Thermodynamic and Kinetic Morphological Wavelengths 14.2 ANISOTROPIC SURFACES 14.2.1 An anisotropic surface's energy per unit area, yA, depends on its inclination, A.. For isotropic s
Trang 1Therefore, 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,
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,
where the amplification factor l/rv = (Bv/Rz)[l - ( ~ T R , / 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
Trang 2346 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,,, = f i A c r i t
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
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
Trang 3C D
Figure 14.6:
be faceted into inclinations corresponding to points Construction for testing whether an B interface and C y-plot of inclination same as in A will prefer Fig C.4a 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 f i is the fraction of the
projected area contributed by facets of type i If fi is the unit normal to the flat
surface and f i ~ , &, and f i 3 are unit vectors normal to type-1 facets, type-2 facets, and along the facet intersections, respectively, as in Fig 14.7a,
Trang 4348 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 f i 3 '* a, x a1 6; = ??I x a,
a1 (a, x a,) (a, x a,) a1 (752 x f i 3 )
so that 6; f i j = 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,
Trang 5Figure 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 t h a t faceting would occur on three facet planes, such as in Fig 14.7b,
and t h a t the counterpart to the tangent circle would be a tangent sphere
Surface morphology of three faceted grains in an annealed alumina
Trang 6350 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
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,
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
Trang 7tractable 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 V r ,
(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" V r [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
and 14.45, the surface positions at an arbitrary time t are
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)
Trang 8352 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 ) i i and <(ii) from Eq 14.47 for ,B = 1/2 and LY = 4 (a)
A polar plot of ~ ( i i ) i i 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 f i n
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]
Trang 9Shape at t = 0 Shape at t = t , Shape at t = 0 Shape,at t 2 > t ,
Figure 14.11: Development of growth shape for an initially circular particle for the v ( f i )
illustrated in Fig 14.10 Rays t f ( f i ) 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
Trang 10354 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, A n 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)
Trang 1114.2
Assume that self-diffusion occurs by a vacancy mechanism and take
Eq 13.3 as the volume diffusion equation
Assume that the diffusion field is in a quasi-steady state and that local equilibrium is maintained at the surface and in the volume at a long dis- tance from the surface, where pv = 0 and p~ has the value characteristic
of a flat surface
Note that one of the solutions to Laplace’s equation is
@ A = p~ - pv = a1 + a2 sin(wz) ewy (14.49)
Solution The height of the surface is given by h = Asin(wx) and the flux equation
is given by Eq 13.3 Therefore,
(14.50)
To evaluate Eq 14.50 we must obtain an expression for @ A by solving the steady-state diffusion equation,
(14.51)
in the volume, subject t o appropriate boundary conditions
y = 0), pv = 0, and from Eq 3.76,
and p~ = p i , so that @ A = p> Because
At the surface (i,e., at
= pz + ySRtcF In the deep interior, pv = 0
(a) At which specific location(s) will the shape of the plate first begin to
change? Explain your reasoning in terms of driving forces for diffusion
Figure 14.13: Portion of infinite plate of thickness h containing a hole of radius R
Trang 12356 CHAPTER 14: SURFACE EVOLUTION DUE TO CAPILLARY FORCES
(b) What role do you expect the initial value of the ratio hlR t o have in de- termining whether the hole in the plate will either shrink and disappear spontaneously or grow spontaneously? Explain your reasoning
Solution
(a) Equation 3.76 demonstrates that the diffusion potential of an atom a t a surface depends on the local surface curvature Consistent with the convention that a convex spherical surface has a curvature +2/R (see Section 14.1), the curvature
of the surface of the flat plate is zero and the initial curvature of the cylindrical surface inside the hole is K I = 1/m - 1/R = -1/R The highest curvature is
at the "rim" o f the hole where the hole intersects the flat surface; the curvature here is K' = 1 / ~ - 1/R + +m Therefore, there is a large diffusion-potential gradient for atoms at the rim of the hole The first shape change would therefore
be rounding o f the sharp edges of the hole The driving force for diffusion would
be reduction of the total surface area, and this would commence by movement of atoms away from the rim o f the hole toward both the flat plate surface and the cylindrical surface of the hole The interior surface of the hole will continue t o evolve at a slower rate, as described in part (6)
(b) Recall that curved interfaces can reduce their area by migration toward the center
o f curvature o f the higher principal curvature Consider two limiting cases, depicted
in Fig 14.14 Case 1, R >> h: assuming complete rounding of the sharp hole edges, as in Fig 14.14a, the curvature o f the rounded hole will be K I = l / ( h / 2 ) -
1/R x 2/h, and the surface tension force will cause the hole t o increase in diameter Case 2 , h >> R: as in Fig 14.146, the hole interior has curvature
K' = l / ( h / 2 ) - 1/R % - l / R , and the surface-tension force acts t o reduce the diameter o f the hole One can make a simple calculation t o investigate this problem further Assume that the hole o f diameter 2R lies somewhere in a fixed area A o f the plate Then the initial total surface area o f the plate and hole (with sharp corners) will be
depicted in cross section In Case 1, the hole expands; in Case 2, it will fill in
Limiting cases of the evolving shape of a plate with a cylindrical hole
14.3 Consider a pillbox-shaped grain ernheddecl in an otherwiw single-crystal sheet (not shown) of thickness h as in Fig 14.15 Such a grain will shrink and
Trang 13r
Figure 14.15: Pillbox-shaped grain in a single-crystal sheet of thickness h
eventually disappear However, if grain boundary grooves develop on the two sheet surfaces and pin the boundary so that it is essentially stationary, the boundary can equilibrate locally and develop a minimum-energy form similar
to that of a soap film held between two rigid circular wires
Show that such an equilibrated boundary would have the form
What happens to the grain when R, decreases to 3/4h? Note: ap =
cosh(P/2) has no solutions when a < 0.75
Solution
(a) One way t o solve this exercise is t o show that the mean curvature of the boundary
is zero when Eq 14.54 is satisfied by inserting ~ ( z ) into Eq 14.29 There is then
no pressure anywhere on the pinned boundary urging it t o change its shape, and
it possesses the shape of minimum energy However, direct consideration of the two curvatures is instructive Figure 14.16 shows a convenient choice for the two orthogonal planes which will be used t o find the mean curvature by the method illustrated in Fig C.2 Consider the curvature at a general point on the boundary such as P in Figs 14.15 and 14.16 The first plane, Plane 1, selected is the constant4 plane in Fig 14.15, which lies in the plane of the paper in Fig 14.16
The second plane, Plane 2 (which must be orthogonal t o the first and intersect it along fi) is indicated by its trace, AB, in Fig 14.16 Using Eq C.5, the curvature
of the boundary intersection with Plane 1 is
Trang 14358 CHAPTER 14: SURFACE EVOLUTION DUE TO CAPILLARY FORCES
If f is the force per unit length, the work required t o expand the boundary radially
by dR, is dW = dB = f 2.rrRgdR,, where d 0 is the change in the energy of the
boundary region lying between z = 0 and z = h/2 So
f= 1 dB 2xR, dR, (14.59)
Integrating over this boundary region yields
Trang 15Using Eqs 14.60 and 14.61 and defining p = h/R,,
~~h 4Rg cosh(2lp) - (2//3) sinh(2/P)
The value of P in Eq 14.65 must satisfy Eq 14.64 The smaller solution of
Eq 14.64 for P gives the smaller boundary energy 9, and therefore putting p = 1.1787 into Eq 14.65 gives the force per unit length acting on the groove as
B
Ftot = 2?rRgF = 2.irRg(0.5296yB) = 3.3273Rgy
14.4 Consider two faces of a faceted crystal advancing at different velocities during crystal growth as in Fig 14.17 The growth rates of facets 1 and 2 are GI and
(a) Find the condition on the velocities under which facet 2 will grow at the
(b) Find the corresponding condition under which facet 1 will grow at the expense of facet 1
expense of facet 2
Figure 14.17: Two facets on a crystal growing at different velocities
Trang 16CHAPTER 14: SURFACE EVOLUTION DUE TO CAPILLARY FORCES
360
14.5
Solution In general, during unit time the faceted interface advances as shown in Fig 14.17 In the situation shown, facets 1 and 2 will both grow with time However, a few simple constructions show that if w2 < w1 cos8, facet 1 will shrink and facet 2 will grow On the other hand, if w1 < w ~ c o s 8 , facet 2 will shrink and facet 1 will grow
Consider the growth from the vapor of a crystal possessing vicinal faces The shape that the crystal assumes depends upon the number and rate at which ledges move across its surfaces When one of the surfaces consists of a series of straight parallel ledges of height, h, running parallel to the z axis, the ledges move parallel to z, causing crystal growth along y Let k = k(z,t) be the
local ledge density at the point z Also, let q = q(x! t ) be the ledge flux equal
to the number of ledges passing the point z per unit time The local slope
of the surface is then (ay/ax), = -hk and the rate of crystal growth along
y is (dyldt), = qh Assume that q = q ( k ) , which is often the case because the flux depends upon both the ledge density and the ledge velocity, which
is, itself, dependent upon the ledge density The rate of growth will then be a function of the inclination of the surface [i.e., 'u = w(fi)], as in Section 14.2.2
(a) Show that the moving ledges obey the equation of continuity,
a characteristic, is shown in Fig 14.18
Figure 14.18: (a) Various polar plots of w(A) for interfaces whose growth velocities,
v(A), are functions of their inclinations, n (b) Shapes at increasing times of a body that was initially spherical and whose w(A)-plot is indicated by A in (a) Growth characteristics
(outward rays) are shown which delineate the paths taken by points on the interface where the inclination remains constant The tangent constructions along the characteristic indicated
by B illustrate this constancy of inclination From Cahn et al (171
Trang 17(c) The evolution of the surface shape as a function of x and t is shown in Fig 14.19
A t a point where the slope is constant, k is constant and, since q = q(lc), q is also constant Because dq/dk = f(k), dq/dk must also be constant Therefore, according t o Eq 14.73, dx/dt = dq/dk = constant The point of constant slope must therefore project as a straight line in the zt-plane Now,
Y
t l
Figure 14.19: Stepped-surface evolution during crystal growth
14.6 Prove that all of the results obtained in Exercise 14.5 for crystal growth (including the basic differential equation, its solution, and the expression for the sink efficiency) also hold for crystal evaporation
Trang 18362 CHAPTER 14: SURFACE EVOLUTION DUE TO CAPILLARY FORCES
Solution The basic differential equation is in a form that holds for both growth and evaporation The diffusion boundary conditions are the same for the two cases, and therefore the solution is equally applicable Finally, the same expression for the efficiency of the surface is obtained (Note that the two changes o f sign encountered in its derivation cancel.)
Trang 19COARSENING OF MICROSTRUCTURES DUE
T O CAPILLARY FORCES
In Chapter 14 we focused on capillarity-driven processes that primarily alter the shape of a body Two types of changes were considered: those driven by reduction of surface area, and those driven by altering the inclination of surfaces In this chapter, changes in the length scales that characterize the microstructure are treated Coarsening is an increase in characteristic length scale during microstructural evolution Total interfacial energy reduction provides the driving force for coarsen- ing of a particle distribution Coarsening plays an important role in microstructural evolution in two principal ways When a particulate phase is embedded in a matrix
of a second phase, flux from smaller to larger particles causes the average particle size to increase as the total heterophase interfacial energy decreases The parti- cles compete for solute and the larger particles have the advantage This process degrades many material properties, depending on the 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
In 1961, the classical theory of particle coarsening was developed at about the same time, but independently, by Lifshitz and Slyozov [l] and Wagner [2] Most of the
Kinetics of Materials By Robert W Balluffi, Samuel M Allen, and W Craig Carter 363
Classical Mean-Field Theory of Coarsening
*
Copyright @ 2005 John Wiley & Sons, Inc
Trang 20364 CHAPTER 15: COARSENING DUE TO 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 theory
Consider a binary system at an elevated temperature composed of A and B atoms containing a distribution of spherical @-phase particles 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 particles develops in the matrix This flux causes the smaller particles to shrink and the larger particles to grow
Figure 15.1: Effect of P-phase particle size on the concentration, Xeq, of component B
in the cy phase in equilibrium with a P-phase particle in a binary system at the temperature
T* assuming that P is pure B (a) Schematic free-energy curves for cy phase and three P-phase particles of different radii, R1 > R2 > R3 The free energies (per mole) of the particles increase with decreasing radius due to the contributions of the interfacial energy which increase as the ratio of interfacial area to volume increases (b) Corresponding phase diagram The concentration of B in the a phase in equilibrium with the &phase particles as determined by the common-tangent construction in ( a ) ; increases as R decreases, as shown
in an exaggerated fashion for clarity ( c ) Schematic concentration profiles in the cy matrix between the three P-phase particles