Thermochemical Processes Episode 7 pps

172 Thermochemical Processes: Principles and Models Table 6.1 Structures of the common metals at room temperature (diameters in angstroms) Metal Structure Metallic diameter Lithium b.c.c 3.039 Sodium b.c.c 3.715 Potassium b.c.c 4.627 Copper f.c.c 2.556 Silver f.c.c 2.888 Gold f.c.c 2.884 Magnesium c.p.h 3.196 Calcium f.c.c 3.947 Zinc c.p.h 2.664 Cadmium c.p.h 2.979 Aluminum f.c.c 2.862 Lead f.c.c 3.499 Titanium c.p.h 2.89 Zirconium c.p.h 3.17 Vanadium b.c.c 2.632 Tantalum b.c.c 2.860 Molybdenum b.c.c 2.725 Tungsten b.c.c 2.739 Iron b.c.c 2.481 Cobalt c.p.h 2.506 Nickel f.c.c 2.491 Rhodium f.c.c 2.689 Platinum f.c.c 2.775 Uranium orthorhombic 2.77 Plutonium monoclinic 3.026 Rare earths Scandium h.c.p 1.641 Yttrium h.c.p 1.803 Lanthanum h.c.p 1.877 where D 0 is a constant for each element, and H Ł 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 H diff is therefore the sum of the energy to form a vacancy H vac 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 Crystal D 0 (cm 2 s 1 ) H vac H diff H 298 sub structure Cu f.c.c. 0–16 124 200 337 Ag f.c.c. 0.04 108 170 285 Mg Hex 1.0 – 134 147 Al f.c.c. 0.05 65.5 123 330 Pb f.c.c. 1.37 54.8 109 195 Fe f.c.c. 0.49 135 284 416 Zr Hex 0.06 – 210 601 V b.c.c. 0.02 424 283 514 Ta b.c.c. 0.02 269 392 781 Mo b.c.c. 0.10 289 386 658 W b.c.c. 1.80 385 586 851 ˛Fe b.c.c. 2.0 154 239 (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 300 500 Rapid 700 Temperature (K) 900 1100 d v Slow Resistivity d v is the resistivity increase due to vacancies added at 1000 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 R vacancy D R 0 exp  H vac RT where R 0 is a constant of the system. Typical values of the energy to form vacancies are for silver, 108 kJ mol 1 and for aluminium, 65.5 kJ mol 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 D 0 1 exp  H 1 RT  C D 0 2 exp  H 2 RT  Examples of this analysis are given in the data as follows: Gold (f.c.c.) D D 0.04 exp  20 500 T T C 0.56 exp  27 500 T T Vanadium (b.c.c.) D D 0.014 exp 34 060 T T C 7.5exp 43 200 T T Rate processes in metals and alloys 175 It can be seen that the divacancy diffusion process leads to a larger value of D 0 , but only a fractional increase in H diff . 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 D ∂ 2 c ∂x 2 D ∂c ∂t with the following boundary conditions: c D c 0 ,xD 0,tD 0; c D 0,x>0,tD 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 c D c 0 p Dt exp   x 2 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 c 0 ,againsta 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 c 0 /2, x D 0, t ½ 0is c D c 0 2 1  erfx/2 p Dt 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 below the originalsurface transfer (x D 0) should be linearwith slope  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 A m B n 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. Rate processes in metals and alloys 177 There is not sufficient experimental evidence to continue this discussion quantitatively at the present time, but the sparse experimental data suggests that for a given compound, the D 0 value is significantly lower than is the case in simple metals. This decrease may be attributed to a low value in the correlation factor which measures the probability that an atom may either move forward or return to its original site in its next diffusive jump. In simple metals this coefficient has a value around 0.8. Diffusion in alloys If samples of two metals with polished faces are placed in contact then it is clear that atomic transport must occur in both directions until finally an alloy can be formed which has a composition showing the relative numbers of gram-atoms in each section. It is very unlikely that the diffusion coefficients, of A in B and of B in A, will be equal. Therefore there will be formation of an increasingly substantial vacancy concentration in the metal in which diffusion occurs more rapidly. In fact, if chemically inert marker wires were placed at the original interface, they would be found to move progressively in the direction of slowest diffusion with a parabolic relationship between the displacement distance and time. At any plane in a Raoultian alloy system parallel to the original interface, the so-called chemical diffusion coefficient D chem , which determines the flux of atoms at any given point, and is usually a function of the local composition so that according to Darken (1948), D chem is given by D chem D X 2 D 1 C X 1 D 2 D chem D J  dc dx  where X 1 , X 2 are the atom fractions of the components and D 1 , D 2 are their self-diffusion coefficients, in this case of each element in alloys of this compo- sition. This equation also shows that as X 2 ! 0, D chem ! D 2 ,orthatina dilute alloy, the diffusion coefficient for the dilute constituent should approach that of its own self-diffusion coefficient in the dilute alloy. In an alloy showing departures from the ideal behaviour, this equation must be modified accord- ingly to D chem D X 2 D 1 C X 1 D 2 1  d ln /d ln X 1  An experimental technique for the determination of D chem in a binary alloy system in which the diffusion coefficient is a function of composition was originally developed by Matano (1932), based on a mathematical development 178 Thermochemical Processes: Principles and Models of Fick’s equations due to Boltzmann. The average diffusion length of particles in a time t can be calculated from Fick’s 2nd law to be given by  x 2 D 2Dt This suggests that x/t 1/2 could be a useful variable transformation. Then the flux accumulation rate can be expressed as ∂c ∂t D dc dz ∂z ∂t D 1 2 x t 3/2 dc dz where z D x t 1/2 and substituting into the general form of Fick’s 2nd law  x 2t 3/2 dc dx D ∂ ∂x  D ∂c ∂x  D 1 t d dz  D dc dz  which is now a single variable differential equation. Integrating the equation in the form  z 2 dc dz D d dz  D dc dz  using as boundary conditions c D c 0 for x<0att D 0 c D 0forx>0att D 0 the result is  1 2  cDc 0 cD0 z dc D  D dc dz  cDc 0 cD0 and substituting for z from the original definition yields the equation  1 2  c 0 0 x dc D Dt  dc dx  cDc 0 since dc/dx D 0atx D 0, and t is the (known) duration of the experiment, which is fixed, and c is a function of x only. This also requires that  cDc 0 cD0 x dc D 0 and the point where x D 0 can be found by equating the area above the curve for c 0 as a function of x when x<0 with that below the curve when x>0. Rate processes in metals and alloys 179 The chemical diffusion coefficient at any concentration C 0 in the experimental diffusion profile is then given by D chem c 0  D 1 2t  dx dc  c 0  c 0 0 x dc where t is known This analysis makes possible the determination of a chemical diffusion coef- ficient from experimental data having made no use of a model, and which takes no account of the atomic mechanism of diffusion, and assumes that the same chemical diffusion coefficient applies to each component of the alloy. If we now place chemically inert markers at the original interface between the two metal samples before the diffusion anneal, more information about the diffusion process can be obtained. Imagine that more atoms of type A move from left to right in the joined samples, J A , than those of type B moving from right to left, J B . Then if both diffusion processes are by atom–vacancy exchange, there must be a corresponding vacancy flow from left to right, J v , equal to the difference between these two fluxes J A ! J B  J v  vacancies A concentration gradient of vacancies will therefore appear across the orig- inal interface. The concentration of vacancies will be less than the equilibrium value on the B-rich side and more than the equilibrium value on the A-rich side. In a classical experiment to determine how this accumulation of vacancies would affect the position of the markers, Kirkendall and Smigelskas showed that when marker molybdenum wires were placed around a brass bar which was then electroplated with a thick layer of copper, the markers moved inward to the centre of the bar while zinc diffused out of the brass and into the copper. Clearly vacancy creation would originally take place on the brass side of the original marker position, and the wires have moved in the direction of the vacancy flux with a t 1/2 dependence on time. The excess vacancies are removed from the interface by diffusional exchange. Otherwise they may tend to aggregate to form pores on the side of the markers containing the more rapidly moving species, the brass side in this case. Darken assumed that the accumulated vacancies were annihilated within the diffusion couple, and that during this process, the markers moved as described by Smigelskas and Kirkendall (1947). His analysis proceeds with the assump- tion that the sum of the two concentrations of the diffusing species (c 1 C c 2 ) remained constant at any given section of the couple, and that the markers, which indicated the position of the true interface moved with a velocity v. 180 Thermochemical Processes: Principles and Models The statement of Fick’s 2nd law then becomes ∂c ∂t D ∂c 1 ∂t C ∂c 2 ∂t D ∂ ∂x  D 1 ∂c 1 ∂x C D 2 ∂c 2 ∂x  c v  D 0(i) Integrating with ∂c/∂t D 0 D 1 ∂c 1 ∂x C D 2 ∂c 2 ∂x  c v D a constant D 0 The constant of integration must be zero because at points on the diffusion couple far from the interface, ∂c 1 ∂x , ∂c 2 ∂x and v D 0 thus v D 1 c  D 1 ∂c 1 ∂x C D 2 ∂c 2 ∂x  substituting for v in (i) ∂c 1 ∂t D ∂ ∂x  D 1 ∂c 1 ∂x  c 1 c D 1 ∂c 1 ∂x  c 1 c D 2 ∂c 2 ∂x  D ∂ ∂x  c 1 D 2 C c 2 D 1 c ∂c 1 ∂x  since ∂c 1 ∂x D ∂c 2 ∂x J 1 DD ∂c 1 ∂x C vc 1 and ∂c 1 ∂t D ∂J 1 ∂x Since X i is a mole fraction equal to c 1 /c D chem D c 1 D 2 C c 2 D 1 c D X 1 D 2 C X 2 D 1 and v D D 1  D 2  dX 1 dx Steady state creep in metals Dislocations are known to be responsible for the short-term plastic (non- elastic) properties of substances, which represents departure from the elastic behaviour described by Hooke’s law. Their concentration determines, in part, not only this immediate transport of planes of atoms through the solid at moderate temperatures, but also plays a decisive role in the behaviour of metals under long-term stress. In processes which occur slowly over a long period of time such as secondary creep, the dislocation distribution cannot be considered geometrically fixed within a solid because of the applied stress. Rate processes in metals and alloys 181 This movement of dislocations leads to the formation of networks of disloca- tions which mutually reduce their mobility, and this produces work-hardening, which is the increased resistance of metals to an applied stress. This effect is decreased in time through the diffusion of vacancies into the dislocation core leading to dislocation climb. The presence of stress-induced diffusion can therefore enhance dislocation climb through the increased atom–vacancy exchange. It is also observed that the process of creep involves some degree of movement of grains relative to one another by grain boundary sliding. An account of the mechanism for creep in solids placed under a compressive hydrostatic stress which involves atom–vacancy diffusion only is considered in Nabarro and Herring’s (1950) volume diffusion model. The counter-movement of atoms and vacancies tends to relieve the effects of applied pressure, causing extension normal to the applied stress, and shrinkage in the direction of the applied stress, as might be anticipated from Le Chatelier’s principle. The opposite movement occurs in the case of a tensile stress. The analysis yields the relationship  1 L dL dt D AD v sV m RTL 2 where L is the length of the grain of a material under a compressive stress, s, and having a molar volume V m . A is a shape factor which has a value between 10 and 40. If this process occurs by grain boundary diffusion as well, Coble (1963) has shown that  1 L dL dt D AD b υsV m RTL 3  1 C D v L 3D b υ  where D b is the grain boundary diffusion coefficient and υ is the boundary width. Note that the dimensionless ratio D b υ/D v L determines whether boundary or volume diffusion predominates in the creep process. Diffusion in interstitial solutions and compounds Although the face-centred cubic structure of metals is close packed, it is still possible for atoms which are much smaller than the host metal atoms to fit into interstitial sites inside the structure, while maintaining the essential properties of metals such as electrical conductivity and heat transport. These interstitial sites are of two kinds. The octahedral interstitial sites have six metal atoms at equal distances from the site, and therefore at the apices of a regular octahedron. The tetrahedral interstitial sites have four nearest neighbour metal atoms at the apices of a regular tetrahedron. A smaller atom can just fit into the octahedral site if the radius ratio is  0.732 ½ r interstitial r metal D½0.414  [...]... interior, and therefore the energy Rate processes in metals and alloys 1 97 Table 6.4 Surface energies of solid elements Element Surface energy (J m2 ) Temp of measurement Hsub (kJ mol 1 ) (298 K) Cu Ag Au Fe Ni Ga In Al Sn W 1.5 1.1 1.4 1.93 1 .72 0 .77 0.63 0.98 0. 67 2.80 1000° C 950° C 1000° C 1 475 ° C 1400° C 20° C 140° C 450° C 200° C 2000° C 3 37 285 368 416 430 272 240 330 301 851 change to bring 1... 2nd edn Oxford University Press (1 975 ) QC185 C.J Smithells Metals Reference Book, 7th edn Butterworth-Heinemann (1996) TN761 S55 C Matano Jpn J Phys., 8, 109 (1932–3) L.S Darken Trans AIME, 175 , 184 (1948) E Smigelskas and E Kirkendall Trans AIME, 171 , 130 (19 47) P Shewmon Diffusion in Solids TMS (1989) C Herring J Appl Physics., 21, 301 (1950) R.L Coble ibid, 34, 1 679 (1963) D.A Porter and K.E Easterling... cementite phase can be produced in dispersion in ferrite depend upon the rate of cooling to Rate processes in metals and alloys 1400 1538 0 .74 1394 dFe 2.43 1493 Liquid Temperature (°C ) 1200 Liquid + Graphite 1153 gFe 9.06 1000 17. 1 Metastable 1600 185 g Fe + Fe3 C 912 g Fe + Graphite 800 74 0 0.096 2. 97 a Fe + Fe3 C 600 aFe a Fe + Graphite 400 0 Fe 5 15 10 Atomic per cent C 20 25 C Figure 6.3 The... decrease in rate as the temperature decreases, and the Gibbs energy of nucleation, compensated in part by the chemical potential driving forces which increase Rate processes in metals and alloys 1 87 Austenite 600 s a ti rm fo 50 Sta rt o ft ra n 70 0 on Austenite + Pearlite lete te comp p le tion m a rm co sfo % an Tr Coarse pearlite Fine pearlite Upper Bainite Temperature (°C) 500 400 300 200 Lower Bainite... Table 6.3 Diffusion data for interstitials in metals Solute D0 (cm2 s 1 ) Hdiff (kJ mol 1 ) Iron Carbon Nitrogen Oxygen 0.004 0.005 0.002 80.0 77 .0 86.0 Vanadium Carbon Nitrogen Oxygen 0.009 0.040 0.025 116 148 123 Molybdenum Carbon Nitrogen Oxygen 0.01 0.004 0.03 172 109 130 Solvent According to the transition state theory, the diffusion process can be described by the equation D D F t exp SŁ exp R HŁ... temperatures less than this because the diffusion coefficient of carbon decreases If the alloy is cooled rapidly to temperatures between 70 0 K and room temperature, martensite, which consists of lens-shaped needles of composition about Fe2.4 C with a 186 Thermochemical Processes: Principles and Models body-centred tetragonal structure dispersed in a matrix of ferrite is produced The c/a ratio of the martensite... the dilute solution of carbon in the metal The diffusion coefficient of carbon in the monocarbide shows a relatively constant activation energy but a decreasing value of the pre-exponential 184 Thermochemical Processes: Principles and Models D0 as the stoichiometric composition is approached The experimental results for TiC may be represented by the equation D D 345 1 x exp 54 000T T cm2 s 1 The value... expected from the equilibrium diagram Cementite is thermodynamically unstable with respect to decomposition to iron and carbon from room temperature up to 1130 K G° D 27 860 24.64T J mol 1 The austenite phase which can contain up to 1 .7 wt% of carbon decomposes on cooling to yield a much more dilute solution of carbon in ˛-iron (b.c.c), ‘Ferrite’, together with cementite, again rather than the stable... the steel composition, and is altered by the presence of alloying elements at a low concentration This is because the common alloying elements such as manganese, nickel and chromium decrease 188 Thermochemical Processes: Principles and Models the diffusion coefficient of carbon, and thus slow down the rate of austenite decomposition at a given temperature In some alloys the general shape of the T–T–T... by direct place-exchange, vacant sites must play a role in the re-distribution, and the rate of the process is controlled by the self-diffusion coefficients Experimental measurements of the 190 Thermochemical Processes: Principles and Models change in resistance of a sample which was originally an ordered compound as the temperature is increased show the kinetics of this transformation since the disordered . c.p.h 3. 17 Vanadium b.c.c 2.632 Tantalum b.c.c 2.860 Molybdenum b.c.c 2 .72 5 Tungsten b.c.c 2 .73 9 Iron b.c.c 2.481 Cobalt c.p.h 2.506 Nickel f.c.c 2.491 Rhodium f.c.c 2.689 Platinum f.c.c 2 .77 5 Uranium. dispersion in ferrite depend upon the rate of cooling to Rate processes in metals and alloys 185 1538 0 .74 1394 1493 9.06 912 0.096 2. 97 740 17. 1 1153 2.43 dFe gFe gFe + Fe 3 C gFe + Graphite aFe +. 2.689 Platinum f.c.c 2 .77 5 Uranium orthorhombic 2 .77 Plutonium monoclinic 3.026 Rare earths Scandium h.c.p 1.641 Yttrium h.c.p 1.803 Lanthanum h.c.p 1. 877 where D 0 is a constant for each element,