Crystallization is a separation process, widely applied in the chemical and pharmaceutical industry. The principle of crystallization is based on the limited solubility of a compound in a solvent at a certain temperature, pressure, etc. A change of these conditions to a state where the solubility is lower will lead to the formation of a crystalline solid.
Crystallization is the process of forming a crystalline material from a liquid, gas, or amorphous solid. The crystals thus formed have a highly regular internal structure, the basis of which is called the crystal lattice. Since the formation of such a highly ordered structure prohibits foreign molecules from being incorporated into the lattice, a solid product of high purity is obtained. The simultaneous formation and purification of a solid product make crystallization an important operation in the process industry.
All crystallization processes are aimed at creating a supersaturated solution or melt. The supersaturation is the driving force under whose influence new crystals are formed and present crystals grow. This is illustrated in figure 1 where a typical cooling crystallization experiment is shown together with the solubility line.
For most substances, the solubility increases with increasing temperature, with sodium chloride being a notable exception. Suppose we start at point A in the diagram, which is undersaturated. Any crystals added to a solution in this region would dissolve. If we now cool to a point between A and B, we enter the meta-stable region where existing crystals will grow, but no new crystals are formed. Cooling further we obtain a labile solution at point B where the spontaneous formation of new crystals, i.e. nucleation, takes place. This dramatically decreases the concentration and point C will be reached. Cooling further, the crystals formed between B and C grow and consume whatever supersaturation we create by cooling, so we stay in the meta-stable region until we reach the end of the crystallization at point D.
The important phenomena that occur during crystallization can be described by the following quantities:
• the nucleation rate B [#/(m3s)]; the number of new crystals formed per unit of time and volume of suspension.
• the growth rate G [m/s]; the rate at which the size of the crystals increases.
Nuclei formation
Nucleation is the first step in the formation of either a new thermodynamic phase or a new structure via self-assembly or self-organization. Nucleation is typically defined to be the process that determines how long an observer has to wait before the new phase or self-organized structure appears. For example, if a volume of water is cooled (at atmospheric pressure) below 0 °C, it will tend to freeze into ice, but volumes of water cooled only a few degrees below 0 °C often stay completely free of ice for long periods. Under these conditions, the nucleation of ice is either slow or does not occur at all. However, at lower temperatures, ice crystals appear after little or no delay. Under these conditions ice nucleation is fast. Nucleation is commonly how first-order phase transitions start, and then it is the start of the process of forming a new thermodynamic phase. In contrast, new phases at continuous phase transitions start to form immediately.
Nucleation is often found to be very sensitive to impurities in the system. These impurities may be too small to be seen by the naked eye but still can control the rate of nucleation. Because of this, it is often important to distinguish between heterogeneous nucleation and homogeneous nucleation. Heterogeneous nucleation occurs at nucleation sites on surfaces in the system. Homogeneous nucleation occurs away from a surface.
Nucleation is usually a stochastic (random) process, so even in two identical systems nucleation will occur at different times.
Crystal Growth
A crystal is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. Crystal growth is a major stage of a crystallization process and consists of the addition of new atoms, ions, or polymer strings into the characteristic arrangement of the crystalline lattice. The growth typically follows an initial stage of either homogeneous or heterogeneous (surface catalyzed) nucleation, unless a “seed” crystal, purposely added to start the growth, was already present.
The action of crystal growth yields a crystalline solid whose atoms or molecules are close-packed, with fixed positions in space relative to each other. The crystalline state of matter is characterized by a distinct structural rigidity and very high resistance to deformation (i.e. changes of shape and/or volume). Most crystalline solids have high values both of Young’s modulus and of the shear modulus of elasticity. This contrasts with most liquids or fluids, which have a low shear modulus, and typically exhibit the capacity for macroscopic viscous flow.
The interface between a crystal and its vapor can be molecularly sharp at temperatures well below the melting point. An ideal crystalline surface grows by the spreading of single layers, or equivalently, by the lateral advance of the growth steps bounding the layers. For perceptible growth rates, this mechanism requires a finite driving force (or degree of supercooling) in order to lower the nucleation barrier sufficiently for nucleation to occur by means of thermal fluctuations. In the theory of crystal growth from the melt, Burton and Cabrera have distinguished two major mechanisms:
Non-uniform lateral distance
The surface advances by the lateral motion of steps which are one interplanar spacing in height (or some integral multiple thereof). An element of surface undergoes no change and does not advance normally to itself except during the passage of a step, and then it advances by the step height. It is useful to consider the step as the transition between two adjacent regions of a surface which are parallel to each other and thus identical in configuration — displaced from each other by an integral number of lattice planes. Note here the distinct possibility of a step in a diffuse surface, even though the step height would be much smaller than the thickness of the diffuse surface.
Uniform normal growth
The surface advances normally to itself without the necessity of a stepwise growth mechanism. This means that in the presence of a sufficient thermodynamic driving force, every element of the surface is capable of continuous change contributing to the advancement of the interface. For a sharp or discontinuous surface, this continuous change may be more or less uniform over large areas each successive new layer. For a more diffuse surface, a continuous growth mechanism may require change over several successive layers simultaneously.
Non-uniform lateral growth is a geometrical motion of steps — as opposed to the motion of the entire surface normal to itself. Alternatively, uniform normal growth is based on the time sequence of an element of the surface. In this mode, there is no motion or change except when a step passes via a continual change. The prediction of which mechanism will be operative under any set of given conditions is fundamental to the understanding of crystal growth. Two criteria have been used to make this prediction:
Whether or not the surface is diffuse: a diffuse surface is one in which the change from one phase to another is continuous, occurring over several atomic planes. This is in contrast to a sharp surface for which the major change in the property (e.g. density or composition) is discontinuous, and is generally confined to a depth of one interplanar distance.
Whether or not the surface is singular: a singular surface is one in which the surface tension as a function of orientation has a pointed minimum. The growth of singular surfaces is known to require steps, whereas it is generally held that non-singular surfaces can continuously advance normally to themselves.
Driving force
Consider next the necessary requirements for the appearance of lateral growth. It is evident that the lateral growth mechanism will be found when any area on the surface can reach a metastable equilibrium in the presence of a driving force. It will then tend to remain in such an equilibrium configuration until the passage of a step. Afterward, the configuration will be identical except that each part of the step will have advanced by the step height. If the surface cannot reach equilibrium in the presence of a driving force, then it will continue to advance without waiting for the lateral motion of steps.
Thus, Cahn concluded that the distinguishing feature is the ability of the surface to reach an equilibrium state in the presence of the driving force. He also concluded that for every surface or interface in a crystalline medium, there exists a critical driving force, which, if exceeded, will enable the surface or interface to advance normally to itself, and, if not exceeded, will require the lateral growth mechanism.
Thus, for sufficiently large driving forces, the interface can move uniformly without the benefit of either heterogeneous nucleation or screw dislocation mechanism. What constitutes a sufficiently large driving force depends upon the diffuseness of the interface, so that for extremely diffuse interfaces, this critical driving force will be so small that any measurable driving force will exceed it. Alternatively, for sharp interfaces, the critical driving force will be very large, and most growth will occur by the lateral step mechanism.
Note that in a typical solidification or crystallization process, the thermodynamic driving force is dictated by the degree of supercooling.