Lecture 7 Grain boundary grooving

Transcription

1 Lecture 7 Grain oundary grooving The phenomenon. A polihed polycrytal ha a flat urface. At room temperature, the urface remain flat for a long time. At an elevated temperature atom move. The urface grow groove along triple junction, where the urface meet grain oundarie. The groove reveal the grain oundarie in the microcope. Atom may move in many way. They may diffue in the lattice, diffue on the urface, or evaporate into the vapor phae. Here we will only conider urface diffuion. Atom diffue on the urface away from the triple junction, making a dent along the junction, and piling two ump neary. The proce conerve the total ma. The proce of grooving wa modeled y Mullin (1957). polihed urface Ψ grain-oundary heating The driver and the paenger. Let γ e the urface energy per unit area, and γ e the grain oundary energy per unit area. The free energy of the ytem i the um of the urface energy and the grain oundary energy: G γ A + γ A. Feruary 21,

2 A the groove grow, the grain oundary area decreae, ut the urface area increae. The net free energy mut decreae. We may ay that the grain oundary i the driver, and the urface the paenger. Local equilirium and the dihedral angle. Atom at the triple junction adjut their poition rapidly compared to diffuion on the urface. During the motion of the urface, the triple junction maintain local equilirium. Local equilirium at the triple junction implie that the urface tenion alance one another, giving the dihedral angle, Ψ : co Ψ 2 γ 2γ. If you like, you may think the grain oundary tenion a a force, pulling the urface down. Thi relation can e derived a follow. Let the triple junction move up y a ditance δ a. The grain oundary area increae y a δ, and the urface area decreae y co( / 2) δa Conequently, the free energy change aociated with the motion of the triple junction i Ψ δg + γ δa 2 γ co δa 2 Ψ In equilirium, δ G 0 for any δ a. Thi lead to γ 2 γ co Ψ. Differential geometry of a curve in a plane. The model i for a mall groove, it depth and width eing much maller than the grain ize, o that the two grain are taken to e infinitely large. The prolem ecome two-dimenional. In the cro-ection normal to the triple junction, the urface i a curve, the grain oundary a traight line, and the triple junction a point. A curve in a plane can e decried in many way. For example, we can precrie the coordinate of a point on the curve, x and y, a function of the curve length. The function x () and y ( ) decrie a et of point that trace the curve. Feruary 21,

3 Conider a tangent line at a point on the urface. Let θ e the angle etween the x-axi to the tangent line. Thu, co θ, inθ. We can locally fit the curve to a circle. Let the radiu of the circle e R. Conider two neary point on the curve, eparated y an arc length d. The tangent line rotate y d θ, o that d Rdθ. The curvature K i defined a the invere of the radiu. We write K θ. We have adopted a ign convention. The olid i eneath the urface. The curvature i poitive when the olid i convex. K +1 / R for a cylindrical olid of radiu R, and K 1 / R for a cylindrical hole of radiu R. When the urface evolve with time, we repreent the family of curve y a function of two variale: y ( x, t) i the height of the urface at location x and time t. At a given time, the curve ha the length, the tangent angle, and the curvature. In the aove expreion, the differential dy / dx ecome the partial differential y /, with time fixed. Atomic flux. The atomic flux i proportional to the gradient of the curvature: K Ω J B. The contant coefficient B i given y B Ωγ D δ kt. For a given urface hape, thi / expreion calculate the atomic flux. To cloe the prolem, we need an expreion of the change in the urface height due to atom relocation on the urface. Ma conervation. Conider the motion of a urface element dx. When the time goe from t to t + t, J ( x t) t numer of atom flow into the element,, J ( x dx t) t +, numer of Feruary 21,

4 atom flow out of the element, and the urface height change from ( x t) conervation require that y, to y( x, t + t). Ma J Ω. Partial differential equation. Collecting the aove equation, we otain a et of partial differential equation to evolve the four unknown function y ( x, t), ( x,t) K (, ) ( ) θ, x t, J x, t : θ K K ΩJ Ω J tanθ,, B,. coθ coθ The initial and the oundary condition. The urface i initially flat: yx,0 ( ) 0. The dihedral angle fixe the lope of the urface at the triple junction: θ ( 0, t) γ 2γ in. At the triple junction, we aume that no ma diffue into or out of the grain oundary. Conequently, at the triple junction the atomic flux on the urface vanihe: ( 0, t) 0 J. The urface remote from the groove remain flat at all time Self-imilar profile. The initial geometry ha no length cale. Inpecting the partial differential equation, we note that the time and the moility et a length cale, ( Bt) 1/. Conequently, the groove grow with a elf-imilar profile. Define the dimenionle coordinate: X ( Bt) 1/ x, Y ( Bt) 1/ y. Decrie the groove profile y a function Y(X). Note that Feruary 21, 2009

5 ( x, t) 1/ 1 1/ 3 / dy [( Bt) Y ( X )] B t Y X y Similarly, make the curvature and the flux dimenionle y multiplying K y ( ), and J y ( / B) 1/ 2 Ω t equation ecome. We will till ue K and J to denote the dimenionle quantitie. The governing Bt 1/ dy tan θ, dθ K, co θ dk J, co θ dj 1 ( Y X tan θ ) Thi i a et of four ordinary differential equation for the function Y ( X ), ( X ) The oundary condition at the triple junction are ( 0) / 2, J ( 0) 0 in θ γ γ. K ( ) ( X ) θ, X, J. Phyically it i evident that θ, K, J hould all vanih at X. The normalized prolem ha only one parameter, γ / γ. The oundary value prolem i olved numerically y a hooting method. We ue the triple junction, X 0, a the initial point. Precried at thi point are the lope θ() 0 and the flux J() 0. We gue value for Y(0) and K(0), and then integrate the ordinary differential equation uing the Runge-Kutta method to a large value of X. The goal i to gue uch value of Y(0) and K(0) that everything vanihe at X. Thi i a nonlinear algeraic prolem for Y(0) and K(0). Small lope approximation. The aove PDE i nonlinear, and wa difficult to olve in Mullin time. He decided to linearize the prolem on the ai of the following phyical conideration. Typically the urface tenion i larger than the grain oundary tenion, o that the grooving urface ha mall lope. The partial differential equation ecome y B Thi equation govern the urface profile y (x,t). Feruary 21,

6 A tated efore, the ize of the groove cale with the time a ( Bt) 1/. The force that drive the groove enter the prolem through the lope at the triple junction, γ / γ. Becaue the governing equation i linear, the groove depth d (i.e., the ditance from the groove root to the plane of the initial flat urface) mut e linear in γ / γ. Mullin' analyi give d ( γ / )( ) 1/ γ 0.39 Bt. The pacing etween the peak of the two ump, w, ha the ame time caling, ut i independent of γ / γ under the mall lope implification. Mullin' analyi give w.6( Bt) 1/. A full nonlinear olution y Sun and Suo confirm that Mullin reult are accurate for practical range of the value of γ / γ. Experiment and extenion. The dihedral angle, the groove width, and the groove depth can e meaured experimentally. From uch meaurement one can deduce the comination γ / γ and M γ. The aolute value of γ and γ have to e determined y ome other experiment. See Toga and Nikolopoulo (199) for a thermal grooving experiment. At an interection etween the free urface and a three-grain junction. The urface form a pit at the point of emergence of the three-grain junction. The urface profile i till elfimilar, all length following the ame time caling a aove. Genin et al. (1992) analyzed the prolem, and found that the pit depth i greater than the groove depth, ut within a factor of 3 for all the configuration conidered y them. Grain-oundary grooving and pitting may reak a polycrytalline thin film on a utrate. If the grain ize i much larger than the film thickne, the aove analyi etimate the time needed for a three-grain junction to pit through the film thickne. However, if the grain ize i Feruary 21,

7 comparale to the film thickne, ma tranported from the groove pile up on the grain, topping the grooving proce. Srolovitz and Safran (1986) and Miller et al. (1990) demontrated that a critical ratio of grain ize and film thickne exit, elow which the grain reach an equilirium configuration without reaking the film. Reference Mullin, W.W. (1957) Theory of Thermal Grooving, J. Appl. Phy. 28, Miller, K.T. and Lange, F.F., and Marhall, D.B. (1990) The intaility of polycrytalline thin film: experiment and theory. J. Mater. Re. 5, Genin, F.Y., Mullin, W.W., and Wynlatt, P. (1992) Capillary intailitie in thin film: a model of thermal pitting at grain oundary vertice. Acta. Metall. 0, Toga, A. and Nikolopoulo, P. (199) Groove angle and urface ma tranport in polycrytalline alumina. J. Am. Ceram. Soc. 77, D.J. Struik, Lecture on Claical Differential Geometry, reprint y Dover. Thi ook decrie curve and urface. Srolovitz, D.J. and Safran, S.A. (1986) Capillary intailitie in thin film. J. Appl. Phy. 60, B. Sun and Z. Suo, A finite element method for imulating interface motion, part II: large hape change due to urface diffuion. Acta Materialia, 5, (1997). Feruary 21,