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POINT DEFECT IN IONIC CRYSTALS AND METALS
The point imperfections, which are lattice errors at isolated lattice points, take place due to imperfect packing of atoms during crystallisation. The point imperfections also take place due to vibrations of atoms at high temperatures. Point imperfections are completely local in effect, e.g. a vacant lattice site. Point defects are always present in crystals and their present results in a decrease in the free energy. One can compute the number of defects at equilibrium concentration at a certain temperature as,
n = N exp [-Ed / kT] (1)
Where n - number of imperfections, N - number of atomic sites per mole, k - Boltzmann constant, Ed - free energy required to form the defect and T - absolute temperature. E is typically of order l eV since k = 8.62 X 10-5 eV /K, at T = 1000 K, n/N = exp[-1/(8.62 x 10-5 x 1000)] ≈ 10-5, or 10 parts per million. For many purposes, this fraction would be intolerably large, although this number may be reduced by slowly cooling the sample.
Fig. 2 Point defects in a crystal lattice corresponding equilibrium concentration of vacancies and interstitial atoms (an interstitial atom is an atom transferred from a site into an interstitial position). For instance, copper can contain 10-13 atomic percentage of vacancies at a temperature of 20-25°C and as many as 0.01 % at near the melting point (one vacancy per 104 atoms). For most crystals the-said thermal energy is of the order of I eV per vacancy. The thermal vibrations of atoms increases with the rise in temperature. The vacancies may be single or two or more of them may condense into a di-vacancy or trivacancy. We must note that the atoms surrounding a vacancy tend to be closer together, thereby distorting the lattice planes. At thermal equilibrium, vacancies exist in a certain proportion in a crystal and thereby leading to an increase in randomness of the structure. At higher temperatures, vacancies have a higher concentration and can move from one site to another more frequently. Vacancies are the most important kind of point defects; they accelerate all processes associated with displacements of atoms: diffusion, powder sintering, etc.
(iii) Frenkel Defect: Whenever a missing atom, which is responsible for vacancy occupies an interstitial site (responsible for interstitial defect) as shown in Fig. 2(c), the defect caused is known as Frenkel defect. Obviously, Frenkel defect is a combination of vacancy and interstitial defects. These defects are less in number because energy is required to force an ion into new position. This type of imperfection is more common in ionic crystals, because the positive ions, being smaller in size, get lodged easily in the interstitial positions. (iv) Schottky Defect: These imperfections are similar to vacancies. This defect is caused, whenever a pair of positive and negative ions is missing from a crystal [Fig. 2(e)]. This type of imperfection maintains charge neutrality. Closed-packed structures have fewer interstitialcies and Frenkel defects than vacancies and Schottky defects, as additional energy is required to force the atoms in their new positions.
Notes : (i) Write your answer in the space given below (ii) Compare your answer with those given at the end of the unit Explain Frenkel and Schottky defects? ………………………………………………………………………………………. …………………………………………………………………………………………. …………………………………………………………………………………………. ……………………………………………………………………………………….
(v) Substitutional Defect: Whenever a foreign atom replaces the parent atom of the lattice and thus occupies the position of parent atom (Fig. 2(d)], the defect caused is called substitutional defect. In this type of defect, the atom which replaces the parent atom may be of same size or slightly smaller or greater than that of parent atom.
(vi) Phonon: When the temperature is raised, thermal vibrations takes place. This results in the defect of a symmetry and deviation in shape of atoms. This defect has much effect on the magnetic and. electric properties. All kinds of point defects distort the crystal lattice and have a certain influence on the physical properties. In commercially pure metals, point defects increase the electric resistance and have almost no effect on the mechanical properties. Only at high concentrations of defects in irradiated metals, the ductility and other properties are reduced noticeably. In addition to point defects created by thermal fluctuations, point defects may also· be created by other means. One method of producing an excess number of point defects at a given temperature is by quenching (quick cooling) from a higher temperature. Another method of creating excess defects is by severe deformation of the crystal lattice, e.g., by hammering or rolling. We must note that the lattice still retains its general crystalline nature, numerous defects are introduced. There is also a method of creating excess point defects is by external bombardment by atoms or high-energy particles, e.g. from the beam of the cyclotron or the neutrons in a nuclear reactor. The first particle collides with the lattice atoms and displaces them, thereby causing a point defect. The. number of point defects created in this manner depends only upon the nature of the crystal and on the bombarding particles and not on the temperature.
Check Your Progress 2 Notes : (i) Write your answer in the space given below (ii) Compare your answer with those given at the end of the unit What are crystal defects and how are they classified? …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… DIFFUSION
Diffusion refers to the transport of atoms through a crystalline or glassy solid. Many processes occurring in metals and alloys, especially at elevated temperatures, are associated with self-diffusion or diffusion. Diffusion processes play a crucial 'role in many solid-state phenomena and in the kinetics of micro structural changes during metallurgical processing and applications; typical examples include phase transformations, nucleation, recrystallization, oxidation, creep, sintering, ionic conductivity, and intermixing in thin film devices. Direct technological uses of diffusion include solid electrolytes for advanced battery and fuel cell applications, semiconductor chip and microcircuit fabrication and surface hardening of steels through carburization. The knowledge of diffusion phenomenon is essential for the introduction of a very small concentration of an impurity in a solid state device: Types of Diffusion (i) Self Diffusion: It is the transition of a thermally excited atom from a site of crystal lattice to an adjacent site or interstice. (ii) Inter Diffusion: This is observed in binary metal alloys such as the Cu-Ni system. iii) Volume Diffusion: This type of diffusion is caused due to atomic movement in bulk in materials. (iv) Grain Boundary Diffusion: This type of diffusion is caused due to atomic movement along the grain boundaries alone.
along the surface of a phase. Diffusion Mechanisms
Diffusion is the transfer of unlike atoms which is accompanied with a change of concentration of the components in certain zones of an alloy. Various mechanisms have been proposed to explain the processes of diffusion. Almost all of these mechanisms are based on the vibrational energy of atoms in a solid. Direct-interchange, cyclic, interstitial, vacancy etc. are the common diffusion mechanisms. Actually, however, the most probable mechanism of diffusion is that in which the magnitude of energy barrier (activation energy) to be overcome by moving atoms is the lowest. Activation energy depends on the forces of interatomic bonds and crystal lattice defects, which facilitate diffusion transfer (the activation energy at grain boundaries is only one half of that in the bulk of a grain). For metal atoms, the vacancy mechanism of diffusion is the most probable and for elements with a small atomic radius (H, N and C), the interstitial mechanism. Now, we will study these mechanisms. (i) Vacancy Mechanism: This mechanism is a very dominant process for diffusion in FCC, BCC and HCP metals and solid solution alloy. The activation energy for this process comprises the energy required to create a vacancy and that required to move it. In a pure solid, the diffusion by this mechanism is shown in Fig. 3(a). Diffusion by the vacancy mechanism can occur by atoms moving into adjacent sites that are vacant. In a pure solid, during diffusion by this mechanism, the atoms surrounding the vacant site shift their equilibrium positions to adjust for the change in binding that accompanies the removal of a metal ion and its valency electron. We can assume that the vacancies move through the lattice and produce random shifts of atoms from one lattice position to another as a result of atom jumping. Concentration changes takes place due to diffusion over a period of time. We must note that vacancies are continually being created and destroyed at the surface, grain boundaries and suitable interior positions, e.g. dislocations. Obviously, the rate of diffusion increases rapidly with increasing temperature.
If a solid is composed of a single element, i.e. pure metal, the movement of thermally excited atom from a site of the crystal lattice to an adjacent site or interstice is called self diffusion because the moving atom and the solid are the same chemical-element. The self-diffusion in metals in which atoms of the metal itself migrate in a random fashion throughout the lattice occurs mainly through this mechanism. We know that copper and nickel are mutually soluble in all proportions' in solid state and form substitutional solid solutions, e.g., plating of nickel on copper. For atomic diffusion, the vacancy mechanism is shown in Fig. 4.
changes positions using an interstitial site does not usually occur in metals for elf-diffusion but is favored when interstitial impurities are present because of the low activation energy. When a solid is composed of two or more elements whose atomic radii differ significantly, interstitial solutions may occur. The large size atoms occupy lattice sites where as the smaller size atoms fit into the voids (called as interstices) created by the large atoms. We can see that the diffusion mechanism in this case is similar to vacancy diffusion except that the interstitial atoms stay on interstitial sites (Fig. 3(b)). We must note that activation energy is associated with interstitial diffusion because, to arrive at the vacant site, it must squeeze past neighbouring atoms with energy supplied by the vibrational energy of the moving atoms. Obviously, interstitial diffusion is a thermally activated process. The interstitial mechanism process is simpler since the presence of vacancies is not required for the solute atom to move. This mechanism is vital for the following cases: (a) The presence of very small atoms in the interstices of the lattice affect to a great extent the mechanical properties of metals. (b) At low temperatures, oxygen, hydrogen and nitrogen can be diffused in metals easily. (iii) Interchange Mechanism: In this type of mechanism, the atoms exchange places through rotation about a mid point. The activation energy for the process is very high and hence this mechanism is highly unlikely in most systems. Two or more adjacent atoms jump past each other and exchange positions, but the number of sites remains constant (Fig. 3 (c) and (d)). This interchange may be two-atom or four-atom (Zenner ring) for BCC. Due to the displacement of atoms surrounding the jumping pairs, interchange mechanism results in severe local distortion. For jumping of atoms in this case, much more energy is required. In this mechanism, a number of diffusion couples of different compositions' are produced, which are objectionable. This is also termed as Kirkendall's effect.
From theoretical point of view, Kirkendall's effect is very important in diffusion. We may note that the practical importance of this effect is in metal cladding, sintering and deformation of metals (creep). 1.3.3 Diffusion Coefficient: Fick’s Laws of Diffusion Diffusion can be treated as the mass flow process by which atoms (or molecules) change their positions relative to their neighbours in a given phase under the influence of thermal energy and a gradient. The gradient can be a concentration gradient; an electric or magnetic field gradient or a stress gradient. We shall consider mass flow under concentration gradients only. We know that thermal energy is necessary for mass flow, as the atoms have to jump from site to site during diffusion. The thermal energy is in the form of the vibrations of atoms about their mean positions in the solid. The classical laws of diffusion are Fick's laws which hold true for weak solutions and systems with a low concentration gradient of the diffusing substance, dc/dx (= C2 – C1/X2 – X1), slope of concentration gradient. (i) Fick's First Law: This law describes the rate at which diffusion occurs. This law states that
i.e. the quantity dn of a substance diffusing at constant temperature per unit time t through unit surface area a is proportional to the concentration gradient dc/dx and the coefficient of diffusion (or diffusivity) D (m2/s). The 'minus' sign implies that diffusion occurs in the reverse direction to concentration gradient vector, i.e. from the zone with a higher concentration to that with a lower concentration of the diffusing element. The equation (2) becomes:
 (3)
where J is the flux or the number of atoms moving from unit area of one plane to unit area of another per unit time, i.e. flux J is flow per unit cross sectional area per unit time. Obviously, J is proportional to the concentration gradient. The negative sign implies that flow occurs down the concentration gradient. Variation of concentration with x is shown in Fig. 5. We can see that a large negative slope corresponds to a high diffusion rate. In accordance with Fick's law (first), the B atoms will diffuse from the left side. We further note that the net migration of B atoms to the right side means that the concentration will decrease on the left side of the solid and increase on the right as diffusion progress.
This law can be used to describe flow under steady state conditions. We find that it is identical in form to Fourier's law for heat flow under a constant temperature gradient and Ohm's law for current flow under a constant electric field gradient. We may see that under steady state flow, the flux is independent of time and remains the same at any cross-sectional plane along the diffusion direction.
Parentheses indicate that the phase is metastable (ii) Fick’s second Law: This is an extension of Fick’s first law to non steady flow. Frick’s first law allows the calculation of the instaneous mass flow rate (Flux) past any plane in a solid but provides no information about the time dependence of the concentration. However, commonly available situations with engineering materials are non-steady. The concentration of solute atom changes at any point with respect to time in non-steady diffusion. If the concentration gradient various in time and the diffusion coefficient is taken to be independent of concentration. The diffusion process is described by Frick’s second law which can be derived from the first law: Equation 4 Fick’s second law for unidirectional flow under non steady conditions. A solution of Eq. (4)given by Where A is constant Let us consider the example or self diffusion or radioactive nickel atoms in a non-radioactive nickel specimen. Equation (4a) indicates that the concentration at x = 0 falls with time as r-12 and as time increases the radioactive penetrate deeper in the metal block [Fig.6 ] At time t1 the concentration of radioactive atoms at x = 0 is c1= A/(Dt1)1/2. At a distance x1 = 0 (Dt1)1/2 the concentration falls to 1/e of c1. At time t2 . the concentration at x = 0 is c2 = A/(Dt2)1/2 and this falls to 1/e and x2 = 2 (Dt2)1/2 . These results are in agreement with experiments.
If D is independent of concentration, Eq. (4) simplifies to
Even though D may vary with concentration, solutions to the differential Eq. 5 are quite commonly used for practical problems, because of their relative simplicity. The solution to Eq.5 for unidirectional diffusion from one medium to another a cross a common interface is of the general form. Where A and B are constant to be determined from the initial and boundary conditions of a particular problem. The two media are taken to be semi-infinite i.e. only one end of each of them, which the interface is defined. The other two ends are at an infinite distance The initial uniform concentrations of the diffusing species in the two media are different, with an abrupt change in concentrations at the interface erf in eqn.5 (a) stands for error function, which is his an integration variable, that gets deleted as the limits of the integral are substituted. The lower limits of the integral is always zero, while the upper limit of the integral is the quantity, whose function is to be determined
A schematic illustration of time dependence of diffusion is shown in fig7. The curve corresponding to the concentration profile at a given instant of time t1 is marked by t1. We can see from fig.7 at a later time t2, the concentration profile has changed. We can easily see that this changed in concentration profile is due to the diffusion of B atoms that has occurred in the time interval t2-t1 The concentration profile at a still later time t3 is marked by t3 . Due to diffusion, B atoms are trying to get distributed uniformaly throughout the solid salutation. From Fig. 7 Its is evident that the concentration gradient becoming less negative as time increases. Obviously, the diffusion rate becomes slower as the diffusion process progress.
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(2)
(4)
(4a)
(5)
(5a) 
(5a)
is a normalization factor. The diffusion coefficient D (m2/s) determines the rate of diffusion at a concentration gradient equal to unity. It depends on the composition of alloy, size of grains, and temperature.