growth rate of phase j

S – supersaturation ratio (¿σ+1 ¿

Sx - solubility ratio of metastable to stable phase

c – concentration of solute

c j ,e - solubility of phase j

f j - functional form of growth kinetics for phase j

31

k j - growth rate constant for crystals of phase j

r j - characteristic size of crystals of phase j

r j ,max - characteristic maximum size scale for crystals of phase j

t – time

t p – competitive time period

u j - normalised size scale for crystals of phase j

χ j - mass fraction of crystals that are phase j

χ j , p - mass fraction of crystals that are phase j at transition point

ϕ j - shape factor for crystals of phase j

ρ j - density of phase j crystals

σ - supersaturation of stable phase ≡(c−ce)/ce

σ j ,i - initial supersaturation of phase j

σ j - supersaturation of phase j with respect to its equilibrium level

σ x – supersaturation, with respect to the stable phase, of a solution saturated with the

metastable phase (σ x=(c2 , e−c1 ,e )/c1 ,e ¿ Sx−1)

ΦOs - Ostwald Ratio

ΦOs, B - Ostwald Ratio as per Black et al18

ΦOs∞ - High supersaturation limit of Ostwald Ratio

Conflicts of interestsThe authors declare no competing financial interests.

32

References

(1) Timofeeva, T. V.; Nesterov, V. N.; Clark, R. D.; Penn, B.; Frazier, D.; Antipin, M. Y., Systematic study of polymorphism in crystalline non-linear optical materials. Journal of Molecular Structure 2003, 647, (1), 181-202.(2) Bernstein, J., Polymorphism in molecular crystals. ed.; Oxford University Press: Oxford, 2007.(3) Cardew, P. T.; Davey, R. J., The Kinetics of Solvent-Mediated Phase Transformations. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 1985, 398, (1815), 415-428.(4) Raza, K.; Kumar, P.; Ratan, S.; Malik, R.; Arora, S., Polymorphism: The Phenomenon Affecting the Performance of Drugs. SOJ Pharm Pharm Sci 2014, 1, (2), 10-20.(5) Mangin, D.; Puel, F.; Veesler, S., Polymorphism in Processes of Crystallization in Solution: A Practical Review. Organic Process Research & Development 2009, 13, (6), 1241-1253.(6) Davey, R. J.; Schroeder, S. L. M.; ter Horst, J. H., Nucleation of Organic Crystals—A Molecular Perspective. Angewandte Chemie International Edition 2013, 52, (8), 2166-2179.(7) Ostwald, W., Studien über die Bildung und Umwandlung fester Körper. 1. Abhandlung: Übersättigung und Überkaltung. Zeitschrift für Physikalische Chemie 1897, 22, 289-330.(8) Chen, C.; Cook, O.; Nicholson, C. E.; Cooper, S. J., Leapfrogging Ostwald’s Rule of Stages: Crystallization of Stable -Glycine Directly from Microemulsions. γ Crystal Growth & Design 2011, 11, (6), 2228-2237.(9) Santra, M.; Singh, R. S.; Bagchi, B., Nucleation of a Stable Solid from Melt in the Presence of Multiple Metastable Intermediate Phases: Wetting, Ostwald’s Step Rule, and Vanishing Polymorphs. The Journal of Physical Chemistry B 2013, 117, (42), 13154-13163.(10) Cheetham, A. K.; Kieslich, G.; Yeung, H. H. M., Thermodynamic and Kinetic Effects in the Crystallization of Metal–Organic Frameworks. Accounts of Chemical Research 2018, 51, (3), 659-667.(11) Andrews, J. L.; Pearson, E.; Yufit, D. S.; Steed, J. W.; Edkins, K., Supramolecular Gelation as the First Stage in Ostwald’s Rule. Crystal Growth & Design 2018, 18, (12), 7690-7700.(12) Stranski, I.; Totomanov, D., Rate of Formation of (Crystal) Nuclei and the Ostwald Step Rule. Z. Phys. Chem. A 1933, 163, 399-408.(13) Volmer, M., Kinetik der Phasenbildung ed.; Dresden annd Leipzig, 1939; Vol. 4.(14) Keller, A.; Goldbeck-Wood, G.; Hikosaka, M., Polymer crystallization: survey and new trends with wider implications for phase transformations. Faraday Discuss 1993, 95, (0), 109-128.(15) Parsons, R.; Robinson, K. M.; Twomey, T. A.; Leiserowitz, L.; Roberts, K. J.; Chernov, A. A.; van der Eerden, J. P.; Bennema, P.; Frenken, J. W. M.; Sherwood, J. N.; Goldbeck-Wood, G.; Shekunov, B. Y.; Lal, M.; Parker, S. C.; Davey, R. J.; Gao, F.; Leusen, F.;

33

Mohammadi, M.; Mehta, R. V.; Alexander, J. I. D.; Woensdregt, C. F.; Rohl, A. L.; Higgs, P. G.; Ungar, G.; Toda, A.; Keller, A.; Maginn, S. J.; Hastie, G., General discussion. Faraday Discuss 1993, 95, (0), 145-171.(16) Bernstein, J.; Davey, R. J.; Henck, J.-O., Concomitant Polymorphs. Angewandte Chemie-International Edition 1999, 38, (23), 3440-3461.(17) Verma, V.; Hodnett, B. K., A basis for the kinetic selection of polymorphs during solution crystallization of organic compounds. CrystEngComm 2018, 20, (37), 5551-5561.(18) Black, J.; Cardew, P. T.; Cruz-Cabeza, A. J.; Davey, R. J.; Gilks, S. E.; Sullivan, R. A., Crystal nucleation and growth in a polymorphic system: Ostwald's Rule, p-aminobenzoic acid and nucleation transition states. CrystEngComm 2018, 20, (6), 768-776.(19) Cardew, P. T.; Davey, R. J. In Kinetic factors in the appearance and transformation of metastable phases, Tailoring of Crystal Growth: Control of Purity, Form and Size Distribution, Manchester, 1982; BACG/ IChemE - NW branch: Manchester, 1982.(20) Nývlt, J., The Kinetics of industrial crystallization. ed.; Elsevier ; Elsevier Science Pub. Co., Inc. [distributor]: Amsterdam; New York; New York, 1985.(21) Ohara, M.; Reid, R. C., Modeling crystal growth rates from solution. ed.; Prentice Hall: Englewood Cliffs, N.J., 1973.(22) Pudipeddi, M.; Serajuddin, A. T. M., Trends in Solubility of Polymorphs. J Pharm Sci 2005, 94, (5), 929-939.(23) Soetaert, K.; Petzoldt, T.; Setzer, R. W., Solving Differential Equations in R. The R Journal 2010, 2, (2), 5-15.(24) Burton, W. K.; Cabrera, N.; Frank, F. C., The growth of crystals and the equilibrium structure of their surfaces. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 1951, 243, (866), 299-358.(25) Bennema, P., Crystal growth from solution — Theory and experiment. Journal of Crystal Growth 1974, 24-25, 76-83.(26) Teychené, S.; Biscans, B., Nucleation Kinetics of Polymorphs: Induction Period and Interfacial Energy Measurements. Crystal Growth & Design 2008, 8, (4), 1133-1139.(27) Zhu, L.; Wang, L.-y.; Sha, Z.-l.; Wang, Y.-f.; Yang, L.-b.; Zhao, X.-y.; Du, W., Interplay between Thermodynamics and Kinetics on Polymorphic Appearance in the Solution Crystallization of an Enantiotropic System, Gestodene. Crystal Growth & Design 2017, 17, (9), 4582-4595.(28) Lu, J.; Wang, X.-J.; Yang, X.; Ching, C.-B., Polymorphism and Crystallization of Famotidine. Crystal Growth & Design 2007, 7, (9), 1590-1598.(29) Kitamura, M.; Hara, T.; Takimoto-Kamimura, M., Solvent Effect on Polymorphism in Crystallization of BPT Propyl Ester. Crystal Growth & Design 2006, 6, (8), 1945-1950.(30) Du, W.; Cruz-Cabeza, A. J.; Woutersen, S.; Davey, R. J.; Yin, Q., Can the study of self-assembly in solution lead to a good model for the nucleation pathway? The case of tolfenamic acid. Chemical Science 2015, 6, 3515-3524.(31) Kitamura, M., Polymorphism in the crystallization of L-glutamic acid. Journal of Crystal Growth 1989, 96, (3), 541-546.(32) Kitamura, M., Controlling factor of polymorphism in crystallization process. Journal of Crystal Growth 2002, 237-239, 2205-2214.(33) Su, W.; Hao, H.; Glennon, B.; Barrett, M., Spontaneous Polymorphic Nucleation of d-Mannitol in Aqueous Solution Monitored with Raman Spectroscopy and FBRM. Crystal Growth & Design 2013, 13, (12), 5179-5187.

34

(34) Sullivan, R. A.; Davey, R. J.; Sadiq, G.; Dent, G.; Back, K. R.; ter Horst, J. H.; Toroz, D.; Hammond, R. B., Revealing the Roles of Desolvation and Molecular Self-Assembly in Crystal Nucleation from Solution: Benzoic and p-Aminobenzoic Acids. Crystal Growth & Design 2014, 14, (5), 2689-2696.(35) Sullivan, R. A.; Davey, R. J., Concerning the crystal morphologies of the and α βpolymorphs of p-aminobenzoic acid. CrystEngComm 2015, 17, (5), 1015-1023.(36) Jiang, S.; ter Horst, J. H.; Jansens, P. J., Concomitant Polymorphism of o-Aminobenzoic Acid in Antisolvent Crystallization. Crystal Growth & Design 2008, 8, (1), 37-43.(37) Stoica, C.; Tinnemans, P.; Meekes, H.; Vlieg, E.; van Hoof, P. J. C. M.; Kaspersen, F. M., Epitaxial 2D Nucleation of Metastable Polymorphs: A 2D Version of Ostwald's Rule of Stages. Crystal Growth & Design 2005, 5, (3), 975-981.(38) Maggioni, G. M.; Bezinge, L.; Mazzotti, M., Stochastic Nucleation of Polymorphs: Experimental Evidence and Mathematical Modeling. Crystal Growth & Design 2017, 17, (12), 6703-6711.(39) Cornel, J.; Kidambi, P.; Mazzotti, M., Precipitation and Transformation of the Three Polymorphs of d-Mannitol. Industrial & Engineering Chemistry Research 2010, 49, (12), 5854-5862.(40) Bennema, P.; Boon, J.; van Leeuwen, C.; Gilmer, G. H., Confrontation of the BCF theory and computer simulation experiments with measured (R, ) curves. σ Kristall und Technik 1973, 8, (6), 659-678.

35

Entry for the Table of Contents

FULL PAPER

Author(s), Corresponding Author(s)*

Peter T. Cardew and Roger J Davey*

Page No. – Page No.

The Ostwald Ratio, Kinetic Phase Diagrams and Polymorph Maps.

Ostwald’s Rule is re-examined and the concept of the Ostwald Ratio revealed leading to

polymorph maps defining regions of appearance of forms during crystallisation.

36