14.1: ISOTROPIC SURFACES 345 Therefore, the energy decreases continuously with time if the Rayleigh instability condition is satisfied, X > Xcrit = 2nR0 (14.28) Any perturbation with a wavelength less than the circumference of the cylinder will not grow. The particular characteristics of morphological evolution are determined by the dominant transport mechanism; their analyses derive from the diffusion potential, which depends on the local curvature. For a surface of revolution about the z-axis, the curvature is given by Eq. (2.16; that is, 1 a2R Substituting Eq. 14.23 into Eq. 14.29 and expanding for small E/R, yields (14.29) (14.30) 14.1.3 Evolution of Perturbed Cylinder by Vapor Transport Suppose that a perturbed cylinder with radius given by Eq. 14.23 evolves by vapor transport in an environment with an ambient vapor pressure in equilibrium with the unperturbed cylinder, Pamb = Peq(, = l/R,). Then, using Eqs. 14.15, 14.16, and 14.17, u, = BV ($ -K) (14.31) According to Eq. 14.25, Rcyl N R,, so Eq. 14.23 shows that v, at z = 0 is approxi- mately dE(t)/dt. Therefore, using Eq. 14.30, (14.32) 1 dt Small perturbations therefore evolve according to E(t) = E(0)et/Tv(’) ( 14.33) where the amplification factor l/rv = (Bv/Rz)[l - (~TR,/X)~]. This first-order kinetic result is consistent with the previous Rayleigh result: only perturbations with wavelengths longer than Xcrit will grow. 14.1.4 Suppose that the perturbed cylinder considered above evolves by surface diffu- sion. A first-order differential equation for the amplitude E(t) follows from in- serting Eq. 14.30 into the surface diffusion relation, Eq. 14.6, and again setting u, = dc(t)/dt at z = 0: Evolution of Perturbed Cylinder by Surface Diffusion (14.34) 39 = - 47r2 BS [l- (i)2] 27rR, E(t) = 1 dt RZ X2 346 CHAPTER 14. SURFACE EVOLUTION DUE TO CAPILLARY FORCES In addition to the Rayleigh result, Eq. 14.34 predicts that a particular perturba- tion wavelength, A,,,, grows the fastest and hence dominates the morphology of the evolving cylinder. This kinetic wavelength maximizes the right-hand side of Eq. 14.34, giving the result A,,, = fiAcrit. 14.1.5 Comparison of surface-diffusion and vapor-transport kinetics in Fig. 14.5 shows a difference in long-wavelength behavior. The amplification factor ~/T(A) in the perturbation growth rate E(t) = ~(0) exp[t/~(X)] is monotonically increasing for vapor transport and approaches BV/ R: asymptotically for long wavelengths. For surface diffusion, ~/T(A) goes to zero for long wavelengths and has a maximum at A = fi (27rR,). For a cylinder with an initial small random roughness, evolution by surface diffusion results in a morphological scale associated with A,,,. For vapor diffusion, no characteristic morphological scale is predicted. Thermodynamic and Kinetic Morphological Wavelengths d& dt Figure 14.5: Behavior of the perturbation-amplitude growth coefficients 1/~" and l/rs for cylinder-pertiirbatiori growth by vapor transport arid surface diffusioii, respectively. For surface diffusion, a fastest-growing wavelength A,,,, determines a morphological scale for the initial instability. Except for the Rayleigh critical wavelength, &,.it, no characteristic length scale appears for vapor transport. 14.2 ANISOTROPIC SURFACES 14.2.1 An anisotropic surface's energy per unit area, y(A), depends on its inclination, A. For isotropic surfaces, the surface energy is simply proportional to the area, but two additional degrees of freedom emerge for the anisotropic case. These correspond to the two parameters required to specify the surface in~lination.~ An anisotropic surface can often decrease its energy at constant area by tilting (i.e., changing its normal). The variation of the interfacial energy with inclination can be represented conveniently in the form of a polar plot (or y-plot), as shown in two dimensions in Fig. 14.6. Here, the energy of each inclination is represented by a vector, r(A)A 3Geometrical constructions for describing anisotropic surfaces are reviewed in Section C.3.1. Some Geometrical Aspects of Anisotropic Surfaces 14.2: ANISOTROPIC SURFACES 347 C D Figure 14.6: be faceted into inclinations corresponding to points B and C. y-plot same as in Fig. C.4a. Construction for testing whether an interface of inclination A will prefer to (i.e., a vector normal to that inclination and of magnitude equal to the interfacial energy at that inclination). If all of these vectors are referred to a single origin, the y-plot is the surface passing through the tips of these vectors. Inclinations of particularly low energies will therefore appear as cusps or depressions in the plot. Conceptually, treatment of the morphological evolution for an anisotropic surface is no different than for an isotropic surface-kinetics requires that s y(A) dA (com- pared to y J dA for an isotropic surface) must decrease monotonically. However, because the evolving surface's geometry is linked to the local surface-energy density through fi, the analysis is considerably more complicated. Furthermore, when a sur- face is sufficiently anisotropic, inclinations fi associated with large energies become unstable and cannot be in local equilibrium-the surface must develop corners or edges. The missing inclinations create points or curves on a surface where surface derivatives will be discontinuous. When the y-plot has cusp singularities, planar facets may appear; such a surface can have portions that are smoothly curved or portions that are flat and these portions are separated by edges or corners where derivatives are discontinuous. For surfaces with the two-dimensional y-plot shown in Fig. 14.6, certain incli- nations will be unstable and will be replaced by other inclinations (facets), even though this increases the total surface area. Whether a certain inclination is un- stable and prone to facet into other inclinations can be determined by a simple geometrical construction using the y-plot [8]. The surface will consist of two differ- ent types of facets, as in Fig. 14.7a. The energy of such a structure per unit area projected on the macroscopically flat surface, "ifac, is where yi is the surface energy of the ith-type facet and fi is the fraction of the projected area contributed by facets of type i. If fi is the unit normal to the flat surface and fi~, &, and fi3 are unit vectors normal to type-1 facets, type-2 facets, and along the facet intersections, respectively, as in Fig. 14.7a, fi = fifiI+ f2h2 (14.36) 348 CHAPTER 14 SURFACE EVOLUTION DUE TO CAPILLARY FORCES Figure 14.7: Morphology of an initially smooth surface that has reduced its energy by faceting. (a) Morphology if two facet inclinations are stable. (b) Morphology if three facet inclinations are stable. If a set of vectors, St, reciprocal to the vectors hi, is introduced so that (14.37) a, x fi3 '* a, x a1 ??I x a, 6; = 6; = n2 = a1. (a, x a,) (a, x a,) a1. (752 x fi3) so that 6; . fij = dij, Eq. 14.35 can be rewritten where Z= yl6p + 726; has the properties Whether faceting will occur can now be settled by a simple geometrical con- struction using the y-plot shown in Fig. 14.6. If the surface to be tested has the inclination fi and the inclinations corresponding to points B and C are chosen as the inclinations for the i = 1 and i = 2 facets, Zmust appear as shown in Fig. 14.6 in order to be consistent with Eq. 14.39. The energy of the surface of average incli- nation fi that is faceted into inclinations corresponding to points B and C is then, according to Eq. 14.38, the projection of Zon a. This energy is smaller than the energy of the nonfaceted interface (indicated by the outer envelope of the y-plot) and the surface will prefer to be faceted. It may also be seen that the energies of all other surfaces with inclinations varying between those at B and C will fall on the dashed circle. All of these surfaces will therefore be faceted. On the other hand, a similar construction shows that all surfaces with inclinations between those at C and D will be stable against faceting into the inclinations at C and D. Points such as those at B and C where the dashed circle is tangent to the y-plot therefore delineate the ranges of inclination between which the surface is either faceted or nonfaceted. The construction indicated in Fig. 14.6 is readily generalized to three dimensions: three facet planes could then be present, as in Fig. 14.7b, and c'then terminates at the point of intersection of three planes rather than two lines. Figure 14.8 shows a three-dimensional y-plot comprised of eight equivalent spher- ical surface regions. The shape of this y-plot is consistent with all surfaces repre- sented by the plot being composed of various mixtures of the three types of facets, 14 2 ANISOTROPIC SURFACES 349 Figure 14.8: The y-plot for a material with a Wulff shape corresponding to a cube when y[100] = y[010] = y[001]. It consists of portions of eight identical spheres, shown here in cutaway view. These spheres share a common point at the origin. but each has a diametrically opposed point directed toward the eight (111) directions. corresponding to the y[lOO], y[OlO], and y[OOl] vectors shown.4 Any interface cor- responding to a vector lying on a groove at the intersection of two spheres, such as yhvl will consist of two types of facets. corresponding to a pair of the vectors y[100], y[OlO], or y[OOl]. Any interface corresponding to a vector going to a spherical re- gion of the plot. such as ypyrl will consist of three types of facets. corresponding to y[lOO], y[OlO], and y[OOl]. Figure 14.9 shows a three-grain junction on the surface of polycrystalline A1203 after high-temperature annealing. Each grain surface has a different inclination Fi ure 14.9: pofjfcrystal. From J M Dynys [9]. 41n Fig. 14.6. which holds in two dimensions, the energies of all faceted surfaces with inclinations between B and C fall on the dashed tangent circle shown. In three dimensions, a comparable construction would show that faceting would occur on three facet planes, such as in Fig. 14.7b, and that the counterpart to the tangent circle would be a tangent sphere. Surface morphology of three faceted grains in an annealed alumina 350 CHAPTER 14: SURFACE EVOLUTION DUE TO CAPILLARY FORCES and exhibits a different facet morphology. Grain 1 remains flat, grain 2 shows two facet inclinations, and grain 3 exhibits three facet inclinations. Other constructions employing the y-plot are reviewed in Section (2.3.1. These include the reciprocal y-plot, which is also useful in treating the faceting problem above, and the Wulff construction, which is used to find the shape (Wulff shape) of a body of fixed volume that possesses minimum total surface energy. 14.2.2 The kinetics of the morphological evolution of anisotropic interfaces can be devel- oped as an extension of the isotropic case. Isotropic interface evolution originates from a diffusion potential proportional to the local geometric curvature (mean cur- vature) multiplied by the surface energy per unit area. The local geometric curva- ture is the change of interface area, 6A, with the addition of volume 6V, K = 6A/6V (see Section C.2.1). Therefore, the local energy increase due to the addition of an atom of volume R is Ryn. The anisotropic analog to the isotropic energy increase is the weighted mean curvature K-, = 6(yA)/6V, developed by J. Taylor [lo]. In the anisotropic case, the diffusion potential is increased by, RK-,, the local energy increase per adatom. It can be shown that K-, = VJsurf f(fi) (14.40) where fis the capillarity vector and Vsurf is the surface divergence operator, similar to the surface gradient introduced in Eq. 14.2.5 Two different types of derivatives are involved in this expression for &,-the first produces {from a derivative in y-space as seen in Eq. C.20; the second derivative used to obtain the divergence is taken along the evolving interface. Rate of Morphological Interface Evolution Evolution by Surface Diffusion and by Vapor Transpott. Although calculation of the morphological evolution for particular cases can become tedious, the kinetic equa- tions are straightforward extensions of the isotropic case [ll]. For the movement of an anisotropic surface by surface diffusion, the normal interface velocity is an extension of Eq. 14.6 which holds for the isotropic case; for the anisotropic case, (14.41) If the surface diffusivity is anisotropic, its surface derivatives must appear as well. For movement by vapor transport of an anisotropic interface that is exposed to a vapor in equilibrium with a very large particle6, the normal interface velocity is an extension of Eq. 14.17: KR~P~~(K = 0) kT 6-l vn = - (14.42) The expression for weighted mean curvature for any surface in local equilibrium is simplified when the Wulff shape is completely faceted [lo, 121. In this case, 5The capillarity vector $ and the weighted mean curvature ny are discussed in more detail in Section C.3.2. 6Weighted mean curvature, which is uniform on a Wulff shape, goes to zero in the limit of large body volumes. 14.2: ANISOTROPIC SURFACES 351 tractable expressions and simulations can be produced for morphological evolution by surface diffusion and vapor transport [13]. However, these models do not include edge and corner energies because they are inadmissible in the Wulff construction- nor do they include nucleation barriers for ledge and step creation, ledge-ledge interactions, and elastic effects associated with edges and corners. Growth Rate for Inclination-Dependent Interface Velocity For a crystalline parti- cle growing from a supersaturated solution, the surface velocity often depends on atomic attachment kinetics. Attachment kinetics depends on local surface struc- ture, which in turn depends on the surface inclination, A, with respect to the crystal frame. In limiting cases, surface velocity is a function only of inclination; the inter- facial speed in the direction of A is given by w(A). The main aspects of a method for calculating the growth shapes for such cases when .(A) is known is described briefly in this section. Given an initial surface, F(t = 0), the surface morphology at some later time, t, can be computed from the growth law w(A) with a simple construction [14, 151. Let r(fl be the time that the growing interface reaches a position r'; therefore, the level set tconst = r(fl could be inverted to give the surface F(tcOnst). The surface normal must be in the direction of the gradient of 7; A = Vr/lVrl, where IVrI must be proportional to [w(A)]-'. Solving for the constant of proportionality, a, as a function of Vr, (14.43) wext(p3 is the homogeneous extension of the surface velocity w(A) from A on the unit sphere to gradients of arbitrary magnitude p" Vr [16]. The extended normal velocity, wext (p3, can be used to construct characteristics that specify the surface completely at some time t [14]. The characteristics are rays that emanate from each position on the initial surface ?(t = 0), given by ?(t) = r'(t = 0) + t V&=t(p3 (14.44) The surface normal A is constant along the characteristics, and therefore the surface velocity u(A) is constant as well (see Exercise 14.5). The characteristics, defined as (14.45) do not depend on the magnitude lVd. Therefore, the time-dependent morphology can be calculated directly from any initial surface r'(t = 0) and a normal velocity .(A) by the following procedure. First calculate ((A) for every point on the initial surface ?(t = 0), then construct rays equal to tf from each point. Using Eqs. 14.44 and 14.45, the surface positions at an arbitrary time t are ?(t) = ?(t = 0) + tf((a) (14.46) The method is illustrated with a simple example in two dimensions. Suppose that the surface has the symmetry of a square and w(k) = w(n1, n2) = w(cos8, sine) 352 CHAPTER 14: SURFACE EVOLUTION DUE TO CAPILLARY FORCES is given by w(h) = 1 + p(n; + cun;n; + n;) (14.47) where a: and ,B are constants. The velocity w(h) and its associated [(h) are illus- trated in Fig. 14.10 for particular values of Q and p.' Figure 14.10: Exaniples of ~(A)ii and <(ii) from Eq. 14.47 for ,B = 1/2 and LY = 4. (a) A polar plot of ~(ii)ii. The magnitude of the plot in each direction, ii = (cos 8, sin O), is the velocity in that direction. (b) <(ii) is plotted parametrically as a function of 8. The vector <(A) = f(8) is generally not in the direction of ii(8). However, t,he surface of the <(O)-plot at any point is always normal to fi(8), as shown in Eq. C.19, which although written for Gii) and y(A), also holds for <(?i) and ~(6). Figure 14.11 shows the shape evolution due to w(h) and its characteristics fin Eq. 14.47 for an initially circular particle. After very long times, the only remaining orientations on the growth shape are those that lie on the interior portion of the f-surface; therefore, the portion of the <-surface with the spinodes (the swallowtail- shaped region) is removed. For morphological evolution during dissolution of a crystal (or disappearance of voids in a crystalline matrix), the same characteristic construction applies, but the sense of the surface normal is switched compared to Fig. 14.11. An example of dissolution is illustrated in Fig. 14.12. The asymptotic growth shapes (Fig. 14.11) are composed of inclinations asso- ciated with the slowest growth velocities, and the fastest inclinations grow out of existence by forming corners. On the contrary, for dissolution shapes (Fig. 14.12), the inclinations associated with the fastest dissolution remain and the slow-speed inclinations disappear into the corners. The asymptotic growth shape is the in- 7((i?L) is related to v(7i) in the same way that the capillarity vector, (, is related to y(6) and is constructed in the sanie way. The Wulff construction applied to v(A) produces the asymptotic growth shape. This and other relations between the Wulff construction and the common-tangent constriiction for phase equilibria are discussed by Cahn and Carter [16]. 14 2 ANISOTROPIC SURFACES 353 Shape at t = 0 Shape at t = t, Shape at t = 0 Shape,at t2> t, Figure 14.11: Development of growth shape for an initially circular particle for the v(fi) illustrated in Fig. 14.10. Rays tf(fi) are drawn from each associated inclination on the initial surface. Fastest-growing inclinations accumulate at 45" and its equivalents and form corners. Figure 14.12: Development of di_ssolution shape for initially circular particle for the w(h) illustrated in Fig. 14.10. Rays tC(-h) are drawn from each associated inclination on the initial surface. The slowest-growing inclinations accumulate at 90" and its equivalents and form corners. terior of the f-surface and the asymptotic dissolution shape is composed of those inclinations between the cusps on the swallowtail-shaped region on the f-surface. Bibliography 1. W.W. Mullins. Solid surface morphologies governed by capillarity. In N.A. Gjostein, editor, Metal Surfaces: Structure, Energetics and Kinetics, pages 17-66, Metals Park, OH, 1962. American Society for Metals. 2. W.W. Mullins. Theory of thermal grooving. J. App2. Phys., 28(3):333-339, 1957. 354 CHAPTER 14 SURFACE EVOLUTION DUE TO CAPILLARY FORCES 3. F.A. Nichols and W.W. Mullins. Surface- (interface-) and volume-diffusion contribu- tions to morphological changes driven by capillarity. Trans. AIME, 233( 10): 1840-1847, 1965. 4. W.W. Mullins. Grain boundary grooving by volume diffusion. Truns. AIME, 5. W.M. Robertson. Grain-boundary grooving by surface diffusion for finite surface slopes. J. Appl. Phys., 42(1):463-467, 1971. 6. M.E. Keeffe, C.C. Umbach, and J.M. Blakely. Surface self-diffusion on Si from the evolution of periodic atomic step arrays. J. Phys. Chem. Solids, 55:965-973, 1994. 7. J.W.S. Rayleigh. On the instability of jets. Proc. London Math. SOC., 1:4-13, 1878. Also in Rayleigh’s Collected Scientific Papers and Theory of Sound, Vol. I, Dover, New York. 8. C. Herring. Some theorems on the free energies of crystal surfaces. Phys. Rev., 9. J.M. Dynys. Sintering Mechanisms and Surface Diffusion for Aluminum Oxide. PhD thesis, Department of Materials Science and Engineering, Massachusetts Institute of Technology, 1982. 10. J.E. Taylor. Overview No. 98. 11-Mean curvature and weighted mean curvature. Acta Metall., 40(7):1475-1485, 1992. 11. J.E. Taylor, C.A. Handwerker, and J.W. Cahn. Geometric models of crystal growth. Acta Metall., 40(5):1443-1474, 1992. 12. A. Roosen and J.E. Taylor. Modeling crystal growth in a diffusion field using fully- faceted interfaces. J. Computational Phys., 114(1):113-128, 1994. 13. W.C. Carter, A.R. Roosen, J.W. Cahn, and J.E. Taylor. Shape evolution by surface diffusion and surface attachment limited kinetics on completely facetted surfaces. Acta Metall., 43(12):4309-4323, 1995. 14. J.E. Taylor, J.W. Cahn, and C.A. Handwerker. Overview No. 98. I-Geometric models of crystal growth. Acta Metall., 40(7):1443-1474, 1992. 15. W.C. Carter and C.A. Handwerker. Morphology of grain growth in response to diffu- sion induced elastic stresses: Cubic systems. Acta Metall., 41(5):1633-1642, 1993. 16. J.W. Cahn and W.C. Carter. Crystal shapes and phase equilibria: A common math- ematical basis. Metall. Trans., 27A(6):1431-1440, 1996. 17. J.W. Cahn, J.E. Taylor, and C.A. Handwerker. Evolving crystal forms: Frank’s characteristics revisited. In R.G. Chambers, J.E. Enderby, A. Keller, A.R. Lang, and J.W. Steeds, editors, Sir Charles Frank, OBE, FRS, An Eightieth Birthday Tribute, pages 88-118, New York, 1991. Adam Hilger. 2 18( 4) : 354-36 1 , 1960. 82 ( 1) : 8 7-93, 195 1. EXERCISES 14.1 Section 14.1.1 treated the smoothing of a sinusoidally roughened surface by means of surface diffusion to obtain Eq. 14.13. Show that the corresponding expression for smoothing by means of crystal bulk diffusion, as in Fig. 3.7, is where w = 27r/X. 0 Use the same small-slope approximations as in Section 14.1.1. (14.48) [...]... and Voorhees studied Sn-rich and Pb-rich solid phases in Pb-Sn eutectic liquid over the range 4 = 0. 6-0 .9 and presented data in support of the volume-fraction effect, as shown in Fig 15 .9 [7] Voorhees’s experimental study of low-volume-fraction-solid liquid+solid Pb-Sn mixtures carried out under microgravity conditions during a space shuttle flight enabled a wider range of solid-phase volume fractions... presence of fine precipitates In single-phase polycrystalline materials, larger grains tend to grow at the expense of the smaller grains as the the total grain-boundary free energy decreases This process is also competitive and often produces unwanted coarse-grained structures 15.1 COARSENING OF A DISTRIBUTION OF PARTICLES 15.1.1 Classical Mean-Field Theory of Coarsening In 196 1, the classical theory of. .. studied without significant influence of buoyancy (flotation and sedimentation) effects [13] The rate of approach to the steady-state particle-size distribution in 0. 1-0 .2 volume-fraction 2 4 - G A a v - o Sn-rich particles a Pb-rich particles Theory - a a o Figure 15.8: Rate constant KD(C$) particle coarsening vs volume fraction of particles, for expressed as a ratio of the rate constant K ~ ( q 5at volume... begins with N = 2, because one-sided grains do not exist Therefore, using Eq 15.38 in Eq 15.37 yields (15. 39) Here, A N / A ,the fraction of the total area occupied by N-sided grains, is constant because of self-similarity It is often convenient to express the growth law given by Eq 15. 39 in terms of a linear grain-size dimension If R,,, is the effective root-mean-square radius of the roughly equiaxed grains,... local velocities of the boundaries are integrated along the perimeter of a grain, an expression for the rate of change of area of the interior grain is obtained: dA dt -= - lB u ds = - (15. 29) Assuming that MB and y and do not depend on orientation and curvature, (15.30) = -2 .irM~y In the relationships above, 0 is the angle that the boundary normal makes with a fixed direction in the plane of the specimen... broader than the classical theory predicts and particles are often larger than the predicted cut-off size, 1.5(R) A study of coarsening in semisolid Pb-Sn alloys verified the t1I3-law kinetics predicted by the mean-field theory (see Fig 15.6) However, some aspects of the classical theory are not observed in Fig 15.6 Limitations of the classical mean-field theory are discussed in Section 15.1.2 To same scale... an elevated temperature composed of A and B atoms containing a distribution of spherical @-phaseparticles of pure B embedded in an A-rich matrix phase, a The concentration of B atoms in the vicinity of each @-phase particle has an equilibrium value that increases with decreasing particle radius, as demonstrated in Fig 15.1 Because of concentration differences, a flux of B atoms from smaller to larger... studies of particle coarsening [5] First, the mean particle size is studied as a function of time For volume-diffusion-limited coarsening, the t1I3-law corresponding to Eq 15.18 is generally observed, in agreement with theoretical predictions The second measured characteristic is the particle-size distribution, including its time dependence The experimental time-dependent evolution of particle-size distributions... constant The point of constant slope must therefore project as a straight line in the zt-plane Now, d y = ($)tdx+ and 2 = ( $)t ($) g)x2 +( (14.74) dt X = -hk + qh-dt dx (14.75) Because all the terms on the right side of Eq 14.75 are constant, i t s projection in the xy-plane must therefore be a straight line Y tl Figure 14. 19: Stepped-surface evolution during crystal growth 14.6 Prove that all of the results... > Because + -d 2 h 6x2 = Aw2 sinwx (14.52) the boundary conditions above and the diffusion equation are satisfied by a solution of the form of Eq 14. 49 with a1 = p i and a2 = ySCLAw2 Therefore, using Eqs 14. 49 and 14.50, R *DXLySRAw3 sinwx = - * D X L y S h W 3 (14.53) kTf kTf Finally, because ( l / h ) ( d h / d t )= ( l / A ) ( d A / d t ) , Eq 14.53 may be integrated t o pro- d -h - at duce Eq . CAPILLARY FORCES theory s essential elements were worked out earlier by Greenwood [3]. This theory is often referred to as the LSW theory of particle coarsening and sometimes as the GLS W. grain-boundary free energy decreases. This process is also competitive and often produces unwanted coarse-grained structures. 15.1 COARSENING OF A DISTRIBUTION OF PARTICLES 15.1.1 In 196 1,. are flat and these portions are separated by edges or corners where derivatives are discontinuous. For surfaces with the two-dimensional y-plot shown in Fig. 14.6, certain incli- nations will