574 CHAPTER 24: MARTENSITIC TRANSFORMATIONS When the shape change is relatively large, the parent phase will no longer be able to accommodate the inclusion elastically, and anticoherency lattice dislocations will be generated to relieve the long-range stress field and reduce the elastic energy. Plastic deformation will therefore occur in the parent phase, and anticoherency dislocations will be added to the interface. These dislocations will generally tend to reduce the mobility of the interface. Because martensite interfaces can be represented as arrays of dislocations, the velocity with which they move will generally be controlled by the same factors that control the rate of glide motion of crystal dislocations. As discussed in Section 11.3, these include dissipative drag due to phonons and free electrons and interactions with a large variety of different types of crystal imperfections which hinder their glide motion. When the martensite forms as enclosed platelets as in Fig. 24.12, additional work must also be done to produce the increase in interfacial area that occurs as the platelets grow. An extensive discussion of the factors involved in the motion of martensite interfaces has been given by Olson and Cohen [9]. As pointed out in Section 11.3.4, there is no clear evidence for the supersonic motion of martensite interfaces. However, velocities on the order of the speed of sound can be achieved in the presence of large driving forces. 24.4 NUCLEATION OF MARTENSITE The homogeneous nucleation of martensite in typical solids is too slow by many orders of magnitude to account for observed results. Calculations of typical values of AGc using the classical nucleation model of Section 19.1.4 (see Exercise 19.3) yield values greatly exceeding 76 kT. Furthermore, nearly all martensitic transformations commence at very sparsely distributed sites. Small-particle experiments [14] have yielded typical nucleation densities on the order of one nucleation event per 50 pm diameter Fe-Ni alloy powder parti~le.~ Thus, nucleation of martensite is believed to occur at a small number of especially potent heterogeneous nucleation sites. The most likely special site for martensitic nucleation is a pre-existing dislocation array, such as a portion of a tilt boundary [9]. The nucleation process involves dis- sociation of the boundary dislocations, so as to produce periodic faults in the parent crystal and thereby provide a mechanism for the lattice deformation. The process of superimposing lattice-invariant deformation onto the deformation that occurs in the dissociation of the original tilt boundary is used to obtain the equivalent of the lattice deformation RB in the crystallographic model of Section 24.2.4. The rate of initiating such a nucleus is limited by the rate at which the dislocations required to form and then expand the configuration can move under the available driving force. The entire process may be free of any energy barrier under sufficiently high driving forces, or else involve local barriers to certain critical dislocation movements which can be surmounted with the assistance of thermal activation. Details of the specific defects required for the mechanism have been worked out for common structural changes (e.g., f.c.c.+ h.c.p., f.c.c.+ b.c.c.) [8, 91. 3Small-particle experiments are carried out by studying nucleation in small particles of the parent phase and are useful in distinguishing between homogeneous and heterogeneous nucleation. If the nucleation is homogeneous, the nucleation rate is simply proportional to the volume of the particle. On the other hand, if it is heterogeneous, the rate goes essentially to zero when the particle size is lower than l/p, where p is the density of heterogeneous nucleation sites. 24 5: EXAMPLES OF MARTENSITIC TRANSFORMATIONS 575 24.5 MARTENSITIC TRANSFORMATIONS IN THREE CONTRASTING SYSTEMS We now describe briefly martensitic transformations in three contrasting systems which illustrate some of the main features of this type of transformation and the range of behavior that is found [15]. The first is the In-T1 system, where the lattice deformation is relatively slight and the shape change is small. The second is the Fe-Ni system, where the lattice deformation and shape change are considerably larger. The third is the FeNi-C system, where the martensitic phase that forms is metastable and undergoes a precipitation transformation if heated. 24.5.1 In-TI System Upon cooling, an In-T1 (19% T1) alloy undergoes an f.c.c. solid solution + f.c.t. solid solution martensitic transformation in which the lattice deformation is relatively slight, corresponding to (24.9) 0 1.0238 OI 0.9881 0 B = [Bij] = 0 0.9881 0 [o and the shape change is correspondingly small. The lattice-invariant deformation is accomplished by means of twinning in this system, so the low-temperature marten- sitic phase consists of twin-related lamellae. If a rod-shaped single crystal of the parent f.c.c. phase is carefully cooled in a small temperature gradient from above the transformation temperature, the transformation can be induced so that the martensite first appears at the cooler end of the specimen as a region separated from the parent phase by a single planar interface that spans the entire cross sec- tion of the specimen. As cooling continues, the single interface advances along the rod until the entire specimen is transformed. Upon subsequent reverse heating, the transformation is found to be reversible and the original single crystal of the parent phase is recovered with a temperature hysteresis of only about 2", as shown in Fig. 24.13, where the progress of the transformation is indicated by measurements of the length change of the specimen. L' ' ' I' ' ' I' 'J w66 68 70 72 74 76 Temperature ("C) Figure 24.13: Temperature dependence of the martensitic transformation in In-20.7 at. % T1. The extent of transformation is revealed by changes of specimen len th caused by the transformation. The dashed line shows the reversible transformation res5ting from continuous cooling and heating. The solid line shows stabilization of the transformation induced during the heating part of the cycle by a hold of 6 h at constant temperature. From Burkart and Read [16]. 576 CHAPTER 24: MARTENSITIC TRANSFORMATIONS f.c.c. 0.2 01 ' ' ' ' ' ' ' ' ' ' '1 66 68 I0 12 14 16 Temperature ("C) Figure 24.14: Temperature dependence of martensitic transformation in In-20.7 at. '% T1 under two different com ressive stresses. Phase fraction of martensite is proportional to the permanent strain whicl? can be determined by the stress-free specimen length. From Burkae and Read [16]. The interface motion is jerky on a fine scale and requires a continuous drop in temperature. This indicates that the interface requires a continuous increase in driving pressure (brought about by increased undercooling) to maintain its motion. This may be taken as evidence that the interface must be accumulating defects due to interactions with obstacles in its path which progressively reduce its mobility. If the heating (or cooling) is interrupted by a hold at constant temperature, the interface becomes stabilized as shown in Fig. 24.13. During the holding period, no further transformation occurs, and then a jump in temperature is required to restart the transformation. This is apparently due to an unidentified time-dependent re- laxation at the interface that occurs during the hold. The extent of transformation therefore depends primarily on the temperature and not on time. The transfor- mation is therefore considered to be athermal to distinguish it from an isothermal transformation, which progresses with increasing time at constant temperature. The transformation can be influenced by an applied stress. As seen in Fig. 24.13, the stress-free transformation to martensite results in a decrease in specimen length. Data in Figs. 24.14 and 24.15 were obtained by applying a series of constant uniax- ial stresses at constant ambient pressure, P. The data show that the transforma- tion temperature increases approximately linearly with applied uniaxial compressive stress. This dependence of transformation temperature on stress state follows from minimization of the appropriate thermodynamic function. For a material under I- 60- 0 0.1 0.2 0.3 Compressive stress (MPa) Figure 24.15: of applied compressive stress. Martensite transformation temperature in In-20.7 at. % T1 as a function From Burkart and Read [16]. 24.5: EXAMPLES OF MARTENSITIC TRANSFORMATIONS 577 uniaxial stress, this function takes the form Guni = Uuni - TS + p v - v, ,+p,uni(l + pas,uni 1 (24.10) where Uuni is the reversible adiabatic work to take a system from a reference state to a state of uniaxial m tress.^ aapp,uni is the applied uniaxial stress above the gauge hydrostatic stress, -P, and E~'~~+~~ is the elastic strain in the axial direction. V, is a reference molar volume, which can be taken to be the molar volume of the parent phase at one atmosphere (i.e.' V, = Vpar). Let the uniaxial strain associated with the martensite transformation be SE;$~, Emeas,uni . The and parent phases, respectively. It is not necessary that EEF'~~~ = par elastic parts of the uniaxial strains in the two phases will be related through their respective elastic constants because the normal components of stress must be equal at the interface. The differential forms of the molar free energy for the parent and martensite phases are and let pF,uni and pas,uni par be the measured uniaxial strains in the martensite meas,uni dGEdrt = - Smart dT + Vmart dP - Vo(l + emart - &girt) dc7app3uni (24.11) This analysis shows that a compressive load decreases the molar free energy- and that a positive &:$, reduces the magnitude of the decrease for the marten- site phase thereby resulting in an increased transformation temperature, consistent with Fig. 24.16. Further analysis shows that the observed shift in transformation temperature results from differences in the Young's moduli of the two phases (see Exercise 24.5). This result is consistent with LeChatelier's principle. dG:i? = - Spar dT + Vpar dP - V, (1 + E~~~'~~~) daaPP,uni t s F C Figure 24.16: Free energy of parent and martensite phases as a function of temperature, illustrating the effect of compressive uniaxial stress on martensite transformation temperature in In-T1 crystals. 4U has the differential dU""' = T dS - P dV + VouaPP,uni dcelas~uni. Considering that this energy change must reduce to the fluidlike P dV work under pure hydrostatic loading, the (1 + cii)-terms must appear because CE.~ = AV/Vo = V/Vo - 1. 578 CHAPTER 24: MARTENSITIC TRANSFORMATIONS Further work found that the transformation in In-TI alloys could be induced isothermally (i.e., without any cooling whatsoever) by the application and removal of a sufficiently large compressive load [16]. This is consistent with the data in Fig. 24.15, which show that there are conditions where the transformation temper- ature on cooling of the stressed specimen is above the transformation temperature of the unstressed specimen on heating, as would be required. 24.5.2 Fe-Ni System Upon cooling, an Fe-Ni (29.3 wt. % Ni) alloy undergoes an f.c.c. solid solution + b.c.c. solid solution martensitic transformation in which the lattice deformation is an order of magnitude larger than in the In-T1 transformation and is B = [Bij] = 0 1.13 0 (24.12) [ l: 1 010 1 The transformation is again found to be reversible and to exhibit hysteresis as shown in Fig. 24.17, which shows a cooling and heating cycle, detected by means of electrical resistivity measurements. However, the hysteresis, corresponding to about 450°C, is much larger than in the In-T1 system, indicating that a much larger pressure is required to drive the transformation. Examination of the morphology of the transformation shows that it is quite different than in the In-T1 case. The martensite now forms as small lenticular platelets embedded in the parent phase, with their habit planes parallel to variants of the invariant plane, as shown in Fig. 24.18. The manner in which the transformation progresses during cooling is also quite different. After forming, each platelet grows very rapidly to a final size and then remains static. As cooling continues, the transformation then progresses by the formation of new platelets. This behavior is attributed to the large lattice deformation, causing a large shape change in this system, which is too large to be accommodated elastically. Instead, plastic flow occurs in the parent phase in the form of the generation and movement of dislocations, and anticoherency dislocations are introduced in the platelet interfaces, causing them to lose their mobility as described in Section 24.3. This explanation is consistent with the large amount of hysteresis observed upon thermal cycling, since this reduction of mobility makes it difficult to reverse the direction of motion of the platelet interfaces. 2.0 G 1.6 - 1.2 - 8 s 0.8 c v) 0.4 lx -100 0 100 200 300 400 500 Temperature ("C) Figure 24.17: Temperature dependence of the martensitic transformation in the Fe-Ni (29.3 wt. %) system during thermal cycle. Extent of transformation revealed by change of specimen electrical resistivity. From Kaufman and Cohen [17]. 24.5: EXAMPLES OF MARTENSITIC TRANSFORMATIONS 579 Figure 24.18: Fe-32 wt. % Ni alloy. From the ASM Metals Handbook, Vol. 8, p. 198. Martensite platelets formed in the f.c.c. -+ b.c.c. transformation in an The phenomenon of stabilization is also observed in this system if the cooling is interrupted and the specimen is held isothermally before cooling is resumed. In this case, the transformation resumes only after the driving force is incremen- tally increased by a significant drop in temperature. Again, the transformation is primarily athermal, depending upon decreases of temperature which provide corre- sponding increases in the driving pressure for the formation of more platelets. Also, a relatively small amount of isothermal formation of martensite is observed if the specimen is rapidly quenched into the temperature range where martensite forms and is then held isothermally [18]. However, the isothermal transformation occurs by the formation of new platelets and not by the growth of existing ones. In general, the result that the platelets form very rapidly (at speeds of the order of the speed of sound) at relatively low temperatures, at rates that are not signifi- cantly temperature-dependent, indicates that the platelet growth is not thermally- activated and occurs only when a sufficiently high driving pressure is available. 24.5.3 Fe-Ni-C System The crystallography of the f.c.c + b.c.t. martensitic transformation in the Fe-Ni-C system (with 22 wt. %Ni and 0.8 wt. %C) has been described in Section 24.2. In this system, the high-temperature f.c.c. solid-solution parent phase transforms upon cooling to a b.c.t. martensite rather than a b.c.c. martensite as in the Fe-Ni system. Furthermore, this transformation is achieved only if the f.c.c. parent phase is rapidly quenched. The difference in behavior is due to the presence of the carbon in the Fe- Ni-C alloy. In the Fe-Ni alloy, the b.c.c. martensite that forms as the temperature is lowered is the equilibrium state of the system. However, in the Fe-Ni-C alloy, the equilibrium state of the system in the low-temperature range is a two-phase mixture of a b.c.c. Fe-Ni-C solid solution and a C-rich carbide phase.5 This difference in be- havior is due to a much lower solubility of C in the low-temperature b.c.c. Fe-Ni-C phase than in the high-temperature f.c.c. Fe-Ni-C phase. If the high-temperature 5The true equilibrium state is the FeNi-C phase plus graphite. However, the carbide phase is so strongly metastable that it can be regarded as an “equilibrium” phase. 580 CHAPTER 24. MARTENSITIC TRANSFORMATIONS f.c.c. Fe-Ni-C parent phase were to be slowly cooled under quasi-equilibrium condi- tions, it would undergo diffusional phase changes resulting in the ultimate formation of the two-phase mixture. However, if the parent phase is rapidly quenched, these phase changes are bypassed and it transforms martensitically to the solid-solution b.c.t. phase, which is therefore a nonequilibrium phase that is metastable to the formation of the equilibrium two-phase mixture. During the quench, the C atoms are trapped in the interstitial positions they occupied in the parent phase, as shown in Fig. 24.3. By comparing these positions with Fig. 8.8a, it may be seen that they are a subset of the complete set of lattice-equivalent interstitial sites that carbon atoms can occupy in the b.c.c. structure.6 Carbon atoms occupying interstitial sites generally act as positive centers of dilation that push most strongly against their nearest-neighbors. The carbon atoms that randomly occupy the sites in Fig. 24.3 push most strongly along the z axis and so produce the observed tetragonality. The b.c.t. phase can be considered as a b.c.c. structure that has been forced into tetrag- onality by quenched-in C atoms that occupy positions inherited from the parent f.c.c. phase. Once the system is cooled to a low enough temperature to preclude any carbide formation due to diffusion, further martensite can be produced by further drops in temperature. The overall transformation on cooling then has many of the fea- tures of the transformation in the FeNi alloy described above. The shape change is large, the martensite forms as embedded lenticular platelets, and the formation is athermal and requires continuously decreasing temperatures to proceed signifi- cantly. However, the transformation is not reversible as in theFe-Ni system. When the Fe-Ni-C martensite is heated, it decomposes by precipitating the more stable carbide phase before it is able to transform back to the high-temperature f.c.c. parent phase. This behavior is typical of steels that are alloys composed mainly of iron and car- bon and, in many cases, additional alloying elements such as nickel, chromium, or manganese. The martensite formed directly after quenching is exceedingly hard but quite brittle. However, it can then be toughened by subsequent heating (temper- ing), which allows some controlled carbide precipitation. Extraordinary mechanical properties can be obtained by this combination of quenching and tempering, and it forms the basis for the heat treatment of steel [15]. Bibliography 1. W.S. Wechsler, D.S. Lieberman, and T.A. Read. On the theory of the formation of martensite. Trans. AIME, 197( 11):1503-1515, 1953. 2. J.S. Bowles and J.K. MacKenzie. The crystallography of martensite transformations I. Acta Metall., 2(1):129-137, 1954. 3. J.S. Bowles and J.K. MacKenzie. The crystallography of martensite transformations 11. Acta Metall., 2(1):138-147, 1954. 4. J.S. Bowles and J.K. MacKenzie. The crystallography of martensite transformations. 111. Face-centered cubic to body-centered tetragonal transformations. Acta Metall., 5. C.M. Wayman. Introduction to the Crystallography of Martensitic Ransformations. 2(2):224-234, 1954. Macmillan, New York, 1964. 6Note that the number of carbon atoms occupying these sites is considerably smaller than the number of sites and that the sites are therefore sparsely populated. EXERCISES 581 6. J.W. Christian. Martensitic transformations. In R.W. Cahn, editor, Physical Metal- lurgy, pages 552-587. North-Holland, New York, 1970. 7. M. Cohen and C.M. Wayman. Fundamentals of martensitic reactions. In J.K. Tien and J.F. Elliott, editors, Metallurgical Treatises, pages 455-468. The Metallurgical Society of AIME, Warrendale, PA, 1981. 8. G.B. Olson and M. Cohen. Theory of martensitic nucleation: A current assessment. In Proceedings of an International Conference on Solid+Solid Phase Transformations, pages 1145-1164, Warrendale, PA, 1982. The Metallurgical Society of AIME. 9. G.B. Olson and M. Cohen. Dislocation theory of martensitic transformations. In F.R.N. Nabarro, editor, Dislocations in Solids, Vol. 7, pages 295-407. North-Holland, New York, 1986. 10. C.S. Barrett and T.B. Massalski. Structure of Metals: Crystallographic Methods, 11. J.M. Ball and R.D. James. Fine phase mixtures as minimizers of energy. Arch. Rat. 12. J.M. Ball. The calculus of variations and materials science. Quart. Appl. Math., 13. J.M. Ball and R.D. James. Theory for the microstructure of martensite and applica- tions. In Proceedings of the International Conference on Martensitic Transformations, pages 65-76, Monterey, CA, 1993. Monterey Institute for Advanced Studies. 14. R.E. Cech and D. Turnbull. Heterogeneous nucleation of the martensite transforma- tion. Trans. AIME, 206:124-132, 1956. 15. R.E. Reed-Hill and R. Abbaschian. Physical Metallurgy Principles. PWS-Kent, Boston, 1992. 16. M.W. Burkart and T.A. Read. Diffusionless phase change in the indium-thallium system. Trans. AIME, 197:1516-1524, 1953. 17. L. Kaufman and M. Cohen. The martensitic transformation in the iron-nickel system. Trans. AIME, 206:1393-1400, 1956. 18. E.S. Machlin and M. Cohen. Isothermal mode of the martensitic transformation. Trans. AIME, 194:489-500, 1952. deformation. Acta Metall., 6:680-693, 1958. Principles and Data. Pergamon Press, New York, 3rd edition, 1980. Mech. Anal., 100:13-52, 1987. 56( 4) : 719-740, 1998. 19. D.S. Lieberman. Martensitic transformations and determination of the inhomogeneous 20. J.F. Nye. Physical Properties of Crystals. Oxford University Press, Oxford, 1985. EXERCISES 24.1 It has been stated that “a martensitic phase transformation can be considered as the spontaneous plastic deformation of a crystalline solid in response to internal chemical forces” [9]. Give a critique of this statement. Solution. According to Eq. 24.1, forward and reverse martensitic transformations can be driven either by internal chemical forces derived from the bulk “chemical” free-energy change, AgB, or by forces due to applied stress. In all cases, the transformation causes a shape change that corresponds to plastic deformation. If we regard transformations that occur due to heating or cooling in the absence of applied stress as spontaneous and transformations that occur due to applied stress as driven then the statement is true. A more inclusive statement might be: “a martensitic phase transformation can 582 CHAPTER 24: MARTENSITIC TRANSFORMATIONS be considered as the plastic deformation of a crystalline solid in response to internal chemical forces and/or applied mechanical forces." 24.2 Find an expression for the cone angle, #l, in Fig. 24.4 in terms of 771 and 773. Solution. equation for the unit sphere, xi2 + zL2 + zL' = 1, equal to Eq. 24.2 to obtain First find the equation for the A'O'B' cone in Fig. 24.4 by setting the (1 - +) z;' + (1 - $) x:' + (1 - 2) xi2 = 0 (24.13) Then, setting z; = 0 yields (24.14) 24.3 Section 24.2.3 claims that the rotation axis in the final rigid-body rotation, R, which rotates a'" -+ a' and I?' + E'in Fig. 24.9 is located at the position ii. By using the stereographic method, show (within the recognized rather low accuracy of the method) that this is indeed the case. 0 The axis of rotation required to bring a''' -+ a' by a rigid-body rotation must lie somewhere on a plane normal to the vector (a''' - a'). 0 Similarly, the axis of rotation required to bring ?' + E'must lie some- where on a plane normal to (?' - Z). 0 These two rotations can therefore be accomplished simultaneously by a single rotation around a common axis lying along the intersection of these two planes. This axis will therefore be parallel to ii= (a'" - a') x (2' - q (24.15) [Too] Figure 24.19: rigid-body rotation, R! in Section 24.f.3. From Lieberman [19]. Stereogram showin the method for locating the rotation axis, 3, for the EXERCISES 583 Solution. First find the poles of the vectors (2’’ - 2) and (E” - Z). The rotation axis, ii, will be the pole of the plane containing these vectors. On a stereogram, this will be the pole of the great circle containing both (a“’ - 2) and (2’ - Z). The vector (5’’ - Z) is perpendicular to the vector (2’’ + a), and they both lie in the same plane. The vector (8’ + Z) lies on a great circle going through both 3’’ and ti and lies midway between them as indicated in Fig. 24.19. Therefore, (5’’ - 2) lies on this same great circle 90” away from (6’ + 3). A similar procedure yields the pole of (E“ - ~7‘). The final step is to locate u’ at the pole of the great circle going through both (a“’ - 2) and (2’ - Z). 24.4 In Section 24.3 we pointed out that martensite platelets (Fig. 24.12) can be accommodated elastically in the parent phase when the lattice deformation and shape change are small. Consider such platelets in a polycrystalline par- ent phase where the platelets have grown across the grains and are stopped at the grain boundaries as in Fig. 24.20. Upon thermal cycling, such a plate will reversibly thicken during cooling and thin during heating due to a “thermo- elastic” equilibrium that is reached between changes in its bulk free energy, AgB, and the elastic strain energy in the system. Approximate the platelet shape by a thin disclike ellipsoid of aspect ratio c/a as in Section 19.1.3 (Eq. 19.23) and show that the platelet thickness, c, and AgB are related by a 2A gB c= A (24.16) where A = constant. Assume an invariant plane strain habit plane and use the elastic-energy expression for an invariant plane strain described in Sec- tion 19.1.3. Figure 24.20: phase. Martensite platelet stopped at grain boundaries in polycrystalline parent Solution. According to Section 19.1.3, the elastic strain energy (per unit volume of platelet) is proportional to c/a. The free energy associated with the platelet can then be written in the usual way as the sum of a bulk term, an elastic energy term, and an interfacial energy term, (24.17) 4 4 C 3 3 a AG = rra2cAgB + rra2c A- + 2.rra27 Here, the interfacial area has been approximated by that of a thin disc. Because a is held constant, the thermoelastic equilibrium requires that aAG/ac = 0, and this leads directly to the condition (24.18) [...]... high-resolution electron microscopy at a (113) symmetric tilt grain-boundary in aluminium Phil Mag Lett., 82:58 9-5 97, 2002 5 J.Q Broughton, A Bonissent, and F.F Abraham The FCC (111) and (100) crystalmelt interfaces-a comparison by molecular-dynamics simulation J Chem Phys., 74(7):402 9-4 039, 1981 6 W.T Read Dislocations in Crystals McGraw-Hill, New York, 1953 APPENDIX C CAPILLARITY AND MATHEMATICS OF. .. all of the species that make up the material possessing total density p For example, an alloy of copper and zinc has five stoiKinetics of Materials By Robert W Balluffi, Samuel M Allen, and W Craig Carter Copyright @ 2005 John Wiley & Sons, Inc 587 588 APPENDIX A: DENSITIES, FRACTIONS, AND ATOMIC VOLUMES OF COMPONENTS chiometric phases-a (pure Cu), p (CuSZng), y (CuZn), E (CuZnS), and 7 (pure Zn)-but... there is a total of s distinct types of sites (s = 2 in NaC1) and there is a total number, of sites of type' j on which are distributed N j atoms (molecules) of component i, the fraction of sites of type j occupied by component i is A.2 ATOMIC VOLUME The atomic volume of component i, Ri, is the volume associated with one atom, molecule, or other entity The total volume, Vtot, is comprised of contributions... meaningful description of interface coherence The Burgers vectors of the dislocations in a semicoherent interface will generally be translation vectors of the DSC-lattice of the reference structure The bicrystal reference structures, which are of most physical relevance, will generally contain interfaces of relatively low energy It is often useful t o describe the dislocation content of coherent and semicoherent... -AS dT + AV d p - vo (Emart - p parw n i - dE;irt) duaPP.uni meas.uni a (24.19) At equilibrium, AGUni 0 and = uapp,uni - meas,uni mear,uni - ( ~ m 3 t - t - d ~ C L ) E m a r t= €par Epar (24.20) if the applied stress is below the elastic limit for each phase and Emaa and Epar are the Young's moduli for each phase.7 At thermodynamic equilibrium subject t o linear elasticity, the Gibbs-Duhem equation... 4% v) Tangent Plane ( 2 = (z,y, z ) , f = (5, r), C) ) (f-.'). (- d2 dZ x -) det du dv Graph Surfaces: Tangent Plane, Surface Normal, and Curvature 2 = f (x, Y) Tangent Plane (2= (a, z ) , f = (5, r), 5)) y, 2 5-x Normal c-z 17-Y a - - 1 aY Mean Curvature c.2: ISOTROPIC INTERFACES AND MEAN CURVATURE 607 due t o its curvature, establishes the Gibbs-Thomson equation, P = YK = Y (f + 4) (C.17) The quantity... between interfaces with the same amount of volume A more precise statement involves those interfaces where the integral of the squared difference are equivalent This is called the L2 n o r m o n functions 608 APPENDIX c: CAPILLARITY AND MATHEMATICSOF SPACE CURVES AND INTERFACES This extension to all of space is used in the derivation of the Cahn-Hoffman c-vectors-a convenient way t o study capillarity... STRUCTURE OF CRYSTALLINE INTERFACES structure in Fig B.7a Because of the good atomic matching across coherent interfaces, the energetic contribution from mismatch is generally small The energy of semicoherent interfaces is minimized when most of interfacial area consists of patches of the coherent reference structure This reduces the core width of the line defects that delineate the coherent regions of the... atomic or molecular weight of each component is M; Crystalline materials have distinct structures with sites distinguished by their symmetry, and it may be important to specify occupancies of particular types of sites Vacant sites must be considered as well A.l.l Mass Density The mass density of material, p , is the amount of mass of the material per unit volume (i.e., kg m-3) For component i, the mass... DEGREES OF F R E E D O M Interfaces that involve a crystalline material may be classified in different ways The broadest system of classification is based on the state 'of matter abutting the crystal: 0 Crystal/vapor interfaces 0 Crystal/liquid interfaces 0 Internal interfaces in solid and/or crystalline materials 'Further information and references may be found in several references [l-31 Kinetics of Materials . regions of the same phase, whereas a heterophase interface separates two dissimilar phases. Crystal/vapor and crys- tal/liquid interfaces are heterophase interfaces. Crystal/crystal interfaces. SYSTEMS We now describe briefly martensitic transformations in three contrasting systems which illustrate some of the main features of this type of transformation and the range of behavior. types are listed in order of increasing complexity. Crystal/vapor and crystal/liquid interfaces both possess two macroscopic geometrical degrees of freedom corresponding to the parameters required