Phase Change Manipulation for Droplet-Based Solid

M. Orme Associate Professor. [email protected]

C. Huang Graduate Student Researcher. Department of Mechanical and Aerospace Engineering, University of California, Irvine, Irvine, CA 92717-3975

Phase Change Manipulation for Droplet-Based Solid Freeform Fabrication

droplet of typical dimensions, 1-10 mm (depending on plasma conditions), falls off the wire and onto the substrate, thereby Solid Freeform Fabrication (SFF) is an emerging technology building the material component. in which a useful metallic component is manufactured from the Another materials synthesis technology which utilizes molten raw material without a mold in one integrated operation. A new technology under development at UCI, termed droplet-based droplets as the deposition element is spray forming. There, a SFF, requires less machining steps than conventional manufac- spray of molten metal is formed by atomizing a fluid column turing techniques, and hence it has the potential to provide of molten metal. Thus, the droplet diameters and speeds vary improved manufacturing quality and significant economic bene- and the precision of fabrication is limited by the angular extent fit (Orme and Muntz, 1992). In droplet-based SFF a useful of the spray cone angle. The spray is then deposited onto a structural component is fabricated by sequentially depositing substrate building a near-net shaped part. Recent developments molten droplets layer by layer. The usefulness of the droplet- in spray forming are described in the review article by Lavernia based SFF technique is determined by the structural characteris- etal. (1992). The droplet-based SFF technique is set apart from other methtics of the material component synthesized. Manipulation of the phase-change characteristics of the sequentially deposited ods of droplet deposition net-form, or near net-form, manufacdroplets during the SFF process offers the potential means to turing in that the droplet streams can be precisely controlled control and improve the microstructure and hence the structural and manipulated. Precise control refers to droplet streams which (Orme, integrity of the part. To this purpose, conditions which cause have an angular stability on the order of 1 micro-radian 7 the newly arriving droplet to locally remelt the previously de- 1994), and speed dispersions as low as 3 X 10 " times the posited and solidified droplet structure are sought. Under these average stream speed (Orme, 1991). The high degree of control conditions, the solid/liquid interface changes direction after so- over the droplet dynamics allows the possibility of precise conlidification and on the onset of remelting, and subsequently trol over the droplet solidification/melting characteristics. Although the droplet generation and deposition processes are again on the onset of resolidification. Therefore, its motion cannot be assumed to vary with time t as tm, as is commonly different in the aforementioned techniques, the numerical method and results presented in this paper are applicable. The assumed (e.g., Stefan solution). objective of this paper is to outline a computationally simple Figure 1 is a conceptual schematic of the droplet-based SFF one-dimensional simulation model that can be used to study technique under development at UCI. In operation, molten dropphase-change manipulation strategies for the droplet-based solid lets on the order of 100 lira, in diameter are formed from capilfreeform fabrication of structural parts. Locally remelting the lary stream break-up and are injected into either a vacuum or previously deposited and solidified droplet material provides a an inert environment. As the droplets are formed, they acquire flexible approach to removing boundaries between successive a charge by passing through a charged electrode (not shown in splats without sacrificing the benefits of rapid solidification. the figure) at the time of droplet break-off from the stream. The model utilizes a coordinate transformation to convert the The intrinsic angular stability of the droplet stream allows the moving boundary into a fixed one, thus simplifying immensely droplets to travel a vertical distance of 0.3 m prior to deposition. the numerical simulations. This model is far more computationThis is practically important since a longer flight allows for ally efficient than competing phase-change models and can be a larger lateral deposition area to be covered by electrostatic used for simulation studies of large-scale problems (requiring deflection. Details about the charging techniques are given elsenumerous repetitions of the same elemental event, i.e., droplet where (Orme et al., 1996). Lateral motion of the x-y table from deposition and remelting) for this and other applications with CAD information coupled with electrostatic deflection allows moving boundaries. subsequent droplet deliveries to build the three-dimensional part microlayer by microlayer without any mold. Since the droplets rapidly solidify, complex structures can be fabricated in the Related Research absence of macroscopic fluid flow. There is a vast body of literature which investigates the probDroplet-based SFF bears similarities with the technology under development by Prinz et al. (1995), Amon et al. (1996), lem of one-dimensional phase change by heat conduction where and Chin et al. (1995) termed Shape Deposition Manufacturing the phase change is assumed to occur at the melting temperature (SDM). In SDM a feedstock wire located directly over the and the molten phase is not in motion (e.g., Carslaw and Jaeger, substrate is melted using a plasma welding torch. A discrete 1962; Goodrich, 1978; Frederick and Greif, 1985; Amon et al., 1996). In most of the investigations only pure melting or pure solidification is possible, but not sequential remelting and solidiContributed by the Heat Transfer Division for publication in the JOURNAL OF fication. In general, the sequential melting and solidification HEAT TRANSFER. Manuscript received by the Heat Transfer Division April 22, problem is difficult because of the moving boundary which is 1996; revision received June 13, 1997; Keywords: Conduction; Materials Pronot known a priori. Many authors have assumed that the intercessing and Manufacturing Process; Moving Boundaries, Associate Technical Editor: A. S. Lavine. face moves with time t as f (where m is usually taken to be

Introduction

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Fig. 1 Conceptual schematic of the droplet-based SFF technique

0.5). In solidification followed by remelting, the solid/liquid interface cannot be described by a simple function of time since the interface motion oscillates in a direction away from the substrate and towards the substrate. The current work is different from existing studies in that the moving boundary problem has been transformed to a fixed boundary problem where the moving interface is found as part of the solution. This eliminates the need to postulate the time-course of the moving boundary that is required by other methods in order to numerically solve the discretized governing partial differential equations. Under the proposed transformation of coordinates, the interface condition becomes a transcendental algebraic equation which provides a unique solution for the interface location as a function of time. Other recent work has concentrated on two-dimensional equilibrium conduction in a droplet which is undergoing deformation (Madejski, 1976; Trapaga et al., 1992; Liu et al., 1993, 1995; San Marchi et al., 1993; Delplanque et al, 1996; Zhao et al., 1996; Waldvogel and Poulikakos, 1997). In several of these works (Madjeski, 1976; Liu et al., 1993,1995; San Marchi et al., 1993; Delplanque et al., 1996), the solid/liquid interface is determined by solving the Stefan problem of solidification, and hence its location varies with the square root of time, i.e., the solid/liquid interface always moves away from the substrate for solidification or towards the substrate for melting. In the work of Trapaga et al. (1992), the enthalpy method was used to simulate the phase-change problem of the deforming splat in an effort to predict the deformation process of the impinging and solidifying splat. The model was also used to predict the temperature profile in the ' 'heat affected zone'' of the substrate, but not the time-course of the solid/liquid interface. This model can be used to estimate the remelting characteristics of the previously deposited material (substrate), though it is very computationally intensive, i.e., simulation of 5 X 10" 3 seconds of real time requires more than 100 hours on a 3100 VAXSTATION due to the complexity of the model. Zhao et al. (1996) theoretically investigated presolidification splat cooling and spreading using the Lagrangian formulation and hence phase change is not considered in their work. The numerical and experimental work of Waldvogel and Poulikakos (1997) present the most complete model of droplet deformation and phase

change to date, in which the heat affected zone of the substrate can be studied. They used the Lagrangian formulation to simulate the motion of the deforming free surface and focused on the determination of single solidified splat shapes as a function of impacting droplet conditions. However, this model is also computationally intensive—they report that an example simulation of one drop deformation requires 54 CPU hours on a HP 9000 Series 735 workstation. The model proposed in this paper provides a quick estimate of the one-dimensional sequential remelting and solidification characteristics of sequentially deposited droplets by use of a transformed interface equation. Comparison of this model with a two-dimensional model such as that provided by Waldvogel and Poulikakos (1997) is inappropriate since the objectives of the two models are notably different. The model given in this work is intended to be applied to the droplet-based SFF application in which a multitude of droplets is deposited sequentially to fabricate the component. For instance, a SFF component with a height of 1 cm requires approximately 1000 droplets to be deposited on the vertical axis. The computational simplicity of this model enables simulation of remelting through 1000 sequentially deposited droplets in approximately 8 minutes on a DEC workstation. As the component grows layer by layer, the droplet temperatures can be adjusted to insure that local remelting occurs. Thus, the study of the manufacturing process of a component is distinct from the study of a single splat. Furthermore, and perhaps more importantly, the model provided in this paper can be applied to other one-dimensional moving boundary problems of finite domain with a sharp moving boundary. Model Development Values used in the simulations of the injecting environment, substrate, and droplet are given in the table below where the subscript °° refers to the ambient inert environment, m refers to the melting point of aluminum, / and ^ refer to the liquid and solid state of the aluminum, and sub refers to the lower face of the substrate. In the table, T is the temperature, L is the latent heat of fusion, Cp is the specific heat capacity, k is the thermal conductivity, and p is the density. Because the rapid solidification time of the droplets is of the order of a microsecond, in most practical circumstances the molten droplets will impinge on a layer of previously delivered droplets which has solidified. Thus, we explore the possibility of remelting the solidified droplets with the thermal energy from the arriving droplets in an effort to drive the two materials to unite, thereby removing the boundaries between the different layers of droplet depositions. Since it is the objective of this work to cause remelting, droplets will necessarily be deposited at high superheats, making the assumption that they spread to flat disks not entirely unrealistic for the conditions sought in the application of droplet-based SFF. In this case, the physical dimension of the splat in the direction normal to the substrate (height) is small compared to the lateral dimension (width) and the remelting depth is small compared to the splat height, hence the problem is approximated as a one-dimensional phase-change situation as

Nomenclature 1 = subscript used to denote conditions through the substrate 2 = subscript used to denote the droplet material in the solid state 3 = subscript used to denote the droplet material in the liquid state a = splat thickness Cp = specific heat capacity Journal of Heat Transfer

k L Da M s(t) sub

= thermal conductivity - latent heat of fusion = substrate thickness = number of droplet layers = solid/liquid interface location = subscript used to denote properties of substrate at lower face Tj = initial droplet impact temperature

T„, = melting temperature of metal T„ = ambient temperature t - time a = thermal diffusivity equal to k/pCp p = density of metal

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Table 1 Material properties used in thermal analysis

k\ ks ksub p\ ps psub

T„ 300 K 933 K Tm L 402J/kg Cp\ 1084J/kgK 895J/gK Cps Cpsub 383J/gK

* i » copper substrate

(2) Fig. 2(a) Nomenclature for one-dimensional solidification and remelting problem

in

s*(t*) < x* < M + 1

* f ^ - - * ? ^ - = 0 dx* dx* , * 9T2

(12)

Oil

(

dx*2

Oil, — Oil

=

1

dt*

(11)

= L*

dx* Tf = - 1 T* = T'f = 0

at

x*=0

at

JC*=iS*(f*)

(3)

(4) (5)

>i..

•. ! • - .

• l-••*J^,•'

dt* at at

x* = -D x* =

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s*(t*)

-L 0

(6) (7)

Fig. 2(b) Nomenclature for transformation from a moving boundary to a fixed boundary

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point of the solid/liquid interface as it travels upward away from the substrate. The solid/liquid interface is difficult to track with the fixed grid approach since it may not necessarily fall on a grid point. To overcome this difficulty, we have transformed the moving boundary problem into a fixed boundary problem. Figure 2(b) illustrates the following problem transformation which is presented in this work:

aluminum, a nozzle diameter of 100 //m, and a stream speed of 8 m/s. Using Lord Rayleigh's relation for droplet production from capillary stream break-up and assuming an inviscid fluid, the droplet production frequency is 17,857 droplets/s (Rayleigh, 1879). Droplet diameters are found from conservation of mass and are 189 /jm. We have chosen a few representative scenarios which may likely be considered in future dropletbased Solid Freeform Fabrication endeavors, though other depo2 s - M(M + 1) M - s (15) sition scenarios may also be employed. V s[s - (M + ])]' s[s - ( M + 1)] Figure 3 illustrates the solid/liquid interface depth versus time for the second splat after it has made contact with the first In Eq. (15) and for the remainder of this paper we have dropped solidified splat which is 10 jim thick. Each curve represents the the asterisk which denotes dimensionless quantities for conve- solid/liquid interface history for a different value of the ratio nience. Under this transformation, which maps the dynamically (Ti - T„,)/(T - T„,i,) where T is the initial liquid droplet m t moving values in the x plane (0, s, M + 1) to the fixed values impact temperature. For the conditions considered in this work, in the 17 plane (0, M, M + 1), the governing equations for the it can be seen that remelting can be initiated by adjusting this liquid and solid state become ratio to values above approximately 1.6. Increasing the ratio to higher values, i.e., increasing either T, or Tsuh, will cause the ^ = (4AV + B2)a^ + 2Aa?T2 in Q