Imperfections In Solids-Books Pdf

IMPERFECTIONS IN SOLIDS
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atom is attracted toward the nearest atom of the other row Thus the relation of shear. stress with displacement can be expressed as follows. where is the amplitude of sine wave and b is the period For small values of x b after. simplification, It is safe to approximate at small values of displacement Hooke s law is applicable Thus. Combing the above two equations maximum shear stress at which slip should occur is. If it is assumed as b a approximate value of shear stress needed for initiation of slip is. Shear modulus G of solids is in the range of 20 150 GPa and thus theoretical shear. strength could be in range 3 30 GPa However actual values of the shear stress required. for plastic deformation is only about 0 5 10 MPa i e about 100 times less than the. theoretical value Because of this discrepancy it must be said that a mechanism other. than shearing of planes must be operative under application of loads thus causing plastic. deformation,3 1 2 Point defects, Defects exists any all solid materials For ease of their characterization defects are. classified on the basis of their geometry which is realistic as defects are disrupted region. in a volume of a solid Defects are,a Point defects zero dimensional. b Line defects single dimensional,c Surface defects two dimensional. d Volume defects three dimensional, Point defects as the name implies are imperfect point like regions in the crystal Typical.
size of a point defect is about 1 2 atomic diameters Different point defects are explained. in the following paragraphs Figure 3 2 depicts various point defects. A vacancy is a vacant lattice position from where the atom is missing It is usually. created when the solid is formed by cooling the liquid There are other ways of making a. vacancy but they also occur naturally as a result of thermal excitation and these are. thermodynamically stable at temperatures greater than zero At equilibrium the fraction. of lattice sites that are vacant at a given temperature T are. where n is the number of vacant sites in N lattice positions k is gas or Boltzmann s. constant T is absolute temperature in kelvins and Q is the energy required to move an. atom from the interior of a crystal to its surface It is clear from the equation that there is. an exponential increase in number of vacancies with temperature When the density of. vacancies becomes relatively large there is a possibility for them to cluster together and. form voids, Figur 3 2 Schematic depiction of various point defects. An interstitial atom or interstitialcy is an atom that occupies a place outside the normal. lattice position It may be the same type of atom as the rest surrounding it self. interstitial or a foreign impurity atom Interstitialcy is most probable if the atomic. packing factor is low, Another way an impurity atom can be fitted into a crystal lattice is by substitution A. substitutional atom is a foreign atom occupying original lattice position by displacing the. parent atom, In the case of vacancies and foreign atoms both interstitial and substitutional there is a. change in the coordination of atoms around the defect This means that the forces are not. balanced in the same way as for other atoms in the solid which results in lattice. distortion around the defect, In ionic crystals existence of point defects is subjected to the condition of charge. neutrality There are two possibilities for point defects in ionic solids. when an ion displaced from a regular position to an interstitial position creating. a vacancy the pair of vacancy interstitial is called Frenkel defect Cations are. usually smaller and thus displaced easily than anions Closed packed structures. have fewer interstitials and displaced ions than vacancies because additional. energy is required to force the atoms into the interstitial positions. a pair of one cation and one anion can be missing from an ionic crystal without. violating the condition of charge neutrality when the valency of ions is equal The. pair of vacant sites thus formed is called Schottky defect This type of point. defect is dominant in alkali halides These ion pair vacancies like single. vacancies facilitate atomic diffusion,3 1 3 Line defects or Dislocations.
Line imperfections one dimensional defects are also called Dislocations They are. abrupt changes in the regular ordering of atoms along a line dislocation line in the solid. They occur in high densities and strongly influence the mechanical properties of material. They are characterized by the Burgers vector b whose direction and magnitude can be. determined by constructing a loop around the disrupted region and noticing the extra. inter atomic spacing needed to close the loop The Burgers vector in metals points in a. close packed lattice direction It is unique to a dislocation. Dislocations occur when an extra incomplete plane is inserted The dislocation line is at. the end of the plane Dislocations can be best understood by referring to two limiting. cases Edge dislocation and Screw dislocation, Edge dislocation or Taylor Orowan dislocation is characterized by a Burger s vector that. is perpendicular to the dislocation line It may be described as an edge of an extra plane. of atoms within a crystal structure Thus regions of compression and tension are. associated with an edge dislocation Because of extra incomplete plane of atoms the. atoms above the dislocation line are squeezed together and are in state of compression. whereas atoms below are pulled apart and experience tensile stresses Edge dislocation is. considered positive when compressive stresses present above the dislocation line and is. represented by If the stress state is opposite i e compressive stresses exist below the. dislocation line it is considered as negative edge dislocation and represented by A. schematic view of edge dislocations are shown in figure 3 3. Figure 3 3 Edge dislocations, A pure edge dislocation can glide or slip in a direction perpendicular to its length i e. along its Burger s vector in the slip plane made of b and t vectors on which dislocation. moves by slip while conserving number of atoms in the incomplete plane It may move. vertically by a process known as climb if diffusion of atoms or vacancies can take place. at appropriate rate Atoms are added to the incomplete plane for negative climb i e the. incomplete plane increases in extent downwards and vice versa Thus climb motion is. considered as non conservative the movement by climb is controlled by diffusion. Screw dislocation or Burgers dislocation has its dislocation line parallel to the Burger s. vector A screw dislocation is like a spiral ramp with an imperfection line down its axis. Screw dislocations result when displacing planes relative to each other through shear. Shear stresses are associated with the atoms adjacent to the screw dislocation therefore. extra energy is involved as it is in the case of edge dislocations Screw dislocation is. considered positive if Burger s vector and t vector or parallel and vice versa t vector. an unit vector representing the direction of the dislocation line A positive screw. dislocation is represented by a dot surrounded by circular direction in clock wise. direction whereas the negative screw dislocation is represented by a dot. surrounded by a circular direction in anti clock wise direction A schematic view of a. negative screw dislocation is shown in figure 3 4,Fifure 3 4 Negative screw dislocation. A screw dislocation does not have a preferred slip plane as an edge dislocation has and. thus the motion of a screw dislocation is less restricted than the motion of an Edge. dislocation As there is no preferred slip plane screw dislocation can cross slip on to. another plane and can continue its glide under favorable stress conditions However. screw dislocation can not move by climb process whereas edge dislocations can not. cross slip, Dislocations more commonly originate during plastic deformation during solidification. and as a consequence of thermal stresses that result from rapid cooling Edge dislocation. arises when there is a slight mismatch in the orientation of adjacent parts of the growing. crystal A screw dislocation allows easy crystal growth because additional atoms can be. added to the step of the screw Thus the term screw is apt because the step swings. around the axis as growth proceeds Unlike point defects these are not. thermodynamically stable They can be removed by heating to high temperatures where. they cancel each other or move out through the crystal to its surface Virtually all. crystalline materials contain some dislocations The density of dislocations in a crystal is. measures by counting the number of points at which they intersect a random cross. section of the crystal These points called etch pits can be seen under microscope In an. annealed crystal the dislocation density is the range of 108 1010 m 2. Any dislocation in a crystal is a combination of edge and screw types having varying. degrees of edge and screw character Figure 3 5 depicts a schematic picture of a common. dislocation,Figure 3 5 Typical dislocation in a crystal.
As shown in figure 3 5 the Burger s vector will have the same magnitude and direction. over the whole length of dislocation irrespective of the character of the dislocation edge. screw or mixed i e Burger s vector is invariant Other geometrical characteristics of. dislocations are, vectorial sum of Burger s vectors of dislocations meeting at a point called node must. t vectors of all the dislocations meeting at a node must either point towards it or away. a dislocation line can not end abruptly within the crystal It can close on itself as a loop. or ends either at a node or at the surface, Dislocations have distortional energy associated with them as is evident from the. presence of tensile compressive shear stresses around a dislocation line Strains can be. expected to be in the elastic range and thus stored elastic energy per unit length of the. dislocation can be obtained from the following equation. where G shear modulus and b Burger s vector, Dislocations in the real crystal can be classified into two groups based on their geometry. full dislocations and partial dislocations Partial dislocation s Burger s vector will be a. fraction of a lattice translation whereas Burger s vector is an integral multiple of a lattice. translation for full dislocation As mentioned above elastic energy associated with a. dislocation is proportional to square of its Burger s vector dislocation will tend to have. as small a Burger s vector as possible This explains the reason for separation of. dislocations that tend to stay away from each other. 3 2 Interfacial defects Bulk or Volume defects and Atomic vibrations. 3 2 1 Interfacial defects, Interfacial defects can be defined as boundaries that have two dimensional imperfections. in crystalline solids and have different crystal structures and or crystallographic. orientations on either side of them They refer to the regions of distortions that lie about a. surface having thickness of a few atomic diameters For example external surfaces grain. boundaries twin boundaries stacking faults and phase boundaries These imperfections. are not thermodynamically stable rather they are meta stable imperfections They arise. from the clustering of line defects into a plane, External surface The environment of an atom at a surface differs from that of an atom in.
the bulk especially the number of neighbors coordination at surface is less Thus the. unsaturated bonds of surface atoms give rise to a surface energy This result in relaxation. the lattice spacing is decreased or reconstruction the crystal structure changes To. reduce the energy materials tend to minimize if possible the total surface area. Grain boundaries Crystalline solids are usually made of number of grains separated by. grain boundaries Grain boundaries are several atoms distances wide and there is. mismatch of orientation of grains on either side of the boundary as shown in figure 3 6. When this misalignment is slight on the order of few degrees 10 it is called low. angle grain boundary These boundaries can be described in terms of aligned dislocation. arrays If the low grain boundary is formed by edge dislocations it is called tilt boundary. and twist boundary if formed of screw dislocations Both tilt and twist boundaries are. planar surface imperfections in contrast to high angle grain boundaries For high angle. grain boundaries degree of disorientation is of large range 15 Grain boundaries are. chemically more reactive because of grain boundary energy In spite of disordered. orientation of atoms at grain boundaries polycrystalline solids are still very strong as. cohesive forces present within and across the boundary. Figure 3 6 Schematic presentation of grain boundaries. Twin boundaries It is a special type of grain boundary across which there is specific. mirror lattice symmetry Twin boundaries occur in pairs such that the orientation change. introduced by one boundary is restored by the other figure 3 7 The region between the. pair of boundaries is called the twinned region Twins which forms during the process of. recrystallization are called annealing twins whereas deformation twins form during. plastic deformation Twinning occurs on a definite crystallographic plane and in a. specific direction both of which depend on the crysta. Imperfections in Solids 3 1 Theoretical yield strength Point defects Line defects and Dislocations 3 1 1 Theoretical strength Atoms are orderly arranged without any disruptions in ideal solids Under mechanical loads all solids tend to reshape themselves This occurs usually by process called slip the translation of one plane of atoms over another under implied shear stresses Consider two

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