ASM Handbook Volume 22 Modeling and Simulation: Processing of Metallic Materials

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ASM Handbook Volume 22 Modeling and Simulation: Processing of Metallic Materials 9. Simulation of PM processes 9.C. Metal Powder Injection Molding Seokyoung Ahn 1, Seong-Taek Chung 2, Seong Jin Park 3,
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ASM Handbook Volume 22 Modeling and Simulation: Processing of Metallic Materials 9. Simulation of PM processes 9.C. Metal Powder Injection Molding Seokyoung Ahn 1, Seong-Taek Chung 2, Seong Jin Park 3, and Randall M. German 4 1 Department of Mechanical Engineering, The University of Texas-Pan American, 1201 W University Dr., Edinburg, TX 78539, USA 2 CetaTech, Inc., TIC 296-3, Seonjin-Ri, Sacheon-Si, Kyongnam, , Korea 3 Center for Advanced Vehicular Systems, Mississippi State University, 200 Research Blvd., Starkville, MS 39759, USA 4 College of Engineering, San Diego State University, 5500 Campanile Drive, San Diego, CA , USA Introduction Plastics formed by injection molding are used everywhere, because of the low overall cost in a high level of shape complexity. Powder injection molding (PIM) builds on the long recognized success of plastic molding by using a high particle content thermoplastic as feedstock. The steps in PIM involve first mixing selected small powders (usually smaller than 20 µm) and polymer binders. The particles are small to aid in sintering densification and often have near spherical shapes to improve flow and packing. The thermoplastic binders are mixtures of waxes, polymers, oils, lubricants, and surfactants. When molten, the binder imparts viscous flow characteristics to the mixture to allow filling of complex tool geometries. A favorite binder system relies on a mixture of paraffin wax and polypropylene, with a small quantity of stearic acid. The combination of powder and binder that works best gives a paste with about the same consistency as toothpaste, with no voids; this often leads to a formulation near 60 volume percent powder and 40 volume percent binders. This mixture is heated in the molding machine, rammed into a cold mold, and when the binder freezes in the mold, the component is ejected. Next, the binder is removed by heat and solvents (some of the binders are water soluble) and the remaining 60% dense powder structure is then sintered to near-full density. The product may be further densified, heat treated, machined, or plated. The sintered compact has the shape and precision of an injection molded plastic, but is capable of performance levels unattainable with polymers. Almost half of all PIM is applied to stainless, steels, but a 1 wide variety of compositions are in production [1]. The equipment used for shaping the compact is the same as used for plastic injection molding, so software for molding machine control is the same as found in plastics. Most molding machines fill a die through a gate from a pressurized and heated barrel (the barrel and gate are connected by a nozzle and runner). A plunger or reciprocating screw generates the pressure needed to fill the die. Computer simulations used for mold filling in plastics are not directly useful for powder injection molding, since inertia, thermal conductivity, and powder-binder separation are new concerns with PIM feedstock. The feedstock enters the barrel as cold granules and during compression to remove trapped air it is heated above the binder melting temperature. Because the feedstock is hot and the die is cold, filling must be accomplished in a split second to avoid premature freezing. After filling the die, packing pressure is maintained on the feedstock during cooling to eliminate shrinkage voids. After sufficient cooling, the hardened compact is ejected and the cycle repeated. It is common to have multiple cavities (4 or more) and cycle times of 15 s, so production rates of 16 parts per minute are often observed. The PIM process is practiced for a very wide range of materials, including most common metal, many ceramics, and cemented carbides. The largest uses are in metal powder injection molding, denoted as MIM. Other variants are CIM for ceramic injection molding and CCIM for cemented carbide injection molding. Simulations used for plastics have been applied to PIM, but the high solid content often makes for differences that are ignored in the plastic simulations. Several situations demonstrate the problems, such as powder-binder separation at weldlines, high inertial effects such as in molding tungsten alloys, and rapid heat loss such as in molding copper and aluminum nitride. Also, powder-binder mixtures are very shear rate sensitive. Thus, the computer simulations to support molding build from the success demonstrated in plastics, but adapt those concepts in new customized PIM simulations for filling, packing, and cooling. Theoretical Background and Governing Equations A typical injection molded component has a thickness much smaller than the overall largest dimension. A typical wall thickness is in the 1 to 3 mm range, while the longest dimension might range near 25 mm with an overall mass near 10 g. There is much variation, but these values offer a glimpse at the typical components [2]. In molding such components, the molten powder-binder feedstock mixture is highly viscous. As a result, the Reynolds number (a dimensionless number characterizing a ratio of inertia force to viscous force) is low and the flow is modeled as a creeping flow with lubrication, as treated with the 2 Hele-Shaw formulation. With the Hele-Shaw model, the continuity and momentum equations for the melt flow in the injection molding cavity are merged into a single Poisson equation in terms of the pressure and fluidity. Computer simulation is usually based on a 2.5-dimensional approach because of the thin wall and axial symmetry. But the Hele-Shaw model has its limitations and cannot accurately describe threedimensional (3D) flow behavior in the melt front, which is called fountain flow, and special problems arise with thick parts with sudden thickness changes, which cause race-track flow. Nowadays, several 3D computer aided engineering simulations exist that successfully predict conventional plastic advancement and pressure variation with changes in component design and forming parameters [3]. For PIM 3D simulation, Hwang and Kwon [4] developed a filling simulation with slip using an adaptive mesh refinement technique to capture the large deformation of the free surfaces, but this is computationally intensive [4-7], so further research is moving toward simplified solution routes [3]. In this section, we will focus on the axisymmetric 2.5-dimensional approach rather than a full 3D approach, because the 2.5-dimensional approach is more robust and better accepted by industry. The post-molding sintering simulation is described in the earlier press-sinter simulation. Filling Stage Powder injection molding involves a cycle that repeats every few seconds. At the start of the cycle, the molding machine screw rotates in the barrel and moves backward to prepare molten feedstock for the next injection cycle while the mold closes. The mold cavity fills as the reciprocating screw moves forward, acting as a plunger, which is called the filling stage. During the filling stage, a continuum approach is used to establish the system of governing equations as follows: Mass and momentum conservation: With the assumption of incompressible flow, the mass conservation, also called continuity equation, is expressed as, u v w + + = 0 x y z (1) where x, y, and z are Cartesian coordinates and u, v, and, w are corresponding orthogonal velocity components. As for the momentum conservation, with lubrication and the Hele-Shaw approximation, the Navier-Stokes equation is modified for molten feedstock during filling stage as follows [8-9], 3 P u = η x z z P v = η y z z P = 0 z where P is the pressure, z is the thickness, and η is the viscosity of PIM feedstock. By combining Equations (1) and (2) with integration in z-direction (thickness direction) gives, (2) where P P S + S = 0 x x y y (3) b 2 z S dz (4) b η Equation (3) is the flow governing equation for the filling stage. This is exactly the same form of steady-state heat conduction equation obtained by substituting temperature T into P and thermal conductivity k into S. In this analogy, S is the flow conductivity or fluidity. As a simple interpretation of this flow governing equation, molten PIM feedstock flows from the high pressure region to the low pressure region, and the speed of flow depends on the fluidity S. During the calculation, the fluidity increases as the thickness of component increases and the viscosity of PIM feedstock decrease for feedstock cooling. After obtaining the pressure field, the velocity components u and v are obtained by integrating Equation (2) in the z-direction (thickness direction). Energy Equation: In accordance with the lubrication and Hele-Shaw approximations during the filling stage, the energy equation is simplified as follows: ρ T t T x T y T z 2 C p 2 + u + v = k + ηγ& 2 (5) where ρ is the molten PIM feedstock s density, C p is the molten PIM feedstock s specific heat, 2 + v z γ& = u z is generalized shear rate and k is the thermal conductivity of and ( ) ( ) 2 the feedstock. In addition, we need a constitutive relation to describe the molten PIM feedstock s response to its flow environment during cavity filling, which requires a viscosity model. Several viscosity models for 4 polymers containing high concentrations of particles are available. Generally they include temperature, pressure, solids loading, and shear strain rate and selected models will be introduced later in this chapter. The selection of a viscosity model depends on the desired simulation accuracy over the range of processing conditions, such as temperature and shear rate, as well as access to the experimental procedures used to obtain the material parameters. Once we have the system of differential equations from continuum-based conservation laws and the constitutive relations for analysis of the filling stage, then we need boundary conditions. Typical boundary conditions during the filling stage are as follows: boundary conditions for flow equation: flow rate at injection point, free surface at melt-front, no slip condition at cavity wall and boundary conditions for energy equation: injection temperature at injection point, free surface at melt-front, mold-wall temperature condition at cavity wall. Note that the only required initial condition is the flow rate and injection temperature at the injection node, which is one of the required boundary conditions. For a more rigorous approach during the filling stage, a few efforts have invoked a full 3D model, and have included fountain flow, viscoelastic constitutive models, slip phenomena, yield phenomena, and inertia effects in governing equations and interface [3, 5-6, 10]. Packing Stage When mold filling is nearly completed, the packing stage starts. This precipitates a change in the ram control strategy for the injection molding machine, from velocity control to pressure control, which is called the switch-over point. As the cavity nears filling, the pressure control ensures full filling and pressurization of the filled cavity prior to freezing of the gate. It is important to realize the packing pressure is used to compensate for the anticipated shrinkage in the following cooling stage. Feedstock volume shrinkage results from the high thermal expansion coefficient of the binder, so on cooling there is a measurable contraction. By appropriate pressurization prior to cooling, then after the gate freezes the component shrinks sufficiently that there are no sink marks (too low a packing pressure) and no difficulty with ejection (too high a packing pressure). For the analysis of the packing stage, it is essential to include the effect of melt compressibility. Consideration is given to the melt compressibility using a dependency of the specific volume on pressure and temperature, leading to a feedstock specific pressure-volume-temperature (pvt) relationship, or the equation of state. Several models are available to describe the pvt relation of PIM feedstock, such as the 2-domain modified Tait model and IKV model. These models predict an abrupt volumetric change for 5 both semi-crystalline polymers used in the binder and the less abrupt volume change for amorphous polymers used in the binder. With the proper viscosity and pvt models, the system of governing equation for the packing stage based on the continuum approach is as follows: Mass conservation: The continuity equation of compressible PIM feedstock is expressed as, ρ + t x ( ρu) ( ρv) ( ρw) + y + z = 0 (6) with the assumption that pressure convection terms may be ignored in the packing stage this becomes: p T T T u v κ β + u + v + + = 0 (7) t t x y x y where κ is the isothermal compressibility coefficient of the material ( ρ ρ p ) and β measures the volumetric expansivity of the material ( ρ ρ T ). Those are easily calculated from the equation of state. Note that the same momentum conservation is used as Equation (2) regardless of the material to be considered as compressible or not. Energy Equation: The energy equation is derived as, ρ C p T t T + u x T + v y 2 T = k 2 z 2 p + ηγ& + βt (8) t That is, the shear rate for the compressible case in packing is for practical purposes the same as for the filling phase. Typical initial and boundary conditions during packing stage are as follows: initial conditions: pressure, velocity, temperature, and density from the results of the filling stage analysis, boundary conditions for equations of mass and momentum conservations: prescribed pressure at injection point, free surface at melt-front, no slip condition at cavity wall, and boundary conditions for energy equation: injection temperature at injection point, free surface at melt-front, and mold-wall temperature condition at the cavity wall, which is interfaced with the cooling stage analysis. 6 Cooling Stage Of the three stages in the injection molding process, the cooling stage is of greatest importance because it significantly affects the productivity and the quality of the final component. Cooling starts immediately upon the injection of the feedstock melt, but formally the cooling time is referred to as the time after the gate freezes and no more feedstock melt enters the cavity. It last up to the point of component ejection, when the temperature is low enough to withstand the ejection stress. In the cooling stage, the feedstock volumetric shrinkage is counteracted by the pressure decay until the local pressure drops to atmospheric pressure. Thereafter, the material shrinks with any further cooling, possibly resulting in residual stresses due to nonuniform shrinkage or mold constraints (which might not be detected until sintering). In this stage, the convection and dissipation terms in the energy equation is neglected since the velocity of a feedstock melt in the cooling stage is almost zero [11-13]. Therefore, the objective of the mold-cooling analysis is to solve only the temperature profile at the cavity surface to be used as boundary conditions of feedstock melt during the filling and packing analysis. When the injection molding process is in steady-state, the mold temperature will fluctuate periodically over time during the process due to the interaction between the hot melt and the cold mold and circulating coolant. To reduce the computation time for this transient process, a 3D cycle-average approach is adopted for the thermal analysis to determine the cycle-averaged temperature field and its effects on the PIM component. Although the mold temperature is assumed invariant over time there is still a transient for the PIM feedstock [14], leading to the following features: Mold Cooling Analysis: Under this cycle-average concept, the governing equation of the heat transfer for injection mold cooling system is written as 2 T = 0 (9) where T is the cycle-average temperature of the mold. PIM Component Cooling Analysis: Without invoking a flow field, the energy equation is simplified as 2 T T ρ C p = k (10) 2 t z Typical initial and boundary conditions applied during the cooling stage are as follows: initial conditions: temperature as calculated from the packing stage analysis, 7 boundary conditions - mold: interface input from the PIM feedstock cooling analysis, convection heat transfer associated with the coolant, natural convection heat transfer with air, and thermal resistance condition from the mold platen, and boundary conditions - component: interface input from the mold cooling analysis. Note that the boundary conditions for the mold and PIM feedstock cooling are coupled to each other. More details on this are given in the Numerical Simulation section below. For a more rigorous approach during the cooling stage, some researchers have included more than two different mold materials with flow analysis that includes the cooling channel details. This enables corresponding heat transfer analysis with any special cooling elements, such as baffles, fountains, thermal pin, or heat pipes [12]. Numerical Simulation As far as the numerical analysis of injection molding is concerned, several numerical packages are already available for conventional thermoplastics. And one may try to apply the same numerical analysis techniques to PIM. However, the rheological behavior of a powder-binder feedstock mixture is significantly different from that of a thermoplastic. Hence, the direct application of methods developed for thermoplastics to PIM requires caution [4, 10]. Commercial software packages, including Moldflow (Moldflow Corp., Framingham, MA), Moldex3D (CoreTech System Co., Ltd., Chupei City, Taiwan), PIMsolver (CetaTech, Sacheon, Korea), and SIMUFLOW (C-Solutions, Inc., Boulder, CO) are available for PIM simulation. Further, several research groups have written customized codes, but generally these are not released for public use. It is well known that powder-binder feedstock mixtures used in PIM exhibit a peculiar rheological feature known as wall slip [10, 15]. Therefore, a proper numerical simulation of the PIM process essentially requires a proper constitutive equation representing the slip phenomena of powder-binder feedstock mixtures [10, 14]. Filling and Packing Analysis For numerical analysis of the filling and packing stages of PIM, both the pressure and energy equations must be solved during the entire filling and packing cycle. This is achieved using the finite element method for Equation (2) while a finite difference method is used in the z-direction (thickness), making use of the same finite elements in the x y plane for solving Equation (3). The finite difference method (FDM) is a relatively efficient and simple numerical method for solving 8 differential equations. In this method, the physical domain is discretized in the form of finite-difference grids. A set of algebraic equations is generated as the derivatives of the partial differential equations and are expressed by finite differences of the variable values at the grid points. The resulting algebraic equation array, which usually forms a banned matrix, is solved numerically. Generally, the solution accuracy is improved by reducing the grid spacing. However, since the FDM is difficult to apply to a highly irregular boundary or a complicated domain typical of injection molding, the use of this method has to be restricted to regular and simple domains, or used with the finite element method (FEM) as a FDM-FEM hybrid scheme [3]. The FEM has been excellent flexibility in treating complex geometries and irregular boundaries, which is a key advantage of this method. It requires discretizing the physical domain into several finite elements. The field variables are represented with shape functions and nodal values over each finite element. Using residual minimization techniques (or, equivalently, variational techniques) such as the Galerkin method, the governing equations are transformed into discretized forms [8, 9]. For three-dimensional simulation of injection molding, the resulting global matrix system from the algebraic equations is a typically large
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