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Crystallographic imperfections:
A perfect crystal, with every atom of the same type in the correct position, does not exist. Allcrystals have some defects. Defects contribute to the mechanical properties of metals. In fact,using the term defect is sort of a misnomer since these features are commonly intentionallyused to manipulate the mechanical properties of a material. Adding alloying elements to a metalis one way of introducing a crystal defect. Nevertheless, the term defect will be used, just this isnot enough to say that crystalline defects are not always bad. There are basic classes of crystaldefects:
1. Point defects or zero-dimensional defects:i) Vacancy, ii) Schottky defects, iii) Interstitialcy, iv) Frenkel defect,v) Compositional defects: a) Substitutional impurity, b) Interstitial impurity;vi) Electronic defects;2. Line defects or one-dimensional defects or Dislocationsi) Edge dislocation; b) Screw Dislocations3. Surface defects or Planar defects: a) Grain boundaries; b) Tilt boundaries; c) Twinboundaries; d) Stacking fault.4. Volume defects or three dimensional defects.
Point Defects
Point defects are where an atom is missing or is in an irregular place in the lattice structure. Pointdefects include self interstitial atoms, interstitial impurity atoms, Substitutional atoms andvacancies. A self interstitial atom is an extra atom that has crowded its way into an interstitialvoid in the crystal structure. Self interstitial atoms occur only in low concentrations in metalsbecause they distort and highly stress the tightly packed lattice structure. A substitutional
impurity atom is an atom of a different type than the bulk atoms, which has replaced one of thebulk atoms in the lattice. Substitutional impurity atoms are usually close in size (withinapproximately
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Point defects:
Point defects are imperfect point like regions in a crystal. The typical size of a point defect is oneor two atomic diameters. These defects are completely local in effect e.g., vacant lattice site. Point imperfections are always present in crystals and their presence results in a decrease
in the free energy.The point defects may be created as follows:i) By thermal fluctuationsii) By quenching from a high temperatureiii) By severe deformation of the crystal lattice e.g., by hammering or rolling. While the latticewill retain its general crystalline nature, numerous defects are introduced.iv) By bombardment of atoms with external high energy particles, e.g., from the beam of thecyclotrons or the neutrons in a nuclear reactor.
The various point defects are discussed below:1. Vacancy: A vacancy is the simplest point defect and involves a missing atom within a metal.
These defects may come up as a result of imperfect packing during the original crystallisation.They may also arise from thermal vibration of the atoms at high temperature.
2. Schottky imperfections:Theseare closely related to vacancies but are found in compoundswhich must maintain a charge balance. They involve vacancies of pair of ions of oppositecharges. This type of defect is dominant in alkali halides.3. Interstitialcy:This type of defect occurs when an extra atom of the same size of different orsame element is added to the crystal lattice particularly when the packing factor is low. Thisresults in atomic distortion. The foreign atom may form added alloying agent or simply animpurity. The vacancy and the interstitialcy are therefore reverse phenomena.4. Frenkel defect: In ionic crystals, the formation of point imperfections is subject to therequirement that the overall electrical neutrality is maintained. An ion displaced from a regularsite to an interstitial site is called a Frenkel defect. As cations are generally smaller ions, it ispossible for them to get displaced into the void space. Anions do not get displaced like this, as thevoid space is too small for their size. A Frenkel imperfection does not change the overall
electrical neutrality of the crystal. The point imperfections in silver halides and CaF 2 are ofFrenkel type. The interstitialcies and Frenkel defects are less in number than Vacancies andSchottky defects, because additional energy is required to force the atoms into the new positionand the closed-packed structures have still fewer interstitialcies and Frenkel defects thanVacancies and Schottky defects. When Ionic crystals do not correspond to exact stoichiometricformulae, defect structures are produced and such defect structures have an appreciableconcentration of point imperfections.5. Compositional defects:
a) Substitutional impurity: A Substitutional impurity (solute) in ant crystalline matrix is a pointdefect. It refers to foreign atom that substitutes for or replaces a parent atom in the crystal.Aluminium and phosphorous doped in silicon are Substitutional impurities in the crystal.
b) Interstitial impurity: An interstitial imperfection is also a point defect. It is a small sizedatom occupying the void spacein the parent crystal without dislodging any of the parent atomsfrom their sites. An atom can enter the interstitial site when it is substantially smaller than theparent atom. In close-packed structures, the largest atom that can fit into the octahedral andtetrahedral voids have radii 0.414r and 0.225r respectively, where r is the radius of the parentatom. Evidence in favour of the above is found when carbon is an interstitial solute in iron. Itoccupies the tetrahedral voids in the high temperature form of iron. The iron atom in FCC crystalhas a radius of 1.29 , whereas the carbon atom has a radius of 0.71 (covalent radius of in
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graphite). The carbon radius is clearly larger than 0.414 x 1.29 =0.53 , which is the size of theoctahedral void. Therefore there are strains around the carbon atoms in the FCC iron, andsolubility is limited to 2 wt %. In the room temperature BCC iron crystals, the voids are stillsmaller and hence the solubility of carbon is very limited, that is only 0.008 wt %.
Trivalent cations such as Fe +3, and Cr+3 can substitute for Al+3 ions in the Al2O3 crystals. If,however, the valency of the Substitutional impurity is not equal to the parent cation, additional
point defects may be created due to such substitution. As an example, a divalent cation Ca+2substituting for a univalent parent ion such as Na+ will, at the same time, create a vacant cationsite in the crystal so that electrical neutrality is maintained.
The presence of a point imperfection introduces distortions in the crystal. If the imperfection is avacancy, the bonds that the missing atoms would have formed with its neighbours are not there.In the case of an impurity atom, as a result of the size difference, elastic strains are created in theregion of the crystal immediately surrounding the impurity atoms. The elastic strains are presentirrespective of whether the impurity atom is larger or smaller than the parent atom. A larger atomintroduces compressive stresses, and corresponding strains around it, while a smaller atom createsa tensile stress-strain field. Similarly, an interstitial atom produces strains around the void it isoccupying. All these factors tend to increase the enthalpy (or the potential energy) of the crystal.
The work required to be done for creating a point imperfection is called the enthalpy offormation (Hf) of point imperfection. It is expressed in kJ mole
-1 or eV /point imperfection.
The concept of equilibriumand kinetics are intimately associated with the basic thermodynamicparameters. Pressure P and Temperature T are familiar intensive parameters ( as these do notdepend on the quantity of materials). As opposed to these, there are extensive parameters (whichdepend on the quantity of materials that comprises the system) such as Internal energy (E),Enthalpy (H), Entropy (S), Free Energy (G) etc.
Internal energy at a temperature T is given by
E = E0 + ,Where E0 is the internal energy of the material at 0 K and Cv is the specific heat at constantvolume.
Internal energy = POTENTIAL ENERGY (vibrational + rotational energy terms) + KINETICENERGY (Translational energy= mv2; m= mass, v= velocity).
The enthalpy or heat content (H) of a material is defined as
H = H0 + , where H0 is the enthalpy at 0 K and Cp is the specific heat at constantpressure.
E & H are related by the expression H = E + PV (Total Heat content = Internal energy +External energy). External energy = PV or pressure-volume energy i.e., Mechanical Energy.
For a condensed systems like liquid and solid state, at atmospheric pressure, PV term isnegligible so that E H. This approximation can be used in most of the problems concerning thesolid materials.
H0 represents the enthalpy at 0 K of the solid material. The gaseous state of the material is takenas the reference zero energy state. To indicate the system has lost energy, H0 is written with anegative sign. As the temperature increases from 0 K, the material absorbs heat from thesurroundings and H increases. The solid melts on reaching the melting point and a further
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quantity of heat H called the enthalpy of fusion is added at the melting temperature. When the
entire solid has melted, the temperature of the liquid may further increase with the absorption ofmore energy.
Al the energy that a system possesses is not available as work during a chemical change. Thatpart of the energy which can become available as work is called the Gibbs free energy (or simply
the Gibbs energy). The part which cannot released as work is called the bound energy. Anotherthermodynamic function called entropy defines the relationship between the total energy and theGibbs energy by the expression
G = H TSWhere, G, H, T & s stand for Gibbs free energy, Enthalpy, absolute Temperature& Entropy of thesystem respectively. The change in free energy may then be expressed as G = H- T. SAs the temperature increases, H increases; but TS increases more rapidly than H and so Gdecreases with increasing temperature.Gibbs energy is used as the criterion of stability. The most stable state of a material is that whichhas the minimum Gibbs energy. For a process to occur spontaneously, the Gibbs energy mustdecrease during the process. Then, for a spontaneous process to occur at constant temperatureand pressure,
We can write G = (H- T. S) < 0Only if there is no change in entropy as in the case of a tilting block or any other commonmechanical equilibrium,G can be replaced by H as a criterion of stability ( S=0)
This means that the stability of a system refers to the lowest potential energy or enthalpy. Severalchemical reactions are known to be endothermic, that is, they absorb heat during the reactionmaking H positive. However they may occur spontaneously indicating that G is negative, TS> H.
The entropy of a system is generally looked upon as the sum of the two entropy terms as under.
Total entropy =Thermal Entropy +Configurational Entropy
Normally, entropy is a measure of the thermal disorder as a major part. The solid state ischaracterised by random vibration of atoms about their mean position. There are two importantparameters in atomic vibration in a solid material; one is frequency of vibration, and the other isii) amplitude of vibration. Frequency of vibration does not considerably vary with temperaturebut the amplitude of vibration varies significantly with temperature. In the liquid state, the atomshave more freedom and also can move past one another. Consequently, the entropy in the gaseous
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state of a substance is much more than in the liquid state and that in the liquid state isconsiderably more than that in the solid state.
At constant pressure, the entropy S of a system is given by S =The entropy of a material at 0 K is zero, in contrast to enthalpy and internal energy terms, whichhave non-zero negative energy terms at 0 K. The entropy increases with increasing temperature.
In addition to the thermal entropy, a system may also possess Configurational Entropy, which isdependent on the configuration of the system. By virtue of the fact that a point imperfection isdistinguishable from the parent atom, the Configurational entropy of a crystal increases from zerofor a perfect crystal to positive values with increasing concentration of the point imperfection.Although mass transport through crystals may result from mechanisms other than vacancymotion, the vacancies and their motions are attractive because unlike other defects, vacanciesexist at equilibrium in all crystals. Suppose the energy Ev is required to create a single vacancy. Insolids, PV changes very little, and energy and enthalpy are essentially the same. The entropy ofthe mixture of a full and vacant lattice sites in a crystal is calculated in exactly the same manneras the Configurational entropy of a binary alloy.
Let us consider that one mole of binary solid solution or an alloy containing NA of component Aand NB atoms of component B. In the crystal structure of the solid solution there are, therefore,(NA +NB) sites for the atoms. There are many ways (say, W) of arranging the atoms on the sites.
In fact there are
alternative arrangements [From permutation and combination of NA & NB].Statistical mechanics following Maxwell-Boltzmann definition shows that the Configurationalentropy due to existence of such alternative arrangements is S =k ln W,
Where, k is Boltzmann constant and W is the number of different configurations of equalpotential energy in which the system can exist. The probability of a system existing in adisordered configuration is almost unity. One mole of a solid contains more than 10 23 atoms. Ifwe mix two different kinds of atoms randomly in a solid, we end up with an extremely largenumber of distinguishable configurations and an appreciable amount of Configurational entropy.Sine w can never be less than 1 , the Configurational entropy may be either zero or positive. It iszero for an absolutely pure solid consisting of the same kind of atoms on all its sites or for aperfectly ordered solid like a compound.
If we consider total number of sites as N in a crystal (assuming N =NA + NB) and replacing NAby n as the number point imperfections & automatically the number of perfect sites in the crystal,if assumed to be NB =(N n); Considering that configurational entropy is zero for a pure orperfectly ordered crystal or before mixing the two components (N-n) & n we may write
S = k ln = k ln = k [ln N! ln n! ln (N-n)!]
Following Sterlings approximation when n 1, ln n! =n ln n nS = k[ N ln N (N-n) ln (N-n) n ln n ]
When we introduce n point Imperfections in one mole of crystal, the change in the free energyG of the crystal can be written as
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G = H T S = n Hf k T [N ln N (N-n) ln (N-n) n ln n],
When, N is Avogadros number. The equilibrium state of the crystal will correspond to theminimum in its free Energy, as shown below in the figure.
Fig: The variation of G (Gibbs free energy) with the number of point imperfections n.Minimum in G also corresponds to the minimum in G (considering negative sign). Then by setting
G =0; we obtain the following expression
(nHf RT[ N ln N (N-n) ln (N-n) n ln n ]) =0Or, Hf =RT [N ln N (N-n) ln (N-n) n ln n]
Or, [N ln N (N-n) ln (N-n) n ln n] =RTH f
Or, N ln N (N-n) ln (N-n) n ln n =RTH f
Or, 0 N ln (N-n) + [n ln (N-n)] n ln n =RTH f
N
N
+ln (N-n)
N
n
1 ln n =RT
H f
Or,nN
N
nN
n
1 +ln (N-n) ln n =RTH f
Or, 1 1 +lnn
nN =
RTH f
lnN
n
=RTH f
N
n
= RT
H f
Or, Nn
= RT
H f
; since N >> n, in a crystal,N n N
Problem:
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1. Calculate the ratio of the number of vacancies in equilibrium at 300 K in aluminiumto that produced by rapid quenching from 800 K . Given that enthalpy of formation(Hf)of vacancies in Aluminium is 68 kJ mole
-1.
Ans: We know that equilibrium concentration of vacancies in any crystal can be calculatedusing the following expression,
Nn
= RTH f
; where n=number of vacancies introduced or created in a crystal
Total number of sites in the crystal =N
R =universal gas constant =8. 314 kJ mole -1 K-1T1 =300 K for the equilibrium concentration of vacancies n1T2 =800 K for the equilibrium concentration of vacancies n2GivenHf =68 kJ mole
-1
nn
2
1 = e RTH f
1
e RT
H f
2
=
H TTfR
2
1
1
11
=
3
1068 8001
3001
314.8
1
=3.989
x 10 8
Problem 2:In aluminium crystals, the number of vacancies triples itself, on increasing thetemperature from 300 to 312.5 K. Calculate the enthalpy of formation of vacancies.Ans: use the formula as above.
N
n
= RT
H f
; where n=number of vacancies introduced or created in a crystalTotal number of sites in the crystal =N
R =universal gas constant =8. 314 kJ mole -1K-1T1 =300 K for the equilibrium concentration of vacancies n1T2 =312.5 K for the equilibrium concentration of vacancies n2
Given, n2
1 =31 Calculate: Hf
Problem 3Arrive at an order of the magnitude of the value for the concentration of vacancies in any
elemental crystal just below its melting point from the following data.
Element HfkJ mole -1 Melting point 0CPb 48 327Ag 106 961Cu 120 1083Ans: 10-4 to 10-5
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DISLOCATIONS
Q 1. a) Define dislocations.b) Explain with the help of sketches two extreme types of dislocations and also c) the role of
dislocations in the mechanical properties of the metals. 15 Marks
Ref: i) Guy, ii) Hayden, Moffat & John Wolf, iii) Raghavan, iv) Rajput & v) Internet docsAnswer:
a) Imperfections or defects in the crystal structure are practically common to most of thematerials which are utilised in imparting different electrical, electronic and mechanical propertiesto materials of our interest. Imperfections are present in the crystals are either natural as a resultof stacking fault during the growth or artificially built in the structure of the crystal and suitablymaneuvered in terms of their types, concentration, orientation or configuration.
The lack of periodicity in parts of the crystals lattice introduced during the growth of the crystalsfrom the melt or during plastic deformations is termed as imperfection or defects. The types ofimperfections envisaged in the crystal lattices are i) point defects, ii) line imperfection, iii) surface
defects & iv) volume defects. Out of all the above imperfections, line imperfections are veryimportant in regard to the mechanical behaviour of metals when subjected to stress, in particularresulting in deformation.
Line imperfections are called dislocations. Line imperfections are one-dimensionalimperfections, as a line is one-dimensional. They are instrumental in affecting the breaking stressand plastic properties and chemical properties of crystalline materials and metals in particular.
The importance of dislocations lies primarily in the fact that they can move.
B)The basic two extreme types of dislocations are observed in the crystals.i) Edge dislocation (or Taylor-Orowan dislocation)ii) Screw dislocation (or Burgers dislocation)
Any general dislocation in a real crystal is a mixture of these two extreme types. These may beregarded as the components of general dislocation.
i) Edge dislocation:An edge dislocation is a defect where an extra half-plane of atoms is introduced mid way throughthe crystal, distorting nearby planes of atoms.
This dislocation has the following properties: A line direction, which is the direction running along the bottom of the extra half
plane, and due to the existence of this type of distortion in the crystal, tensile,compressive and shear fields, may be present.
Burgers vector which describes the magnitude and direction of distortion to thelattice. In an edge dislocation, the Burgers vector (BV), denoted by the
symbol b, is perpendicular to the line direction. The BV characterizes adislocation.
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Burgers vector is described by drawing a rectangle in the region beinginvestigated by connecting an equal number of atoms on opposite sides of thecentre of dislocation. In case of an edge dislocation in a real crystal, thecircuit traced as above fail to close. The circuit so traced is known asBurgers circuit.
In a real crystal however, the atoms just above the edge incomplete planeare squeezed together and are in a state of compression. J ust below the edge,the atoms are pulled apart and are in a state of tension.
The edge dislocations of opposite nature are represented by the symbol &denoting the insertion of extra incomplete atomic plane from the top and
bottom of the crystal respectively. A pure edge dislocation exhibits both climb and glide motion.
Unlike that in a perfect crystal, the atoms in real crystals, having dislocations are shifted from theequilibrium positions and the bond lengths are drifted from the equilibrium value. The bondlengths are stretched to above the normal value. The distorted configuration extends all along theedge into the crystal.
The plane over which the displacement of atoms is done to cause a dislocation is called the slipplane and the region on the slip plane over which the displacement of atoms terminates is calledthe slipped part of the slip plane.
The stresses caused by an edge dislocation are complex due to its inherent asymmetry.
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ii) Screw dislocation:
When a part of the crystal from the top or bottom is shifted by one interatomic distance to the leftor right with respect to the bottom or top part of the crystal respectively, screw dislocation results
in a crystal. The screw dislocation occurs at the boundary between the displaced and un-displacedparts of the slip plane as shown in the figure.
The screw dislocation is characterized by the following general criteria:
The Burgers circuit is drawn in the region under investigation by tracing the equalnumber of atoms on opposite sides of the centre of dislocation and the Burgers circuit
fails to close, if any screw dislocation exists in the region. Burgers vector is determined by the step needed to close the circuit and the step isdenoted by b and has the same direction as the direction of dislocation line i.e. Burgersvector is parallel to the screw dislocation line.
The atomic bonds in the region immediately surrounding the screw dislocation lineundergo shear distortion and the dislocation line appears to be the vertical pillar of aspiral staircase, when compared.
Screw dislocations are represented by or depending on whether the Burgersvector and dislocation lines parallel or anti-parallel.
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The identification of a screw dislocation may be more precisely described as under: Screw dislocation may originate from a partial slipping of section of crystal plane. Shear stresses are associated with this type of dislocation and the extra energy is stored as
elastic strain energy along the dislocation. The successive planes of atoms are transferred into the surface of the helix of screw. A screw dislocation has its displacement of Burgers vector parallel to the linear
imperfection but there is distortion of the plane. In case of a screw dislocation, the complete plane of atoms normal to the dislocation line
does not exist. Rather all the atoms lie on a single surface spirally from one end to theother with dislocation line as the axis of spiral. The displacement of the atoms from theiroriginal position in a perfect crystal is given by the equation:
2b
r , where r =the displacement along the direction;
=the angle measured from some axis perpendicular to the dislocation line;If = 2, the displacement becomes b, indicating that b is the measu re of the strength ofscrew dislocation. A screw dislocation does not exhibit climb motion but glide motion only.Mixed dislocations:Any general dislocation line in a real crystal can be resolved into two extreme types ofdislocation such as edge & screw dislocations. Due the combined effect of two different typesof dislocations having burgers vector each being perpendicular to the other, the resultantdisplacement of the mixed dislocation is a curved line rather than the linear lines as in thecase of pure edge or screw dislocations.Dislocations have certain other geometrical characteristics: When the vector sum of the Burgers vector of different dislocations present in a crystal
meet at point, called the node, is zero. This is analogous to Kirchhoffs law for electricalcurrents, meeting at a junction.
A dislocation line cannot end abruptly within the crystal. It either ends at a node or at thesurface. Alternatively, it can close on itself as a loop.
c) Role of dislocations in the properties of materials and metals in particular: (5)
The dislocations, as it is believed, originate mainly when a crystal is stressed, but some may beproduced during the solidification of the metal, due to impurity atoms and thermal vibrations. Ingood crystals the normal density of dislocation lines is around 108 / cm2 whereas in deformedcrystals it may be as high as 1012/cm2. There are certain salient points in relation to the role ofdislocation in the properties of materials and metals in particles.
i) Burgers vector; ii) Glide; iii) Climb; iv) Cross-slip; v) Jogs; vi) Width of dislocation
Role and significance of dislocation:It is necessary to point out that i) The passage of dislocation through a crystal lattice requiresfar less a stress than the theoretical shear stress and ii) The movement of the dislocationthrough the lattice produces a slip or slip band at the free surface.
The cause of mobility of dislocations and their multiplication during deformations areexplained by the following considerations: When the metals undergo plastic deformation, some fraction of the deformation energy
(around 5%) is stored internally as strain energy around the dislocation produced and the
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rest is dissipated as heat. Due to the dislocation produced by plastic deformation, someamount of compressive, tensile and shear lattice strains are imposed into the vicinity ofthe edge dislocation region; whereas for a screw dislocation, lattice strains are purelyshear only.
Strain fields that exist around dislocations, which are influential in determining themobility of dislocations and their ability to multiply are important for explaining the
properties, mechanical or thermal behaviour of materials and metals in particular. The edge dislocation plays the role in the mechanism throughslip for the simplest mode
of mechanical deformation. The screw dislocation plays an important role in the crystal growth.As we know that plastic deformation in materials occurs through two basic modes called slipand twinning, so the dislocations movement plays the key role in such deformationmechanism. Both edge & screw dislocations are important in describing the cold-workedstate and its elimination by annealing process. The dislocation interactions are possiblebetween the edge, screw and mixed dislocations and for a variety of orientations of all ofthem. These strain fields and associated forces are important in strengthening mechanism ofmetals.
The presence of dislocations in crystals influences the following different mechanicalbehaviours of metals under stress:
Plastic deformation through slip & twinning. Shear strength of metals. Plastic deformations initiated by critically resolved shear stresses derived from the
applied tensile stress on the metallic articles. Influence of other imperfections on the dislocation movements in a crystal during
deformation. The effect of temperature on the stress to move a dislocation and hence the
mechanical behaviour of the metals. Multiplication of dislocations during deformations. Work-hardening and dynamic recovery due to applied stress. The effect of grain size, solute impurities, precipitate particles etc on the
dislocation motion and hence mechanical behaviour of the metals. Strengthening mechanismof metals.
Q2. What is a dislocation? (5) Describe the role of dislocations in explaining the mechanicalbehaviour of metals? (10)Ans:
The physical definition of line imperfections may be categorised as under:
A linear disturbance of the atomic arrangement, which can move very easily on the slipplane through the crystal, is known as dislocation. The line defects can therefore beconsidered as the boundary between the two regions of a surface which are perfectthemselves but out of register with each other.
In case of crystals it arises when one part of the crystal shifts or slips relative to the restof the crystal such that the displacement terminates within the crystal. However, if thedisplacement does not terminate within the crystal but continues throughout the crystal, itmay not introduce any defect in the crystal.
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Mathematically, dislocations are a type of topological defect, sometimes called a soliton. There are two extreme primary types of dislocations, namelyedge dislocationsandscrew
dislocations. Mixed dislocations are intermediate between these and they have thecharacteristics of both. There may be huge number of dislocations of both extreme typesand mixed dislocations in a crystal.
Screw dislocation
Edge dislocation
The tensile and compressive strains exit around an edge dislocation and shear strainaround a screw dislocation. Dislocations have distortional energy associated with them. Ifwe consider these strains as elastic strain, the elastic strain energy E per unit length of a
dislocation of Burgers vector b is given by E =2
2b, where is the shear modulus of
the crystal.
Two dislocations of opposite orientation, when brought together, can cancel each other(this is the process of annihilation), but a single dislocation typically cannot "disappear"on its own but by deformative forces to move the dislocation out of the boundary surfaceof the crystals.
The mathematical theory explains why dislocations behave as somewhat stable particles.They can be moved about, but maintain their identity as they move, and ultimately moveout of the crystal by proper heat treatment.
Thermodynamically, line imperfections are less stable than point imperfections, becausein the case of line imperfections, the enthalpy of the crystal increases much more rapidlydue to distortion around the dislocation than the entropy ( as the different regions of thecrystal are perfect themselves but out of register with each other.)
In a crystal, the density of dislocation is measured by counting the number of points atwhich they intersect at random cross-section of the crystal. These points, called etch pits;can be seen under the microscope, after chemical etching of a pre- polished surface.
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Q2. Role of dislocations in the properties of materials and metals in particular: (10)
The dislocations, as it is believed, originate mainly when a crystal is stressed, but some may beproduced during the solidification of the metal, due to impurity atoms and thermal vibrations. Ingood crystals the normal density of dislocation lines is around 108 / cm2 whereas in deformedcrystals it may be as high as 1012/cm2. In addition to the grain boundaries, each grain also
possesses small angle tilt boundaries separating crystallites with only a small mis-orientation withrespect to each other to yield sub-grains. Precipitates or second phase particles in the grain forminter-phase boundaries. All these boundaries are the preferred sites where dislocations aregenerated because the existing mis-arrangement in the atomic structure takes the shape ofdislocation on the application of stress.
There are certain salient points in relation role of dislocation in the properties of materials andmetals in particles.
Burgers vectorThe burgers vectors for CsCl and NaCl tend to be large (3.95 ) to make the ionic structureelectrically stable as the smaller BV would render the insertion of an incomplete plane halfway from body corner of CsCl to the body center and make the structure electrically
charged; but that for copper crystals, the same situation does not arise and the BV istherefore 2.55 for a full dislocation. Dislocation movementDislocations move through the atomic planes inside the crystals by glideandclimb motionandJ ogs in case of edge dislocation andcross-slip process in case of screw dislocation.
Role and significance of dislocation:
The edge dislocation plays the role of the mechanism for the simplest mode ofmechanical deformation, slip.
The screw dislocation plays an important role in the crystal growth.Both are important in describing the cold-worked state and its elimination by annealing process
It is necessary to point out regarding the role of dislocation that:i) The passage of dislocation through a crystal lattice requires far less a stress than the
theoretical shear stress.ii) The movement of the dislocation through the lattice produces a slip or slip band at the
free surface. To explain the properties, mechanical or thermal behaviour of materials and metals in
particular, several characteristics of dislocations are important, which include strain fieldsthat exists around dislocations, which are influential in determining the mobility ofdislocations and their ability to multiply.
When the metals undergo plastic deformation, some fraction of the deformation energy(around 5%) is stored internally as strain energy around the dislocation produced and therest is dissipated as heat. Due to the dislocation produced by plastic deformation, someamount of compressive, tensile and shear lattice strains are imposed into the vicinity ofthe dislocation region.
Whereas for a screw dislocation, lattice strains are purely shear only. These lattice distortionsmay be considered to be strain fields that radiate from the dislocation line.
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The strain fields surrounding dislocations in close proximity to one another may interactsuch that forces are imposed on each dislocation by the combined effect of all of itsneighbouring dislocations.
The dislocation interactions are possible between the edge, screw and mixed dislocations and fora variety of orientations of all of them. These strain fields and associated forces are important instrengthening mechanism of metals.
X ray studies show that during plastic deformation, the crystalline structure of metal is not lost,even though more imperfections are introduced.Plastic deformation in metals occurs by two basic modes called i) Slip & ii) Twinning. Slip is ashear deformation and steps are created during plastic deformation by slip but the orientation ofall parts of the crystals remains the same before and after slip. Twinning, on the other hand,changes the orientation of the twinned parts.
The slip mode of deformation is most common mode in many crystals at ordinary and elevatedtemperatures. At low temperature, the mode of deformation changes over to twinning in a numberof cases.Real crystals deform at a much lower stress than predicted by theoretical model calculations.
This discrepancy can be explained only on the basis of presence of dislocations present in the
crystals. The measured critically resolved shear stresses (CRSS) values should be associated withthe stress required to move a dislocation that is already present in a real crystal and not with thestress required to shear a perfect crystal.Normally, dislocations are ways present in real crystals. Whiskers are very thin but are free fromany dislocations. Such crystals can withstand stresses much higher tan the ordinary crystals,without undergoing plastic deformation. If, however, dislocation is introduced accidentally, forexample, at the surface, the crystal abruptly loses all its strength and there is a big drop in thestress required to cause further strain, which is permanent. The example of a copper whisker isshown in the figure below.
Figure. : A copper whisker deforms plastically, as soon as dislocation is introduced, with a
big drop in the stress required to cause further deformation
The stress to move a dislocation is an important aspect in determining the behaviour of metalsunder stress. In case of a crystal having broken bonds in the atomic planes of the crystal lattice ofa metal, the dislocations are called narrow dislocation, and in other cases where bonds are onlydistorted above below the slip planes around the edge dislocation called wide dislocation. Innarrow dislocations, the row of atoms has to move a full one atomic distance in contrast to thefractional movement of row of atoms in case of a wide dislocation. Hence, the narrow
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dislocations are far more difficult to move than wide dislocation. The width of dislocation isrelated inversely to the stress required to move the dislocation.
The nature of the chemical bonding in a crystal determines the extent of the relaxation and thewidth of a dislocation. In covalent crystals, the bonding is strong and directional; the relaxation issmall resulting in narrow dislocation. The Peierls-Nabbaro (P-N) stress is correspondingly veryhigh. The application of a tensile stress usually results in a brittle fracture by crack propagation,
before the stress required moving a dislocation is attained. Hence covalent crystals such adiamond, silicon are brittle. In metals, the bonds are non-directional and weaker than covalentbonds and hence the dislocations are wide and stress required to moving a dislocation is less.
Therefore, the metals exhibit considerable plastic deformation and are said to be ductile. A copperwire can be cold-drawn to wire hundred times its original length without breaking, in spite ofwork hardening that occur during drawing. In contrast, transition metals like iron have somecovalent character in the metallic bonds due to d-orbitals, which is directional, andcorrespondingly, the transition metal crystals are harder than copper and can not be cold-workedto the same extent as copper.In ionic crystals, the bonds are non-directional and somewhat weaker than covalent but strongerthan metallic bonds. Therefore some amount of plastic deformation may be expected in ioniccrystals. However, this plastic deformation occurs under some special circum stances when the
surfaces of the ionic crystals are free from cracks that can cause brittle fracture.
Q3. Differences between edge and screw types of dislocations (5)
Sl.No.
Edge dislocation Screw dislocation
1.
2.
3.
4.
5.
6.
This dislocation arises due to introductionor elimination of an extra plane of atomsinto the crystal lattice.
Tensile, compressive and shear stress fieldsmay be present around this dislocation.
Region of lattice distortion extend along anedge of an incomplete plane inside thecrystal.
Burgers vector is perpendicular to thedislocation direction line.
These dislocations are formed duringcrystallisation or deformation.
An edge dislocation can glide and climb.
A screw dislocation provides for easy crystalgrowth because additional atoms and unitcells can be added to step of the screw.Only shear strain fields are present.
Region of lattice distortion extend along intwo separate planes at right angles to oneanother inside the crystal.
Burgers vector is parallel to the direction ofdislocation line.
These dislocations are also formed duringcrystallisation or deformation.
A screw dislocation can glide but not climb.
Problem 1Calculate the dislocation energy per m3 of copper with a dislocation density of 1010 m-2. Theshear modulus of copper is 45 GN m-2and the lattice parameter is 3.61 .
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Ans: Elastic strain energyEper unit length of a dislocation of Burgers vectorb is approximatelygiven by
E2
2
b
Given, =45 GN m-2 = 45 x 109 N m-2and lattice parameter a = 3.61 = 3.61 x 10-10mSince copper is monoatomic and for a monoatomic FCC crystal Burgers vectors of dislocations is type, the magnitude of the Burgers vector
b= 3.61 x2
011222
= 3.61 x 10 -10 x
2
1= 2.55 x 10 -10 m
Dislocation Energy per unit length
E
2
2
b =
2
m10x2.55xmN10x452-202-29
=146.89 x 10 -11 J m-1
As it is given that the dislocation densityofcopper is 1010 m-2
The dislocation energy per m3 is then given by =14.7 J m-3.
Problem 4:If there are 1010 per m2 of edge dislocation in a simple cubic crystal, how much would each ofthese climb down on an average when the crystal is heated from 0 to 1000 K? The enthalpy offormation of vacancies is 100 kJmole-1. The lattice parameter is 2 . The volume of one mole ofthe crystal is 5.5 x 10 -6 m3. (5.5 cm3).
Ans: At equilibrium, there are no vacancies in the crystal at 0 K. To maintain equilibriumconcentration the number of vacancy that must be created on heating from 0 to 1000 K is givenby
N = N exp(-RTH f ) = 6.023x106 x exp[(-100 x 1000) /(8.314 x 1000)=3.60 x 108 mole-1 =
=6.54 1023 m-3 .
If the edge dislocations in the crystal climb down, atoms will be added to the extra plane andthese atoms coming from the other parts of the crystal would create vacancies.
As one step is 2 (2 x 10 -10 m), or 5 x 109number of atoms will be required for 1 m ofthe dislocation line to climb down by one step. The average amount of climb down is then
=1.31 x 104
steps =2.62 x 10-6
m.