In this article we will discuss about the process of nucleation and growth of liquid metals.
Suppose the temperature of the liquid metal is brought below its freezing temperature, so that solid metal x is now a stable phase. How does the solid appear, and the liquid disappear? It would be unrealistic to assume that such a change occurs all at once by simultaneous cooperative motion of every atom in the liquid to transform completely too crystalline structure of metal x.
Even the cooling curve of a pure metal shows a horizontal line, i.e., a period of time over which this change occurs. In fact, the process of solidification must begin on a very small scale, by forming first a crystalline configuration of perhaps only a few hundred atoms of the liquid, called the nuclei.
Then the nuclei grow by more and more atoms leaving the liquid metal and joining the crystalline configuration as illustrated by Fig. 5.3 in stages. The growing crystallites of the solid eventually impinge on one another until all the liquid is solidified. The result is the polycrystalline solid metal.
Thus, the process of solidification takes place in two steps one after the other:
(i) Nucleation:
It is the formation of small stable particles of solid metal x from liquid metal, called nuclei.
(ii) Growth:
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It is the increase in size of these stable nuclei particles by atoms joining them from liquid metal.
A nucleus is essentially a small group of atoms that have taken up arrangement in definite space lattice, is stable and capable of further growth. The process is called nucleation, and it involves the formation of a new phase within the pre-existing liquid phase.
When such a metal particle has solidified, a new interface is created between the particle and the liquid. This new interface has a positive energy, i.e., energy has to be supplied during solidification to create this interface. The creation of this interface acts as a barrier to the process of freezing.
Suppose the liquid is cooled below its freezing temperature by an amount ∆T without freezing (called supercooling). The liquid has a free energy that is higher than the solid by the amount ∆g per unit volume.
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This extra free energy is expected to contribute to the creation of interface between the solid and the liquid phase, and thus, to the stability of the nuclei. Liquid metals, thus, can be cooled to temperatures far below their equilibrium freezing points before solidification begins. Table 5.1 illustrates the maximum supercooling obtained in some metals.
The difference in volume free energy helps to create the interface between the tiny solid formed and the liquid. But a tiny solid particle has a large surface area to its volume ratio, i.e., the volume free energy available is lesser than required to create the surface area, and thus, the tiny solid is unstable. For example, as illustrated in Fig. 5.4. When a solid sphere of radius r is formed, an interface of area 4πr is created between the solid and the liquid, then, the ratio-
When the solid particle is really tiny, i.e., as r approaches the value of zero, this ratio (equation 5.7) becomes infinite. This large requirement of surface energy effectively prevents the initial solidification of tiny particles. Thus, during the nucleation stage, surface energy is the dominant factor.
As r increases, this ratio becomes smaller. When the rate of volume free energy-change supplied just balances the requirements for the creation of surface, the particle is said to be nucleated, as then it becomes stable. Once the particle has nucleated, it grows with a continuous decrease of total energy. This is the process of growth, as now, the surface energy is no more the controlling factor.
Types of Nucleation:
The process of nucleation can be of two types:
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(i) Homogeneous Nucleation:
When the solid begins to nucleate throughout the bulk of the liquid without preference for any point, i.e., when the probability of nucleation is same everywhere within the volume of the liquid phase, it is called homogeneous nucleation.
(ii) Heterogeneous Nucleation:
When the solid nucleates preferentially at certain sites in the liquid phase, i.e., the probability of nucleation occurring at certain preferred sites in the liquid is much more than that at other sites, it is called heterogeneous nucleation.
A metal solidifies when the liquid cools to below its freezing temperature, because the free energy associated with the crystalline solid is then less than the energy of the liquid. The energy difference between the liquid and the solid is the volume free energy, ∆g per unit volume. This energy is negative (below Tm) and is the driving force for solidification.
If it is assumed that solid nucleus formed is a sphere (any other shape can also be assumed) with a radius, r. The volume free energy change is 4/3 r3. ∆g for the nucleus Fig. 5.4 illustrates the creation of interface between the solid and the liquid when a nucleus solidifies, having an energy of ϒ per unit area associated with it.
The surface energy required when a surface area 4r2 of sphere is created is 4r2. ϒ. This term is always positive and inhibits the process of nucleation.
Thus, the process of nucleation is controlled by two factors:
(i) Free energy change associated with the change L → solid metal, i.e., volume free energy change.
(ii) Surface energy of the new solid formed.
The total free energy change, ∆f, can thus be:
Each term in equation (5.8) is plotted with the radius of tiny solid in Fig. 5.5. (a). It also shows that total free energy change, ∆f passes through a maximum with increase in the value of r. Initially, as the solid particles start to form, the total free energy increases as the surface energy term is dominant.
At the maximum, the variation with r of the surface energy term and the volume energy term exactly balance each other. Thereafter, the variation in the volume energy term becomes dominant. As this term is negative, there is a continuous decrease in the total free energy change beyond the maximum. At the temperature (< Tm) for which this curve, Fig. 5.5 (a) has been drawn, the critical radius, r* (denoted by asterisk) is the value of r corresponding to the maximum of ∆f*.
At this critical stage, the total free energy of the system decreases at the same rate whether the nucleus grows, or shrinks from this size. The value of r* is therefore obtained by equating to zero the derivative of ∆f with respect to radius, r; thus,
and the equivalent critical free energy is obtained by putting the value from equation 5.9 in equation 5.8 as-
This value, ∆f* is known as the nucleation barrier. The solid particles of size less than r* tend to redissolve (reduce in size) as it lowers the energy. The particles of radius larger than r* lower the free energy by growing in size. Sub-critical particles (r < r*) are called embryos, and of larger size (r > r *) are called nuclei.
The critical sized particle is called critical nucleus, which can grow, or redissolve as free energy decreases in both cases. Fig. 5.5 (b) illustrates the variation of ∆f as a function of r for different temperatures. At T = Tm, ∆g = 0, the curve ∆f versus r increases continuously, without passing through a maximum. At T < Tm, ∆g < 0, and the curve has a maximum. The maximum value of ∆f decreases with increasing driving force, i.e., with increasing magnitude of ∆g. As ∆g becomes more negative with the decrease of the temperature, critical condition occurs at smaller values of ∆f* and r*.
This can be obtained by putting the value of ∆g from equation 5.6 in 5.9 and 5.10 respectively as:
Thus, as the undercooling increases, the critical radius for nucleation decreases (and also the critical total energy, ∆f*), which means, there is large increase in the rate of formation of critical nuclei, which can give more fine-grained solid metal.
The process of nucleation is that unit step in which the critical-sized particle gains at least one more atom from the liquid phase to just become super-critical. Because, once the size is greater than r*, the particle does not shrink as it then leads to increase of free energy. But when it grows in size, there takes place decrease in the free energy.
Any further addition of atoms to the supercritical particle is considered to be the subsequent process of growth. Thus, the rate of nucleation, l(dN/dt) can be approximated as the product of the number of critical-sized particles and the frequency with which these become super-critical.
In homogeneous nucleation, the nucleus forms from a cluster of atoms in the liquid. The liquid metal has short-range order. On a statistical average, there is a definite groupings of atoms around any given atom, i.e., a statistical distribution of clusters of different sizes exists in the liquid metal.
As the temperature is lowered, the sizes of clusters increase because of the reduction in thermal energy. The Maxwell-Boltzmann statistics can be used to estimate the number of critical-sized particles in the liquid phase.
Let the total number of particles per unit volume of the liquid phase be n0, then the number of critical-sized particles n* is approximately given by:
where, ∆f* is activation energy pertaining to critical nucleus size. If m* is the number of atoms in the liquid phase facing the critical sized nucleus across the interface as illustrated in Fig. 5.6. If any one of these m* atoms jumps from the liquid phase to the critical nucleus particle, it becomes supercritical, i.e., it is said to be nucleated.
The frequency v’ with which any of the m* atoms can jump the interface to join the solid particle is:
where, ∆Hd is the activation energy for diffusion across the interface, and v is the lattice vibration frequency (~ 1013 sec-1 above the Debye temperature). The factor m* is geometrical, and depends on the shape and size of critical sized nucleus. Now, the rate of homogeneous nucleation given as number of stable nuclei formed per unit volume per unit time, I (= dN/dt)
Here, nO can be approximated to as the number of atoms in the liquid phase per unit volume (~ 1029 m-3).
Thus, nO vm* is around 1043 m-3sec-1. If a graph is plotted between I and temperature, the result is the curve in Fig. 5.7. As ∆f* is a function of temperature below Tm, and its value at Tm (∆T = 0) in infinite, and thus, from equation 5.15, the nucleation rate, I is zero at Tm (the equilibrium temperature).
Below Tm, ∆f* has a finite value, and thus, the rate of nucleation, I, increases with the decreasing temperature (as ∆f* decreases) and attains a maximum value at some degree of undercooling. Below this temperature, the term ∆Hd dominates. As the temperature drops, the diffusion becomes smaller, the rate of nucleation decreases. It becomes zero again at zero kelvin.
Volume Strain Energy Factor:
During solidification if a change in volume occurs, it is accommodated by flow in liquid phase. If transformation occurs entirely within a solid, the volume change introduces elastic strains in the phases. The elastic strain energy is also always positive and thus, tends to inhibit transformation. The equation for free energy change, ∆f shall include the strain energy term, where ɛ is strain energy per unit volume, such as-
A greater amount of supercooling is thus, needed for nucleation to occur. However, nucleation in a solid phase seldom occurs homogeneously, because of presence of structural defects in solid phase such as vacancies, impurities, dislocations, grain boundaries, etc.
Superheating of Metals:
Although liquid metals are easily supercooled, but, metals do not normally superheat, i.e., heated above their melting points without melting. Fig. 5.8 (a) illustrates a solid metal in contact with vapour phase. If a small quantity of liquid metal forms on the solid surface, it tries to cover the maximum possible surface because the liquid metals always wet their own solids.
Here, solid-vapour (‘A’ in Fig. 5.8 a) interface has been replaced by two interfaces, solid-liquid and liquid-vapour. The process of wetting requires their surface energies to have the relationship.
where ϒSV, ϒLV and ϒSL surface energies of solid-vapour, liquid-vapour, and solid-liquid interfaces respectively. This equation means, the formation of liquid nucleus decreases the surface free energy term.
There is no surface energy barrier on melting (reverse of solidification), and thus, no superheating is required, and does not occur. Both the terms, the volume free energy term and surface energy term favour the process of melting. In Gold, the values of surface energies are (as ergs cm-2) solid-vapour: 1400; solid-liquid: 132; liquid-vapour = 1128. In fact, the gain in surface energy 140 ergs cm-2 means that a thin films of liquid the melting point is reached.
Problem:
Find out the size of the critical nucleus for homogeneous nucleation if the tiny solid formed is a cube (Take ‘a’ as length of cube edge).
Solution:
Total free energy change depends on volume free energy change and the surface formed. If ‘a’ is the side of the cube, then-
where, ∆g is volume free energy change per unit volume and ϒ is the interface energy per unit area At the maximum in the curve,
(ii) Heterogeneous Nucleation:
The formation of bubbles of CO2 in a glass of coke takes place on certain select points, frequently on the glass wall, or on surface of straw-pipe, and not inside the volume of the coke. This is heterogeneous nucleation, where the probability of nucleation occurring at certain preferred sites is much greater than that at other sites.
If appropriate values are put in the equation (5.15), it demonstrates that a metal must be supercooled approximately 100° below its freezing temperature before homogeneous nucleation occurs due to barrier presented by the surface energy requirements of the nuclei. Actually, homogeneous nucleation is a rather difficult process, and almost never occurs in industry.
Although it can be produced in laboratory, but the solidification of all industrial metals do not occur by this nucleation, since the undercooling is normally only
several degrees. Thus, in practice, catalyzed nucleation of liquid must be occurring either on a nucleating substance floating in the liquid such as inclusions of foreign particles, or on the walls of the container as preferred nucleation sites. Actually, heterogeneous nucleation is a much easier process.
If a solid embryo (s) forms on the mould-wall as a spherical cap as illustrated in Fig. 5.9 (a), and under quasi- equilibrium conditions, Fig. 5.9 (b), the following is the equation balancing the interfacial energies-
where, θ is the contact angle (between embryo surface and mould wall), ϒlm, ϒsm and ϒls are surface tensions (interfacial energies) between liquid and mould-wall, solid and mould-wall, and liquid and solid respectively. The contact angle θ is dependent only on these three surface energies, and thus, the shape of the embryo remains invariant as a spherical cap as illustrated in Fig. 5.9 (a) after some time.
An expression for total free energy change, ∆f, can be written in terms .of the volume free energy change (using volume of the spherical cap-shaped solid formed) and the surface energies of the interfaces involved as was done for homogeneous nucleation.
Let us write some of these terms separately:
As the solid embryo forms, high energy mould-liquid interface is replaced by low-energy mould-solid interface, i.e., surface area formed ( r2 sin2 θ) between solid embryo and mould replaces an equivalent area of liquid and mould interface but their surface energies are different.
Thus, there are three surface energy terms, two of them are positive (one the curved cap and the second, solid-mould) and one is negative (as it has been replaced).
If ∆g is the free energy change per unit volume, then the expression for the total free energy change, ∆fhet, for hetrogeneous nucleation is:
If we proceed as was done for homogeneous nucleation, then the expressions for the critical radius and for critical total free energy for the heterogeneous nucleation are:
If the equation (5.26), and equation (5.10) for the homogeneous nucleation are compared, then, we can write as:
As the contact angle changes from 0° to 180°, the term in the bracket in the equation (5.27) changes from 0 to 1, i.e., some important conclusions can be made for the following cases:
(i) If the solidifying solid is such that it makes only a point contact with the mould wall, i.e., θ = 180°. As cos 180° = -1, the equation (5.27) reduces to-
∆f*het = ∆*homo.
i.e., the free energy change for heterogeneous nucleation is as large as for the homogeneous nucleation, i.e., the presence of mould wall, or any such surface (θ = 180°) does not help to nucleate the solid. Such a surface has no role to play in the process of solidification. Nucleation is basically homogeneous.
(ii) If the solid completely wets the mould wall, i.e., solid forms a thin film on the mould wall (as if wetting it completely), the angle θ = 0°. As cos 0° = 1, ∆f*het = 0. There is no energy barrier to the heterogeneous nucleation, and it can start just almost at freezing temperature.
(iii) If θ = 90°, then cos 90° = 0, and thus ∆f*het.= 1/2 ∆f*homo The hemi-spherical-shaped solid is still effective as the energy barrier is half of homogeneous nucleation.
Even equation (5.25) can be used to make some conclusions. As contact angle, θ, decreases, the value of r*het decreases, which indicates that the volume of the heterogeneous nucleus becomes smaller and, hence, requires fewer atoms for its formation. At θ = 0°, the volume (r*) becomes zero, so that one expects nucleation to occur without any supercooling.
The above conclusions help to provide a base for choosing an inoculant, i.e., a heterogeneous nucleating agent for nucleation to be deliberately induced. If the contact angle, θ, is small, heterogeneous nucleation becomes easier. The contact angle θ can be minimised by choosing an inoculant which forms a low energy solid-mould interface (low ϒsm).
ϒsm can be made small if the solid and inoculant (or mould wall) have same or similar crystal structures and their lattice parameters are almost equal, i.e., there should be fairly good crystal matching at this interface. It, thus, gives a clue about the nature of inoculant to be chosen.
Perhaps, the most famous example involves making of artificial rain, i.e., seeding the rain-bearing clouds with fine crystals of silver iodide or sodium chloride. The crystals of these salts have atomic planes which make good matching with the ice crystal to result in low energy interface to have a small contact angle with ice.
Clouds consist of small drops of water (plus water vapour). When the cloud temperature is below 0°C, water droplets get super-cooled. The ice crystals nucleate in the cloud and grow. When the ice crystals are sufficiently large, they fall as snow or are converted to rain, depending on the temperature of atmosphere.
Making the supercooled cloud to release its water content as rain is essentially a nucleation problem, i.e., clouds are seeded with these salt particles (by air-spraying them) for the nucleation to occur with minimum of super-cooling. If the rate of nucleation of ice crystals can be increased, the size of the product particles can be decreased.
This as well can minimise the damage (due to small sizes) caused by hailstorm. It is well known that artificial diamonds are produced from graphite by seeding with nickel. Diamond cubic structure is just like FCC crystal structure of nickel with almost similar lattice parameters such as 3.57 and 3.52 A° respectively.
The rate of the heterogeneous nucleation can be obtained on similar arguments as for homogeneous nucleation. The number of critical embryos can be given by-
where nm is the number of atoms in the liquid phase surrounding the mould wall. The value of nm is much smaller than n, the total number of atoms in the assembly (as in homogeneous nucleation). Thus, the rate of heterogeneous nucleation is,
where, v is the frequency of jump, and ∆Hd is enthalpy of diffusion across liquid-solid interface. The pre-exponential term (1026 m-3 s-1) here is much smaller than in homogeneous nucleation (1042 m-3, s-1), but even then, the rate of heterogeneous nucleation is much higher than the rate of homogeneous nucleation because the ∆f*het < ∆f*homo.