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.
2. Surface and Interfacial Tension
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.
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}/h_{n} = constant, where g _{n} is the surface tension of crystal face n, and h_{n} 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 

3. The Equilibrium of Infinitely Large Phases, Supersaturation, Supercooling, and Metastability.
Consider the phase diagram of a single component (simple molecule) system in PT coordinates.
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 P_{0}, 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 
†Assuming ideal systems and entropy/enthalpy of the transitions are independent of T.
P_{0} is the equilibrium pressure of the infinitely large condensed phase, C and C_{0} are the real and equilibrium concentrations of the solute, the D H’s represent the (+ve) enthalpy of the transition, the T_{trans} are the equilibrium temperatures for the transition with the subscripts sat, b and m refering to saturation, boiling and melting, respectively, D T = T_{trans}–T, 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/P_{0})_{, }since all these quantities give a measure of the driving force for the particular phase transition concerned.
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 GibbsThomson equations.
4. Equilibrium of Finite Phases
4.1 Laplace Equation
(2)
shows that the pressure of a liquid droplet, P_{l}, is always greater than the surrounding vapour pressure, P_{v}, 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 P_{l} = P_{v}.
4.2 GibbsThomson Equation
(3)
where: v_{l} is the molecular volume of the liquid droplet. For the condensation of a vapour to a liquid, D m = kT ln(P/P_{0}), 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 GibbsThomson equation to water droplets at 293 K gives the following values of P/P_{0}:
r /mm 
10^{3} 
10^{4} 
10^{5} 
10^{6} 
P/P_{0} 
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 GibbsWulff theorem shows that:
(6)
where: v_{c} is the molecular volume of the crystal. Eq. (6) also has the familiar form of the GibbsThomson equation.
The GibbsThomson 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.
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:
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 v_{c} 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) 
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: V_{c }= 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 h_{n} are the heights of the pyramids and A_{n} are the surface areas of the corresponding crystal faces. Substituting for V_{c}^{* }and h_{n} 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 r^{2} 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 preexponential 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 ~10^{25} 10^{35} cm^{3}s^{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^{3}sec^{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
5.5.1 Steadystate nucleation rate
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
5.5.3 Statistical mechanical terms
The replacement term.
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 ^{3}v_{c}^{2}/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
. (analogous to Eq. 7)
6.1 Determination of n(i*), the equilibrium concentration of critical nuclei
n(i*) is given by:
(18)
where: n_{0} is the adatom (adsorbed atom) concentration on the substrate and D G_{het}* 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 G_{het}* depends upon the shape of the critical nucleus upon the substrate.
There is also a statistical contribution to D G_{het}*, 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 n_{s}. This term, D G_{conf} is given by:
(19)
Including this D G_{conf} term in Eq. (18) gives:
(20)
where: D G_{het}* is the Gibbs free energy of formation of the critical nucleus, excluding the D G_{conf} contribution..
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 GibbsThomson equation once more and
. (23)
where: D G_{hom} is the Gibbs free energy of formation for homogeneous nucleation of this system and V_{het}* and V_{hom}* are the volumes of the heterogeneous and corresponding homogeneous critical nuclei, respectively.
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 3dimensional 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 GibbsWulff law.
Maximising D Gi with respect to r leads to:
(25)
and
. (26)
Notice that in this case, D G_{het}* 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 G_{het}* equal to half the edge free energy terms.
6.2.2.(ii) Anisotropic 3dimensional 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
For direct impingement, the rate, W_{d}*, 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, W_{s}*, is:
. (27)
where: l* = circumference of nucleus on substrate surface, a_{0} = adatom jump distance, D G_{des}= free energy of activation for desorption and D G_{sd} = 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 preexponential factor. As with the homogeneous nucleation case, W _{het} is also not very dependent upon the supersaturation.

Nucleation Process 
Typical W _{het} values (cm^{2}s^{1)} 

Condensation from the vapour 
10^{17} 

Crystallisation from sol. 
10^{20} 

Crystallisation from the melt 
10^{22} 
Set the critical supersaturation level for heterogeneous nucleation to the rate at which , i.e. the rate is 1 nucleus cm^{2}sec^{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 nonaccommodation,
(iii) quasiadsorption of impingement atoms, (this is where an adatom reevaporates before
equilibrating with the surface),
(iv) timedependent 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.
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.
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 CaCO_{3}) 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.
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:
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:
Different epitaxial theories have been developed to account for different overgrowth thicknesses and the manner in which any lattice mismatch is accommodated.
If the overgrowthsubstrate 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 overgrowthovergrowth 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 substrateovergrowth 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 overgrowthsubstrate interaction becomes so weak, compared to the overgrowthovergrowth 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.
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 wellknown modes of epitaxial growth:
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.
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 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, T_{r}.
The atomically rough crystal faces (i.e. S, K and F faces at temperatures above T_{r}) grow by a different mechanism to the atomically flat faces (F faces at temperatures below T_{r}).
8.2 Classification of Crystal Surface Sites
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:
8.3.1 Continual growth mechanism for atomically rough faces
(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}.
Atomically flat F faces (i.e. F faces at temperatures below T_{r}), however, cannot grow by this mechanism.
8.3.2 Layer growth of flat faces

8.3.2.(i) 2D nucleation growth
The growth of a defectless, atomically smooth crystal face is a periodic process involving:
Similarly to the 3D nucleation case, the rate of 2D nucleation is given by:
(32)
where: W _{2D }is the preexponential factor for 2D nucleation and D G_{2D}* 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 nonlinear 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 G_{2D}*/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 T_{r} is known as kinetic roughening, to distinguish it from the thermodynamic roughening transition that occurs at T_{r}.
8.3.2.(ii) Screw dislocation growth
Experiments showed that growth of faceted crystals could occur at relative supersaturations, a (P/P_{0}), 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.

Consequently, we can distinguish different growth regimes for an F crystal face as shown below.
8.5 Crystal growth experiments
Electrodeposition experiments:
A single crystal with a specific orientation can be placed in contact with the electrolyte solution, and so the growth of both screwdislocationfree 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 T_{r} grows by entirely different mechanisms, it would be advantageous to know T_{r}. Is there a parameter that will give an indication of T_{r}?
Þ D S_{trans}.
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 d_{hkl} 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
The ability to influence the rate of crystal growth, and the resulting morphology can have important consequences, e.g.
9.1 Equilibrium crystals and macroscopic crystals
The equilibrium form of a crystal is governed by:
. (GibbsWulff law, Eq. 1)
However, the equilibrium shape is only likely to occur for submicroscopic 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, E_{att}, i.e.
R µ E_{att}. (39)
The attachment energy is defined as the interaction energy per molecule between a slice (hkl) and the crystal face (hkl) so E_{att} = S E_{i} where E_{i} = interaction energy per molecule of a slice of thickness, d_{hkl}, with the i_{th} under/overlying slice.
Crystal faces with the lowest E_{att} 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: A_{hkl} = surface area of a molecule in the crystal face (hkl). Since A_{hkl} = v/d_{hkl}, Eq. (40) becomes:
(41)
i.e. .
9.3 The role of specific additives. Nucleation promotion and crystal growth inhibition
The addition of ‘tailormade’ additives to a supersaturated melt/solution can strongly influence the resulting crystal morphology. Tailormade 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 tailormade 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.
The crystallisation of polymers introduces some specific factors that will now be considered.
Single crystals of polymers can be obtained from dilute solution. Such crystals, or lamellae, have lamellar thicknesses, l, only in the range of typically 510 nm. Xray 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.
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.
Any theory on the crystallisation of polymers must account for the following main experimental observations.
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:
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 A_{eg 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 GibbsThomson 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 T^{0}_{m} is the theoretical melting point of the extended crystal and D H_{melt} is the enthalpy of melting per monomer. Eq. (45) is known as the HoffmanWeeks equation. It shows that:
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 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
In this model, growth occurs in 3D and hence to limit the lamellar thickness, l, certain ‘pinning rules’ are introduced, namely:
This model relies on computer simulations.
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 
MorphologyT 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.