24 Theoretical aspects of crystal nucleation and growth
Prof. P. N. Kotru
24.1 Introduction
Crystal growth is a centuries old practice of mankind. Be it metals, common salt, sugar industries and several such activities all use crystal growth. Crystals have been admired by man for as long as he has appreciated beauty. However, the importance of crystals for the development of scientific knowledge and also for technological developments has been realized somewhere a little more than half a century back. At one time large well formed crystals were available to man from natural sources only. Most of these were gems or items in the museum and so were not freely available for scientific study or any other purpose from the utility point of view. The development of methods for producing large crystals heralded in a new era of intensive scientific investigations on crystals, which caused major development to take place in inorganic chemistry, metallurgy, ceramics, geophysics and also the fast expansion of solid state physics and materials science.
The production of crystals in industries is taken as a new agriculture that has flourished in the last half a century to new heights which has resulted into a materials revolution , whether it is an industry or a scientific laboratory , commercial production or preparation of crystalline materials for technological applications or intensive scientific study , the field of crystal growth has achieved the status of being a thrust area of study and research which is in tune with the requirements of mankind for better life.
Laboratory grown crystals find applications in frequency controlled oscillators ( quartz ) ; polarisers ( calcite , sodium nitrate ) , transducers( quartz, Rochelle salts, ammonium dihydrogen phosphate ) ; grinding and cutting ( diamond ) ; radiation detectors (anthracene , potassium chloride ); infra- red optics ( lithium fluoride ) ; bearings (aluminium oxide ) ; transistors (germanium , silicon ) ; magnetic devices ( garnets , ferrites ) ; strain gauges ( silicon) ; ultrasonic amplifiers (cadmium sulphide ); masers and lasers ( ruby ,gallium arsenide , calcium tungstate etc.) ; lenses ( fluorite ); melting crucibles ( magnesium oxide ) ; tunnel diodes ( gallium arsenide ); substrates for integrated circuits ; high Tc superconducting applications ; computer applications (ferrites) and several other device applications of materials based on electronic, optical , magnetic , mechanical and thermal characteristics. Good quality single crystals are essential for different scientific and commercial purposes. Modern solid state electronics has achieved its ever expanding applications in technological evelopment because of crystal growth revolution. The list given here in respect of utility of crystals is not an exhaustive one but it certainly is an indicator that crystal growth is of paramount importance for the development of scientific research, technology and industrial growth.
Inspite of the fact that there has been an unprecedented growth of scientific research in the area of crystal growth over the last more than half a century, crystal growth is still considered to be more an art than science, although an art based on scientific principles. Much of crystal growth is still an art and technique rather than science. Prediction of optimum conditions for the successful growth of good quality single crystals of a new material cannot be done, but a lot of effort has gone into building up theory, experience and data over the past several years.
We shall now discuss some basic theoretical and experimental aspects of crystal growth.
24.2 Nucleation
The complex subject of crystal nucleation and growth has been the focus of several specialists, investigators and scientists. However, in practical terms, it is yet partly an art. The knowledge gained so far provides a reasonably good foundation for making intelligent guessing about new or unobservable crystal growth situations, but is far from being a foundation for making scientific predictions.
In having some basic understanding of the theory of nucleation and crystallization we shall consider mainly crystallization from the melt, while making some limited reference to crystallization from solution or vapour wherever applicable.
To have some idea about nucleation, we shall consider crystallization of pure liquids. In this regard, supercooling is an essential requirement. If a liquid were not supercooled, it would never get crystallized. Obviously, it means that super cooling is pre–requisite for crystallization. In crystallization from solution, the word supersaturation is used instead of supercooling.
Let us try to understand what we mean by a supercooled liquid. Ordinarily, a liquid is supposed to freeze and change its state from liquid to solid at the freezing point. Existence of a liquid below the freezing point is called supercooled liquid. This state of supercooled liquid is an unstable state. It is also termed as metastable state. The “instability” or so-called “metastability” can be better understood by referring to figure 24.1 which represents a P ─ T diagram. In this particular P-T diagram, the three curves are extended through the triple point as indicated by dotted lines. In this figure P-T and F-T diagrams are placed one above the other in order to be able to make off- hand comparison.
Figure 24.1: P-T and F-T diagrams
Figure 24.1 is a free energy ‘F’ versus temperature ‘T’ diagram alongwith a pressure ‘P’ versus temperature ‘T’ diagrams, revealing relationship between the two thermodynamic quantities. The extensions of the curve past the intersection points as indicated by dotted lines represent metastable states. If a system containing liquid and vapour up to and beyond the triple point, the system is expected to follow the extension of the curve rather than change to crystal and gas (vapour). Looking at the free energy F versus the temperature diagram too, which for immediate comparison is placed just below P- T diagram, the metastable states are represented by the extensions of the curves past the intersections as illustrated in F – T diagram. It is also clearly indicated by the diagram as to under which conditions an unstable system can change to a better stable system.
The initiation of the transformation of an unstable to a more stable state is called nucleation. It may also be defined as spontaneous appearance of little order domain in the midst of randomness.
Let us consider a familiar example of cooling of water with time. The ideal cooling curve appears to be as shown in figure 24.2 .However, less ideal cooling curve of water is of the type as shown in figure 24.3
Figure 24.2: Ideal cooling curve of water
Figure 24.3: Less ideal cooling curve for water showing super-cooling
Let us perform a familiar experiment of making observations and recording temperature while a liquid is cooling. One would notice a pattern of the type shown in figure 24.4 which represents the cooling curve of a liquid. Note the temperature accurately at the instant when crystallization just starts spontaneously. One would find that the temperature at which nucleation just starts is always lower than the melting point (M. P.). The temperature at which this happens is taken as “Nucleation Temperature” (N.T.). In this situation one can say that the liquid has supercooled and is in a state of metastability. All liquids supercool and T, as shown in figure 24.4, is the supercooling at the nucleation temperature. Supercooling of water has also been represented by less ideal cooling curve for water in figure 24.3.Referring to figure 24.4 which represents cooling curve of a liquid , nucleation has occurred at the point A and the process that takes place between the points A and B and also between the points B and C is called crystal growth. The initiation of transformation from metastable state to a more stable state takes place at the point A and this process of transformation continues along the route A → B → C of the curve.
Figure 24.4: cooling curve of a liquid
24.2.1 Types of Nucleation.
Nucleation is of two types:
1. Homogeneous nucleation and
2. Heterogeneous nucleation
If nucleation takes place in absolutely pure materials it is called as homogeneous nucleation. Nucleation taking place as a result of foreign particles (impurities) is called heterogeneous nucleation.
24.2.1.1 Theory of Homogeneous Nucleation
In order to explain homogeneous nucleation we must understand as to why should pure water be supercooled 400C before freezing. The answer to this is provided by considering the effect of surface energy on the total energy of very small particles.
So, we proceed to calculate free energy in the process of formation of the very first tiny crystal in the supercooled liquid. There are three free energy terms which are as follows:
i) Free-energy of the supercooled liquid ,
ii) Free-energy of the crystal,
iii) Free-energy of the interface between the liquid and crystal
By an interface we mean a boundary that separates the two phases of material such as a crystal and a liquid. If one of the phases happens to be gas, the interface is often called a surface.
In order to determine free energy, we shall take the example of a system which consists of supercooled liquid and a tiny spherical crystal of radius ‘r’ as shown in a schematic diagram of figure 24.5. Here, we consider system of volume V to be under isothermal conditions. Let ‘F’ represent the free energy of the whole system. The entire system can be taken to be composed of following three parts:
(i) Supercooled liquid of volume ( V- 4/3. π r3 ) ,
(ii) Crystal of volume 4/3. π r3
(iii) The interface between the crystal and supercooled liquid.
Figure 24.5: schematic diagram showing a small crystal with a supercooled liquid around it
The free energy of the whole system F is obviously given by the sum of the free energies of these
parts:
The interfacial free energy σ is determined by work required to be done in creating unit area of new
interface which is always positive.
Therefore F= fL(V- 4/3 .π r 3 ) + fC (4/3 .π r 3 ) + σ (4 π r 2 )……..24.1
The subscript L and C refer to liquid and crystal respectively.
Now, suppose the crystal grows in size and its radius increases by Δr. The increase in size of the crystal will bring about a change in the total energy of the system because the volume of the supercooled liquid will decrease whereas the volume of tiny crystal will increase and the work required to create unit area of new interface will also be more. Let the change in the free energy of the system be represented by ΔF. Therefore, free energy of the system on account of increase in the radius of crystal by Δr is given by:
F + ΔF = fL [ V- 4/3 .π ( r + Δr)3 ] + fC[ 4/3 .π (r + Δr)3 ] + 4πσ ( r + Δr )2 ……………24.2
Subtracting equation24.2from 24.1 and combining terms we get:
F = ─ 4/3. π fL [ ( r + r )3 ─ r3 ] + 4/3 . πfC [ ( r + r )3 ─ r3] + 4 πσ [ ( r + r )2 ─ r2 ]………24.3
This equation relates the change in free energy of the system brought about by change in the radius of a spherical crystal.
By nucleation we mean the first appearance of a crystal when it did not exist before. So, in order to obtain the expression for free energy of formation of crystal of radius r, we may put r in equation no. 24.3
As shown in figure 24.1 regarding Free energy-temperature diagram, it is quite clear that (fC ─ fL) is always negative on account of the fact that the liquid is supercooled.
Based on equation 24.4, if a graph is drawn suggesting dependence of change in free energy ΔF on crystal formation of radius Δr, it will be as is shown in figure 24.6 From this figure it becomes clear that
Figure 24.6: Free energy of formation of a spherical crystal ΔF as a function of radius of the
From figure 24.6, it is clear that if a spherical crystal gets nucleated and grows from zero size (zero radius) to a radius approaching 3σ , the free energy increases which is not thermodynamically
(fL─fC)
favourable. The free energy attains maximum value when the spherical crystal increases in size from zero radius to r = 2σ/(fL─ fC) .Once the crystal reaches to this size, any further growth or the dissolution decreases energy the free of the system which is thermodynamically favourable. The radius 2σ is,(fL ─ fC)
therefore called as critical radius r* which if crossed over decreases the free energy and so thermodynamically favourable for growth into a crystal. However, in the event of its inability to cross over would lead to decrease in the size of the crystal. In other words, it would favour dissolution and not the growth as after this cross over the free energy decreases. The energy corresponding to r* is put as F* which is a quantity called as Free–energy barrier or activation energy of nucleation. The existence of free–energy barrier or the activation energy of nucleation is the reason why liquids supercool.
Strictly speaking, according to the laws of thermodynamics, nucleation is impossible because the growth of an embryo crystal causes increase in free energy. It may be noted that σ is always positive (i.e., greater than zero). This is what is ideally expected. However, in reality nucleation does take place. This contradiction between theory and practice can be explained on the basis of the following argument.
When thermodynamics is applied to any situation, the two important thermodynamic quantities, like temperature and pressure, are treated to be uniform which could be true on a macroscopic scale. However, the macroscopic manifestation is due to atoms and molecules composing the system which are in motion in all possible directions. Because of this all the thermodynamic parameters change locally and so cannot actually be treated as uniform at a local level. There can be fluctuations in free energy which may be sufficient enough to cross the free energy barrier and eventually become cause for the nucleation to occur.
Here, we consider a system which is in a metastable equilibrium (say for example supercooled liquid), any change in the system would increase the free energy. If nucleation takes place and the crystal grows to a critical size, it is an unstable equilibrium and so even a small change in either direction would lead to a spontaneous and larger change. As such, the crystal either would grow further or would get dissolved depending upon the direction in which change takes place.
24.2.2 Nucleation Rate
Nucleation rate is determined by the number of nuclei formed per cubic centimetre per second. In liquids there are several groupings of molecules which are transitory and in a state of continuous flux. Such transitory groupings of molecules are technically called as “sub–critical nuclei” with size less than the critical radius r*. Because of free energy fluctuations, these sub–critical nuclei form and dissolve because it lowers the free energy, thus making it thermodynamically a favourable process. Application of Boltzmann distribution yields an expression for “stationary distribution of sub–critical embryos” as:
N( r ) =Σ/eFr/kT ______________ , ………… 24.7
Where N (r) stands for number of embryos of radius r,
Σ is a constant,
Fr stands for free energy of formation of a sub–critical embryo of radius r and k is a Boltzmann constant.
If one has to determine nucleation rate it is required to calculate the concentration of nuclei of critical size per unit volume and multiply the same by the rate at which a nucleus of critical radius gains one more molecule per unit time. If nucleation temperature is known, it leads to determination of nucleation rate as a function of temperature.
Let ф be the rate at which a critical nucleus gains one molecule and η be the rate of nucleation at temperature T. The rate of nucleation at temperature T is given by:
η =ф Σ /eF*/kT _____________ ……………………..24.8,
where F* stands for free energy of formation of a here proceed as per this equation, further growth will nucleation is completed. critical nucleus. When various parameters given take place and we can say that the process of nucleation is completed.
24.2.2.1 Dependence of Nucleation Rate ‘η’ on Temperature T
From figure 24.6, we know
r* = 2σ/fL ─ fC
It is well known that at the melting point fC = fL
So, r * = ∞
It means that at the melting point, the critical radius r* is infinite, F* is infinite and η = 0.
At 00K, the molecules have no kinetic energy and so such molecules cannot move. As such, the critical nucleus is not in a position to gain one more molecule and so η = 0 again.
Nucleation does occur at appreciable rates at some temperatures. So, ‘η’ versus ‘T’ curve must, therefore, have a minimum. If a liquid is cooled so fast past this maximum that no nuclei gets time to form, it becomes a glass.
We know that silicate materials are found in amorphous state in nature. The above explanation is an answer to the question as to why silicate materials almost always form glass. For silicate materials, the factor ф given in equation 24.8 is usually extremely small. The melts of silicate materials are highly viscous (with high viscosity) and very low diffusion coefficients. These three factors put together makes it very difficult to cool a molten silicate at such a slow rate as to make it possible for nucleation to occur at all with the result that they almost always form glass and fail to get crystallize. Man has limitations over time, temperature and pressure. However, nature has no such limitations. In nature cooling may last for thousands and millions of years and the process may pass through geological times which are impossible to happen in the laboratory experiments on growth by a researcher. Thus opening of the possibility of nucleation and growth becomes impossible for such materials. In contrast to this , ф is very large for water and as a result crystallization can be conveniently done and only the most extreme measures will have to be taken , if at all one is able to do that, can glass be made.
24.2.1.2 Heterogeneous Nucleation
Nucleation in the case of absolutely pure materials is called homogeneous nucleation which has already been explained in the above section. However, if the nucleation is by foreign particles, the process is called heterogeneous nucleation. In the discussion on nucleation theory, it is nucleation temperature which plays a very important role. In increasing nucleation temperature, one may do so by lowering the effective value of V which reduces the critical radius r* and the activation energy F*.
Considering the understanding of heterogeneous nucleation, let us assume the crystal to be part of a sphere and calculate critical size and activation energy for nucleation on the substrate (which in this case means an impurity or any foreign particle that may act as a seed). To make this calculation as applicable to heterogeneous nucleation, we consider an ideal system as shown in figure 24.7.
Figure 24.7: Ideal system for calculations of heterogeneous calculations
Different parts of the system are:
(i) Substrate
(ii) Crystal cap
(iii) Supercooled liquid
These parts are identified in the figure.
Let σCL represent the interfacial free energy between the liquid and nucleating crystal. Since we are considering heterogeneous nucleation, it is supposed that nucleation is taking place at the surface of some foreign (impurity), solid material σLS and σCS represent interfacial free energies between substrate and liquid, and between substrate and nucleating crystal respectively.
Let θ be the contact angle as shown in figure 24.7. The three interfacial energies (one may also call it the surface tension) are related to each other through an equation which may be expressed as:
σLS =σCLcosθ + σCS ………………………………….24.9
This equation represents the balancing of surface tension at the surface intersection. There is an outward pull along the substrate which is represented by σLS and the inward pull due to the component of σCL along the line σCS(i.e., σCLcosθ ) and σCS . The balancing of outward pull along the substrate (i.e., σLS ) and the inward pull given above (i.e., σCLcosθ + σCS ) yields equation 24.9
Let us now find out the terms required in the expression for the free energy of formation of the crystal with the help of model shown in figure 24.7.
(i) The volume of the nucleating crystal = ⅓ π r3 ( 2 ─ 3 cos θ + cos3θ )……24.10
(ii) The area of liquid to crystal contact = 2π r2 (1 ─ cosθ )…………………..24.11
(iii) The area of crystal to substrate contact = π r2 ( 1 ─ cos2θ )…………………24.12
The free energy of formation of the crystal in the light of equations 24.10, 24.11, &24.12as represented by F is given by:
F=(fC ─ fL)[⅓ πΔ r3(2 ─ 3 cosθ + cos3θ )]+σCL[2πΔr2(1─cosθ)]+(σCS─σLS)[πΔr2(1─ cos2θ)] ….24.13
In equation 24.13, the contact area with surface-energy σLS and with surface-energy σCS have been taken as same.
For a given substrate material interfacial free energies σLS and σCS and the contact angle cosθ are constant. The above equation takes the same form as the equation for homogeneous nucleation.
Therefore,
Equation 24.14 assumes a form which is a comparable one for homogeneous nucleation. Actually, in this case of θ = 180⁰, cosθ = ─ 1 means that the embryo crystal has no contact with the substrate.
In the case of θ <180⁰, the nucleation temperature (N. T.) on the substrate should be less than homogeneous nucleation temperature (N.T.). However, in the case of θ = 0⁰ , cosθ = 1 and so the entire right hand side of equation 24.13 becomes zero and as such the equation leads to the value of F as zero . It means that F would be minimum if θ is zero and that would be the case if the substrate were a crystal of the material which is nucleating. This is crystal growth. Further, as the contact angle between the substrate and the nucleating crystal decreases, heterogeneous nucleation temperature (N.T.) decreases. Also, the result of the equation suggests that the nucleation temperature is higher if the bond between nucleating crystal and the substrate is tighter (i.e., the bond is stronger).
In heterogeneous nucleation, substrate plays a very significant role. In practice, it is observed that crystals which have identical structures with very little mismatch between their lattice parameters are excellent nucleating agents for the other. It is on account of the fact that the substrate material makes the strongest and tighter bond with the crystal to be nucleated.
References.
- Knight,C.A. ; “ The Freezing of Supercooled Liquids”, Van Nostrand ,1967.
- Pamplin,B.R.;” Crystal Growth”, Pergamon Press,N.Y.,1980.
- Brice, J.C. : “ The Growth of Crystals From Liquids”,North-Holland, Amsterdam,1973.
Suggested Reading.
1.Hartman, P.: “ Crystal Growth- An Introduction”,North Holland,Amsterdam,1973.
2. Brice,J.C.: “ The Growth of Crystals From Melt”,North Holland, Amsterdam,1965.
3. Gilman, J.J.: “ The Art & Science of Growing Crystals”,Wiley, N.Y.,1963.