Crystallization mechanisms in solution

14

Journal of Crystal Growth 90 (1988) 14—30 North-Holland. Amsterdam

CRYSTALLIZATION MECHANISMS IN SOLUTION R. BOISTELLE and J.P. ASTIER 2 — CNRS, Campus Luminy, Case 913, F-13288 Marseille Cedex 09, France

CRMC

Received 29 January 1988

Crystallization from solution is a sequence of events which occur more or less consecutively but are rarely completely unconnected. The present contribution is a survey of nucleation, growth, phase transition, habit modification and ripening. Its aim is to provide a theoretical basis to biochemists who intend to approach crystallization.

1. Introduction

2. Supersaturation and activity

The aim of this survey is to provide biochemists who intend to tackle crystallization with an uncomplicated theoretical basis. But no special attention is given to protein crystals, it being assumed that nucleation and growth mechanisms are the same for any crystalline material. Usually, the different steps of the growth process are discussed as if crystallization were a sequence of unconnected events. Actually, all steps are interdependent since the system in which crystallization occurs is in continuous evolution. As nucleation and growth proceed, the overall supersaturation of the solution decreases. Nucleation and growth kinetics are decelerated. Accordingly, the system tends towards equilibrium and the thermodynamic factors take precedence over the kinetic factors. New polymorphic modifications and phase transitions may occur, since the crystals have formed at different times and there is a certain difference in their size. Ostwald ripening can take place, the largest crystals growing at the expense of the smallest ones. Finally, due to variation of supersaturation and the presence of impurities, there are sometimes important habit changes. These different points are discussed in the following sections.

Once the proper solvent has been found, supersaturation may be achieved in different ways. The most common methods are: cooling or heating the solution, according to whether solubility increases or decreases with increasing temperature; evaporating the solvent; mixing solutions containing soluble species giving rise to the precipitation of sparingly soluble ones; changing the PH; dialyzing the solution. There are many variants of these methods. The pH can be adjusted by diffusing a vapor over the solution, or more simply by addition of some acidic or alkaline component. The precipitant is either a poor solvent or another solution containing some dissolved compound. The addition of the precipitant can be achieved instantaneously, or more slowly by diffusion through a gel. Evaporation can be controlled by equilibrating the solution against different reservoirs. Only a few exampies are given here and the list is far from exhaustive. The driving force for nucleation and growth is the difference between the chemical potential of a —

— —

—

—

0022-0248/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

R. Boistelle, J. P. Aslier / Crystallization mechanisms in solution

molecule in the supersaturated and saturated solution, respectively. If C is the actual concentration before any crystallization and C~the concentration at equilibrium, i.e. the solubility, the driving force per molecule can be written as I.~ji=k~Tln(C/C

5),

C (au) -

(1)

where kB is the Boltzmann constant and T the absolute temperature. Several quantities are used for defining the ratios /3 C/CS and a (C C5)/C5, and the supersaturation. The most common ones are the difference C C5. The two former quantities are dimensionless since they are normalized with respect to solubility. However, their values depend on the units chosen for the concentrations. On the other hand, C C5, which represents the amount of solute which may precipitate, is much more dependent on the concentration units (molar fractions, molarities, gram per liter, etc.). In most cases it is preferable to use /3 or a, the comparison between supersaturations in different systems being simpler. It is noteworthy that the same values of /3 can be obtained for different values of C C5. This occurs, for example, when different temperature or pH ranges are available for crys=

15

=

—

—

—

—

tallization. As a specific case, let us consider a solution where supersaturation is promoted by decreasing the pH either from pH~or pH2 or from pH2 to pH3 (fig. 1). Taking into account the corresponding that in both situations concentrations, /3 10, whereasit Cappears C 5 is 900 or 90 in the former in thedifferent latter case, respectively, Accordingly, we or expect nucleation and growth rates when crystallization occurs in the two solutions. Solubility curves are not always as =

—

pH3

pH2

more or less dissociated, activities are preferred to concentrations for calculating the supersaturation. In order to illustrate this point, we may consider an imaginary substance RX in an aqueous solution. When dissolution takes place we have, for example, (2) and when equilibrium is achieved, the solubility product of the substance is: 2 + ~ a (X2 ~ (3) K51, a (R —

=

and X2. the superwhere the symbol a 2~ refers to theThen, activities of the free divalent ionsbeRdefined as saturation must /3

=

a(R2~)a(X2)/K 5~,

simple as for the case shown in fig. 1. Many examples involving proteins are given in a recent review paper [1] where the theory of protein solubility is extensively discussed. In some cases, it is ~L/kBT in /3 which is taken as the definition of supersaturation. Since in many solutions the supersaturation is small, it is also possible to approximate in /3 by a. However, the discrepancy becomes important as soon as the supersaturation exceeds 15%. In electrolytic solutions where the solute is =

pH1

Fig. 1. Schematic representation of a solubility curve showing that large and small concentration differences may correspond to the same supersaturation ratios when crystallization occurs between pH1 and pH2 or between pH2 and pH3.

(4)

where the ionic product in the numerator relates the actual activities of the ions in solution. For any chemical species i, activity and concentration are related by (5) where ~ is the aciivity coefficient, which is defined by 2/(1 + Br,I~’2),

—log y,

=

Az~P”

(6)

16

R. Boistelle. J.P. Astier

/ Crystallization mechanisms in solution

with

0.75

(7)

J=~>2C,z~.

0.30

050 0.25

In the Debye—HUckel equation (6), the constants A and B are 0.5317 and 0.3334 at 25°C when i is expressed in A; C,, z, and , are the concentrations, charges and radii in solution of the species i under consideration. Values of r, have been tabulated [2] for many different substances. All uncharged species have z U and

0 0

_____________________________

log [C]

0.15

[~]

-5

=

coefficient y, being equal to 1 as can be seen in eq. (6). When there are several, or many, dissolved species rthe 0, ionic sointhat strength solution, they Ido the ofnot calculation thecontribute solution, of their in theincreasing activities activity must be made by an iteration procedure. In our simple example, there is no particular reason why the substance RX must be fully dissociated in 2~and X2 do not form any solution and why R complex with H + or OH—. To simplify the problem, we allow only one soluble complex RX and only one complex with the solvent HX. In that case, the following equilibria and mass balances must be taken into account:

10

=

[OH][H~]/[H2O]

[Hx]/[H~][x 2~][X2]

=

=

[RX]/[R Mass balances CR

=

=

CH

=

~ K K 2~, 3~.

=

2}

(8a) (8b) (8c)

/

pH~

8 (I) and the 10 ion and Fig. 2. 4Variation of the ionic strength complex concentrations in a solution containing 50 mmol/l (NH andtotal NaH2PO4. The(NH pH is adjusted by ammonia, [N], 4),S04 being the ammonia 3 +NH~) in the system; calculations based on data from refs. [2—4].

whether the supersaturation changes in the presence of impurities. Two main cases are possible: the impurity only increases the ionic strength I of the solution without forming complexes with the solute or it increases I and simultaneously forms soluble the solute (by trapping 2~or complexes X2 in thewith example described above). Since R the concentrations and consequently the ac.

IR2~]+ [RX],

(9a)

[x2] + [RX] + [HX], [H~] + [HXJ [OH]. —

(9b) (9c)

When the set of eqs. (9) is solved for the concentrations, the resulting expressions are inserted in the set of eqs. (8) which is solved by iteration, taking as initial values the dissociation constants K~ and the thermodynamic stability constants Ka. Once the concentrations of the different species are known, it is possible to calculate activities and supersaturation. Of course, by definition, a(H~) 10pH, In real systems, the solutions are rarely pure as in the example given above and we may wonder =

tivities of the free ions decrease, the supersaturation (eq. (4)) must decrease proportionally. Instead of calculating the variation in supersaturation of a given substance, we show here the variation of the concentrations in a solution containing only two salts, and one whose pH is adjusted by addition of ammonia. When 50 mmol/l of (NH 4)2S04 and NaH2PO4 are dissolved in water, the pH of the solution is about 4.50. Several ions and complexes form. The dissociation constants of the chemical species involved can be found from ref. [3]; those concerning amino acids might be found from ref. [4]. Fig. 2 shows the concentration curves of the ions and complexes as a function of pH. Some concentrations change by several orders of magnitude. Others, like that of SO~, not given in fig. 2,

R. Boistelle, J.P. Astier

/

17

Crystallization mechanisms in solution

remain more constant: 4.7 x 102 to 6.1 x iO~ for pH 4.5 and 10, respectively. The total ammonia concentration ~which includes all species containing NH3 or NH~,changes slowly between =

pH 4.5 and 8, but the amount required to raise the pH to 10 increases drastically above this value. Despite this fact, the ionic strength of the solution varies significantly only between pH 6 and 8. Indeed, above the latter value, there is no contribution of [NH3] to I, since NH3 is an uncharged species. Now, if any solute is added to the solution and if it dissociates and forms bonds with the components of the solution, all equilibria are shifted and the concentrations modified, This example, which describes a very simple system, perfectly illustrates how difficult it is to adjust the ionic strength and the other parameters such as supersaturation without knowing precisely what happens in the solution, =

crystals. Such a mechanism, however, requires energy, i.e. the activation free energy for nucleation. It is furthermore necessary to exceed a critical value of this energy if embryos are to develop into nuclei and crystals. 3.1. Activation free energy for nucleation

=

3. Nucleation When sufficient supersaturation is reached, nuclei form according to two principal mechanisms. When the nuclei occur in the bulk of On the solution, nucleation is termed homogeneous. the other hand, when nuclei form onto solid substrates, nucleation is called heterogeneous. The nucleation theones for vapor or solution are presented elsewhere in detail [5,6]. For this reason we will emphasize here only some essential points. Other aspects concerning specifically the nucleation and growth of protein crystals were discussed

In order to nucleate a cluster of volume V and area S, the energy required is ~G

=

—

_VkBT ln /3 + S-y,

(10)

where Q is the volume of a molecule inside the crystal and ‘y the interfacial free energy between nucleus and solution. Hence, the first term of eq. (10) is a volume term, V/Q being the number of molecules inside the nucleus. The second term, the surface term, represents the excess energy expended in creating the nucleus surface. It is supposed here that y has the same value over the whole nucleus surface, or that the nucleus is limited by crystallographically equivalent faces. If we consider a spherical nucleus, eq. (10) becomes 3/3Q)kBT in /3+4lTr2y, (11) ZIG= —(417r This variation of ~G with r is shown in fig. 3. Due to the competition between volume and surface terms ZIG passes through a maximum at a certain value of r. In other words, when the nucleus reaches a critical radius .

.

.

.

.

.

*

r

2f2y

— —

12

k 8T ln /3’

in a recent review paper [7]. In a solution where the mean energy provided by the supersaturation is constant, there must be some energy fluctuations in order to obtain embryos which then transform into nuclei and crystals. In the first step of the process, the molecules, which randomly diffuse through the solution, meet and coalesce in the form of small aggregates. Hence, formation is most often the resultthe of the additionofofembryos monomers to aggregates already existing: schematically, a monomer plus a monomer give a dimer, a dimer plus a monomer give a trimer and so on, up to a critical size above which the nuclei transform into

the corresponding critical activation free energy for nucleation is 16 ~2 3 ZIG* 2’ (13) 3(kBT ln /3) =

or 2y) (14) LIG* i(4~rr* which shows that the energy required to get a nucleus, stable at the supersaturation /3, is one third the energy required to create its surface. At the critical value r * (fig. 3), the nucleus is stable. =

18

R. Boislelle, J.P. Astier

/

Crystallization mechanisms in solution

J

Surface term

LX 6

:

________________________________

\

Volume

term

solution). Solution of the equations for the nucleus size and activation energy for nucleation implies that the critical radius for heterogeneous nucleation is the same as that for homogeneous nucleation (eq. (12)). However, since the nucleus develops onto the substrate, the sphere is incomplete and less molecules are required to form it (fig. 4). On the other hand, the critical activation energy for heterogeneous nucleation is now the product of the activation energy for homogeneous nucleation and a term depending on the value of a since 3a). (14) ZIG~1=ZIG*(~ cos a + ~ cos The influence of a is easy to demonstrate, using three peculiar values. For a 180°, the term in parentheses is one and ZIG ~, ZIG *, The nucleus has no affinity for the substrate. For a 90 °, ZIG~’et ZIG */2 and for a 0, ZIG~et 0. The smaller a, the smaller the energy required for —

\

=

=

Fig. 3. Activation free energy for nucleation as a function of the nucleus size. The critical value ~G* must be reached for creating a nucleus of cntical radius r *

If one molecule is withdrawn from it (r r *), it grows Spontaneously, both processes taking place with an energy gain, Eq. (13) shows that the activation energy for nucleation decreases with increasing supersaturation and temperature, and with decreasing interfacial free energy. In other words, we may expect a larger nucleation rate at larger /3 and T, and smaller values. Finally, in eq. (13) the dimensionless quantity l6ir/3 is the so-called shape factor of the nucleus. For a cubic-shaped nucleus, the shape factor is 32 with V 8r3 and S 24r2 in eq. (10). Homogeneous nucleation arises mainly in pure solutions when the supersaturation is rather high. In contrast, heterogeneous nucleation occurs most frequently when the supersaturation is low and when the solute molecules have some affinity for solid substrates which may be the wails of the crystallization vessel, the surface of the stirrer or any other solid like dust and paper particles. With such a nucleation mechanism, the nucleus develops onto the substrate with which it makes a contact angle a. Fig. 4 shows this special situation for a cap-shaped nucleus, a situation which in volves three areas and surface free energies (substrate/nucleus, substrate/solution and nucleus/ ‘~‘

=

=

=

=

—~

—~

forming the nucleus. Thus the substrate catalyzes the nucleation and the nucleus may form at very low supersaturation. In some cases, the nuclei never form in the bulk of the solution but always on the walls of the crystallizer, especially when these walls exhibit a certain surface texture. There are also some special cases of heterogeneous nucleation where the nuclei which transform later into crystals are more or less oriented on the substrate. There may be a textural orientation: the nuclei all have the same contact plane with respect to the substrate but are not oriented with respect to each other. In other cases there is also an azimuthal orientation which generates a perfect orientation of all nuclei: there is an epitaxy between deposit and substrate.

Solution

F

a

Substrate -______________________

Fig. 4. Cap shaped nucleus of radius r forming on a solid substrate, with which it forms the contact angle a.

R. Boistelle, J.P. Astier

/

19

Crystallization mechanisms in solution

3.2. Nucleation rate

I

J(au) The nucleation rate J for homogeneous nucleation can be defined as the number of nuclei formed per unit volume and unit time. Schematically, J may be written as J

=

K0 exp( _ZIG*/kBT),

(15)

where the kinetic coefficient K0 is K0=N0v0,

Metastable

Spontaneous

Region

Nucleation

(16)

with N0 the solubility of the material expressed as the number of molecules per unit volume, and v0 the frequency at which the critical nuclei become supercritical and transform into crystals. From inspection of eqs. (13), (15) and (16), it clearly appears that the nucleation rate depends principally on three parameters: N0, ~y and /3. Other things being equal, nucleation is more rapid in solutions where the solubility is high. This is easy to understand since there are more monomer encounters when they are closer to each other, This point was ytouched upon in fig.independent 1. In addition, the parameter is not completely of

*

/3 .

Fig. 5. Nucleation rate J versus supersaturation /1. Below the critical value $ * the solution remains metastable, whereas above /1* the nucleation rate increases drastically.

It is sometimes possible to advantageously use measurements of induction periods. If t, is the real time which elapses between the moment when the supersaturation is achieved and the moment when 2/3)’ nuclei the plotline, of iflnwet, assume versus that (1n after should occur, give a straight

N 0. When the solubility increases, by changing the solvent or the solution composition, the interfacial energy between the crystal and the solution decreases. In other the greater the affinity of the solvent for words, the crystal, the larger N 0, the smaller y and the higher the nucleation rate. Nucleation occurs therefore at relatively low supersaturation. Conversely, the difficult materialand is sparingly soluble, nucleationwhen is more occurs only when high supersaturations are reached. In that case, the supersaturated solution may stay in a metastable state over very long periods. But as soon as a critical supersaturation /3 * is exceeded, the nucleation rate becomes catastrophic (fig. 5). In small systems it is particularly important to adjust very precisely the /3 value around /3*, When /3 ~ /3*, the induction period for nucleation can be much too long (several days, weeks or even more). When /3>> /3*, the induction period is very short but the number of nuclei can be too large by several orders of magnitude. Sometimes, when /3 is extremely large, nucleation even results in the formation of an amorphous or very poorly crystalline phase.

only one nucleus forms per unit solution volume 1 cm 3)• Since, for a spherical nucleus (eqs. (13) and 2y3/3(k~T)3, (15)), theit should slope be of possible the line is to get t,

(Jt, —

=

16~7S2

the value of y since all other parameters are known. Obviously, y is obtained with marginal accuracy but in many cases the order of soluble magni2 for very tude is good: few erg/cm materials, and abetween 50 to 100 erg/cm2 for sparingly soluble ones. The nucleation rate, however, depends not only on the value of the supersaturation but also on the way and velocity by which it is achieved. As an example, we carried out some experiments on lysozyme nucleating from solutions buffered by sodium acetate at pH 4.5, using the hangingdrop method. The droplets are equilibrated against reservoirs at 6%, 7.5%, 10% and 15% NaCI, the initial concentrations in the droplets being one half of these values, The initial droplet volume and protein concentration are 16 ~.tland 10 mg/mI, respectively. The evolution of the NaC1 and lysozyme concentrations measured by refractometry and spectrophotometry are given in fig. 6. Equi=

20

R. Boistelle, J.P. Astier

[NaCI]°~

10 15

5

/

Crystallization mechanisms in solution

saturation of the protein becomes concentration too high

C~p mg/mF)

150

strength.

/~ ~

Instead

of changing the ionic strength of the the droplet. All other things being equal, the vapor reservoir it may be useful to change the radius r of pressure p of the droplet is now affected by the term 2y/r. Fig. 7 gives an example of the varia-

15

~~_—~‘

10 0

1(hrs)

20

40

60

80

.

20

40

60

l(hrs)f~

80

Fig. 6. Variation of sodium chloride and protein concenirations in droplets equilibrating against reservoirs at 6%, 7.5%, 10% and 15% of salt. The highest possible protein concentration (20 mg/ml) is achieved only in the case of the two droplets of lowest ionic strength where nucleation is delayed.

librium between droplet and reservoir is achieved when kBT ln(p/p

5)

=

for above the which chosensuperionic

2Qy/r,

(17)

where p is the pressure inside the droplet and p5 the saturation vapor pressure over the reservoir, Since p.~decreases with increasing reservoir concentration, the driving force for evaporation is the highest in the droplet when the final concentration of salt is the highest (15% in the present case). Accordingly, the initial slope of the corresponding concentration curve is the greatest. It is also noteworthy that the plateaus of the curves should not exceed the reservoir concentrations. Actually, the limits are slightly exceeded at the beginning of the plateaus partially due to experimental uncertainty. The shapes of the protein curves are quite different. Only in the droplets of smallest final NaCl concentrations (6% and 7.5%) does the protein concentration reach the highest possible value of 20 mg/ml. Nucleation is delayed, but later the formed crystals are of good size and quality. On the other hand, we always observed the precipitation of an amorphous phase almost immediately in the droplets at 15% NaCl. The droplets at 10% NaC1 are in an intermediate situation. Obviously. in the present case it is more suitable to crystallize at lower ionic strength and higher protein concentration. However, there is also an upper limit

tion in concentration in droplets of different sizes. We have carried out these experiments by equilibrating droplets against reservoirs at 10% and 20% of sodium chloride. The initial droplet concentrations were one half of these values. The smaller the radius of the droplet and the higher the evaporation rate, the earlier the nucleation and the larger the number of crystals. When supersaturation is achieved by diffusion of a salt through a gel towards a solution containing the protein, nucleation is also faster with the gel of highest ionic strength. In fig. 8 we have displayed our results in the case of gels at 20% polyacrylamide and 0.8% bisacrylamide containing 5%. 10% or 15% NaCl, In all experiments, the nucleation of lysozyme, inside the solutions in contact with the gels, occurs at nearly the same NaCI concentration (between the arrows on the

[NoCI] %

20

-

15 -

10

£ 5

:

B 12 -16 • 20

(hrs) 0 ________________________________ 50 100 150 Fig. 7. Variation of the salt concentration in droplets of initial volumes of 8, 12. 16 and 20 91. The smaller the droplet, the greater its evaporation rate.

R. Boistelle. J.P. Astier

/

Crystallization mechanisms in solution

magnitude [11]. On the other hand, the shapes of the nucleation rate curves are not significantly

[r’JaCl]% 15

10

affected by thesupersaturation impurities. Thevalues curves(fig. are5).shifted toward larger

__—~°

/ ../~

21

In other words, we may say that the impurities stabilize the supersaturation. The induction penods are longer and the width of the metastable zone increases. Another general trend is that the

~__—.~ _—~

6

0 ,

50

100

150

t

(hrs)

200

Fig. 8. Vanation of the salt concentration in solutions in contact with gels at 6%, 10% and 15% of sodium chloride. The nucleation of lysozyme occurs at nearly the same ionic strength but it is earlier (arrows), and the nucleation rate is larger with the gels of largest ionic strength.

curves). Due to the more rapid supersaturation increase in the system at 15% NaCl, nucleation is earlier and the nucleation rate faster. Accordingly, the crystals remain smaller. 3.3. Nucleation in the presence of impurities

The term impurity is used here for designating either real impurities of the system or additives put into the system in order to obtain some specific effect. It must be emphasized that the first potential impurity for the solute, i.e. for the nucleus and the crystal, is the solvent. As discussed above, the solvent, or more generally the solution, influences the interfacial free energy ‘y. Since any adsorption onto the nucleus decreases y [8—10),we should observe an increase of the nucleation rate when impurities are present in the solution. But this would be the sole result of the thermodynamic effect which is actually in competition with a kinetic effect. Indeed, nucleus development proceeds only through a few growth sites. Therefore, only a few adsorbed impurity molecules are necessary for completely inhibiting the nucleus growth and keeping it at a subcritical size (r
sorption and integration kinetics of the solute. When the number of solute molecules hitting the .

nucleus is very high, the solute prevents the impurities from entering and blocking the growth sites. Conversely, if the amount of impurity is large, the supersaturation must be enhanced to keep the nucleation rate at a suitable value.

4. Growth When a nucleus develops and transforms into a crystal, it exhibits different faces, the growth mechanisms and growth rates of which depend not only on external factors (supersaturation, impurities, etc.), but also on internal factors (structure, bonds and defects). Therefore, before discussing the growth rate equations, it is important to first consider the influence of the crystal structure on the growth mechanisms. For doing this we summarize here the Periodic Bond Chain (PBC) theory [12—15].This theory assumes that growth is the result of the consecutive formation of strong bonds between growth units. A PBC is an uninterrupted chain of strong bonds which repeat periodically through the crystal. For complicated structures there are three types of PBCs, the complete PBC being the one which has the same composition as the crystal and no electrostatic dipole moment perpendicular to the direction along which it runs. According to this concept, three types of faces may be found in a crystal (fig. 9): F faces (flat): they have PBCs in at least two different directions in a slice of thickness dhk/ (interreticular distance); —

22

R. Boistelle, .J.P. Astier

/

Crystallization mechanisms in solution

relaxation time for desonption of the growth unit, F

and face. consequently the larger the rate of growth of the Hereafter we consider only the case of layer growth (F faces) since the large growth rates of K

K F

S

Fig. 9. Schematic representation of a crystal exhibiting Oat (F). stepped (S) and kinked (K) faces. Front face exhibits a polygonized growth spiral. whereas top face exhibits a two-dimensional nucleus,

—

S faces (stepped): they contain only one PBC in

the slice dhk,; K faces (kinked): they contain no PBC in the slice dhk/. Since K faces contain only kinks, i.e. growth sites, they grow by direct incorporation of atoms or molecules, Since F faces are flat, the number of kinks is very small in contrast with the K faces. Such F faces grow either by a two-dimensional nucleation mechanism or by a spiral-growth mechanism. The layers spread parallel to the crystal face. Finally, the S faces grow by a one-dimen—

sional nucleation mechanism. Accordingly, we may predict that the growth rate of the F faces should be much smaller than the growth rate of the K faces. The S faces are intermediate between these two limiting cases. Since it is only necessary to know the relative values of the bond energies for finding the PBCs and classifying the faces, it should be possible to determine the relative growth rates of the faces. This was shown [14] by introducing the concept of reduced growth rate. This is an expression of the growth rate where all factors which are independent of the crystal face are not considered. In short, the growth rate of any face is supposed to be proportional to the attachment energy of a growth unit deposited on a terrace of the face under consideration. Implicitly, it is understandable that the higher this energy, the greater the

and S faces result in their disappearance from the crystal growth morphology. 4.1. Growth by 2D nucleation

In the case of perfect crystals, or when there are no defects in a crystal face, growth takes place by two-dimensional nucleation [16,17]. In order to create a 2D nucleus, the adsorbed molecules, or more generally the growth units, must diffuse to the face and cluster. Once the nucleus has exceeded a critical size it becomes stable and exhibits some kink positions where the growth units can then be more readily incorporated. Such a situation is depicted in the upper part of fig. 9. In further steps, the 2D nucleus spreads across the surface. When a complete layer has formed, the crystal has grown by one monomolecular layer. Two models are likely for such a mechanism. In the mononuclear model there is only one nucleus which spreads across the surface at the same time. It is supposed that the whole crystal face is covered before the next nucleus forms. In such a case, the growth rate of the face is R=J2dS,

(18)

where J2 is the 2D nucleation rate, d the height of the layer and S the area of the face. In the polynuclear mechanism (fig. 10) several nuclei spread on the face at the same time. Each

~~II~~7 Fig. 10. Birth and spread

model: several two-dimensional

nuclei spread across the crystal face.

R. Boistelle, J. P. Astier

/ Crystallization mechanisms in solution

______

nucleus expands until it encounters another nucleus or the edges of the faces. Eqs. (19)—(21), describing the growth models by 2D nucleation, are somewhat different according to the theory from which they are derived. For the mononuclear mechanism we have for instance. R

k/3~2exp( —ZIG/k~T),

=

23

______

___________________________________________

(19)

whereas for the polynuclear mechanism R

k$~6[exp($)

=

—

1I2~3exp(—ZIG/3knT), (20) Fig. 11. Sequence of steps. generated by a polvgonized growth spiral, spreading across the basal face of an octacosane crystal.

or R

k/3”3( /3

=

—

1)2”31n /3”2exp( —ZIG/kBT), (21)

where k is a kinetic constant depending on ternperature and ZIG is the activation free energy for creating the 2D nucleus. If the nucleus is a square, ZIG 4X2/k~T ln ~, (22) =

where A is the edge free energy (per molecule) of the nucleus, Growth by a two-dimensional nucleation mechanism is rather rare. Moreover, when it occurs, it is often difficult to control and adjust the growth rate at a given value. As in the case of three-dimensional nucleation, there is a critical value of the supersaturation, typically 30% to 50% below which there is no growth at all and above which growth becomes very rapid. 4.2. Growth by a spiral growth mechanism

When a screw dislocation emerges on a crystal face, it provides a step or a sequence of steps which can spread over the surface (fig. 11). Since the steps are always present, adsorbed solute molecules are more easily trapped than in the previous case. Growth is much more regular. The steps expand by rotating around the emergence point of the dislocation and when a step reaches the face edges, the crystal thickness has increased

The spiral growth mechanism was first developed in the so-called BCF theory [22] and later generalized and adapted to solution growth [23—25]. Some relevant equations have been presented in a review of crystal growth from nonaqueous solutions [26], the theories being fully described in ref. [19], with special attention to diffusion problems [27]. Here we will only recall some important points. Because the steps of the growth spirals are equidistant and move with a lateral velocity v (fig. 12), the growth rate of the face is: (23)

R=vd/y.

The distance between the steps is strongly dependent on supersaturation and to some extent on the spiral shape since y =fAa/k~T ln /3, (24) where a is the length of a growth unit and f the dimensionless shape factor of the spiral (1= 19 for a circular spiral). The expression of v is much more cumbersome than that of y. The most general equation for R is:

R

=

NoAD~ax

{n + ~

+

AA 5~

by one monolayer. Actually, the height of the step is often greater than d, the thickness of a monolayer, especially in impure solutions where step bunching is often observed,

-

+Ah_Y_coth(_2’~_) [2x5 2x5

—

i}}

,

(25)

24

R. Boistelle, J.P. Astier

_____

/ Crystallization

Iii

mechanisms in solution

_____

Fig. 12. Profile of a face growing by a spiral growth mechanism.

where D~is the coefficient for volume diffusion, A the length for entering the crystal surface from the solution, and A5 the length for exchanging growth units between step and surface, Accordingly, the four terms in the denominator are lengths. They represent a kind of impedance for the adsorption reaction, the impedance in the unstirred layer of thickness ~, the impedance for entering the steps, and finally the impedance for surface diffusion (x5 being the mean free path of the adsorbed molecules). It is impossible to solve such an equation and only the limiting cases can be considered. As an example, let us consider the normal cases A >> x,,, since the mean free path of surface diffusion is probably small in solution. We may also imagine that A >> iS since the thickness of the boundary layer in supersaturated solutions is rather limited. Finally, if A,, x.,, then eq. (25) reduces to relatively simple expressions. If the supersaturation in small, we have y >> x5 and

mental values with those associated with the different mechanisms, when it is possible to estimate the latter enthalpies with enough accuracy. Finally, it is noteworthy that eq. (25) contains N0, the solubility, parameter included in n,0 in eqs. (26) and (27). Here, we also find that, all other things being equal (driving force included), the growth rate of a crystal is much larger in solutions where the solubility is high. Octacosane, which is about 80 times more soluble than hexatriacontane, grows about 80 times faster [28]. 4.3. Growth in the presence of impurities

When growth takes place in the presence of

where n50 is the number of adsorbed growth units, per unit area, at crystal—solution equilibrium, and D5 is the surface diffusion coefficient. At high supersaturation, we have .v ~ x, and

Impurities, the growth rates of the crystal can be greatly affected. More precisely, each face is affected in a specific way according to the affinity of the impurity for the face. Several parameters in the growth rate equations are modified by impurity adsorption [19] according to whether adsorption takes place in the kinks, the steps or on the flat surfaces between the steps. The corresponding adsorption models have been reviewed and discussed several times [29—321.We only recount here the general trends. At least theoretically, impurity adsorption

R

should increase the growth rate of a face since adsorption decreases the edge free energy A (eq.

‘~<

R

=

2n50D5Q

kBT

2

(26)

=

(27)

n50D5Q ~, x~2

These two equations and 2 orare R often bii. oversimplified The kinetic coeffireduced to R ba cients h depend on temperature. From a plot of lnh versus 1/T it is possible to get the growth enthalpy. It is sometimes possible to rule out some growth mechanisms by comparison of the experi=

=

(24)). But, as in the case of three-dimensional nucleation, theimpurity kinetic factor is much more affected by the than the thermodynamic factor. It is sufficient to block a few kinks in order to slow down the growth rate by several orders of magnitude. Impurities or additives may be active at very low concentration and the general trend is

R. Boistelle, f.P. Astier

/

Crystallization mechanisms in solution

that their efficiency increases with increasing concentration. On the other hand, here also, their efficiency decreases with increasing supersaturation. In general, when adsorption is reversible and desorption rapid, the shape of the growth rate curve is not appreciably modified from that obtained in a pure solution. The curves are shifted toward higher supersaturations, but remain linear, quadratic or exponential as a function of supersaturation. There is, however, a special case, relatively charactenstic of long chain compounds, that pertains when the impurity is a long chain molecule, possessing many adsorption sites. Adsorption on the crystal is then never completely reversible. The impurity is always anchored at the crystal surface at several points. In that case the growth rate curves are often very different from the curves in a pure solution. This can be first observed by measuring the displacement ZIl of the face as a function of time at constant supersaturation. There are periods where the crystal grows, followed by periods where it is blocked. This is the result of the progressive poisoning of the surface by the impurity. Either fresh and clean 2D nuclei must form or impurities must desorb in order for growth to be possible. Hence, a much smaller mean value R ZIl/t is obtained (dashed line in fig. 13). When the mean values of R are plotted as a function of /3, it often happens that the growth rate is zero or nearly zero in a wide zone of supersaturation. The width of the zone increases with increasing impurity concentration [32]. Some critical supersaturation must be exceeded for each impurity concentration (fig. 14). When growth is hindered to such a degree, there may be a signifi=

—

—

— — — —

—

-

— — — — —

t

Fig. 13. Mean growth rate ~l/t (dashed line) in an impure system where growth is sometimes completely inhibited over long periods,

25

R

*

/31

/32 Fig. 14. Shapes of the growth rate curves when the crystal faces are poisoned by strongly adsorbed molecules: critical supersaturations must be exceeded in order that growth can take

place.

cant impurity incorporation, the impurity being alone or inside fluid inclusions.

5. Phase transitions and Ostwald ripening In most systems, especially in closed systems, there is a continuous decrease of the supersaturation when crystallization occurs. Both the nucleation and the growth kinetics decrease, factors that contribute to phase transitions. The term phase is used to designate several types of solids. When the phases have exactly the same chemical composition but different crystal structures they are called polymorphs. From mineralogical and crystallographic points of view the term polymorph or polymorphic modification should be employed only in this case. Graphite and diamonds are probably the most famous polymorphs since both are built up by carbon atoms and only carbon atoms. In the case of protein crystals the term polymorph rarely corresponds strictly to this definition, In some cases, the crystal composition differs only by the number of solvent molecules present. Solvates are phases where the ratio of solute over solvent molecules is constant. Hydrates are the most common solvates. In some cases, especially when an amorphous phase precipitates, the number of solvent molecules in the precipitate is not clearly defined and changes when the crystallinity increases with time.

26

R. Boistelle, f.P. Astier

/

Crystallization mechanisms in solution

In solution, when several phases coexist, the only stable phase is the one which has the lowest solubility. However, according to Ostwald’s rule of stages [33], it is rarely the most stable phase which nucleates first. On the contrary, if a solu-

is therefore the solubility of a large crystal of infinite size: a few tens of micrometers on the laboratory time scale. If the crystal is not too small, =

tion is supersaturated with respect to two phases at the same time, it is often the phase of highest solubility, i.e the less supersaturated one, which crystallizes first. When growth proceeds, the solute concentration decreases until it reaches the solubility curve of the crystallized phase. At this moment the solution is supersaturated only with respect to the phase of lowest solubility. If no crystals of this latter phase form, then the first precipitated phase remains in a metastable state. On the other hand, when a crystal of the new phase forms, the solute concentration decreases once more and the first phase becomes unstable. It dissolves more or less rapidly, the generated solute molecules contributing to the growth of the stable phase. The transformation kinetics of these so-called solution-mediated phase transformations have been discussed in a recent paper [34). In some special cases, there are epitactic relationships between the new stable phase and the old metastable phase onto which it grows [351. Another consequence of the decrease of supersaturation in a closed system is Ostwald ripening, In contrast with phase transitions, it concerns only crystals of the same composition and structure, i.e. crystals of the same phase. As we have seen in the section devoted to nucleation, a nucleus, a crystal here, is stable only if it has a critical size which depends on the supersaturation of the solution. In a system where there are several or many crystals, these crystals have different sizes according to the time at which they formed and the velocity at which they grew. Consequently, only one class of crystals obeys eq. (12), with r r* for the mean residual supersaturation /3. Replacing the volume S2 by the ratio rn/p (molecular weight over density), eq. (12) may be rewritten as

C

5(1

+

2my/k~Tpr),

(29)

which is another way to show that the solubility of a small crystal is larger than the solubility of a large one. If we consider now the crystal which has the critical radius r 6 the difference in solubility between this crystal and any other is 2myC5 I I c. kBTP (30) —

=

(~—

—

—~-)~

A crystal of size r r * must grow, since neither of them is in equilibrium with the solution. According to the theories [35—37],the rate-determining step of Ostwald ripening is often (but not only) volume diffusion, assuming that dissolution and growth kinetics and greater than diffusion kinetics, Here we only mention a few points. Let us consider two crystals of radii r1 and r in a solution at constant temperature T and bulk concentration C.. Since they are not of infinite size and since equilibrium must be achieved, the crystals must (slightly) dissolve in order to restore the proper concentrations. At the end of this process they are surrounded by spherical halos of concentration Cr and Cr respectively (fig. 15). When the distance X between the crystals is not too large, the two concentration spheres intersect with a cross section of area A. Since there is a

=

ln(~r/~s) 2my/k~Tpr, =

(28) Fig. 15. Schematic representation of the solute transfer from

which is the Gibbs—Thomson—Freundlich equation where the subscript r indicates that the equality is fulfilled only for crystals of radius r. E’5

the smallest to the largest crystal during Osiwald ripening. The crystals are separated by the distance X and their concentra-

tion spheres interact with the cross section A.

R. Boistelle, J.P. Astier

/

concentration difference around the crystals, there is also a concentration profile. According to Fick’s law, the mass q of solute transferred per unit time from the smallest to the largest crystal is =

dt

D A (C C,), X D A 2myC5 “X k8Tp r1

(31)

—

‘

=

(±.

dt

—

~),

27

Crystallization mechanisms in solution

(32)

where D~is the coefficient for volume diffusion. Since the finally crystalobtain, of radius r has qa by surface area 2, we replacing r:

while the mean size increases. With increasing time, only one crystal should remain in the systern. Another consequence of this process is habit modification. During growth, supersaturation is generally large as compared to that which exists when ripening takes place. The growth shape slowly transforms into an equilibrium shape which may be completely different. We will return to this supersaturation effect in the next section. 6. Crystal habit

47rr

dr di’

— —

D~myC~A 1 ~i 2irp2k~T~ ~

—

r/

which is the growth rate of the crystal of size r. Actually, in a real system, where there are many crystals, several diffusion lengths and crystal radii must be taken into account and eq. (33) is difficult to solve. With some important simplifications [36,37], and assuming that the critical size r * corresponds to the average size ~ of the crystals in the system, then the highest possible growth rate of the crystals which ripen is D~rnyC~1

ldr\ ~~Jma~

=

2k~Tp2

‘~‘

(34)

Integration of eq. (34) yields: 3D~myC —

5~

=

—

ta).

(35)

As described above, the crystal habit, or overall shape of the crystal, depends on internal factors (structure and bonds) and on external factors (supersaturation and solution composition). It is the consequence of the relative growth rates of the faces: the smaller the growth rate, the more extensive the development of the face, In order to extend a face, the only possibility is to selectively decrease its velocity. This is schematically represented in fig. 16. Even faces which are generally not present on the crystal morphology (K and S faces) may develop, provided that their growth rates become smaller than the growth rates of the other faces. There are different ways to modify the relative growth rates. In the following we summarize the most common possibilities. The first variable which can be used is supersaturation, since the different crystal faces do not have the same behavior in highly or weakly super-

4k BTP

Since r0, the crystal radius at time zero, is often negligible, we may say that the crystal radius increases proportional to the cubic root of time. It is noteworthy that, in general, Ostwald ripening is very fast for crystals of radius less than 1 ~.tm,fast for crystals around 1 ~tm, but very slow as soon as the crystals reach about 100 ~tm. We have, however, only discussed the case of a true isothermal ripening. When temperature fluctuations [38] are applied to the system the ripening kinetics increase drastically, since dissolution and growth rates are accelerated. The main consequence of Ostwald ripening is the disappearance of most crystals, especially the smallest ones. The number of crystals decreases

saturated solutions. When supersaturation is high, the growth units are more rapidly integrated into the faces where their adsorption energy is the

______

Fig. 16. Schematic representation of a habit change; the faces of lowest growth rates develop at the expense of the faces of greatest growth rates.

R. Boistelle, J.P. Astier / Crystallization mechanisms in solution

28

____

IT

Fig. 17. Dendrites or needle-like crsst,ils of harks ts-amylase grown at high supersaturation.

largest. This is especially true when there is an important anisotropy in the bond energies along the different crystallographic directions. The general trend is that needles grow longer while platelets grow wider and thinner with increasing supersaturation. As with many other materials, barley a-amylase crystals are sensitive to supersaturation. They develop as thin intricate dendrites or large crystals according to whether supersaturation is large (fig. 17) or small (fig. 18). The general experimental conditions were described in a recent paper [39]. In our experiments the starting concentrations of barley a-amylase were 3 and 8 mg/mi respectively. In the case of closed and unstirred systems it is, however, difficult to know the exact supersaturation near the crystal. Fig. 19 shows the pronounced concentration differences around crystals growing in a droplet. In this cxperiment we have grown the crystals of barley

f_—.

\

‘

--

-

Fig. 19. Concentration differences around crystals of barley

a-amylase, seen with phase contrast microscopy. The supersaturation close to the crystals is much smaller than that existing in the bulk.

a-amylase using the hanging drop method. When the supersaturation is small, solvent and impurity effects may play a more important role. Actually, the solvent is always present and it must be pointed out that it should be considered as the primary impurity for the crystal. Indeed, the crystal—solvent interaction energies crucially depend not only on the chemical nature of the solvent but also on the nature of the uppermost atoms in the crystal faces. Thus, it is very easy to get octahedra when sodium chloride forms in formamide, which is quite impossible in pure water except only under very special conditions [40]. Since water and formamide are miscible in any proportion, either of them can be considered as the solvent or the impurity. In addition, in both solvents and in any mixture of them,, there is a critical supersaturation which is required for growing octahedra instead of cubes. When the solvent is not pure, which is the case of buffered solutions, it is unlikely that the growth rates are not affected by the solution cornposition. In fig. 2, we have shown the concentration variations of the species containing ammonia sulfate and phosphate. Depending on the pH, the equilibria are displaced and the crystal surface charges are modified. Accordingly, there is no reason why NH3 and HPO,~,for example, must

Fig.

lii. Single crystal of h,irle~ n-arnslase supersaturation.

grown at low

adsorb on the same faces as NH~ and H2PO4 From the example just given, it follows that solution components may also play an important role

R. Boistelle, J.P. Astier

/ Crystallization mechanisms in solution

as habit modifiers, even if they cannot be considered sensu stricto as impurities or additives, Additives are often employed as habit modifiers and, like the buffer components, their efficiency can be greatly affected by the pH of the solution. Nearly inactive in the acidic range, they become very active in the basic range (or conversely). For instance, phosphonic and carboxylic acid derivatives are more active at high pH due to their higher degree of ionization as demonstrated in the case of calcium sulfate [41]. Another typical example concerns the influence of glycine on sodium chloride. Glycine is a good habit modifier only in the range 3.5
29

the impurity molecule can be better fitted to the nature of the crystal face.

Acknowledgements The authors are indebted to R. Haser for helpful discussions about protein crystals and to Mrs. M.C. Toselli and D. Destre for drawing the figures and for typing the manuscript.

References [1] T.A. Arakawa and SN. Timasheff, in: Methods in Enzymology, Vol. 114, Part A, Eds. H.W. Wyckoff, C.H.W. Hirs and S.N. Timasheff (Academic Press, New York, 1985) p. 49. [2] J. Kielland, J. Am. Chem. Soc. 59 (1937) 1675. [3] Vol. R.M.4:Smith and A.E. Martell, (Plenum, Critical Stability Constants, Inorganic Complexes New York, 1976). [41A.E. Martell and R.M. Smith, Critical Stability Constants, Vol. 1: Amino Acids (Plenum, New York, 1974). [51AC. Zettlemoyer, Nucleation (Dekker, New York. 1969). [6] F.F. Abraham, Homogeneous Nucleation Theory (Academic Press, New York, 1974). [7] G. Feher and Z. Kam, in: Methods in Enzymology, Vol. 114, Part A, Eds. H.W. Wyckoff, C.H.W. Hirs and SN. Timasheff (Academic Press, New York, 1985) p. 77. [8] B. Mutaftschiev, Critical Rev. Solid State Sci. 6 (1976) 157. [9] IN. Stranski, Bull. Soc. Franc. Mineral. Cnst. 79 (1956) 359. [10] R. Lacmann and IN. Stranski, in: Growth and Perfection of Crystals, Eds. R.H. Doremus, B.W. Roberts and D. Tumbull (Wiley, New York, 1958) p. 427. [11] H. Naono and M. Miura, Bull. Chem. Soc. Japan 38 (1965) 80. [12] P. Hartman and W.G. Perdok, Acta Cryst. 8 (1955) 49. [13] P. Hartman, in: Crystal Growth: An Introduction. Ed. P. Hartman (North-Holland, Amsterdam, 1973) p. 367. [14] P. Hartman and P. Bennema, J. Crystal Growth 49 (1980) 145. [15] P. Hartman, Geol. Mijnbouw 61(1982) 313.

[16] R. Kaischew, Acta Phys. Hung. 8 (1957) 75. [17] W.B. Hillig, Acta Met. 14 (1966) 1868. [18] G.H. Gilmer and P. Bennema, J. Appl. Phys. 43 (1972) 1356. [19] M. Ohara and R.C. Reid, Modeling Crystal Growth Rates From Solution (Prentice Hall, Englewood Cliffs, NJ, 1973). [20] B. Simon, A. Grassi and R. Boistelle, J. Crystal Growth 26 (1974) 77. [21] H.E. Lundager Madsen and R. Boistelle, J. Crystal Growth 46 (1979) 681.

30

R. Boi ste/Ic, J.P. Astier

/

Crystallization mechanisms in solution

]22] W.K. Burton, N. Cabrera and F.C. Frank, Phil. Trans. Roy. Soc. London A243 (1951) 299. [23] A.A. Chernov, Soviet Phys.-Usp. 1(1961)126. [24] P. Bennema and G.H. Gilmer, in: Crystal Growth: An Introduction, Ed. P. Hartman (North-Holland, Amsterdam, 1973) p. 263. [25] OH. Gilmer, R. Ghez and N. Cabrera. J. Crystal Growth 8 (1971) 79. [26] R. Boistelle, in: Interfacial Aspects of Phase Transformations, Ed. B. Mutaftschiey (Reidel, Dordrecht, 1982) p. 531.

[27] F. Rosenberger, Fundamentals of Crystal Growth I (Springer, Berlin, 1979). 128] R. Boistelle. in: Current Topics in Materials Science, Vol. 4, Ed. E. Kaldis (North-Holland, Amsterdam, 1980) p. 413. [29] R. Kern, Bull. Soc. Franc. Mineral. Crist. 91(1968) 217. [301 R.L. Parker, Solid State Phys. 25 (1970) 151. [31] R.J. Davey J, Crystal Growth 34 (1976) 109.

[321R. Boistelle, in: Interfacial Aspect of Phase Transforma[33] [34]

1351 [36] [37]

lions, Ed. B. Mutaftschiev (Reidel, Dordrecht, 1982) p. 621. W. Ostwald, Z. Physik. Chem. 22 (1897) 289. PT. Cardew and Ri. Davey, Proc. Roy. Soc. (London) A398 (1985) 415. R. Boistelle and C. Rinaudo, J. Crystal Growth 53 (1981) 1. M. Kahlweit, Advan. Colloid Interface Sci. 5 (1975) 1. OW. Greenwood, Acta Met. 4 (1956) 243.

[38] H.H. Hohmann and M. Kahlweit, Ber. Bunsenges. Physik.

Chem. 76 (1972) 933. [39] B. Svensson, R.M. Gibson, R. Haser and J.P. Astier, J. Biol. Chem. 262 (1987) 13682. [40] M. Bienfait, R. Boistelle and R. Kern, in: Adsorption et Croissance Cristalline, Colloq. Intern. CNRS, No. 152 (CNRS, Paris, 1965) p. 577. [41] ME. Tadros and I. Mayes, J. Colloid Interface Sci. 72 (1979) 245.