In this article we will discuss about:- 1. Introduction to Diffusion of Metals 2. Atomic Models of Diffusion of Metals 3. Factors 4. Fick’s Laws 5. Homogenisation.
Contents:
- Introduction to Diffusion of Metals
- Atomic Models of Diffusion of Metals
- Factors Effecting Diffusion of Metals
- Fick’s Laws of Diffusion of Metals
- Homogenisation and Diffusion of Metals
1. Introduction to Diffusion of Metals:
Consider a single phase, like a still gas, liquid, or a solid. Although, there is no overall motion of the phase, individual atoms or molecules are nevertheless moving about it under the influence of thermal energy (as the phase present at room temperature is much above the absolute zero). At absolute zero, the probability of a jump by an atom or molecule is zero. In a solid metal too, atoms are jumping continually from one position in crystal lattice to a neighbouring position.
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In a pure metal, as all atoms are alike, it is difficult to detect any evidence of diffusion, unless some radioactive isotopes of this metal are present in the lattice, and then, the rate of self-diffusion, i.e., rate at which the atoms diffuse among themselves, can be measured. If there is a segregation of atoms of alloying element or impurity element in a solid solution in a region, just like a cored solid solution, then the concentration gradient and the thermal energy can cause these individual atoms to move in a direction of depleted region.
Thus, if a rod of an alloy, which has concentration gradient of solute (Fig. 1.56 (a) and (b)-initial profile), is heated for a few hours at a temperature where atomic movement is fast (that is, below the freezing temperature of alloy), the solute atoms get redistributed until the rod becomes uniform in composition as illustrated by final profile in Fig. 1.56 (b) and in 1.56 (c).
Thus, diffusion is the mechanism by which matter is transported through a matter, also called hetro-diffusion. It is defined as the mass flow process by which atoms (or molecules) change their positions relative to their neighbours in a given phase under the influence of thermal energy and a gradient.
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Thermal energy (that is. the temperature of a phase is higher than absolute zero) is a necessity for the mass flow as the atoms have to jump from site to site during diffusion. The thermal energy causes vibrations of the atoms about their mean positions in the solid, whose amplitude increases with the rise of temperature.
Actually, the thermal energy has mainly two effects to account for the large increase in the rate of diffusion with the rise of temperature- increase in jump frequency of atoms and also increase in the concentration of vacancies, which is required for the main mechanism of diffusion called vacancy mechanism.
The gradient could be concentration gradient, an electric or magnetic field gradient, temperature gradient or even, a stress gradient. Diffusion, under a concentration gradient, is more relevant in heat treatment and is thus, discussed here.
The presence of solute atoms in a crystal lattice causes strain in the lattice, because of different sizes of atoms of solute and solvent. The strain is at a minimum level if solute atoms are evenly dispersed throughout the crystal. Hence, there is a tendency for diffusion to occur until the concentration gradient of solute atoms within the crystal lattice is zero.
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The mass flow, or diffusion is of great importance in most heat treatment processes and thus, some understanding of mechanisms, laws of diffusion and simple solutions of the equations involved, will help to grasp better many heat treatments including carburising, nitriding, annealing, age-hardening, oxidation, decarburisation, etc.
2. Atomic Models of Diffusion of Metals:
The basic step in the diffusion process is the transfer of atom from one site to an adjacent site in a crystal lattice thus, called lattice diffusion.
This transport of atoms may occur in many ways:
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In the interstitial diffusion, solute atoms, which are small enough to occupy interstitial sites, just as carbon, boron, nitrogen, oxygen, hydrogen atoms (five of them) in iron, jump from one interstitial site to the neighbouring interstitial site (Fig. 1.57) as one step. Because of solute atoms being small, these move easily through the crystal lattice of parent metal, but still must possess an exceptional large amount of energy before it can jump out of its interstice.
The neighbouring interstitial site is usually vacant in normal alloys (normally these are dilute alloys), that is, the interstitial diffusion is not dependent on the probability of its adjacent site being vacant.
The activation energy for interstitial diffusion is simply the energy barrier at site 3 [Fig. 1.57 (a)] along the path from interstitial site 1 to the interstitial site 2 as the interstitial atom has to push its way through the parent atoms with its outer electron cloud having a minimum overlap with the parent atoms at mid-point [point 3 in Fig. 1.57 (a)], requiring a momentary barrier energy, called the enthalpy of motion.
As these small interstitial atoms can diffuse without the aid of creating a vacancy, thus, interstitial diffusion is 4 to 6 orders of magnitude faster than the substitutional atom diffusion. For example, the diffusion coefficient of carbon in FCC-iron (austenite) at 1000°C is 3 x 10-11 m2 /sec, while of nickel in FCC-iron (austenite) at 1000°C is 2 x 10-16m2/sec, which is much less.
Some of the common types of mechanisms of diffusion, in the substitutional solid solutions, are illustrated in Fig. 1.58. The most commonly perceived is the vacancy diffusion [Fig. 1.58 (a)] in which atoms diffuse by interchanging positions with the neighbouring vacant sites, that is, the solute atom must have a vacant site in its neighbourhood. Thus, here, the probability that the adjacent site is vacant is to be taken into account.
As is known that the number of vacancies are in thermal equilibrium in a crystal and is given by:
where, N is the total number of atomic sites,
ΔHf is the enthalpy of formation of vacancies.
The expression, exp(-Δ Hf/RT) also includes the probability that a given atomic site is vacant. In vacancy diffusion, apart from this probability, we have to consider as we did for interstitial diffusion, the probability that the vibrating substitutional atom has sufficient energy to move its way through to the vacant site, and which is given by exp ( – Δ Hm/RT), where Δ Hm is the enthalpy of motion for the jump of a substitutional atom. The product of these two probabilities gives the probability that a vacant site is available at the same time as a vibration of sufficient amplitude takes place.
The diffusion coefficient is given by:
where, v is the frequency of vibrations, and δ is the jump distance.
The diffusion coefficient in interstitial diffusion is given by:
where, Δ Hm is the enthalpy of motion for interstitial atom. The absence of probability factor involving Δ Hf for interstitial diffusion makes it, generally much faster than substitutional diffusion by vacancy mechanism. That is why, the segregation of carbon by itself, in steels does not normally occur.
In interstitialcy mechanism, the configuration where two atoms share a common lattice site and moves through the lattice, or we can say that the diffusing solute atom (say atom A in Fig. 1.58 b i) is only temporarily interstitial and is in lattice site by diffusion, but makes a solvent atom B in self-interstitialcy [Fig. 1.58 (b ii)] and this process continues. We can say that atom A [Fig. 1.58 (b i)] is in dynamic equilibrium with other atoms in the lattice positions. However, the energy to form such an interstitial is many times than that to produce a vacancy, and thus, interstitialcy mechanism is less likely to take place.
In the direct exchange mechanism [Fig. 1.58 (c)], adjacent atoms exchange places but severe local distortion is involved during motion, making it very unlikely. In the ring mechanism, three or four atoms in the form of a ring [Fig. 1.58 (d)] move simultaneously round the ring and thus, interchange their positions.
The diffusion of solute atoms with the aid of vacancy is a common mechanism, but then, vacancies may associate preferentially with solute atoms in solution, that is, the binding of vacancies to the solute atoms increases the effective vacancy concentration near these atoms, and thus, the mean jump rate of the solute atoms is much increased.
Diffusion is a structure-sensitive property, and it has been experimentally proved that, D, the diffusion coefficient increases with the lattice irregularity, that is, it is aided by line defects as well as surface defects. Diffusion along them, is sometimes, called short-circuit diffusion. As there are no constraints on one side of the free-surface of a crystal, the enthalpy of motion of an atom diffusing along free surface is considerably less than that for an atom moving within the lattice of the crystal.
In a poly-crystalline material, the grain boundary regions are not as closely packed as inside the crystal, and thus, ΔHm for the grain boundary diffusion is smaller than that for lattice diffusion.
The measurements show that free surface and grain boundary forms of diffusion, have lower activation energies than for lattice diffusion by the following variations:
The diffusion along dislocations (line defects) called pipe diffusion is faster than lattice diffusion, and that is why, diffusion is faster in the cold worked state than in annealed state of a metal.
A lower activation energy of diffusion such as for grain boundaries and external surfaces does not necessarily mean that diffusion along these high diffusivity paths always dominate over lattice diffusion, but on the contrary the reverse is true particularly at high temperatures.
According to the Fick’s Laws of diffusion, the material transported by a diffusion process for a given compositional gradient also depends on the effective area through which atoms diffuse. Since the total free-surface area or grain boundary area (for commonly used grain sizes of materials) to the volume ratio of any poly-crystalline metal is usually very small, say, for example, the fraction of the total cross-sectional area of a typical metal used for grain boundary diffusion in only 10-5.
Thus the ratio, DG.B/Dlattice must be around 105 before the amount of material transported along grain boundaries is comparable to that diffused by lattice diffusion in the grain. At high temperatures, this ratio is much less than 105, and thus, lattice diffusion is dominant. Though in particular cases, like, sintering and oxidation, grain boundary and surface diffusion become dominant. The grain boundary diffusion becomes more competitive if the grain size is very fine (ultra-fine grains) and lower is the temperature.
As the temperature decreases, the diffusion rate along grain boundaries and free surfaces decreases much less rapidly than diffusion rate through the lattice, that is, these become comparable. At still lower temperatures, both lattice diffusion and grain boundary diffusion may be ineffective, and then significant atomic transport can occur along pipe of disturbed lattice surrounding a dislocation line. If a void inside a metal is connected to the surface of the specimen by a dislocation line, the counter flow of vacancies and atoms is able to occur more easily.
3. Factors Effecting Diffusion of Metals:
The diffusion coefficient, D, is normally not a constant, but is a (strong) function of temperature, concentration and crystal structure. Also, in addition to the lattice diffusion which has been mainly considered, significant amount of diffusion may take place by other methods like along the grain boundaries, surfaces, or dislocations.
Moreover the diffusion may take place not only by concentration gradient, but also by local state of stress, electric, or magnetic field, or temperature gradient. The effect of concentration on diffusion coefficient in dilute solutions, or over a small range of concentration is not much. For example, Fig. 1.61 illustrates the effect of carbon concentration on diffusion coefficient for carbon in iron at 927°C.
The crystal structure also effects the diffusion coefficient. BCC crystal structure has atomic packing factor of 0.68 with a void fraction of 0.32, whereas FCC has a void fraction of 0.26. At a given temperature, diffusion and self-diffusion occurs about one hundred times more rapidly in ferrite (BCC) than in austenite (FCC). Even with in a crystal structure (except in cubic metals), such as in bismuth (rhombohedral), the ratio of self-diffusion constants is about one thousand for diffusion parallel and perpendicular to the c-axis.
The rate of diffusion is faster in a distorted crystal structure due to elastic strains, or extensive cold working. As grain boundary diffusion is faster than that within the grains, thus diffusion is faster in fine grained materials, particularly when the grain sizes are in the ultra-fine grain range.
4. Fick’s Laws of Diffusion of Metals:
The driving force for the process of diffusion is the concentration gradient (to be more correct-activity gradient), and is used here for discussion. Fick’s first law states that the amount (J) of matter (under investigation) moving across a unit area of plane (normal to the x-axis and the x-axis is parallel to the direction in which the concentration gradient is operating) in unit time is proportional to the concentration gradient (dc/dx) at the same instant,
where, the negative sign means, that, flow of matter occurs down the concentration gradient; dc/dx is the concentration gradient in the x-direction; D is the diffusion coefficient (m2/sec) and is a constant characteristic of a system and depends on the nature of the diffusing species, the matrix in which it is diffusing and very strongly on the temperature at which diffusion occurs. On a rough estimate, D is doubled for every twenty degrees rise of temperature.
The diffusion coefficient, D as a function of temperature is given by:
where, D0 is the pre-exponential constant (m2/sec.),
Q is the activation energy for diffusion (KJ/mole).
Table 1.14 gives values of some D0 and Q. These values are approximate as diffusion coefficient varies significantly with the composition of the matrix.
Fick’s first law describes the mass flow under the steady state conditions, i.e., the flux is independent of time and remains the same at any cross sectional plane along the diffusion direction, i.e., the concentration gradient remains constant. Fick’s first law could be used to estimate one type of oxidation of metals at high temperatures showing parabolic relationship between the time of oxidation and the amount of oxide formed.
Here, it is assumed that diffusion of only the metal atoms takes place and the difference between the metal concentration, C0 at the oxide surface and the concentration Cm at the metal surface has a constant value, ΔC. See Fig. 1.59.
Thus, at any distance x of the oxide layer, the rate of transfer of metal atoms per unit area is given by:
where, K’ = – 2D’ ΔC, but, it is positive quantity as the minus sign gets cancelled by the minus sign of the concentration difference, AC. We see that the thickness of the oxide layer increases as the square root of the time of oxidation, which relationship in graphical form is given as in Fig. 1.59 (b).
The equation 1.59, or as simplified:
may be regarded as fundamental to most diffusion processes, when the atoms migrate in one direction.
Fick’s Second Law (Diffusion Transport Equation):
This law is an extension of the Fick’s first law, but is applicable to more practical problems, as it is for non-steady state flow, that is, at any given instant, the flux is not the same at different cross-sectional-planes along the diffusion direction x. At the same cross-section, the flux is not the same at different times. Thus, the concentration-distance profile changes with time.
This law of transient-state diffusion, based on the matter being conserved within the system, is expressed as:
If it is assumed that ‘D’ is independent of the concentration, then
Even when, ‘D’ may vary with concentration, the simple solutions, thus, obtained of the differential equation (1.62), are quite commonly and satisfactorily used for practical problems.
The equation (1.62) has been solved for unidirectional diffusion from one medium to another across a common interface as:
where, A and B are constants, which are found out from the initial state, and boundary conditions of a particular problem. The diffusion direction x is perpendicular to the common interface and the origin of x is at this interface. Two mediums are taken to be semi-infinite, i.e., only one end of each of them, which is at the interface is defined and the other is infinite. The initial uniform concentration of the diffusing species in the two mediums is different with a sudden change in concentration at the interface.
The ‘erf’ in equation (1.63) means ‘error function’, which is a mathematical function defined as:
where, η is an integration variable which gets deleted as the limits of the integral are substituted.
The solution of an infinite source (or even sink) in a semi- infinite slab, where the supplying concentration at the surface remains fixed (i.e., an infinite source).
Where, Cs is the concentration at the surface, C0 is the original or base concentration, and Cx is the concentration at a distance, x, into the slab This shall be used in application, like carburising, decarburising, etc.
5. Homogenisation and Diffusion of Metals:
Cast products invariably have segregation, or chemical heterogeneity. Diffusion can change a chemically inhomogeneous solid solution to a homogeneous solid solution. As diffusion is very slow at room and low temperatures, castings are heated to high temperatures for long times, the process called is homogenisation or homogenising annealing.
The diffusion coefficient of the alloying element in the solvent matrix if is high, then, homogenisation is achieved easily but, otherwise, it may take very long time unless, some cold working is done to reduce as much as possible, the spacing between segregations.
For example, even at moderate temperatures, zinc diffuses rapidly in copper, it produces effective homogenisation, but nickel diffuses so slowly in copper that it is difficult to remove segregation in cast cupro-nickels. However, if cupro-nickels are plastically deformed, which decreases the distance between segregations, that is, the diffusion distance, effective homogenisation can be obtained at the same temperature.
Segregation occurs in steels too, depending on the amount and type of elements present in it. Homogenising annealing, if carried out, at relatively high temperatures, can eliminate segregation. A special case of chemical heterogeneity in steels needs some consideration called banding in steels [Fig. 1.62 (a)] and its elimination by diffusion. Banding in steels is a chemical heterogeneity in rolled steels, showing a microstructure consisting of alternate bands (layers) of ferrite and pearlite.
It is due to interdendritic segregation of alloying elements, mainly Mn during the solidification of ingots and, is difficult to avoid in cast steels. Banding is very prominent in microstructures of steels having ferrite and pearlite in almost equal proportions, for example in steel having 0.25% carbon and 1.5% manganese. During rolling, the segregated areas are elongated and compressed into narrow bands. The bands of pearlite (dark) coincide with Mn segregation.
As Mn reduces the activity of carbon in austenite, thus carbon segregates with the manganese. After hot rolling, as austenite transforms, pearlite forms at the segregation of carbon and manganese. Banding in plate steels is not always harmful, but it does cause anisotropy in tensile ductility and notch- impact energy, and it is easier to propagate a crack parallel to bands in ferrite than normal to them.
The elimination of banding is a classical problem of diffusion. The mathematical treatment of this diffusion problem becomes simple, if it is assumed that alloy concentration (here Mn) varies sinusoidally along a straight line normal to the band about the average value as schematically illustrated in Fig. 1.62 (b). The diffusion of Mn from rich region to the deficient region can eliminate/reduce banding. Then, the relationship holds,
where, C is the variation of Mn from the average concentration at a point x, Cm is initial maximum variation from the average concentration of Mn, x is the distance in meters, and l is the distance between a region of maximum concentration of Mn and a neighbouring region of minimum concentration. DMn, the diffusion coefficient is assumed to be independent of concentration. Now, using equation (1.66) for the concentration, than a solution of Fick’s Second law, equation (1.62) is,
where, Cm is a constant, and sin πx/l is the sinusoidal variation of concentration and as the maximum is needed of this function for the present conditions of the problem, and thus, can be set equal to unity. Thus, the factor, exp 
controls the decrease in degree of inhomogeneity with increasing time, τ, and this factor decreases from unity towards zero with increasing time.
This means that time required to produce a given degree of homogenisation increases with the square of diffusion distance, I and is inversely proportional to diffusion coefficient, DMn. The diffusion coefficient, DMn is not very high. Table 1.16 illustrates its value at three different temperatures.
In a steel slab, the value of l, the diffusion distance can be taken to be safely as 4 mm. The time required for 90% homogenisation comes to be as illustrated in table 1.16, which is very large, almost as a waste of energy and money. Thus, some heat treating operations should be carefully planned in advance not to become too expensive.
Now, if this steel slab is reduced by plastically deforming it so that l, the diffusion distance becomes 0.02 mm, then the time required for 90% homogenisation at 1200°C is about 26 hrs., which is high but much less than given in table 1.16 i.e. the diffusion is greatly accelerated by reducing ‘l’. Thus, if homogenisation is being planned, first reduce the spacing between the segregations as much as possible by plastic deformation and then homogenising annealing is done. [For 90% homogenising, the relationships is which can be solved for setting the time of annealing.]
Thus, equation 1.67 can be modified for solving some problems, if it is assumed that variation in concentration of an element is sinusoidal.
