The three thermally active and overlapping stages during annealing are: 1. Recovery 2. Recrystallisation 3. Grain Growth.
The first two stages of annealing are caused by the stored cold- worked energy. Grain growth, the third stage of annealing, occurs if annealing is continued after recrystallisation has completed. In this stage, the recrystallised grains grow in size at the expense of other normally smaller grains.
The removal of cold-worked state occurs by annealing. There is no change in composition or crystal structure during annealing. The driving force for recovery and recrystallisation is the stored cold-worked energy, whereas, the driving force for grain growth is the energy stored in grain boundaries.
Stage # 1. Recovery:
It is the initial stage of the annealing cycle of a cold worked metal before recrystallisation occurs. Fig. 7.6 also indicates the changes in some properties of cold worked polycrystalline metal, here nickel, on annealing. During recovery, a small part of the stored cold-worked energy decreases to cause substantial decrease in electrical resistivity, and only a slight lowering of the hardness without a visible change in optical microstructure.
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Recovery is defined as the process of annihilation and the rearrangement of defects within the deformed metal without the movement, or migration of high angle boundaries; as illustrated schematically by no visible change in optical microstructure. In a given cold worked metal, the individual properties recover at different rates and attain various degrees of completion.
The relaxation processes occurring during recovery are of two basic types:
1. Annihilation of excess point defect, in particular vacancies.
2. Rearrangement of dislocations, and in process some annihilation of them.
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The relaxation processes, or the structural changes during recovery occur more, or less simultaneously throughout the deformed matrix. The first relaxation process starts at lower temperatures during annealing, which is followed by the second process at higher temperatures, but before the process of recrystallisation takes place.
At lower temperatures, the excess point defects, generated during the process of cold working, are annealed out, i.e., reduce their number to their equilibrium value in various ways. The most important point defect is a vacancy, which may have a finite mobility even at relatively low temperatures.
Fig. 7.7 illustrates how vacancies as well as interstitialcy atoms sink at grain boundaries, or at edge dislocations, or annihilate each other. The annealing out of point defects causes decrease in electrical resistivity of a material as free valence electrons are less scattered. Thus, the rate of recovery at a constant temperature can be studied by measuring the changes in electrical resistance as a function of time.
Fig. 7.8 illustrates changes in electrical resistance of copper, which had been cold worked at 4.2°K and then annealed isothermally at various higher temperatures. It illustrates the kinetics of the recovery (rate of recovery can be measured by the slope of the curve at a temperature) during isothermal annealing.
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Recovery is initially very rapid, and more so when the annealing temperature is high. With increasing time at the constant temperature, the recovery (i.e., the reactions during recovery) becomes slower, and eventually stops at a plateau, typical of the annealing temperature. The greater the initial cold work (i.e., greater the stored energy), the more rapid is the initial rate of recovery. Decreasing the grain size increases the rate of recovery (as the dislocation density increases).
At slightly higher temperatures of recovery, the rearrangement of dislocations takes place and in the process, mutual annihilation of dislocations of opposite signs takes palace. The rearrangement of the dislocations is assisted by the thermal activation (i.e., by heating), which causes slip, cross-slip and climb of dislocations over small distances.
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Fig. 7.9 illustrates some methods of two opposite type dislocations coming together to annihilate each other. Also the long dislocation dipoles now disintegrate to lower the elastic strain energy by forming long prismatic loops, which later break up into isolated loops as illustrated in Fig. 7.10.
Thus, during recovery, annihilation of dislocations occurs without substantial decrease of dislocation density, but the most important structural change is the reduction in surplus point defects, in particular, the vacancies. The atoms of impurities and alloying elements in solid solution alloys having attraction for vacancies and dislocations, decrease the migration of vacancies, and even rearrangement of dislocations, and thus, reduce the rate of recovery.
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One of the most important recovery processes, which leads to rearrangement of the dislocations, with a resultant lowering of the lattice strain energy, is Polygonisation. After local annihilation of opposite signed dislocations in early stages of recovery, excess dislocations of the same sign are left over in the metal.
The picture could look like Fig. 7.11 (a), where a single crystal/of a metal has been cold worked by bending in one direction, which produces excess edge dislocations of same sign on parallel slip planes.
The stored elastic strain energies of these dislocations get reduced further when they arrange themselves in sub-boundaries, Fig. 7.11 (b). Fig. (c) Illustrates that strain fields are additive when edge dislocations of same sign accumulate on the same plane. However, if same dislocations are arranged in a vertical sequence, Fig. 7.11 (d), the tensile field of upper dislocations and compressive field of lower dislocations partly cancel each other.
The decrease in the total strain energy, thus, is the driving force for polygonisation. Polygonisation is the process of arranging excess edge dislocations in the form of tilt boundaries, and the excess screw dislocations in the form of twist boundaries, with the resultant lowering of the elastic strain energy.
The curvature to the crystal, Fig. 7.11 (a), can be attributed to the presence of excess positive edge dislocations (as all of them have lower half planes missing). On heating the bent crystal further during annealing, the edge dislocations form a number of tilt boundaries, and thus, convert the curved bent surface of the crystal to a number of planar surfaces tilted with respect to one another, Fig. 7.11 (b).
The small almost perfect crystal segments are called sub-grains, or, polygons, and the tilt boundaries are called sub-boundaries. The sub-grains lie inside a grain. There is very small difference in orientation of atoms in the neighbouring sub-grains.
The process of polygonisation is illustrated in Fig. 7.12. The climbs as well as slip of dislocations are essential for polygonisation. The important movement is the climb process which involves the diffusion of vacancies to, or, from the edge of the half planes of dislocations. The number and diffusion of vacancies increase with temperature.
The climb of dislocation is a thermally-activated process. Slip also becomes easier at higher temperatures as the critical resolved-shear-stress for slip decreases with the rise of temperature. Screw dislocations can join sub-boundaries by cross-slip which is also facilitated by thermal-activation. Thus, the rate of polygonisation increases rapidly with increase of temperature.
Sub-boundaries can be observed under optical microscope after appropriate etching as chain of etch-pits where each pit corresponds to emergence of the dislocation on polished surface. More often sub-boundaries are seen in micro-sections as a network of fine lines inside the grains, the grain boundaries being revealed as thicker lines. Electron micrographs have been observed to show regular hexagonal networks, and planar sub- boundaries. In iron alloys, three sets of dislocations give rise to hexagonal network of dislocations as illustrated in Fig. 7.13.
The rate of polygonisation depends on the nature of material, the amount of cold deformation, temperature of recovery (also prehistory of heating), the amount and nature of impurities, etc.
Metals of low stacking-fault energy such as copper, α -brass, silver, polygonise less readily as compared to aluminium having high stacking-fault energy. Extended dislocations in aluminium have smaller stacking- faulted width, which gets constricted to allow climb, or cross-slip (depending on nature of the constricted dislocation) required for polygonisation. Under the creep conditions at high temperatures, metals with very low stacking-fault energy may climb to polygonise, but more often polygonisation may not take place.
The presence of solute atoms in a metal reduces the rate of polygonisation. The formation of Cottrell atmospheres, which decrease the climb and slip, may take place or solute may reduce the stacking-fault energy. In general, a pure metal can be polygonised in a shorter time than an alloyed metal or impure metal.
When dislocations have aligned into sub-boundaries with inside of the sub-grains almost free of dislocations, the sub-grains then tend to grow larger as the time, or/and temperature is increased. The driving force for the growth of the sub-grains is the decrease in elastic strain energy. The strain energy of the combined boundary is less than of the two or more separate sub-boundaries added together. The angle of misorientation of sub-grains across the new boundary must increase in the process.
Two experimentally established mechanisms of growth are:
(i) Migration of sub-boundaries
(ii) Coalescence of sub-grains
In the first mechanism, dislocations move out of tilt boundary to join those in adjacent tilt boundary, so that two or more sub-boundaries combine to form a single boundary as illustrated in Fig. 7.14.
The growth of sub-grains by coalescence of neighbouring sub-boundaries has been experimentally seen in aluminium under electron microscope. Fig. 7.15 (a) illustrates the structure of two neighbouring-sub-grains having slight difference in orientation before coalescence; (b) illustrates the rotation of right hand sub-grain to have the same lattice orientation as in sub-grain on left side.
This process is also accomplished if both the sub-grains had rotated a definite angle to obtain the same lattice orientation eliminating the common sub-boundary. The end result in either case is illustrated in (c) i.e., after coalescence. The final sub-grain structure after migration of some sub-boundaries is illustrated in (d).
Coalescence, i.e., disappearance of common sub-boundary requires a movement of dislocations out of it and into the other boundaries of the two sub-grains. The process needs that dislocations climb cooperatively so that the spacing between them remains equal and that the boundary retains its low energy consideration.
The process of coalescence, i.e., of climb is strongly dependent on temperature. Large-scale diffusion of atoms from the hatched regions, Fig. 7.15 (b) becomes difficult when sub-grains become large sized, and thus coalescence almost stops. It has been seen that group coalescence also occurs, i.e., instead of two sub-boundaries, a number of neighbouring sub-boundaries may disappear simultaneously.
Sub-grains may grow of size around 10 μm, though the growth of sub-grains has to be within the limit of original deformed grain, and the new boundary should be a low angle boundary having the difference of orientation not exceeding 10- 15° (normally 1°).
Polygonisation is probably one of the important mechanisms of recovery, but it may not always occur as recovery takes place by other processes. A H.C.P single crystal (such as zinc or cadmium) or of cubic structure, when deformed by easy glide (slip on a single slip system) may not contain excess dislocations of one sign necessary for polygonisation, and still recovers completely its softness without recrystallisation.
Such materials have been seen not to recrystallise even when heated close to their melting points. Zinc crystal after substantial plastic strain (but deformation was on single basal plane), when rested for 24 hours at room temperature, underwent complete recovery, attaining the flow stress of the undeformed crystal.
The recovery of yield stress, R, defined as-
R = (σm – σ)/(σm – σ0) …(7.1)
where, R is fraction of recovery; σ = flow stress before annealing; σ = flow stress after recovery; σ0 = initial flow stress or flow stress after complete annealing. The rate of recovery can be obtained by-
dR / dt = a/t
The kinetics are similar to that in Fig. 7.8, and the activation energy of 20000 cal/mole was obtained. Actually, this recovery process can be symbolised by typical Arrhenius-plot, to be expressed by-
1 / t = Ae-Q/RT …(7.2)
where, Q is the activation energy; t is the time needed for a fraction of recovery of flow stress; A is constant; T is absolute temperature. If recovery has occurred to the same extent at two different temperatures, then,
Problem 1:
If assumed that activation energy for recovery of yield point of zinc crystal is constant to be 20000 cal/mol. The deformed crystal recovers one fourth of its yield point is 5 minutes at 0″C, how much time shall it take at 27°C?
Solution:
Using the given data:
T = 5 minutes, Q = 20000 cal/mole, R=2.
= 0.275 minutes.
Dynamic Recovery:
The low temperature recovery process basically is the reduction of number of point defects, particularly the vacancies to their equilibrium number. At high temperatures, the recovery process in deformed poly-crystalline materials is essentially the process of movement of dislocations to sub-boundaries i.e. the process of polygonisation and annihilation of excess dislocations.
If the process of movement of dislocations into sub-grains or cell boundaries takes place during cold working, then the recovery process accompanying it is called dynamic recovery. Though it may take place at low temperatures in some pure metals as it is being simultaneously stressed, but can be quite intensive at high temperatures as movement of dislocations increases with the rise of temperature of metal (or working).
As the critical resolved-shear-stress decreases with the rise of temperature, the dynamic recovery decreases the rate of work-hardening. The process of polygonisation takes place readily in metals of high stacking-fault energy, but metals with low stacking-fault energy may, under the working stress, constrict to polygonise.
In metals with very low stacking-fault energy, the dislocations cannot constrict and thus cannot jump or cross-slip, and are seen to be lying along their slip planes. It is suggested that then, thermally activated cross-slip is the main mechanism in dynamic recovery.
Stage # 2. Recrystallisation:
Fig. 7.6 (c) illustrates that major part of the stored strain energy (due to large peak in the curve) of a cold-worked metal is released over a small range of temperatures in the second overlapping stage of annealing, and when the optical microstructures reveal, Fig. 7.6 (d), nucleation and the growth of equiaxed strain-free grains at the cost of original deformed grains.
As the time, or temperature of annealing increases, the deformed grains are completely replaced by new strain-free grains. It is not only that shape, size and orientation of the new grains are different than the original cold worked grains, but also the dislocation density decreases from heavily deformed state of 1011-1012 cm-2 to 106 to 10 cm-2.
This process, for which the driving force is the remaining stored cold-worked strain energy, is called recrystallisation. Recrystallisation is the process of formation of new strain-free grains from deformed grains in a solid body by the movement of high angle boundaries. Unlike recovery, the process of recrystallisation makes the mechanical and physical properties of the deformed metal to return completely to those of the annealed state.
Mechanical properties like hardness, yield strength, tensile strength, percentage elongation change drastically over a very small temperature range to become typical of the annealed material. Although the physical properties like electrical resistivity, undergo appreciable decrease during recovery, but also decrease sharply during recrystallisation.
The kinetics of recrystallisation resembles a phase transformation, i.e., recrystallisation is a nucleation and growth process. The strain-free nuclei form and begin to grow in the deformed metal, when the temperature is high enough, and gradually absorb the whole of the deformed matrix.
The isothermal recrystallisation studies can be made by putting around twenty samples of the cold-worked metal in a constant-temperature bath. Specimens are kept for different times and then removed, and their percentages of recrystallisation are determined by microscopic examination, to yield a typical curve as illustrated in Fig. 7.17.
The important features of the curve are:
1. There is an incubation period.
2. The kinetics of recrystallisations is such that recrystallisation starts slowly (slope of the curve increases slowly) and gradually reaches to a maximum rate, and then rate of recrystallisation slows down again (near the completion of recrystallisation).
Nucleation in the process of recrystallisation means new strain-free grains form by the growth of specific lattice domains (sub-grains, or cells) already present in cold-worked state, or which get developed during the process of recovery. Electron microscopic studies, however, suggest that the fundamental process in recrystallisation is the migration of boundaries separating the cold worked matrix from a region that is essentially free of dislocations.
Such a mobile boundary has to be a high angle boundary as it is capable of migrating quickly (as compared to low angle boundary, or, sub-boundary). A large angle boundary (θ ≈ 30-40°) has large lattice irregularities, or even gaps. The atoms on such a boundary can easily transfer their links from one grain to the other. When such a boundary migrates through the cold-worked structure, it leaves behind a recrystallised structure.
A sub-grain can act as a nucleus for growth in recrystallisation if it has lesser dislocations than its neighbours; has large misorientation angle (~ 15°) with the neighbour so that the boundary is mobile; and the interfacial energy required for increasing its size is less than the volume free energy released, when strained-part is replaced by strain-free part.
The following three models have been suggested to create a nucleus with at least one of its boundary with neighbour to be a mobile high angle boundary. Microscopic examinations show that favoured sites for the growth of sub-grains are grain boundary edges (where three grains meet in a line), grain boundaries, inclusions, deformation-bands, intersection of twins, or surfaces of the materials.
These are also sites of heavy distortion, or high dislocation density, or severe lattice bending (i.e., have an excess of dislocations of one sign, which can easily polygonise to produce sub-grains), or are in the vicinity of sharp changes in orientation. These facts illustrate that recrystallisation nuclei are formed (on heating) only from strongly curved regions of the lattice, where polygonisation can readily occur. Thus, polygonisation is a necessary preliminary stage to recrystallisation.
The process of recrystallisation can be picturised as- Polygonisation of cold worked metal in bent lattice regions creates sub-grains where the strain energy is lower than the surrounding matrix. These sub-grains grow initially fast till the angle between them increases to a few degrees, and then the growth slows down.
The three models for nucleation of recrystallisation in a metal are:
1. By the motion of pre-existing grain boundaries between neighbouring grains.
2. By the motion of sub-grain boundary.
3. By coalescence of sub-grains.
The experimental fact that grain boundary surfaces are quite effective sites for the nucleation of strain-free grains is the basis of the first model. Neighbouring grains frequently have two quite different sub-grain sizes, Fig. 7.18 (a). Large sized sub-grains (have low density of dislocations) grow into the grain having smaller cells.
The grain boundary bulges as illustrated in Fig. 7.18 (b), to form a protrusion, or ‘tongue’. The moving large angle boundary ‘sweeps out’ the lattice defects from the grain having smaller cells. A recrystallisation nucleus is created.
When the sub-grain sizes across a grain boundary, Fig. 7.18 (c) are almost of same size, the sub-grains might have first grown in size by coalescence, so that it is a large portion of almost perfect structure which then moves in adjacent grain having increased density of dislocations.
Fig. 7.18 (c) shows grains before coalescence and Fig. 7.18 (d) after coalescence of sub-grains and migration of a portion of high angle boundary. This model has been experimentally verified in iron, nickel, etc. Inclusions are also quite common sites for nucleation for recrystallisation (photomicrograph Fig. 7.19).
There are regions of high dislocation density around inclusions, where cell structure can develop better, and that the misorientation across the cell walls are greater than average in the neighbourhood of the inclusions, promoting the migration of sub- boundary.
The second model is based on the growth of sub-grains by the migration of sub-boundaries. The driving force as discussed earlier in recovery, is that a boundary of misorientation, θ3 formed from two sub-boundaries, θ1 and θ2 has less energy than the sum of energies of θ1 and θ2, where θ3 is greater than either θ1 or θ2, i.e., the growth of sub-grain increases the misorientation angle in between two new subgrains.
As this process continues to occur by a succession of migration of sub-boundaries, a high angle boundary gets developed (this probably happens in the incubation period of recrystallisation), and then, the nucleus acts as a recrystallised grain with a high angle boundary surrounding it.
In the third model, the coalescence of sub-grains leads to the formation of recrystallised grain. As discussed earlier and illustrated in Fig. 7.15, the coalescence of two sub-grains by the rotation of atleast one of the sub-grains, results in a sub-grain which has increased misorientation with its other neighbours.
Fig. 7.20 illustrates that when, say, four sub-grains coalesce, the size of the sub-grain thus formed increases considerably, and its orientation with one of its neighbours may become large enough to make the boundary between them as mobile large angle boundary. Fig. 7.21 illustrates such a recrystallised grain in aluminium.
Once by growth, the almost perfect sub-grain becomes large in. the deformed matrix and the low angle boundary becomes a large angle boundary, its rate of migration increases. At this moment, the recrystallised nucleus has just nucleated (the strain-free nucleation of grain has just occurred). In all these models, diffusion plays an important role such as for dislocation climb, volume diffusion (for coalescence), etc.
Thus, the formation of recrystallised nuclei is a thermally-activated process, i.e., it takes place faster at higher temperatures. A deformed metal having non-uniform distribution of dislocations, polygonise faster to produce large sub- grains by growth in some areas to produce recrystallised nuclei easily.
Stage # 3. Grain Growth:
When the recrystallisation is complete, i.e., the deformed grains have been replaced by strain-free grains, and if heating is continued to higher. temperatures, or for longer times, the grain boundaries slowly migrate and produce a uniform increase of grain size at the cost of neighbouring recrystallised smaller grains. This process is called as grain-growth.
The driving force for grain growth is the energy associated with the grain boundaries, i.e., when the grain size increases, the total grain boundary area decreases, and thus, the total energy of the polycrystalline metal are lowered. Grain growth can also occur in any ordinary metal or alloy, even when strain-free, if heated to higher temperatures.
The two dimensional model of grains in a metal just after the process of recrystallisation is illustrated in Fig. 7.28 (a), which illustrates irregular grains of different sizes and with different number of sides. One triple point of it is exploded in Fig. (b). The surface tension of grain boundaries in a metal or in a single-phase alloy can be approximately taken to be equal. At the triple point, these forces should be balanced, provided, Fig. 7.28 (b),
ϒ = 2 ϒ cos (θ / 2) … (7.6)
or cos (θ / 2) = 1 / 2
or θ = 120°
i.e., under the equilibrium conditions, the three grains should meet at angles of 120° at the triple point i.e., the grain should be hexagonal. Thus, the stable equilibrium requires the two dimensional model of grains to consists of regular straight-side hexagons. In order to attain the equilibrium of surface tensions, i.e., to make θ = 120° of Fig. 7.28 (b), the grain boundaries migrate to bend with simultaneous displacement of triple point from X to Y as illustrated in Fig. 7.28 (c) where dotted lines are original positions of the boundaries.
As a curved boundary has more energy and under the surface tension force, the boundary straightens by moving in the direction of its centre of curvature, Fig. (d). The straightening of the boundaries takes place, but which again disturbs the stable equilibrium, i.e., angle θ becomes less than 120°. This process is repeated again and again resulting in slow disappearance of grain 1, while grain 2 and 3 gradually grow. On the whole average size of grains increases.
But just after recrystallisation, the two dimensional model of grains consists of polygons of 3, 4, 5, 6 and more sides (Fig. 7.28 a). As illustrated in Fig. 7.28 (c), the grain boundaries bend to have equilibrium due to the surface tension forces at the triple points, and thus, grains with less than six sides have their boundaries convex outwards, and those with more than six sides convex inwards as illustrated in Fig. 7.28 (a).
As illustrated in Fig. 7.29 (a) the boundary migrates towards its centre of curvature, and thus, grains with concave boundaries grow at the expense of grains with convex boundaries, which eventually are consumed. It is possible that grain-growth starts in a portion of metal, where recrystallisation has already taken place, but recrystallisation is still occurring in other portions of the metal.
During grain growth, the number of sides of the growing grains may decrease, Fig. 7.30, or increase, Fig. 7.31. Geometrical coalescence, Fig. 7.32 may take place in metals having strong preferred orientation (i.e., in highly textured metal), where two grains such as 1 and 2 (originally far placed) of nearly similar orientation meeting and forming one grain during grain growth, is more likely to occur.
The grain growth taking place with increasing time at a constant temperature is small as compared to with increasing temperature at constant time (normally one hour at a temperature).
During grain growth, the average grain size uniformly increases and thus, the grain boundary area per unit volume decreases. The corresponding decrease in the grain boundary energy per unit volume becomes the driving force for the grain growth. For a spherical grain (assumed) of radius, R, the grain boundary energy per unit volume
where, D is grain diameter. The instantaneous rate of growth, dD/dt can be taken to be proportional to the grain boundary energy per unit volume of the metal, which in turn can be taken to be inversely proportional to D, i.e.
where, t is time; K is constant of proportionality. Integration of this equation results in,
D2= Kt + C …(7.11)
and if, Do is the average grain size at the start (when t = 0), then, after the evaluation of the integration constant, C, equation (7.11) gives,
D2 – D02 = Kt …(7.12)
If the initial diameter of grains is very small as .compared to the present diameter, then (putting D0 = 0),
D = Bt1/2 …(7.13)
This is parabolic growth law. It is observed experimentally only in a few ideal cases. In pure metals and single phase alloys, an analogous and empirical equation governing the grain growth is,
For small initial grain size (D0 = 0)
D = Ctn …(7.15)
where, D is the grain diameter in millimeter after time t minutes, and n and C are independent of time, but vary with composition and temperature. The value of C increases steadily with the increase of temperature. The value of exponent, n, in most cases is smaller than the value of 1/2 as predicted by the grain growth equation (equation 7.12).
It approaches the value of 0.5 as an upper limit with increasing temperature as well as with increasing purity of the metal. Fig. 7.33 illustrates that as aluminium content decreases in copper, the value of n increases from 0.1 to higher values and almost becomes 0.5 when the temperature is 600°C.
The solute atoms (of impurities as well as alloying elements) is solid solution state are elastically attracted towards dislocations to form Cottrell atmospheres, and towards the grain boundaries to form grain-boundary-atmospheres. The migration of grain boundaries as required for grain growth is effectively decreased due to grain-boundary-atmospheres.
As the content of the solute decreases, there is increasing less hindrance for the migration of the grain boundaries. At higher temperatures of annealing, the grain-boundary solute atmospheres are broken up by thermal vibrations, so that ‘n’ approaches the value of 0.5.
Zener-Paranjpe Effect of Inclusions (Pores) on Grain Growth:
The presence of impurity atoms, or alloying atoms in the form of second phase inclusions inhibit the grain growth in metals and alloys, and has been usefully employed to control the grain growth. The inclusions should be fine sized, uniformly dispersed and insoluble (at that temperature) in the metal or alloy. The inclusions could be products formed during metal-refining such as oxide, silicate or sulphide or even intentionally added elements as carbides, or nitrides, etc.
A spherical inclusion (as a simple case) is taken to be present at the grain boundary, Fig. 7.34 (a), and this is also the minimum energy state for the inclusion and the grain boundary. If the boundary moves away from the inclusion, 7.34 (c), the missing segment of the boundary (due to inclusion) has to be created, which requires extra energy.
Suppose the boundary has moved slightly to the right as illustrated in Fig. 7.34 (b), the surface tension tries to keep the boundary normal to the surface of the inclusion due to which the boundary takes a curved shape. The total length of the line of contact between the inclusion and the boundary is 2 r cos θ, where, θ is the angle between the equilibrium position of the boundary and the surface, where it meets the boundary in new position.
The horizontal component of the surface tension is σ sin θ. The product of this component and the length of the line of contact gives the pulling force of the boundary on the particle. By Newton’s law, this force is also the drag force of the particle on the grain boundary,
F = 2 σ a r sin θ cos θ …(7.16)
This force has a maximum value,. when θ = 45°; sin θ cos θ = 1 / 2 , i.e.,
fmax. = r σ…(7.17)
If Ns is the number of inclusions per unit area of the boundary, then the total drag force on the boundary is, NS r σ. As surface tension is the driving force for the grain growth, then as a grain of radius, R, shrinks, the surface tension per unit area,
= (2 σ)/ R … (7.18)
At a time, when the boundary is unable to pull itself away from its inclusions, the drag force must be equal to this force,
NS r σ = 2 σ / R, i.e.
the condition for pinning the grain boundary is
Ns r R = 2 …(7.19)
It has been assumed that the inclusions are uniformly distributed throughout the matrix. The number of inclusions which hold back a grain boundary of area A, are those whose centres lie inside a volume with in an area A, and whose thickness equals twice the radius of the inclusions, i.e., in a volume of 2 r A. If Nv is the number of inclusions per unit volume, then this volume holds particles equal to 2 Nv Ar. The number of particles per unit area = 2 Nv r. If F is the volume fraction of the second phase, then
where, 4/3 r3 is the volume of one inclusion. Putting this in the equation relating the drag force and driving force,
r / R = (3 /4) F
or R = 4 r / 3 F …. (7.21)
Quite a few assumptions have been made in driving this equation. However, it explains important facts that R, the equilibrium grain size becomes smaller, if r, the size of the inclusion is decreased, and when F, the volume fraction of the inclusions is increased.
The presence of a large number of fine second-phase-inclusions uniformly-dispersed keeps the grain size fine of the material during heating at high temperatures. In high speed steel (18/4/1), presence of 1% vanadium as finely and uniformly dispersed vanadium carbide keeps the steel fine grained even when heated up to 1260-1290°C. In steels, non-metallic inclusions formed during the deoxidation of the liquid steel tend to restrain grain growth.
Fig. 7.35 illustrates that the inherently-fine-grained steels resist grain growth as these have been deoxidised by the addition of aluminium, which produces finely and uniformly dispersed AIN inclusions. Silicon-killed steels, called inherently-coarse-grained steels, show continuous increase of grain size with increase of temperature as these steels do not have finely dispersed undissolved inclusions.
Even in inherently fine-grained steels, above a certain high temperature = 1050°C, grain coarsening occurs because these dispersed inclusions get dissolved in the matrix at that temperature, and then the rate of growth may be so high that the grain size may become even larger than the slowly growing inherently-coarse-grained steels. Similar effects are illustrated in Fig. 7.37.
Problem 4:
If AlN (0.01 per cent by volume) inclusions are present in steel to keep the grain size as small as 0.1 mm., what should be the size of these inclusions if it is assumed that AlN is uniformly present?
Solution:
% Volume of AlN = 0.01%
or, Volume fraction of AlN = 0.01 / 100
= 10-4
R = 0.1 mm = 10-4 m
As, R = 4 r / 3 F
Or r = (3 / 4) RF
= 3 / 4 × 10-4 × 10-4
= .75 × 10-8 m
Thermal Grooves:
The formation of thermal grooves on metal surfaces is another mechanism which is sometimes effective in impeding the grain growth. It has been seen experimentally in thin strips at high temperatures that the grain growth rate decreases when the grain size becomes larger than one-tenth of thickness of strip, and is drastically reduced when the grain size is two, or three times the thickness.
Most grain boundaries pass completely through the metal strip, and are perpendicular to the free surface of the strip, Fig. 7.36 (a). At these high temperatures, thermal grooves may form on the metal surface where grain boundaries meet the surface.
Three surfaces meet on a line at Y as illustrated in Fig. 7.36 (a) (two free surfaces-one on left of Y and another on right of Y and one grain boundary). In order to balance the vertical component of grain boundary energy, a groove must form with a dihedral angle, θ, Fig. 7.36 (b), so that,
ϒB = 2 ϒS cos θ / 2 … (7.22)
where, ϒB is the grain boundary energy, and ϒS is the surface energy of free surface. The formation of groove takes place, probably by the diffusion of atoms to free surface out of grooved regions at these high temperatures.
Once a groove has formed, it anchors the end of the grain boundary. Suppose the grain boundary has moved to the right-away from the groove. Fig. 7.36 (c), it requires the extra energy to create extra surface of the grain boundary for this step. Thermal grooves, thus, act as barriers to such motion of grain boundaries, i.e., the grain-growth, thus, stops. The curve in Fig. 7.37 for temperature 650°C, illustrates that thermal grooves stop grain growth.
Problem 5:
Thermal grooves are present at high temperatures on the surface of a metal where the grain boundary is perpendicular to the free surface. What is the angle at the bottom of the thermal groove if it is given that the ratio of energy of free surface to the grain boundary is 3.
Solution:
According to Fig. 7.36 (b), and the equation under equilibrium conditions is:
ϒB = 2 ϒS cos θ / 2
Given, ϒS / ϒB = 3,
∴ cos θ / 2 = 1 / 6
Or θ = 161°