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Applications of self-defined arrays for pattern-forming alloy solidification
Computational Materials Science 11 Ž1998. 27–34
Applications of self-defined arrays for pattern-forming alloy solidification K. Williamson a
a,)
, A. Saigal
b
Department of Mechanical Engineering, Old Dominion UniÕersity, Norfolk, VA 23529, USA b Department of Mechanical Engineering, Tufts UniÕersity, Medford, MA 02155, USA Received 27 April 1997; accepted 9 May 1997
Abstract In this paper, we consider a pattern-forming solidification problem involving self-defined arrays ŽSDAs.. These SDAs originate from an iterated function system ŽIFS. and are imposed on the moving boundary so that microstructures evolve as prescribed shape attractors during iterated advance of a phase boundary. SDAs are generated from iterated application of a geometric rule requiring the pth iterate of a given array to be defined in terms of the Ž p y 1.th iterate. In this study, we use a geometric rule based on the Cantor middle-third fractal set to generate an SDA for an isolated dendrite microstructure. We apply this notion of an SDA to a phase change problem where fractal mass-flow creates the SDA geometry and results in a solidification microstructure. q 1998 Elsevier Science B.V.
1. Introduction Iterated function systems ŽIFSs. provide powerful tools to describe the geometries found in nature. These methods are related to iterated transform algorithms which are used to generate fractal images like the Julia set w1,2x. In a more general sense, these algorithms are related to dynamical systems which possess various kinds of attracting sets. For iterated function systems, the attracting set is an image which is encoded by a set of transformations which reveal the image after a prescribed number of iterations w3x. As a dynamical system, alloy solidification belongs
)
Corresponding author.
to a select group of processes which form complex self-organized patterns for a variety of restabilized equilibrium states w4,5x. The characteristic behavior of such systems allow them to undergo stability transitions which take them from quiescent states to dynamically restabilized states having time-dependent spatial patterns. Since the general characteristics of self-organized solidification patterns are known and have been induced experimentally in transparent model systems, the objective of this study is to find an IFS which defines the attracting set for these images. The majority of self-organized solidification experiments have been carried out in undercooled melts where experimental evidence confirms theory that the steady state growth velocity and the tip radius of the dendrite pattern are related to the bath undercooling. In fact, theoretical predictions for the operating point
0927-0256r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 7 - 0 2 5 6 Ž 9 7 . 0 0 0 4 9 - 9
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K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
Ži.e. unique tip radius and tip velocity. have matched experimental results for the restricted case of large undercoolings and a isolated dendrite geometry w6x. Unfortunately, the same models do not explain dendrites when there is a positive temperature gradient Žno undercooling. in the liquid ahead of the interface. The model presented in this paper describes an isolated dendrite growing into a positive temperature gradient with mass-flow as a shaping condition on the phase boundary. In this approach, we formulate the phase change problem using mass-flow as an iterated function system leading to an isolated dendrite shape attractor for the phase boundary.
2. Basic pattern-forming diffusion Dendritic solidification as we normally analyze it, looks at the dimensionless temperature field w7,8x: u Ž x , t . s T Ž x , t . y T` r Ž Lcy1 . ,
Ž 1.
where uŽ x, t . is the deviation of the local temperature T Ž x, t . from a colder temperature T` at infinity. L and c are the latent and specific heats, respectively. Here we ignore convection and write heat conservation in terms of the diffusion equation given as u t s Du x x ,
Ž 2.
where D is the diffusivity in the liquid. To simplify the picture, we ignore diffusion in the solid and focus here on the so called ‘one sided model’. At the phase boundary, we introduce the Stefan Žor continuity. condition which requires the latent heat of solidification to move the phase boundary. This is given by Vnormal s Du n ,
Ž 3.
where n is the unit normal to the interface. Supplementing the continuity condition at the phase boundary is a thermodynamic condition requiring an adjustment in melting temperature due to a curved surface. This Gibbs–Thompson condition for melting temperature, u s , of a curved portion of interface is given by us s u Ž x , t . y d 0 k ,
Ž 4.
where uŽ x, t . is the temperature of an adjacent flat portion of interface. The term, d 0 , is a capillarity term involving surface tension and k is the radius of curvature at the curved portion of the interface. If we model curvature as a small bump Žpositive curvature. on an otherwise flat surface, the bump sees accelerated heat diffusion and rapid phase change with respect to other parts of the interface since it has an advantage of being surrounded by colder liquid. This advantage is offset by a corresponding Gibbs– Thompson reduction in melting temperature. All pattern forming aspects of the phase change process is governed by this dynamical balance between diffusion tending to maximize the bump and capillarity effects tending to minimize it. This basic characteristic of destabilizing forces being dynamically balanced by stabilizing forces is the hallmark of other pattern-forming systems like Taylor–Couette flow and the Raleigh–Bernard convection cell w9x. The basic model outlined above demonstrates the essence of the Mullins–Sekerka planar instability and suggests a needle crystal shape solution if capillary forces fail to restrict diffusive tendency to maximize the arc length of the bump. Ivantsov’s solution w10x confirms the needle crystal geometry and shows that it grows as a paraboloid having an isothermal surface for constant velocity. This solution forms the basic foundation of our steady state approach and has been extended to include morphological stability, microscopic solvability and shape anisotropy theories which suggest length scales so that tip radius and tip velocity can be defined independently. The motivation for the current model is to relax the steady state constant velocity requirement and examine a shape solution for an isolated dendrite as a shape attractor emerging from mass flow formulated as an IFS. Our feeling is that the shape solution from an unsteady heat flow model incorporating mass-flow as an IFS should approach the steady-state shape as the limiting attractor. This approach suggests that steady state analyses provide solutions for a type of instantaneous shape transformation which is rarely observed in real systems. More common is the iterated shape selection. This type of iterated shape selection is the familiar growth pattern seen in ice daggers on tree branches. These daggers increase length iteratively and form shape under the influence of gravity. One length serves as the starting point for
K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
29
the next incremental length increase. Here the essential details lie in the iterative sequence Žor IFS. leading to the final shape and not in the shape itself. It is this notion of iterated shape selection which motivates our current model.
3. A fractal mass flow boundary condition Temkin w10x showed that a phase change interface is made up of seven or eight atomic layers and follows a ‘mountain range’ structure as it minimizes free energy during transformation from liquid to solid. An approximation of the mountain range structure is shown in Fig. 1. As indicated by Mandelbrot w11x and others during the past decade, mountain range structures do follow fractal scaling, therefore, imposing a fractal condition on mass-flow during alloy phase change could provide insight into the pattern forming characteristics of the system. An implicit assumption for fractal mass flow is that a destabilized interface has its fractal structure amplified during phase change. This assumption imposes a self-similarity criterion so that the interface appears the same regardless of scale. In terms of the mushy-zone, this requires the destabilized interface to maintain the same appearance on both macroscopic and microscopic scales. For a one-dimensional planar interface, fractal mass flow leads to a destabilized interface having a fractal dimension w12x which is less than one. From a side profile, the appearance of such an interface is cellular, but its mean interfacial length projected into the liquid is less than the planar case due to interstices between so-called cellular fingers. These interstices are cre-
Fig. 1. Schematic of Temkin’s multi-layered interface with ‘mountain range’ structure.
Fig. 2. Patial phase diagram showing decrease in melting temperature due to mass flow.
ated by mass-flow and the result of this mass-flow condition is that the interface is discretized into progressively smaller segments as it advance into the liquid ahead of it. Because the discretization is selfsimilar and recursive, the geometry is fractal. We imagine the mass-flow phenomena which destabilizes the interface by a tip-splitting process whereby large cells bifurcate into smaller cells as predicted by the marginal stability hypothesis w13x. At the point where cells become stable, mass-flow operates on the growth processes of individual cellular fingers to create a microstructure having a defined fractal dimension equal to zero. We claim that this microstructure is the dendrite and it is the shape attractor resulting from an IFS involving mass-flow and contractive mappings which iteratively reduce the length of the phase boundary from macro-scale to micro-scale. This scaling phenomena coupled with established notions of a critical radius for nucleation w14x suggest that contractive mappings due to mass flow must cease at an interfacial length somewhere near the critical radius. Given our assumption that the phase boundary is a fractal structure, the existence of dendrites provides a transitional link between solidification structures having dimension less than one Žcellular arrays. and a structure having dimension zero Žliquid.. Alloy phase change is governed by the phase diagram which requires liquid at concentration c 0 to freeze out solid at composition kc 0 . As shown in Fig. 2, this leads to a solute pile-up region ahead of
30
K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
the phase-change interface where concentration approaches c 0rk. Since the melting temperature is reduced by concentration, phase change is delayed in the solute pile-up region. Naturally, uniform solute pile-up across the entire interface delays phase change; therefore, a mass flow boundary condition is needed to identify solute poor regions along the interface where phase change is not delayed. Chalmers w15x confirms this position and argues that lateral distribution of solute along the interface is necessary for cellular microstructure. Solute trapped in interstices between cellular fingers delays phase transition so that the phase boundary in these intercellular regions moves slower than the remaining parts of the phase boundary. Given directional solidification along x for an interface of length r 0 along y, we can describe a mass flow boundary condition with a single fractal constant. Assuming extremum conditions involving solute-pileup and a lateral distribution of solute at composition c 0 and c 0rk, we impose a mass flow boundary condition where phase change occurs in a way that regions 0 - y - Bn and A n - y - r 0 see concentration c 0rk while region Bn - y - A n sees concentration c 0 . As we will see in Section 4, data points A n and Bn are determined by exponential scaling of an IFS fractal constant while solute concentrations c 0 and c 0rk control phase change by creating independent melting temperatures TM s T0 and T M s T0 y e , respectively.
4. The self-defined array Self-defined arrays ŽSDAs. w16x capture the essence of transformations which simultaneously translate and downsize the interface as phase change is delayed by mass flow in two of the three regions discussed above. Here SDAs are iterated function systems which describe discrete regions along the phase boundary where mass-flow influences phase change by suppressing melting temperature. Because motion of the phase boundary is iterative mass-flow is also iterative and the SDA changes by a self-referential rule based on the fractal geometry of a massflow condition. Here the SDA generates a set of boundary points having spatial coordinates w X Ž t ., Y x
which change as the phase boundary advances and is redefined by mass-flow. SDAs are important for describing pattern-forming solidification because they define a fractal geometry which is self-similar on all scales given a phase boundary of arbitrary length r 0 . In this model we define the SDA using two spatial variables for data points at each time increment. As an approximation for planar uniformity, we specify that at any time, t, SDA data points Y s A n and Y s Bn must have the same X Ž t . values. With these rules as outlined above, the SDA at t s 0 describes the interface as X X Ž0. X Ž0. X Ž0. X Ž0.
Y r0 r0 0 0
where the interface is defined in the region 0 - y r 0 . These boundary points are repeated in the initial array because they bifurcate as a result of mass-flow and create new data points. An initial instability brought on by the competition between undercooling and surface tension destroys the planar geometry, and triggers fractal mass flow at time 1 y D t. This decreases the length of the interface because phase change is delayed phase change in mass the two mass flow regions as shown in Fig. 3. The SDA at t s 1 is given as X X Ž1. X Ž1. X Ž1. X Ž1.
Y r0 r 0 y r 0rN r 0rN 0
and the phase boundary unaffected by mass flow is defined in the region r 0rN - y - r 0 y r 0rN. Note that the SDA describes a contractive mapping Žor downsizing transformation. as the phase boundary translates from X Ž t s 0. to X Ž t s 1.. This requires that phase transition is delayed mass-flow regions 0 - y - r 0rN and r 0 y r 0rN - y - r 0 . Following an IFS formulation the SDA at t s 1 is used as an input to generate the SDA at t s 2. The new SDA Žor second iterate. describes the phase
K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
31
factor which plays the same role as complex constants used to generate the attractor for fixed points in fractal generating algorithms like Julia sets. As will be discussed below, N sets a stability range for the microstructure. Clearly N s 2 is a stability boundary and the system maintains the planar solution for a degenerate SDA which attracts to a fixed Y s r 0r2. In general, N is important in defining fractal dimension given as w11x Df s
Fig. 3. Self defined data points for the needle crystal shape attractor.
boundary after a second contractive mapping at t s 2 y D t. The SDA at t s 2 is X X Ž2. X Ž2. X Ž2. X Ž2.
Y r 0 y r 0rN r 0 y r 0rN y r 0 Ž N y 2.rN 2 r 0rN q r 0 Ž N y 2.rN 2 r 0rN
As before, the SDA at t s 2 is used as an input to generate the SDA at t s 3. The third SDA iterate describes the interface after a third contractive mapping at t s 3 y D t. The SDA at t s 3 is X Y X Ž3. r 0 y r 0rN y r 0 Ž N y 2.rN 2 X Ž3. r 0 y r 0rN y r 0 Ž N y 2.rN 2 y r 0 Ž N y 2. 2rN 3 X Ž3. r 0rN q r 0 Ž N y 2.rN 2 q r 0 Ž N y 2. 2rN 3 X Ž3. r 0rN q r 0 Ž N y 2.rN 2 The iteration sequence is infinite. Here we stop at three iterations for brevity. For actual dendrites, we expect that these mappings scale to a nucleation limit as discussed previously. The essence of the SDA formulation discussed above can be captured by a few key points. First, all Y values in the array are generated by manipulating two constants, r 0 and N. Here N is a fractal scale
log Ž j . log Ž 1rN .
,
Ž 5.
where j represents the number of pieces an initial line segment Žor interface. is discretized into. For a Cantor middle-third fractal, Eq. Ž5. defines fractal dimension as 0.63 for N s 3 and j s 2 because two line segments result after middle-third segments are removed at each iterate. For our dendrite fractal, j s 1 and Eq. Ž5. defines a fractal dimension of zero, for any value of N. This fractal is the geometric inverse of the Cantor middle-third fractal since middle-third segments are iterated and outer-third segments are removed. Finally, Y values in successive SDAs follow from a self-referential rule requiring that new values evolve as an arithmetic sum involving exponential terms r 0rN p and r 0 Ž N y 2. prN pq1. Here p is the order of the iteration and the factor 2 represents a symmetry condition for two mass flow regions. The SDA for the dendrite microstructure follows a general form given by X X Ž t s n q 1. X Ž t s n q 1. X Ž t s n q 1. X Ž t s n q 1.
Y An A nq 1 Bnq 1 Bn
where A n q Bn s A nq1 q Bnq1 s r 0 . This relationship establishes a self-referential rule so that self-defined values A n and Bn change by the same factor. Ultimately, it also creates the self-similar characteristics of the microstructure so that each Y value in the SDA is given as A0 s r0
B0 s 0
B1 s B0 q r 0rN
A1 s A 0 y r 0rN p
A nq1 s A n y r 0 Ž N y 2 . rN pq1 p
Bnq 1 s Bn q r 0 Ž N y 2 . rN pq1
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K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
As before, the region Bn - y - A n defines a phase change region which is not affected by mass flow. The mass flow regions 0 - y - Bn and A n - y - r 0 have phase change delayed so that associated phase boundaries lag behind the boundary in region Bn - y - A n . Fractal constant N sets proportionality between the three regions and satisfies the inequality 2 - N - 5. This proportionality restriction sets a stability criterion for the dendrite microstructure, and brings in familiar nonlinear characteristics which suggest that certain solutions Žor patterns. fall within a narrow stability window w17x.
5. Applications for self-defined arrays SDAs allow us to formulate a pattern-forming boundary condition for the phase change problem. This is a natural extension of SDAs since X Ž t . values track location of the phase boundary and are solutions to the free boundary problem. Corresponding Y values in the SDA set boundary points for three independent regions along the phase boundary. These regions are governed by an iterated mass-flow condition which generates a fractal microstructure. As shown in Fig. 4, this nonlinear mass-flow creates a fractal boundary and imposes a static bifurcation in melting temperature at the phase boundary. Recog-
Fig. 4. Modified Stefan problem for melting temperature bifurcation due to fractal mass-flow condition.
nizing that the melting temperature bifurcates allows us to superpose two linear phase change problems at the fractal interface. Each solution corresponds to a branch on the bifurcation shown in Fig. 4. The upper branch gives a fast boundary solution X F Ž t . which is recorded in the SDA and corresponds to T M s T0 ŽFig. 2.. The lower branch of the bifurcation corresponds to TM s T0 y e and gives a slow boundary solution X S Ž t . which is insignificant except that if defines the phase boundary separation X F Ž t . y X S Ž t . at each iterated advance of fractal phase boundary. Given a melting temperature which is bifurcated by a fractal mass-flow condition as outlined above, the pattern-forming phase change problem is formulated as Tx x s 0 in 0 - x - X Ž t . , t ) 0 T Ž X Ž t . , t . s Ž T M . upper s T0
Bn - y - A n
Ž 6. Ž 7.
T Ž X Ž t . , t . s Ž T M . lower s T0 y e 0 - y - Bn and A n - y - r 0
Ž 8.
r LX X Ž t . s kTx Ž X Ž t . , t .
Ž 9.
X Ž 0. s 0
Ž 10 .
T Ž 0, t . s TA
Ž 11 .
where formulation follows a quasi-stationary approximation w18x, k is the conductivity, X Ž t . is the location of the phase change boundary, L is the latent heat, T0 is melting temperature of liquid at concentration c 0 , e is amount melting temperature is decreased after mass flow to concentration c 0rk in the liquid, r is the density and TA is the face temperature of the one-dimensional phase change domain. As shown in Fig. 5, the phase boundary is non-uniform, and is a two phase region which has a fractal microstructure as defined by the SDA. Eqs. Ž6. – Ž11. follow standard Stefan formulation, except Eqs. Ž7. and Ž8. are modified to account for mass flow. For a fully posed pattern-forming problem, the solution seeks to determine the location of the fast boundary X F Ž t ., the phase front separation, and the unique dendrite microstructure for a prescribed value of N. Naturally, the space-time solution domain is discretized so that X F Ž t . for p time segments corresponds to p iterates of the SDA.
K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
33
Similarly, in slow boundary regions 0 - y - Bn and A n - y - r 0 TS Ž x , t . s TA q Ž TM y TA . y e
x XS Ž t .
Ž 14 .
and X S Ž t . s Ž 2 krr L . Ž T0 y TA . P t y 2 kt err L
Fig. 5. One-dimensional pattern-forming Stefan problem with fractal mass-flow condition.
The solution to the quasi-stationary problem for the fast boundary region Bn - y - A n is given by x TF Ž x , t . s TA q w T0 y TA x Ž 12 . XF Ž t . and X F Ž t . s Ž 2 krr L . Ž T0 y TA . P t
1r2
,
Ž 13 .
where TF Ž x, t . is the temperature profile in the solid behind the fast boundary located at X F Ž t ..
1r2
, Ž 15 .
where TS Ž x, t . is the temperature profile in the solid behind the slow boundary located at X S Ž t .. As discussed previously, the difference in phase boundary location leads to a phase boundary separation DF s X F Ž t . y X S Ž t .. This boundary separation is defined in terms of the Stefan number ŽSt s cDTrL. and thermal diffusivity Ž a s krr c . so that
F Ž St . s w 2 a P St P t x
1r2
Ž 16 .
and DF s F Ž St F . y F Ž St S .
Ž 17 .
This phase boundary separation defines the ledge length shown in Fig. 6. As these results suggest, mass-flow generates phase boundary microstructure by a two step process. First it creates phase boundary separation along the primary solidification direction and then maps the resulting pattern onto progressively smaller lengths of the fast boundary. This creates a self-similar fractal microstructure which is described by a SDA for an isolated dendrite shown in Fig. 6.
6. Conclusions
Fig. 6. Affine transformation of planar interface to dendrite shape attractor.
This study shows how iterated function systems ŽIFS. in the form of self-defined arrays ŽSDAs. may be applied as boundary conditions to define subdomains along a phase boundary. If these subdomains follow a fractal geometry and are created by a mass-flow condition which suppresses melting temperature, superposed free boundary problems describe pattern-forming phase change for a static bifurcation in melting temperature. The emerging microstructural pattern is encoded in the SDA and is stable in a numerical range 2 - N - 5.
34
K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
References w1x J. Hutchinson, Indiana Univ. Math. J. 35 Ž1981.. w2x M. Barnsley, Fractals Everywhere, Academic Press, CA, 1988. w3x E. Jacobs, R. Boss, Naval Res. Rev. 3 Ž1993.. w4x P. Glansdorf, I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley-Interscience, New York, 1971. w5x H. Levine, Proceedings of 2nd Woodward Conf., San Jose State Univ., Springer-Verlag, New York, 1989. w6x L. Liu, J. Kirkaldy, Acta Metall. 43 Ž1995. 8. w7x J. Langer, Rev. Modern Phys. 52 Ž1980. 1. w8x E. Ben-Jacob, N. Goldenfeld, J. Langer, G. Schon, Physica 12D Ž1984. 245. w9x P. Hohenberg, M. Cross, Fluctuations and Stochastic Phenomena in Condensed Matter, Lecture Notes in Physics, vol. 268, Springer-Verlag, New York, 1968.
w10x D. Temkin, in: O. Ovisienko ŽEd.., Growth and Imperfections of Metallic Crystals, English translation by Consultants Bureau, New York, 1968. w11x B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Co., New York, 1991. w12x M. McGuire, An Eye For Fractals, Addison-Wesley Publishing Co., New York, 1991. w13x W. Kurtz, R. Trivedi, Solidification Processing, Academic Press, CA, 1987. w14x D. Woodruff, The Solid–Liquid Interface, Cambridge University Press, 1973. w15x B. Chalmers, Principles of Solidification, John Wiley and Sons, New York, 1964. w16x K. Williamson, A. Saigal, Comput. Mater. Sci. 6 Ž1996. 343. w17x J. Langer, Metall. Trans. A 15A Ž1984.. w18x V. Alexiades, A. Solomon, Mathematical Modeling of Melting and Freezing Processes, Hemisphere Publishing, 1991.
Applications of self-defined arrays for pattern-forming alloy solidification K. Williamson a
a,)
, A. Saigal
b
Department of Mechanical Engineering, Old Dominion UniÕersity, Norfolk, VA 23529, USA b Department of Mechanical Engineering, Tufts UniÕersity, Medford, MA 02155, USA Received 27 April 1997; accepted 9 May 1997
Abstract In this paper, we consider a pattern-forming solidification problem involving self-defined arrays ŽSDAs.. These SDAs originate from an iterated function system ŽIFS. and are imposed on the moving boundary so that microstructures evolve as prescribed shape attractors during iterated advance of a phase boundary. SDAs are generated from iterated application of a geometric rule requiring the pth iterate of a given array to be defined in terms of the Ž p y 1.th iterate. In this study, we use a geometric rule based on the Cantor middle-third fractal set to generate an SDA for an isolated dendrite microstructure. We apply this notion of an SDA to a phase change problem where fractal mass-flow creates the SDA geometry and results in a solidification microstructure. q 1998 Elsevier Science B.V.
1. Introduction Iterated function systems ŽIFSs. provide powerful tools to describe the geometries found in nature. These methods are related to iterated transform algorithms which are used to generate fractal images like the Julia set w1,2x. In a more general sense, these algorithms are related to dynamical systems which possess various kinds of attracting sets. For iterated function systems, the attracting set is an image which is encoded by a set of transformations which reveal the image after a prescribed number of iterations w3x. As a dynamical system, alloy solidification belongs
)
Corresponding author.
to a select group of processes which form complex self-organized patterns for a variety of restabilized equilibrium states w4,5x. The characteristic behavior of such systems allow them to undergo stability transitions which take them from quiescent states to dynamically restabilized states having time-dependent spatial patterns. Since the general characteristics of self-organized solidification patterns are known and have been induced experimentally in transparent model systems, the objective of this study is to find an IFS which defines the attracting set for these images. The majority of self-organized solidification experiments have been carried out in undercooled melts where experimental evidence confirms theory that the steady state growth velocity and the tip radius of the dendrite pattern are related to the bath undercooling. In fact, theoretical predictions for the operating point
0927-0256r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 7 - 0 2 5 6 Ž 9 7 . 0 0 0 4 9 - 9
28
K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
Ži.e. unique tip radius and tip velocity. have matched experimental results for the restricted case of large undercoolings and a isolated dendrite geometry w6x. Unfortunately, the same models do not explain dendrites when there is a positive temperature gradient Žno undercooling. in the liquid ahead of the interface. The model presented in this paper describes an isolated dendrite growing into a positive temperature gradient with mass-flow as a shaping condition on the phase boundary. In this approach, we formulate the phase change problem using mass-flow as an iterated function system leading to an isolated dendrite shape attractor for the phase boundary.
2. Basic pattern-forming diffusion Dendritic solidification as we normally analyze it, looks at the dimensionless temperature field w7,8x: u Ž x , t . s T Ž x , t . y T` r Ž Lcy1 . ,
Ž 1.
where uŽ x, t . is the deviation of the local temperature T Ž x, t . from a colder temperature T` at infinity. L and c are the latent and specific heats, respectively. Here we ignore convection and write heat conservation in terms of the diffusion equation given as u t s Du x x ,
Ž 2.
where D is the diffusivity in the liquid. To simplify the picture, we ignore diffusion in the solid and focus here on the so called ‘one sided model’. At the phase boundary, we introduce the Stefan Žor continuity. condition which requires the latent heat of solidification to move the phase boundary. This is given by Vnormal s Du n ,
Ž 3.
where n is the unit normal to the interface. Supplementing the continuity condition at the phase boundary is a thermodynamic condition requiring an adjustment in melting temperature due to a curved surface. This Gibbs–Thompson condition for melting temperature, u s , of a curved portion of interface is given by us s u Ž x , t . y d 0 k ,
Ž 4.
where uŽ x, t . is the temperature of an adjacent flat portion of interface. The term, d 0 , is a capillarity term involving surface tension and k is the radius of curvature at the curved portion of the interface. If we model curvature as a small bump Žpositive curvature. on an otherwise flat surface, the bump sees accelerated heat diffusion and rapid phase change with respect to other parts of the interface since it has an advantage of being surrounded by colder liquid. This advantage is offset by a corresponding Gibbs– Thompson reduction in melting temperature. All pattern forming aspects of the phase change process is governed by this dynamical balance between diffusion tending to maximize the bump and capillarity effects tending to minimize it. This basic characteristic of destabilizing forces being dynamically balanced by stabilizing forces is the hallmark of other pattern-forming systems like Taylor–Couette flow and the Raleigh–Bernard convection cell w9x. The basic model outlined above demonstrates the essence of the Mullins–Sekerka planar instability and suggests a needle crystal shape solution if capillary forces fail to restrict diffusive tendency to maximize the arc length of the bump. Ivantsov’s solution w10x confirms the needle crystal geometry and shows that it grows as a paraboloid having an isothermal surface for constant velocity. This solution forms the basic foundation of our steady state approach and has been extended to include morphological stability, microscopic solvability and shape anisotropy theories which suggest length scales so that tip radius and tip velocity can be defined independently. The motivation for the current model is to relax the steady state constant velocity requirement and examine a shape solution for an isolated dendrite as a shape attractor emerging from mass flow formulated as an IFS. Our feeling is that the shape solution from an unsteady heat flow model incorporating mass-flow as an IFS should approach the steady-state shape as the limiting attractor. This approach suggests that steady state analyses provide solutions for a type of instantaneous shape transformation which is rarely observed in real systems. More common is the iterated shape selection. This type of iterated shape selection is the familiar growth pattern seen in ice daggers on tree branches. These daggers increase length iteratively and form shape under the influence of gravity. One length serves as the starting point for
K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
29
the next incremental length increase. Here the essential details lie in the iterative sequence Žor IFS. leading to the final shape and not in the shape itself. It is this notion of iterated shape selection which motivates our current model.
3. A fractal mass flow boundary condition Temkin w10x showed that a phase change interface is made up of seven or eight atomic layers and follows a ‘mountain range’ structure as it minimizes free energy during transformation from liquid to solid. An approximation of the mountain range structure is shown in Fig. 1. As indicated by Mandelbrot w11x and others during the past decade, mountain range structures do follow fractal scaling, therefore, imposing a fractal condition on mass-flow during alloy phase change could provide insight into the pattern forming characteristics of the system. An implicit assumption for fractal mass flow is that a destabilized interface has its fractal structure amplified during phase change. This assumption imposes a self-similarity criterion so that the interface appears the same regardless of scale. In terms of the mushy-zone, this requires the destabilized interface to maintain the same appearance on both macroscopic and microscopic scales. For a one-dimensional planar interface, fractal mass flow leads to a destabilized interface having a fractal dimension w12x which is less than one. From a side profile, the appearance of such an interface is cellular, but its mean interfacial length projected into the liquid is less than the planar case due to interstices between so-called cellular fingers. These interstices are cre-
Fig. 1. Schematic of Temkin’s multi-layered interface with ‘mountain range’ structure.
Fig. 2. Patial phase diagram showing decrease in melting temperature due to mass flow.
ated by mass-flow and the result of this mass-flow condition is that the interface is discretized into progressively smaller segments as it advance into the liquid ahead of it. Because the discretization is selfsimilar and recursive, the geometry is fractal. We imagine the mass-flow phenomena which destabilizes the interface by a tip-splitting process whereby large cells bifurcate into smaller cells as predicted by the marginal stability hypothesis w13x. At the point where cells become stable, mass-flow operates on the growth processes of individual cellular fingers to create a microstructure having a defined fractal dimension equal to zero. We claim that this microstructure is the dendrite and it is the shape attractor resulting from an IFS involving mass-flow and contractive mappings which iteratively reduce the length of the phase boundary from macro-scale to micro-scale. This scaling phenomena coupled with established notions of a critical radius for nucleation w14x suggest that contractive mappings due to mass flow must cease at an interfacial length somewhere near the critical radius. Given our assumption that the phase boundary is a fractal structure, the existence of dendrites provides a transitional link between solidification structures having dimension less than one Žcellular arrays. and a structure having dimension zero Žliquid.. Alloy phase change is governed by the phase diagram which requires liquid at concentration c 0 to freeze out solid at composition kc 0 . As shown in Fig. 2, this leads to a solute pile-up region ahead of
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K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
the phase-change interface where concentration approaches c 0rk. Since the melting temperature is reduced by concentration, phase change is delayed in the solute pile-up region. Naturally, uniform solute pile-up across the entire interface delays phase change; therefore, a mass flow boundary condition is needed to identify solute poor regions along the interface where phase change is not delayed. Chalmers w15x confirms this position and argues that lateral distribution of solute along the interface is necessary for cellular microstructure. Solute trapped in interstices between cellular fingers delays phase transition so that the phase boundary in these intercellular regions moves slower than the remaining parts of the phase boundary. Given directional solidification along x for an interface of length r 0 along y, we can describe a mass flow boundary condition with a single fractal constant. Assuming extremum conditions involving solute-pileup and a lateral distribution of solute at composition c 0 and c 0rk, we impose a mass flow boundary condition where phase change occurs in a way that regions 0 - y - Bn and A n - y - r 0 see concentration c 0rk while region Bn - y - A n sees concentration c 0 . As we will see in Section 4, data points A n and Bn are determined by exponential scaling of an IFS fractal constant while solute concentrations c 0 and c 0rk control phase change by creating independent melting temperatures TM s T0 and T M s T0 y e , respectively.
4. The self-defined array Self-defined arrays ŽSDAs. w16x capture the essence of transformations which simultaneously translate and downsize the interface as phase change is delayed by mass flow in two of the three regions discussed above. Here SDAs are iterated function systems which describe discrete regions along the phase boundary where mass-flow influences phase change by suppressing melting temperature. Because motion of the phase boundary is iterative mass-flow is also iterative and the SDA changes by a self-referential rule based on the fractal geometry of a massflow condition. Here the SDA generates a set of boundary points having spatial coordinates w X Ž t ., Y x
which change as the phase boundary advances and is redefined by mass-flow. SDAs are important for describing pattern-forming solidification because they define a fractal geometry which is self-similar on all scales given a phase boundary of arbitrary length r 0 . In this model we define the SDA using two spatial variables for data points at each time increment. As an approximation for planar uniformity, we specify that at any time, t, SDA data points Y s A n and Y s Bn must have the same X Ž t . values. With these rules as outlined above, the SDA at t s 0 describes the interface as X X Ž0. X Ž0. X Ž0. X Ž0.
Y r0 r0 0 0
where the interface is defined in the region 0 - y r 0 . These boundary points are repeated in the initial array because they bifurcate as a result of mass-flow and create new data points. An initial instability brought on by the competition between undercooling and surface tension destroys the planar geometry, and triggers fractal mass flow at time 1 y D t. This decreases the length of the interface because phase change is delayed phase change in mass the two mass flow regions as shown in Fig. 3. The SDA at t s 1 is given as X X Ž1. X Ž1. X Ž1. X Ž1.
Y r0 r 0 y r 0rN r 0rN 0
and the phase boundary unaffected by mass flow is defined in the region r 0rN - y - r 0 y r 0rN. Note that the SDA describes a contractive mapping Žor downsizing transformation. as the phase boundary translates from X Ž t s 0. to X Ž t s 1.. This requires that phase transition is delayed mass-flow regions 0 - y - r 0rN and r 0 y r 0rN - y - r 0 . Following an IFS formulation the SDA at t s 1 is used as an input to generate the SDA at t s 2. The new SDA Žor second iterate. describes the phase
K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
31
factor which plays the same role as complex constants used to generate the attractor for fixed points in fractal generating algorithms like Julia sets. As will be discussed below, N sets a stability range for the microstructure. Clearly N s 2 is a stability boundary and the system maintains the planar solution for a degenerate SDA which attracts to a fixed Y s r 0r2. In general, N is important in defining fractal dimension given as w11x Df s
Fig. 3. Self defined data points for the needle crystal shape attractor.
boundary after a second contractive mapping at t s 2 y D t. The SDA at t s 2 is X X Ž2. X Ž2. X Ž2. X Ž2.
Y r 0 y r 0rN r 0 y r 0rN y r 0 Ž N y 2.rN 2 r 0rN q r 0 Ž N y 2.rN 2 r 0rN
As before, the SDA at t s 2 is used as an input to generate the SDA at t s 3. The third SDA iterate describes the interface after a third contractive mapping at t s 3 y D t. The SDA at t s 3 is X Y X Ž3. r 0 y r 0rN y r 0 Ž N y 2.rN 2 X Ž3. r 0 y r 0rN y r 0 Ž N y 2.rN 2 y r 0 Ž N y 2. 2rN 3 X Ž3. r 0rN q r 0 Ž N y 2.rN 2 q r 0 Ž N y 2. 2rN 3 X Ž3. r 0rN q r 0 Ž N y 2.rN 2 The iteration sequence is infinite. Here we stop at three iterations for brevity. For actual dendrites, we expect that these mappings scale to a nucleation limit as discussed previously. The essence of the SDA formulation discussed above can be captured by a few key points. First, all Y values in the array are generated by manipulating two constants, r 0 and N. Here N is a fractal scale
log Ž j . log Ž 1rN .
,
Ž 5.
where j represents the number of pieces an initial line segment Žor interface. is discretized into. For a Cantor middle-third fractal, Eq. Ž5. defines fractal dimension as 0.63 for N s 3 and j s 2 because two line segments result after middle-third segments are removed at each iterate. For our dendrite fractal, j s 1 and Eq. Ž5. defines a fractal dimension of zero, for any value of N. This fractal is the geometric inverse of the Cantor middle-third fractal since middle-third segments are iterated and outer-third segments are removed. Finally, Y values in successive SDAs follow from a self-referential rule requiring that new values evolve as an arithmetic sum involving exponential terms r 0rN p and r 0 Ž N y 2. prN pq1. Here p is the order of the iteration and the factor 2 represents a symmetry condition for two mass flow regions. The SDA for the dendrite microstructure follows a general form given by X X Ž t s n q 1. X Ž t s n q 1. X Ž t s n q 1. X Ž t s n q 1.
Y An A nq 1 Bnq 1 Bn
where A n q Bn s A nq1 q Bnq1 s r 0 . This relationship establishes a self-referential rule so that self-defined values A n and Bn change by the same factor. Ultimately, it also creates the self-similar characteristics of the microstructure so that each Y value in the SDA is given as A0 s r0
B0 s 0
B1 s B0 q r 0rN
A1 s A 0 y r 0rN p
A nq1 s A n y r 0 Ž N y 2 . rN pq1 p
Bnq 1 s Bn q r 0 Ž N y 2 . rN pq1
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K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
As before, the region Bn - y - A n defines a phase change region which is not affected by mass flow. The mass flow regions 0 - y - Bn and A n - y - r 0 have phase change delayed so that associated phase boundaries lag behind the boundary in region Bn - y - A n . Fractal constant N sets proportionality between the three regions and satisfies the inequality 2 - N - 5. This proportionality restriction sets a stability criterion for the dendrite microstructure, and brings in familiar nonlinear characteristics which suggest that certain solutions Žor patterns. fall within a narrow stability window w17x.
5. Applications for self-defined arrays SDAs allow us to formulate a pattern-forming boundary condition for the phase change problem. This is a natural extension of SDAs since X Ž t . values track location of the phase boundary and are solutions to the free boundary problem. Corresponding Y values in the SDA set boundary points for three independent regions along the phase boundary. These regions are governed by an iterated mass-flow condition which generates a fractal microstructure. As shown in Fig. 4, this nonlinear mass-flow creates a fractal boundary and imposes a static bifurcation in melting temperature at the phase boundary. Recog-
Fig. 4. Modified Stefan problem for melting temperature bifurcation due to fractal mass-flow condition.
nizing that the melting temperature bifurcates allows us to superpose two linear phase change problems at the fractal interface. Each solution corresponds to a branch on the bifurcation shown in Fig. 4. The upper branch gives a fast boundary solution X F Ž t . which is recorded in the SDA and corresponds to T M s T0 ŽFig. 2.. The lower branch of the bifurcation corresponds to TM s T0 y e and gives a slow boundary solution X S Ž t . which is insignificant except that if defines the phase boundary separation X F Ž t . y X S Ž t . at each iterated advance of fractal phase boundary. Given a melting temperature which is bifurcated by a fractal mass-flow condition as outlined above, the pattern-forming phase change problem is formulated as Tx x s 0 in 0 - x - X Ž t . , t ) 0 T Ž X Ž t . , t . s Ž T M . upper s T0
Bn - y - A n
Ž 6. Ž 7.
T Ž X Ž t . , t . s Ž T M . lower s T0 y e 0 - y - Bn and A n - y - r 0
Ž 8.
r LX X Ž t . s kTx Ž X Ž t . , t .
Ž 9.
X Ž 0. s 0
Ž 10 .
T Ž 0, t . s TA
Ž 11 .
where formulation follows a quasi-stationary approximation w18x, k is the conductivity, X Ž t . is the location of the phase change boundary, L is the latent heat, T0 is melting temperature of liquid at concentration c 0 , e is amount melting temperature is decreased after mass flow to concentration c 0rk in the liquid, r is the density and TA is the face temperature of the one-dimensional phase change domain. As shown in Fig. 5, the phase boundary is non-uniform, and is a two phase region which has a fractal microstructure as defined by the SDA. Eqs. Ž6. – Ž11. follow standard Stefan formulation, except Eqs. Ž7. and Ž8. are modified to account for mass flow. For a fully posed pattern-forming problem, the solution seeks to determine the location of the fast boundary X F Ž t ., the phase front separation, and the unique dendrite microstructure for a prescribed value of N. Naturally, the space-time solution domain is discretized so that X F Ž t . for p time segments corresponds to p iterates of the SDA.
K. Williamson, A. Saigalr Computational Materials Science 11 (1998) 27–34
33
Similarly, in slow boundary regions 0 - y - Bn and A n - y - r 0 TS Ž x , t . s TA q Ž TM y TA . y e
x XS Ž t .
Ž 14 .
and X S Ž t . s Ž 2 krr L . Ž T0 y TA . P t y 2 kt err L
Fig. 5. One-dimensional pattern-forming Stefan problem with fractal mass-flow condition.
The solution to the quasi-stationary problem for the fast boundary region Bn - y - A n is given by x TF Ž x , t . s TA q w T0 y TA x Ž 12 . XF Ž t . and X F Ž t . s Ž 2 krr L . Ž T0 y TA . P t
1r2
,
Ž 13 .
where TF Ž x, t . is the temperature profile in the solid behind the fast boundary located at X F Ž t ..
1r2
, Ž 15 .
where TS Ž x, t . is the temperature profile in the solid behind the slow boundary located at X S Ž t .. As discussed previously, the difference in phase boundary location leads to a phase boundary separation DF s X F Ž t . y X S Ž t .. This boundary separation is defined in terms of the Stefan number ŽSt s cDTrL. and thermal diffusivity Ž a s krr c . so that
F Ž St . s w 2 a P St P t x
1r2
Ž 16 .
and DF s F Ž St F . y F Ž St S .
Ž 17 .
This phase boundary separation defines the ledge length shown in Fig. 6. As these results suggest, mass-flow generates phase boundary microstructure by a two step process. First it creates phase boundary separation along the primary solidification direction and then maps the resulting pattern onto progressively smaller lengths of the fast boundary. This creates a self-similar fractal microstructure which is described by a SDA for an isolated dendrite shown in Fig. 6.
6. Conclusions
Fig. 6. Affine transformation of planar interface to dendrite shape attractor.
This study shows how iterated function systems ŽIFS. in the form of self-defined arrays ŽSDAs. may be applied as boundary conditions to define subdomains along a phase boundary. If these subdomains follow a fractal geometry and are created by a mass-flow condition which suppresses melting temperature, superposed free boundary problems describe pattern-forming phase change for a static bifurcation in melting temperature. The emerging microstructural pattern is encoded in the SDA and is stable in a numerical range 2 - N - 5.
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References w1x J. Hutchinson, Indiana Univ. Math. J. 35 Ž1981.. w2x M. Barnsley, Fractals Everywhere, Academic Press, CA, 1988. w3x E. Jacobs, R. Boss, Naval Res. Rev. 3 Ž1993.. w4x P. Glansdorf, I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley-Interscience, New York, 1971. w5x H. Levine, Proceedings of 2nd Woodward Conf., San Jose State Univ., Springer-Verlag, New York, 1989. w6x L. Liu, J. Kirkaldy, Acta Metall. 43 Ž1995. 8. w7x J. Langer, Rev. Modern Phys. 52 Ž1980. 1. w8x E. Ben-Jacob, N. Goldenfeld, J. Langer, G. Schon, Physica 12D Ž1984. 245. w9x P. Hohenberg, M. Cross, Fluctuations and Stochastic Phenomena in Condensed Matter, Lecture Notes in Physics, vol. 268, Springer-Verlag, New York, 1968.
w10x D. Temkin, in: O. Ovisienko ŽEd.., Growth and Imperfections of Metallic Crystals, English translation by Consultants Bureau, New York, 1968. w11x B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Co., New York, 1991. w12x M. McGuire, An Eye For Fractals, Addison-Wesley Publishing Co., New York, 1991. w13x W. Kurtz, R. Trivedi, Solidification Processing, Academic Press, CA, 1987. w14x D. Woodruff, The Solid–Liquid Interface, Cambridge University Press, 1973. w15x B. Chalmers, Principles of Solidification, John Wiley and Sons, New York, 1964. w16x K. Williamson, A. Saigal, Comput. Mater. Sci. 6 Ž1996. 343. w17x J. Langer, Metall. Trans. A 15A Ž1984.. w18x V. Alexiades, A. Solomon, Mathematical Modeling of Melting and Freezing Processes, Hemisphere Publishing, 1991.