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172 Thermochemical Processes: Principles and Models Table 6.1 Structures of the common metals at room temperature (diameters in angstroms) Metal Lithium Sodium Potassium Copper Silver Gold Magnesium Calcium Zinc Cadmium Aluminum Lead Titanium Zirconium Vanadium Tantalum Molybdenum Tungsten Iron Cobalt Nickel Rhodium Platinum Uranium Plutonium Rare earths Scandium Yttrium Lanthanum Structure b.c.c b.c.c b.c.c f.c.c f.c.c f.c.c c.p.h f.c.c c.p.h c.p.h f.c.c f.c.c c.p.h c.p.h b.c.c b.c.c b.c.c b.c.c b.c.c c.p.h f.c.c f.c.c f.c.c orthorhombic monoclinic h.c.p h.c.p h.c.p Metallic diameter 3.039 3.715 4.627 2.556 2.888 2.884 3.196 3.947 2.664 2.979 2.862 3.499 2.89 3.17 2.632 2.860 2.725 2.739 2.481 2.506 2.491 2.689 2.775 2.77 3.026 1.641 1.803 1.877 where D0 is a constant for each element, and HŁ is called, by analogy with the Arrhenius equation for gas reaction kinetics, the ‘activation energy’. Since the quantity which is involved relates to a condensed phase under constant pressure in this instance, it is more correct to call this term an ‘activation enthalpy’. The pre-exponential term has a fairly constant range of values for most metals of approximately 0.1 to 1, and the activation energies follow the Rate processes in metals and alloys 173 same trend as the heats of vaporization. The latter observation gives a hint as to the nature of the most important mechanism of diffusion in metals, which is vacancy migration. It is now believed that the process of self-diffusion in metals mainly occurs by the exchange of sites between atoms and neigh-bouring vacancies in the lattice. The number of such vacancies at a given temperature will clearly be determined by the free energy of vacancy forma-tion. The activation enthalpy for self-diffusion Hdiff is therefore the sum of the energy to form a vacancy Hvac and the energy to move the vacancy. It has been found that the heat of formation of vacancies is approximately half the total enthalpy of activation for diffusion, for the lower values of activation energy, rising to two-thirds at higher energies (see Table 6.2) and hence it may be concluded that it is roughly equal to the enthalpy of vacancy movement. This contribution can be obtained by measurements of the elec-trical resistance of wire samples which are heated to a high temperature and then quenched to room temperature. At the high temperature, the equilib-rium concentration of vacancies at that temperature is established and this concentration can be retained on quenching to room temperature (Figure 6.2). Table 6.2 Data for diffusion coefficients in pure metals Element Cu Ag Mg Al Pb Fe Zr V Ta Mo W ˛Fe Crystal structure f.c.c. f.c.c. Hex f.c.c. f.c.c. f.c.c. Hex b.c.c. b.c.c. b.c.c. b.c.c. b.c.c. D0 (cm2 s 1) Hvac Hdiff 0–16 124 200 0.04 108 170 1.0 – 134 0.05 65.5 123 1.37 54.8 109 0.49 135 284 0.06 – 210 0.02 424 283 0.02 269 392 0.10 289 386 1.80 385 586 2.0 154 239 298 sub 337 285 147 330 195 416 601 514 781 658 851 (415) The total electrical resistance at room temperature includes the contribution from scattering of conduction electrons by the vacancies as well as by ion-core and impurity scattering. If the experiment is repeated at a number of high temperature anneals, then the effects of temperature on the vacancy contribu-tion can be isolated, since the other two terms will be constant providing that 174 Thermochemical Processes: Principles and Models Rapid Slow dv dv is the resistivity increase due to vacancies added at 1000 K 300 500 700 900 1100 Temperature (K) Figure 6.2 The increase in electrical conductivity when a metal sample is heated to a high temperature and then quenched to room temperature, arising from the introduction of vacant sites at high temperature the temperature at which the resistance is measured is always the same. The energy to form vacancies is then found from the temperature coefficient of this contribution Rvacancy D R0 exp Hvac where R0 is a constant of the system. Typical values of the energy to form vacancies are for silver, 108kJmol 1 and for aluminium, 65.5kJmol 1. These values should be compared with the values for the activation enthalpy for diffusion which are given in Table 6.2. It can also be seen from the Table 6.2 that the activation enthalpy for self-diffusion which is related to the energy to break metal–metal bonds and form a vacant site is related semi-quantitatively to the energy of sublimation of the metal, in which process all of the metal atom bonds are broken. At high temperatures there is experimental evidence that the Arrhenius plot for some metals is curved, indicating an increased rate of diffusion over that obtained by linear extrapolation of the lower temperature data. This effect is interpreted to indicate enhanced diffusion via divacancies, rather than single vacancy–atom exchange. The diffusion coefficient must now be represented by an Arrhenius equation in the form D D D0.1/exp RT 1 C D0.2/exp RT 2 Examples of this analysis are given in the data as follows: (f.c.c.)D D 0.04exp 20500T C 0.56exp 27500T VanadiumD D 0.014exp 34060T C 7.5exp 43200T Rate processes in metals and alloys 175 It can be seen that the divacancy diffusion process leads to a larger value of D0, but only a fractional increase in Hdiff. The measurements of self-diffusion coefficients in metals are usually carried out by the sectioning technique. A thin layer of a radioactive isotope of the metal is deposited on one face of a right cylindrical sample and the diffu-sion anneal is carried out at a constant temperature for a fixed time. After quenching, the rod is cut into a number of thin sections at right angles to the axis, starting at the end on which the isotope was deposited, and the content of the radioisotope in each section is determined by counting techniques. The diffusion process in which a thin layer of radioactive material is deposited on the surface of a sample and then the distribution of the radioactive species through the metal sample is analysed after diffusion, obeys Fick’s second law ∂2c ∂c ∂x2 ∂t with the following boundary conditions: c D c0, x D 0, t D 0; c D 0, x > 0, t D 0 D can be regarded as a constant of the system in this experiment since there is no change of chemical composition involved in the exchange of radioactive and stable isotopes between the sample and the deposited layer. The solution of this equation with these boundary conditions is 2 ! c D pDt exp 4Dt The procedure in use here involves the deposition of a radioactive isotope of the diffusing species on the surface of a rod or bar, the length of which is much longer than the length of the metal involved in the diffusion process, the so-called semi-infinite sample solution. An alternative procedure which is sometimes used is to place a rod in which the concentration of the isotope is constant throughout c0, against a bar initially containing none of the isotope. The diffusion profile then shows a concentration at the interface which remains at one-half that in the original isotope-containing rod during the whole experiment. This is called the constant source procedure because the concentration of the isotope remains constant at the face of the rod which was originally isotope-free. The solution for the diffusion profile is with the boundary condition c D c0/2, x D 0, t ½ 0 is c D c0 .1 erf.x/2pDt// It follows that a plot of the logarithm of the concentration of the radioactive isotope in each section against the square of the mean distance of the section 176 Thermochemical Processes: Principles and Models belowtheoriginalsurfacetransfer(x D 0)shouldbelinearwithslope .1/4/Dt. Since t, the duration of the experiment, is known, D may be calculated. Diffusion in intermetallic compounds Inter-metallic compounds have a crystal structure composed of two inter-penetrating lattices. At low temperatures each atomic species in the compound of general formula AmBn occupies a specific lattice but at higher temperatures a second-order transition involving a disordering of the atoms to a random occupation of all atomic sites takes place. The mean temperature at which the order–disorder transformation takes place depends upon the magnitude of interaction energy of A–B pairs. This is exothermic, which brings about the low-temperature order, the more so the higher the transition temperature. In the ordered state an atom cannot usually undergo a vacancy exchange with an immediately neighbouring site because this is only available to the other atomic species. Thus in the CuAu inter-metallic compound having the NaCl crystal structure, a copper atom can only exchange places with a site in the next nearest neighbour position. In disordered CuZn, nearest neighbour sites can be exchanged as in the self-diffusion of a pure metal. It is quite probable that ordered metallic compounds have a partial ionic contribution to the bond between unlike atoms resulting from a difference in electronegativity of the two metals. Miedema’s model of the exothermic heats of formation of binary alloys uses the work function of each metallic element in determining the ionic contribution to bonding in the solid state, instead of Pauling’s electronegativity values for the gaseous atoms which are used in the bonding of heteronuclear diatomic molecules. However, in the solid metallic state the difference in valency electron concentration in each pair of unlike atoms, adds a repulsive (endothermic) term to the heat of formation, and thus reduces the resultant value. This repulsive component has been found to be proportional to the bulk modulus, B, where B D s/.V/V/ which is the relative volume change in response to an applied stress, s. Vacancies on each site will therefore carry a virtual charge due to the partial transfer of electrons between the neighbouring atoms, and vacancy interaction between the two lattices should therefore become significant at low tempera-tures, leading to divacancy formation at a higher concentration than is to be found in simple metals. These divacancy paths would enhance atomic diffu-sion in two jumps for an A atom passing through a B site to arrive at an A site at the end of the diffusive step. Above the order–disorder transforma-tion the entropy contribution to the Gibbs energy of formation outweighs the exothermic heat of formation, and thus any atom–vacancy pair can lead to diffusion in the random alloy. ... - tailieumienphi.vn
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