The control of crystallisation is of fundamental importance in many biological and industrial processes; for instance in biomineralisation, specific mineral polymorphs are selectively crystallised, whilst in oil extraction, the precipitation of barium sulphate crystals within oil pipes is a serious problem. Controlled crystallisation can only be achieved through an understanding of the crystallisation process at the molecular level.

Crystallisation is a condensation process involving the creation of a crystalline daughter phase from a parent phase.

• The formation of any new phase from a bulk parent phase requires the creation of an interface between the two phases.
• The creation of this interface requires an amount of work, dependant upon the surface/interfacial tension of the interface.

To extend the area of the interface, molecules from the bulk interior must be brought to the surface, this requires work to be done against the cohesive forces in the bulk.

Þ In general, the extension of any interface between two homogenous phases will require work.

• The surface tension (or interfacial tension), g , is defined as the work required to create unit area of interface; units Nm-1.
• The surface tension may be considered to arise due to a degree of unsaturation of bonds that occurs when a molecule resides at the surface and not in the bulk.
• The term surface tension is used for solid/vapour or liquid/vapour interfaces.
• The term interfacial tension is more generally used for the interface between two liquids, two solids or a liquid and solid.

• The surface tensions of liquids and gases are isotropic, so the lowest energy form will be the one that minimises the surface area (hence liquid droplets/gas bubbles are spherical).
• However in crystals, the surface free energy is generally anisotropic. Consequently, the equilibrium crystal shape will not generally be spherical, but can be determined by the Gibbs-Wulff theorem.

In 1878, Gibbs proposed that for the equilibrium shape of a crystal, the total surface Gibbs function of formation should be a minimum for a constant volume of crystal, i.e.:

(1)

where: An = area of the nth face and g n = surface tension of the nth face, which is assumed constant over the whole face and independent of the crystal shape.

Wulff (1901) stated that g n/hn = constant, where g n is the surface tension of crystal face n, and hn is the distance from a point in the crystal known as Wulff’s point. We can construct the equilibrium shape of a crystal by the following procedure.

 Wulff plot Draw vectors normal to all possible crystallographic faces from an arbitrary point. The distances µ to g n, are marked on the vectors. Planes normal to the vectors are constructed through the marks. The resulting closed polyhedron is the equilibrium form.

Consider the phase diagram of a single component (simple molecule) system in P-T co-ordinates.

• The crystal phase is stable at high P and low T.
• The liquid phase is stable at high P and high T.
• The vapour phase is stable at high P and low T.
• Two (infinitely large) phases are in equilibrium along the lines and the three (infinitely large) phases are simultaneously at equilibrium at the triple point O.

At equilibrium, the chemical potentials of the phases are equal.

If we increase the pressure on a gas initially at pressure Pa and temperature Ta , then at pressure P0, the gas will be in equilibrium with its crystal. Any further increase in the pressure, say to pressure Pb , will make the gas unstable with respect to the bulk, infinitely large crystal, i.e. m v > m c, the chemical potential of the gas will be higher than that of the infinite crystal.

The difference between the chemical potentials, D m = m v - m c, represents the thermodynamic driving force for the phase transition (in this case crystallisation) to occur, and is known as the supersaturation.

3.1 Different expressions for the supersaturation

 Process Expressions† for the supersaturation, D m Condensation of vapour to liquid or crystal Crystallisation from solution Crystallisation from the melt Formation of gas bubbles in a liquid Electrocrystallisation of metals

Assuming ideal systems and entropy/enthalpy of the transitions are independent of T.

P0 is the equilibrium pressure of the infinitely large condensed phase, C and C0 are the real and equilibrium concentrations of the solute, the D H’s represent the (+ve) enthalpy of the transition, the Ttrans are the equilibrium temperatures for the transition with the subscripts sat, b and m refering to saturation, boiling and melting, respectively, D T = TtransT, z is the valence of the neutralising ions, e is the elementary electric charge and h is the overpotential.

Hence supersaturations can also be expressed in terms of D P, D C, D T or a (relative supersaturation, e.g. P/P0), since all these quantities give a measure of the driving force for the particular phase transition concerned.

3.2 Metastable phases

Obviously, at the initial stages of the phase transition, the new phase cannot be described as infinite; rather the new phase will consist of either small crystallites, droplets or bubbles. These small clusters have large surface area to volume ratios, and since the creation of surfaces/interfaces requires work, they are unstable relative to their bulk phases.

The effects of size on the equilibrium conditions of these small clusters are denoted by the Laplace and Gibbs-Thomson equations.

4. Equilibrium of Finite Phases

4.1 Laplace Equation

(2)

shows that the pressure of a liquid droplet, Pl, is always greater than the surrounding vapour pressure, Pv, by the amount 2g /r, known as the Laplace or capillary pressure.

When the phase boundary is flat (r ® Ą ), the capillary pressure is zero and Pl = Pv.

4.2 Gibbs-Thomson Equation

(3)

where: vl is the molecular volume of the liquid droplet. For the condensation of a vapour to a liquid, D m = kT ln(P/P0), so we have:

(4)

The equation illustrates that the vapour pressure of a liquid droplet increases with decreasing droplet radius. The physical reason for this is evident when one considers that an atom on a curved convex surface will be more weakly bound than one on a flat surface, i.e. more unsaturated bonds occur (see schematic diagram below).

Application of the Gibbs-Thomson equation to water droplets at 293 K gives the following values of P/P0:

 r /mm 10-3 10-4 10-5 10-6 P/P0 1.001 1.011 1.114 2.95

(Walter J. Moore, "Physical Chemistry", p 482)

If we consider the reverse situation of a gas bubble in a liquid, then:

(5)

i.e. the equilibrium vapour pressure of a gas bubble in a liquid decreases with the bubble radius. This may be understood since the convex liquid surface results in the surface atoms being bound more firmly (fewer unsaturated bonds), hence the vapour pressure decreases.

For the formation of an equilibrium crystal, a more detailed analysis of the Gibbs-Wulff theorem shows that:

(6)

where: vc is the molecular volume of the crystal. Eq. (6) also has the familiar form of the Gibbs-Thomson equation.

The Gibbs-Thomson equation illustrates the difficulty in creating a daughter phase from a bulk parent one since there exists an energy barrier to the formation of the new phase (which is due to the relatively large surface energy that is required to create the tiny nuclei).

Consider a water vapour cooled so that the vapour is supersaturated with respect to liquid water.

• Initial stage of condensation requires the aggregation of water molecules to form a tiny droplet, or nucleus, that then grows by additional vapour molecules impinging upon it.
• However, the Gibbs-Thomson equation indicates that this nucleus, and others like it, will have a vapour pressure many times that of bulk liquid.

Þ vapour will not be supersaturated with respect to these nuclei and so these nuclei will tend to evaporate.

The water vapour is in a metastable state, due to the existence of the energy barrier to the formation of the new phase.

So how does the new phase form?

In any thermodynamic system, statistical fluctuations about the normal state occur. Consequently, the new phase will be able to form provided a positive free energy fluctuation occurs that is equal to, or larger than, the magnitude of this energy barrier.

Homogeneous nucleation refers to nucleation of a species within a single component phase.

Heterogeneous nucleation refers to nucleation of a species upon a foreign substrate.

Crystal growth theories are concerned with the growth of the crystal nuclei to macroscopic dimensions.

5. Classical Homogeneous Nucleation Theory

(see e.g. Gibbs, Volmer, Becker and Döering, Frenkel)

In nucleation, only aggregates of a size greater than a critical value will be stable and grow to become the condensed phase; aggregates of this size are termed critical nuclei.

In classical nucleation theory, the rate of nucleation, J, is considered to result from a two step process:

1. the formation of a near equilibrium concentration, n(i*), of critical nuclei and
2. the impingement at a rate, W*, of the condensable species upon these nuclei, which causes them to become essentially free-growing and thus eventually bulk, condensed phase.

Thus:

. (7)

5.1 Determination of n(i*), the equilibrium concentration of critical nuclei

(8)

5.2 Determination of D G*, the Gibbs free energy of formation of the critical nucleus

The free energy of formation, D Gi, of a cluster from i monomers in the vapour is:

(9)

For a spherical cluster, Eq. (9) becomes:

(10)

where: r is the radius of cluster and vc is the molecular volume of the cluster.

 A graph of D Gi against r shows a maximum at r = r*, the radius of the critical nucleus, so that differentiating D Gi with respect to r and equating to zero, i.e.: gives: (11) i.e. the Gibbs-Thompson equation.

Substituting Eq. (11) into (10) gives the Gibbs free energy of formation of the critical nucleus in terms of the supersaturation as:

. (12)

or in surface free energy terms as:

(13)

Equation (13) is universal, as shown below.

The Gibbs free energy of formation of the critical nucleus for a crystal is given by:

(14)

where: Vc = volume of the crystalline critical nucleus. The volume of any crystal can be considered as the sum of the volumes of pyramids constructed on the crystal faces with a common apex in an arbitrary point (the Wulff point) within the crystal, hence:

(15)

where hn are the heights of the pyramids and An are the surface areas of the corresponding crystal faces. Substituting for Vc* and hn in Eq. (13) using Eqs. (15) and (6) gives:

i.e. Eq. (13) as before.

5.3 Determination of W*, the impingement rate

The impingement rate W*, is given by:

(16)

where: R* = adsorption flux, i.e. the frequency with which condensing molecules reach unit area of the critical nucleus and s* = surface area of the nucleus, i.e. 4p r2 for a spherical nucleus.

5.4 Overall Classical Homogeneous Nucleation Rate

Combining Eqs. (7), (8), and (12), the homogeneous nucleation rate can be written as:

or

(17)

where: W is known as the pre-exponential factor. W is not very dependent upon the supersaturation, compared with the exponential factor, and therefore one can consider W as nearly a constant. Typical values of W are ~1025 -1035 cm-3s-1.

 The form of Eq. 17 is such that J remains negligibly small until the supersaturation reaches a critical value, the Ostwald metastable limit, at which point J suddenly and dramatically increases. Hence, this critical supersaturation level can be arbitrarily set as the rate at which , i.e. the rate is 1 nucleus cm-3sec-1 (or any other suitable experimentally detectable limit) with little loss of accuracy.

The main assumptions of classical nucleation theory are:

i) equilibrium conditions and

ii) the validity of ascribing the macroscopic properties such as surface tension to the critical nucleus.

Hence classical nucleation theory is most applicable to systems at low to moderate supersaturations.

5.5 Improvements to classical homogeneous nucleation theory

Zeldovich factor, Z, gives instead of the "equilibrium" concentration of critical nuclei, the steady state concentration.

5.5.2 Applicability of macroscopic surface tension to small nucleus sizes

• For spherical particles the surface tension will tend to decrease with decreasing size.
• For anisotropic crystals (where the critical nucleus will not in general be spherical), edge and corner effects may also become important for very small nuclei.

5.5.3 Statistical mechanical terms

The replacement term.

5.6 Atomistic approaches

At high supersaturations, where the critical nuclei may contain only a few atoms or molecules, the assumptions of classical nucleation theory are not valid.

 In the atomistic approach, macroscopic terms such as the surface tension are avoided, and the free energy of the critical nucleus (which contains relatively few atoms) is computed directly, nowadays by computational techniques such as vibrational frequency calculations, Monte Carlo simulations and Molecular Dynamics simulations.

5.7. Homogeneous nucleation experiments

One way to help ensure homogeneous nucleation occurs in the case of crystallisation from a solution/melt is to disperse the solution/melt as droplets.

Typically, nucleation experiments measure the nucleation rate variation with supersaturation. A plot of lnJ vs. 1/D m 2 will give a straight line of gradient -16p g 3vc2/3kT and an intercept of lnW , provided classical nucleation theory is applicable. Hence g and W may be determined.

Alternatively, the critical homogeneous supersaturation limit may be found by experiment, and g and W can then be calculated by setting lnJ to zero.

6. Classical Heterogeneous Nucleation Theory

• Nucleation occurs upon a foreign substrate,
• at a greater rate than homogeneous nucleation.
• Similarly to the homogeneous case, the heterogeneous nucleation rate, Jhet, may be expressed by:

. (analogous to Eq. 7)

6.1 Determination of n(i*), the equilibrium concentration of critical nuclei

n(i*) is given by:

(18)

where: n0 is the adatom (adsorbed atom) concentration on the substrate and D Ghet* is the Gibbs free energy of formation of the critical nucleus.

6.2 Determination of D G*, the Gibbs free energy of formation of the critical nucleus

The value of D Ghet* depends upon the shape of the critical nucleus upon the substrate.

• Isotropic nuclei are cap-shaped,
• Anisotropic nuclei may be disc-shaped or polygonal, depending upon the extent of anisotropy.

There is also a statistical contribution to D Ghet*, which is independent of the nucleus size or shape, that accounts for the distribution of the clusters and single adatoms among the adsorption sites of density ns. This term, D Gconf is given by:

(19)

Including this D Gconf term in Eq. (18) gives:

(20)

where: D Ghet* is the Gibbs free energy of formation of the critical nucleus, excluding the D Gconf contribution..

6.2.1 Cap-shaped nuclei

The contact angle, q , may be determined by treating the interfacial tensions as forces and balancing them in the plane of the substrate, leading to Young's equation:

. (21)

Employing the classical technique of considering the cluster as bulk crystal with terms included to account for its surface, gives:

. (22)

Substituting for g sx-g sx using Young’s Eq., this gives:

where:. Maximising D Gi with respect to r then leads to:

i.e. the Gibbs-Thomson equation once more and

. (23)

where: D Ghom is the Gibbs free energy of formation for homogeneous nucleation of this system and Vhet* and Vhom* are the volumes of the heterogeneous and corresponding homogeneous critical nuclei, respectively.

• For q = 180° , nucleation does not occur on the substrate and the homogeneous value of
D G* is obtained.
• If q = 0° , complete wetting occurs and this model then predicts no nucleation barrier. For q = 0° , though, the critical nucleus will be a 2-D monolayer disc and so Eq. (23) will not be applicable for such cases of very strong adsorption of adatoms onto the substrate.

6.2.2 Disc-shaped nuclei

Such a disc shape may occur if either:

1) adatom adsorption on to the substrate is very strong, producing a monolayer 2D disc or

2) the nucleus is 3-dimensional and anisotropic.

The Gibbs free energy of formation for either shape will be:

. (24)

For the monolayer disc, h is a constant equal to the height of the atom perpendicular to the substrate surface, whilst for the anisotropic, less strongly adsorbed nucleus, h will be determined according to the Gibbs-Wulff law.

6.2.2.(i) Monolayer discs

Maximising D Gi with respect to r leads to:

(25)

and

. (26)

Notice that in this case, D Ghet* is equal to half the edge free energy terms, since the critical nucleus is a 2D, as opposed to a 3D, equilibrium shape. Any other 2D critical nucleus equilibrium shape will also have D Ghet* equal to half the edge free energy terms.

6.2.2.(ii) Anisotropic 3-dimensional nuclei

TRY YOURSELVES!

6.3 Determination of W*, the rate of impingement

The impingement of atoms onto the growing nucleus may occur by either:

1) direct impingement of atoms from the vapour onto the nucleus surface (as in
homogeneous nucleation) or

1. adatom impingement by the surface diffusion of adatoms to the nucleus periphery.

For direct impingement, the rate, Wd*, is the same expression as for the homogeneous nucleation case, i.e.:

(i.e. Eq. 16)

where: s* = surface area of the critical nucleus.

For adatom impingement, the rate, Ws*, is:

. (27)

where: l* = circumference of nucleus on substrate surface, a0 = adatom jump distance, D Gdes= free energy of activation for desorption and D Gsd = Gibbs free energy of activation for surface diffusion.

For nucleation from the vapour, the adatom impingement rate > direct impingement rate, hence the impingement rate is given by Eq. (27). For nucleation from solution/the melt, however, the reverse is often true, in which case the impingement rate will be given by Eq. (16).

6.4 Overall Classical Heterogeneous Nucleation Rate

Combining Eqs. (7) and (20), the classical heterogeneous nucleation rate becomes:

or:

(28)

where: W het is the pre-exponential factor. As with the homogeneous nucleation case, W het is also not very dependent upon the supersaturation.

 Nucleation Process Typical W het values (cm-2s-1) Condensation from the vapour 1017 Crystallisation from sol. 1020 Crystallisation from the melt 1022

Set the critical supersaturation level for heterogeneous nucleation to the rate at which , i.e. the rate is 1 nucleus cm-2sec-1.

6.5 Improvements to Classical Heterogeneous Nucleation Theory

Include terms to account for the following factors if they are significant:

(i) the steady state concentration of critical nuclei, i.e. include Z factor,

(ii) thermal non-accommodation,

(iii) quasi-adsorption of impingement atoms, (this is where an adatom re-evaporates before
equilibrating with the surface),

(iv) time-dependent nucleation,

(v) correction for small critical nuclei at high supersaturations,

(vi) the critical nucleus may have some relatively large additional energy terms due to strain
or dislocations, if the lattice dimensions of the substrate and critical nucleus differ greatly.

6.6 Atomistic Approach

As in homogeneous nucleation, the atomistic approach is more applicable at high supersaturations, where the critical nuclei may contain only a few atoms or molecules.

6.7 Heterogeneous nucleation experiments

Heterogeneous nucleation experiments also typically measure the nucleation rate variation with supersaturation.

• lnJhet vs. 1/D m 2 will give a straight line for any 3D nucleus. The gradient will be
-sg 2vc2/kT (where s is a factor depending upon the shape of the critical nucleus and the interfacial tensions of the system) and the intercept will be lnW het (provided classical nucleation theory is applicable). Hence the value of the g and W het may be determined.
• Alternatively, the critical heterogeneous supersaturation limit may be found by experiment, and g and W het can then be calculated by setting lnJhet to zero.

7. Epitaxy

The oriented growth of a crystalline material on a different material is termed epitaxy.

The epitaxial growth of thin films is the basis of the fabrication of numerous modern devices.

Epitaxy is also important in biomineralisation, where growth of an inorganic mineral (such as CaCO3) occurs on an organic matrix. The influence of the substrate can often be sufficient to induce the crystallisation of not only a specific orientation, but also a specific crystal polymorph.

A great variety of methods for epitaxial deposition of different materials have been invented. These include Chemical Vapour Deposition(CVD), Molecular Beam Epitaxy (MBE), Liquid Phase Epitaxy (LPE) and Atomic Layer Epitaxy (ALE) among others.

7.1 Epitaxial theories

• Heterogeneous nucleation theories consider whether a particular body will induce nucleation of another.
• Epitaxial theories consider the conditions under which a single overgrowth orientation minimises the interfacial energy of the system.

Epitaxial growth differs from normal crystal growth (i.e. the growth of a material upon the same material) in that the substrate and overgrowth crystals have different chemical potentials. This difference arises in two fundamental ways:

1. the substrate and overgrowth may differ in the nature and strength of their chemical bonds
2. the substrate and overgrowth may differ in their lattice structures.

The lattice mismatch (or misfit), m, between the substrate and overgrowth in a particular direction is defined as:

(30)

where: a and b are the lattice parameters of the substrate and overgrowth, respectively.

The chemical bonding between the substrate and overgrowth will determine the maximum interaction that can occur between the two, whilst the lattice mismatch will determine the spatial variation of the interaction.

Factors aiding epitaxy:

• strong substrate-overgrowth interaction
• small lattice mismatch
• high compressibility substrate/overgrowth

Different epitaxial theories have been developed to account for different overgrowth thicknesses and the manner in which any lattice mismatch is accommodated.

7.1.1 Misfit accommodation

If the overgrowth-substrate interaction is sufficiently strong, an overgrowth monolayer may be constrained to match the crystal structure of the substrate completely, i.e. a coherent, or pseudomorphic, monolayer occurs. For this situation, the misfit is accommodated entirely by the homogeneous strain of the overgrowth.

With either increased overgrowth-overgrowth bonding, or misfit, the coherent monolayer will become unstable, and misfit dislocations are then introduced to relieve this homogeneous strain. The misfit dislocations comprise regions of good fit between the substrate and overgrowth, producing the maximum substrate-overgrowth interaction and regions of bad fit, "the dislocation", in which this interaction is poor.

As the monolayer extends in height, the interaction between the substrate and subsequent overgrowth layers decreases, whilst the energy required to homogeneously strain the overgrowth into coherency remains practically constant per additional layer. Hence for any initially pseudomorphic monolayer, there will be a critical height at which the coherent structure becomes unstable and misfit dislocations are introduced to relieve the homogeneous strain, until eventually all this strain has disappeared and the misfit is accommodated entirely by misfit dislocations.

Finally, when the overgrowth-substrate interaction becomes so weak, compared to the overgrowth-overgrowth forces, each adatom will then be located at its usual crystal lattice position. There is no strain whatsoever, and the misfit is accommodated entirely by a misfit vernier.

7.2 Modes of epitaxial growth

The dependence of the chemical potential, m , of the overgrowth on its thickness, n, constitutes the main difference of the epitaxial growth from the usual crystal growth of a material, and leads to three well-known modes of epitaxial growth:

1. Volmer-Weber mechanism of island growth
2. Frank-van der Merwe mechanism or layer-by-layer growth
3. Stranski-Krastonov mechanism, or layer-by-layer followed by formation of 3D islands.

8. Crystal Growth

The mechanism of crystal growth is unambiguously determined by the structure of the crystal face.

8.1 Classification of Crystal Faces

The equilibrium shape of crystals is bounded by the crystal faces with the lowest specific surface energies.

• Faces with the lowest surface tensions will typically be the most close-packed, be atomically flat, and have low Miller indices. Why?

A crystal face at a small angle to one of these low energy faces, however, will not be atomically flat; but will consist of terraces and steps. In fact, crystal faces can be divided into three groups.

 F (flat) faces. These are parallel to at least two dense atomic rows. S (stepped) faces. These are parallel to one dense atomic row. K (kinked) faces. These are not parallel to any dense atomic rows.

F faces are generally atomically flat. However, as temperature is increased and entropic factors become more important, F faces undergo a surface roughening transition at a critical temperature known as the roughening temperature, Tr.

The atomically rough crystal faces (i.e. S, K and F faces at temperatures above Tr) grow by a different mechanism to the atomically flat faces (F faces at temperatures below Tr).

 Position of atom No. of saturated bonds Within Face (1) 5 Within Step (2) 4 Within Kink (3) 3 Upon Step (4) 2 Upon face (5) 1

The detachment of these atoms will lead to a change in surface energy of the crystal for all but the atom at the kink position. At this position, the atom has 3 saturated bonds, and 3 unsaturated bonds. In all crystals, the kink site has the following properties:

• It determines the equilibrium of an infinitely large crystal with the ambient phase. Why?
• Its chemical potential is equal to the chemical potential of the crystal.
• Its desorption energy of the kink site is equal to the crystal enthalpy of evaporation.
• If a system is supersaturated, it will be favourable for an adatom arriving at the kink site to be incorporated into the crystal.

8.3 Crystal growth mechanisms

8.3.1 Continual growth mechanism for atomically rough faces

1. If a face is an S face, K face, or F face at a temperature above Tr, it is atomically rough and can grow continuously. Why?

• Growth rate is fast and is simply proportional to the flux of atoms from the parent phase, i.e. the growth is diffusion controlled. The rate of crystal growth is given by:

(31)

where: l is the kinetic coefficient. Hence the growth rate depends linearly upon the supersaturation. The kinetic coefficient is proportional to the surface roughness (in terms of the probability of finding a kink site), and to the exponent of the activation energy, D U, for incorporation of a building unit into the lattice. Typical values of R are 10-4 - 10-1 cm sec-1.

• Crystals that grow by the continual growth mechanism tend to have rounded morphologies. Why?

Atomically flat F faces (i.e. F faces at temperatures below Tr), however, cannot grow by this mechanism.

8.3.2 Layer growth of flat faces

1. Consider an atomically flat crystal face belonging to perfect defectless crystal. A single adatom on such a face is bound more weakly than an adatom in a cluster of adatoms on the surface. Consequently, there is an energy barrier to the formation of each new crystal layer. This situation is the 2D analogue of homogeneous nucleation, and hence the rate of growth of this face will be determined by the frequency of the formation of 2D critical nuclei.
2.

 If the F crystal face is not perfect, and in particular if screw dislocations are present, then these screw dislocations represent a non-vanishing source of steps which alleviate the necessity of a 2D nucleation growth mechanism. Instead the rate of crystal growth is determined by the rate of the lateral movement of the steps.

8.3.2.(i) 2D nucleation growth

The growth of a defectless, atomically smooth crystal face is a periodic process involving:

1. formation of a 2D nucleus (the rate determining step)
2. lateral spreading of this nucleus until the crystal face is completely covered. Further growth then requires the formation of a 2D nucleus again.

Similarly to the 3D nucleation case, the rate of 2D nucleation is given by:

(32)

where: W 2D is the pre-exponential factor for 2D nucleation and D G2D* is the Gibbs free energy for formation of the 2D critical nucleus. It can be shown that (TRY IT!):

(33)

Thus R becomes:

(34)

The rate has a non-linear dependence on the supersaturation. Once more, the form of Eq. (34) is such that growth will not proceed at a finite rate until a critical supersaturation is reached.

At high supersaturations, the face may grow by a multilayer mechanism due to the formation of multiple 2D nucleation sites. The exponential now becomes (-D G2D*/3kT).

With increasing supersaturation, the 2D critical nucleus becomes increasingly small in size, so that only a few atoms are needed to form each critical nucleus. The density of 2D critical nuclei may then become so large that arriving atoms can be incorporated practically at any site.

Þ The surface becomes atomically rough so growth can now proceed continuously (R µ D m ).

This roughening of the crystal face, which occurs at very high supersaturations well below Tr is known as kinetic roughening, to distinguish it from the thermodynamic roughening transition that occurs at Tr.

8.3.2.(ii) Screw dislocation growth

Experiments showed that growth of faceted crystals could occur at relative supersaturations, a (P/P0), as low as 0.01%, far below the critical supersaturation limit for 2D nucleation growth.

Þ a different growth mechanism must be operating.

Burton, Cabrera and Frank (BCF) developed the theory of screw dislocation crystal growth and found that:

(35)

where: s is the supersaturation parameter D m /kT, s c is the characteristic supersaturation of the system and C is a rate constant. Two limiting cases can be distinguished:

At low supersaturations, s c >> s , so tanh(s c/s ) ® 1, hence:

(36)

which is the BCF parabolic law.

At supersaturations sufficiently higher than s c, tanh(x ® 0) = x, hence:

(37)

and the BCF linear growth rate is obtained.

 The parabolic law holds up to the characteristic supersaturation, s c, beyond which a linear relationship is gradually established. The density of kinks on the surface is then so high that the face becomes rough and so a continuous growth mechanism occurs.

8.4 Growth regimes

• S and K faces always grow by the continual growth mechanism, so R µ D m .
• F faces will grow by this mechanism at temperatures above Tr or if the supersaturation is high, since the F faces are then atomically rough.
• At temperatures below Tr and at lower supersaturations, F faces grow by either the 2D nucleation or spiral growth mechanism and a non-linear growth rate is obtained.

Consequently, we can distinguish different growth regimes for an F crystal face as shown below.

8.5 Crystal growth experiments

Electrodeposition experiments:

1. Electric current, I, gives the growth rate, R.
2. The amount of material deposited is given by the quantity of electricity that has flowed.
3. The supersaturation D m is given by the overpotential, h .

A single crystal with a specific orientation can be placed in contact with the electrolyte solution, and so the growth of both screw-dislocation-free crystal faces and faces with dislocations can be studied.

1) Constant overpotential

2) Constant current

Spiral growth mechanism

Plot of I vs. overpotential squared is a straight line, unless the crystal face is rough.

MBE (Molecular Beam Epitaxy) is a powerful method for the investigation of crystal growth processes, particularly when used in conjunction with a surface analytical technique such as RHEED (Reflection High Energy Electron Diffraction).

RHEED experiments can be used to determine whether a 2D nucleation, layer by layer, growth mechanism is operating. This mechanism is characterised by a periodic variation in the growth rate, and hence the RHEED intensity will oscillate with a time period corresponding to the growth of one monolayer.

In general, good agreement is observed between experiment and crystal growth theories.

8.6 Determination of roughening temperatures

Since an F face below and above Tr grows by entirely different mechanisms, it would be advantageous to know Tr. Is there a parameter that will give an indication of Tr?

Þ D Strans.

• For D Strans/R < 1.2, the face is atomically rough for all D m values.
• For D Strans/R > 3.3, the face is atomically flat for all D m values.
• For 1.2 < D Strans/R < 3.3, the lower the D m value, the more likely the face is to be flat.

In addition, an expression for the transition temperature has been given as (Chui and Weeks 1978, Fischer and Weeks1983):

(38)

where: g hkl is the surface tension of crystal face (hkl), and dhkl is its interplanar spacing (Chui and Weeks 1978, Fischer and Weeks1983).

8.7 Morphological instability and constitutional supercooling

Consider a chance fluctuation that gives rise to a protrusion on the crystal interface. Under stable conditions, further growth of this protrusion will be unfavourable and so the interface will return to a smooth shape. Under unstable conditions, further growth of the protrusion will be favoured, and so the interface will become distorted. This is known as morphological instability.

Constitutional supercooling occurs when the supersaturation or supercooling increases with distance into the parent phase. Consequently, the rate of growth of any protrusion will be greater than the growth rate of the rest of the crystal interface and hence morphological instability occurs. Constitutional supercooling typically occurs at high supercoolings /supersaturations due to the liberated enthalpy of crystallisation raising the temperature of the interface, and hence reducing the supercooling there.

8.7.1 Typical morphologies that can occur under unstable conditions

9. Crystal morphology

The ability to influence the rate of crystal growth, and the resulting morphology can have important consequences, e.g.

• In oil recovery, crystallisation of barium sulphate crystals in the pipes reduces the oil flow-rate, hence impeding the growth rate of these crystals is highly desirable.
• In the semiconductor industry, layers of a particular morphology are often required.
• Biomineralisation relies on the ability of an organic matrix in the organism to promote the crystallisation of specific morphologies or polymorphs of inorganic minerals.

The equilibrium form of a crystal is governed by:

. (Gibbs-Wulff law, Eq. 1)

However, the equilibrium shape is only likely to occur for sub-microscopic crystals. As a crystal grows, its surface area to volume ratio decreases so the difference in the surface energy of various crystal forms becomes less significant. Kinetic factors, such as the relative growth rates of individual crystal faces, will dominate.

9.2 Predicting crystal morphologies

For surface controlled growth, Hartman and Bennema (1980) have shown that the growth rate, R, of a face increases with increasing attachment energy, Eatt, i.e.

R µ Eatt. (39)

The attachment energy is defined as the interaction energy per molecule between a slice (hkl) and the crystal face (hkl) so Eatt = S Ei where Ei = interaction energy per molecule of a slice of thickness, dhkl, with the ith under/overlying slice.

Crystal faces with the lowest Eatt values will become dominant in the resulting crystal habit.

The surface tension represents to a first approximation (ignoring any entropic and surface reconstruction effects), the degree of unsaturation of bonds per unit area it may be seen that:

(40)

where: Ahkl = surface area of a molecule in the crystal face (hkl). Since Ahkl = v/dhkl, Eq. (40) becomes:

(41)

i.e. .

• Equilibrium form: faces with the lowest g values dominate the crystal habit.
• Microscopic and macroscopic crystals grown under surface control, faces with the lowest Eatt values dominate the resulting habit.
• For continuous growth (i.e. diffusion controlled growth), the crystal habits will tend to be rounded, as all crystal faces are rough and predicted to grow at the same rate.
• At very high supersaturations, morphological instability can occur and then dendritic, Hopper, cellular etc. growth forms may occur.

The addition of ‘tailor-made’ additives to a supersaturated melt/solution can strongly influence the resulting crystal morphology. Tailor-made additives are ‘designer’ impurities that have one part that resembles the crystallising species, and another part that differs significantly from it. Hence, these additive have sufficient molecular compatibility to be adsorbed onto specific crystal faces, but once incorporated, further growth of the face is impeded due to the difference between the additive and crystallising species.

An effective tailor-made additive that impedes the crystal growth of a specific crystal face, may conversely also be an efficient nucleator of that face if the additive is surface active.

10. Polymer crystallisation

The crystallisation of polymers introduces some specific factors that will now be considered.

10.1 Chain folding

Single crystals of polymers can be obtained from dilute solution. Such crystals, or lamellae, have lamellar thicknesses, l, only in the range of typically 5-10 nm. X-ray diffraction showed that the polymer chains run along (or almost along) this thin dimension of the lamellae, and since the polymer chains are >> 10 nm long, this means the polymer chain must fold back and forth many times in forming the lamellae.

Consequently, polymer crystallisation occurs via a chain folding mechanism.

• The crystallisation rate of polymers from solution is virtually independent of the polymer concentration (i.e. R µ Cx, where x is typically only ~ 0.3 for high molecular weights).

This is due to the connectivity of a polymer chain; any portion of a polymer chain will have other polymer chain segments around it, and hence it effectively always experiences a certain local concentration of other polymer segments that is independent of the overall bulk polymer concentration.

10.2 Growth theories

Any theory on the crystallisation of polymers must account for the following main experimental observations.

1. l µ 1/D T .
2. Crystals grow laterally at a constant rate given by R µ exp(-K/TD T), where K is a constant for a particular system and D T regime.
3. At low D T, curved crystals develop. At high D T, faceted crystals occur.

Since the extended chain crystal is not observed in polymer single crystals, this demonstrates that polymer crystallisation theories must be kinetic in origin. The growth rate is regarded as the sum of two opposing factors:

1. the thermodynamic driving force that favours large l
2. a barrier term which favours small l.

Two main theories have developed: the surface nucleation models (e.g. Hoffman, 1960) and the entropic barrier models (e.g. Sadler & Gilmer, 1984).

10.2.1 Surface nucleation models

Consider a lamellar crystal containing i monomers (each of volume v), of an infinitely long polymer grown from the melt.

 The Gibbs free energy change of formation of this lamellar is given by: (42) Since the fold surface area is far greater than the edge surface area, the contribution S Aeg e can be neglected, so: (43) At the melting point of this lamellae, D G = 0, and so: (44)

Eq. (44) has the familiar form of the Gibbs-Thomson equation and shows that l µ 1/D T (since D m µ D T), in agreement with experiment. Eq. (45) can be rewritten as:

(45)

where T0m is the theoretical melting point of the extended crystal and D Hmelt is the enthalpy of melting per monomer. Eq. (45) is known as the Hoffman-Weeks equation. It shows that:

• a melting point depression occurs due to small lamellar thicknesses,
• the equilibrium, lowest energy form of a polymer crystal is the extended chain (i.e. unfolded) crystal.

This model is entirely analogous to the 2D nucleation mechanism of crystal growth for small molecules. As before, different growth regimes can be distinguished depending upon the rate, i, at which the nuclei form and the spreading rate, g.

 Regime I. Mononucleation: i << g, so R = biL L = lateral length of crystal, b = stem thickness. Regime II. Polynucleation: i g and R = b(2ig)1/2. Regime III. i > g The ‘nucleation rate’ is so high that lateral spreading is not required. R µ i.

In all three regimes, the rate depends on i, which has an exponential dependence on TD T. g » constant. Hence R µ exp(-K/TD T) in all cases, as required, with different values of K for each growth regime.

But the surface nucleation models do not predict curved faces at low D T. Why not?

10.2.2 Entropic barrier models

• Rounded crystal faces at low D T Þ crystal faces are rough. However, unlike the crystal growth of small molecules, there still exists a barrier to growth for these rough polymer crystal faces (i.e. R µ exp(-K/TD T), and not R µ D T).
• In the entropic barrier models, it is considered that this barrier to crystallisation arises since a polymer chain segment can attach itself in different ways to the crystal face, but not all of these configurations will favour further crystal growth. Hence a configurational entropy barrier exists in polymer crystallisation, and since the number of unfavourable chain segment configurations will increase with l, the entropic barrier also increases with l and hence with 1/ D T.

In this model, growth occurs in 3D and hence to limit the lamellar thickness, l, certain ‘pinning rules’ are introduced, namely:

1. there is a very low probability of chains becoming extended at the growing surface before becoming incorporated into the crystal,
2. once a chain is incorporated into a crystal in a folded state, it is unable to escape and become extended.

This model relies on computer simulations.

10.2.3 Comparison of models

 Property Surface Nucleation Models Entropic Barrier Models Thin lamellar crystals 3 3 l µ 1/D T 3 3 R µ exp(-K/TD T) 3 3 Morphology-T dependence 7 3 Applicability to all systems ? 3

At present, the entropic barrier models appear to predict the experimental observations better. However, debate is still in progress, and various improvements to both theories have been suggested. In addition, a theory proposing segment incorporation based on ‘size, shape and force’ recognition has also been suggested, which leads to molecular weight segregation during growth.