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Quantitative texture prediction of epitaxial columnar grains in additive manufacturing using selective laser melting
Accepted Manuscript Title: Quantitative texture prediction of epitaxial columnar grains in additive manufacturing using selective laser melting Authors: Jian Liu, Albert C. To PII: DOI: Reference:
S2214-8604(16)30074-4 http://dx.doi.org/doi:10.1016/j.addma.2017.05.005 ADDMA 173
To appear in: Received date: Revised date: Accepted date:
19-4-2016 3-5-2017 3-5-2017
Please cite this article as: Jian Liu, Albert C.To, Quantitative texture prediction of epitaxial columnar grains in additive manufacturing using selective laser melting (2010), http://dx.doi.org/10.1016/j.addma.2017.05.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Quantitative texture prediction of epitaxial columnar grains in additive manufacturing using selective laser melting7 Jian Liu, Albert C.To* Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15261 *
Corresponding author ([email protected])
Highlights
A texture prediction method was proposed for epitaxial columnar grains in SLM. The texture prediction method was combined with the melt pool prediction. Process and microstructure were linked quantitatively for the metal SLM AM process. Texture evolution with the number of layers for SLM AlSi10Mg was simulated. The simulated texture showed pattern and intensity similar to experiment results.
Abstract Metal additive manufacturing (AM) such as selective laser melting (SLM) has the powerful capability to produce very different microstructural features, hence different mechanical properties in metals using the same feedstock material but different values of process parameters. However, the relation between processing-microstructure is mostly investigated by experiments, which is expensive and time-consuming since the parameter space is quite large. The lack of a reliable theoretical model of the processingmicrostructure relationship of AM material is preventing AM technology from being widely adopted by the manufacturing community. Hence, the goal of this work is to establish the link between the microstructure (texture) and the process parameters (laser power, scanning speed, preheat and scanning strategy) of a metal SLM process. To achieve the above goal, a quantitative semi-empirical method is proposed to predict the texture of the epitaxial columnar grains grown from polycrystal substrates. Combined with the melt pool prediction by the Rosenthal solution, the processing and microstructure were linked together quantitatively. The proposed method is used to estimate the texture evolution with the number of layers for EOS-DMLS-processed AlSi10Mg (unidirectional scanning direction in one layer and no rotation of scanning direction between layers). The texture reaches a steady state after five layers, and the steady state texture has similar pattern and intensity to that obtained from the experiment using the same process parameter values and scanning strategy.
Keyword: additive manufacturing; selective laser melting; texture; microstructure
1. Introduction Selective laser melting (SLM) is a powder bed fusion metal AM technique that utilizes a moving laser beam to melt the powder particles in each layer selectively, which leaves behind metallic layers joined together to form the desired geometry [1, 2]. Previously, optimization of the process parameters was predominately for the goal of minimizing the porosity [3]. For metallic structural components, the
microstructure such as grain size, shape and texture also significantly influence the properties, including elastic modulus, strength, anisotropy, plastic deformation behavior, fatigue and crack propagation properties. In general, AM has the ability to produce very different microstructural features in the metals deposited using the same feedstock material but different values of process parameters [4-7], which cannot be achieved by traditional metal part fabrication techniques due to their inability to control heat transfer conditions at very small length scales. The possibility of controlling the microstructure of metals using SLM has spurred intense research in predicting microstructure from the given process parameters. This work proposes a new model that takes the key SLM process parameters as input and predicts the grain texture in a quantitative manner. The proposed model builds upon previous works in predicting the columnar-to-equiaxed transition (CET) through melt pool modeling as a function of the process parameters. For instance, Gaumann et al. [8] integrated the modified Rosenthal melt pool model [9] and Hunt’s CET model to obtain processing–microstructure maps which define processing windows for different morphologies, specifically in their case, to ensure columnar mode in the laser repair of single crystal (SX) components. Beuth and co-workers [10] translated the solidification map of Ti–6Al–4V in the 𝐺-𝑅 (thermal gradient-solidification rate) space [11] to a process map in the 𝑃-𝑉 (beam power-velocity) space using finite element analysis to obtain the G and R values. Finite element analysis was performed for the deposition of single beads of Ti-6Al-4V via electron beam (EBM) wire feed AM processes. The local 𝐺 and 𝑅 not only depend on the process parameters, but also vary up to an order of magnitude along the melt pool. The results presented were only for the top of the melt pool, which is a critical location. Dehoff et al. [6] developed electron beam scan strategies to achieve site-specific control of the texture of Inconel 718 based on principles of columnar-to-equiaxed transitions. An analytical model for fusion welding melt pool prediction compiled by Grong [12] was used to guide the choice of process parameters to produce controlled spatial variation of G and R that will generate different solidification grain morphology according to the solidification map of Inconel 718 [13]. Above are some examples in integrating the melt pool prediction together with the CET model. Note the research is limited to a qualitative evaluation of the overall grain morphology by evaluating the predicted 𝐺 𝑛 /𝑅 ratio at a critical point of the melt pool. No model to predict the detailed grain texture was proposed. The effect of process parameters on the texture is the subject of research in this paper. To the best of the authors' knowledge, there has been no research yet to quantitatively model and predict the texture of the solidified metal when the base metal is polycrystalline which is the case in powder-bed SLM. The objective of this work is to develop a quantitative semi-empirical method to estimate the texture of the epitaxial columnar solidified microstructure on a polycrystal (PX) substrate. Combined with the melt pool prediction by the Rosenthal solution, the process parameters and microstructure can be linked together quantitatively. This paper is organized as follows. Section 2 reviews the necessary background for the formulation of the model including the melt pool models, the CET model and the epitaxial columnar growth on single crystal (SX) substrates. After introducing the Rosenthal melt pool model and discussing the local epitaxial columnar dendrite growth from both SX and PX bases, Section 3 presents the proposed method to estimate the texture in detail. Section 4 presents the results and discussion of melt pool profile and the texture evolution with the number of layers, of AlSi10Mg processed by the EOS DMLS system. This is followed by conclusion and future works in Section 5.
2. Background Inherent to the SLM and other high-energy beam based process is the formation of a pool of molten metal (i.e. melt pool) directly below the heat source with melting occurring ahead of the heat source and solidification behind it. The solidified microstructure depends on local solidification conditions at the trailing edge of the melt pool. Hence, the melt pool geometry (i.e. size and shape) and thermal profile (i.e. thermal gradient and cooling rate, etc.) need to be first predicted as a function of the process parameters. Then the microstructure can be predicted using the melt pool profile based on solidification theory. The following provides the background information necessary to discuss formulation of the model. The temperature field around the moving heat source can be predicted by simple analytical models [14] or complex numerical simulations [15-19]. Experimentally, the temperature field can be measured using the IR or near-IR cameras [20, 21]. Once the temperature field is known, the melt pool shape can be obtained as the liquidus isotherm, i.e., the surface at which 𝑇 = 𝑇𝑚 (𝑇𝑚 is the melt temperature of the material and the undercooling of the solidifying front is ignored). The local thermal variables such as thermal gradient 𝐺 = |𝜵𝑇| and cooling rate 𝑇̇ = 𝜕𝑇/𝜕𝑡 can also be calculated along the boundary of the melt pool. SLM is a rather complex process controlled by mass transfer (powder) and by heat and fluid flow in the melt pool. For this reason, experimental measurement of the complex physical phenomena is difficult because the process is both highly transient and localized. Numerical simulations consider these physical phenomena (mass, energy and momentum conservation) and solve the governing equations using numerical techniques within a certain spatial and temporal domain. The finite element thermal analysis [15-17] and coupled flow-thermal modeling and simulation usually performed at the powder scale [18, 19] are two representative groups of numerical simulations. Those complex numerical simulations are usually time-consuming and not suited for applications such as real-time control [22] and the exploration of the entire process parameter space for process optimization [23]. For computational efficiency, analytical solutions or correlation models in simple mathematical form are useful because essential physics are taken into account without the need for numerical simulation. It usually deals with the quasi-steady state solution in the moving coordinate system attached to the heat source. Rosenthal [14] first proposed the analytical solution of the temperature field relative to a point heat source that is moving on a semi-infinite solid substrate. In order to obtain a simple close form solution, that work considered only the conduction of heat along with other simplifications such as 1) temperature-independent thermal properties, 2) no latent heat, beside the assumptions of point source and semi-infinite boundary condition. These solutions can be used to predict the temperature field at a distance far from the heat source, but the prediction of temperature in the vicinity of the heat source is only a rough approximation. For example, the transverse cross-sections of the melt pool predicted by Rosenthal solution will always be a semi-circle, which will have an impact on the prediction of columnar solidification texture. Cline and Anthony [9] modified Rosenthal’s theory to include a two-dimensional (2D) surface Gaussian distributed heat source with a constant distribution parameter and found an analytical solution. Their solution is a significant step to improve the temperature prediction in the near heat source regions [24]. A similar solution has been derived by Eagar and Tsai [25]. Although the analytical solution for 3D double ellipsoidal heat source has also been derived [24], it is more useful for
the deep penetration welding. For simplicity, only the original Rosenthal model will be considered in this work. With the melt pool geometry and thermal profile, and based on the recent solidification theory, the resulting solidified microstructure such as the grain size, shape and texture can be predicted. The important solidification parameters include thermal gradient 𝐺, cooling rate 𝑇̇ and solidification rate 𝑅. In general, the solidification morphology can be planar, cellular, columnar dendritic or equiaxed dendritic, depending on the cooling rate and the 𝐺 𝑛 /𝑅 values at the solid-liquid interface [26]. The CET model considers the competition between (1) the columnar dendrite growth of the seed crystal provided by the partly melted base metal, and (2) the nucleation and growth of equiaxed grains within the undercooled liquid ahead of the tip of the dendrite. The analytical model for the CET model was first developed by Hunt in 1984 [27]. The volume fraction of equiaxed grains was obtained by integrating the growth rate from the time that nucleation starts until the time that the columnar front reaches the equiaxed grains [8]. In Hunt's simplified CET model, different 𝐺 𝑛 /𝑅 ratio corresponds to different volume fraction ∅ of equiaxed grains. Hunt proposed that the structure is fully equiaxed if ∅ > 0.49 (columnar dendrites will be completely blocked), fully columnar if ∅ < 0.0066 and a mixture of columnar and equiaxed if 0.0066 < ∅ < 0.49. Hunt's simplified CET model can be represented as a microstructure/solidification map in the 𝐺-𝑅 space in the form of the two curves of 𝐺 𝑛 /𝑅 which corresponds to ∅ = 0.49 and ∅ = 0.0066, respectively [28]. Such microstructure/solidification maps in the 𝐺-𝑅 space have been experimentally calibrated for many materials such as Ti–6Al–4V [11], Ni-based super alloy Inconel 718 [13] and CMSX-4 [8]. In the case of a single crystal substrate as in the case of laser repair of SX superalloy components, epitaxial columnar growth will result in a SX solidification microstructure, which is preferred. Hence, the CET has been the focus in early studies of laser processing of single crystal components to ensure columnar growth and to avoid equiaxed growth mode [28-30]. Also, since the thermal gradient direction varies along the boundary when base metal orientation is constant, it is possible for the changes of dendrite growth direction. This phenomenon is known as oriented to misoriented transition (OMT) and has also been observed and modeled in the repair of single-crystal turbine parts using the laser-based deposition process [31, 32]. When the substrate is a polycrystal, the grain size, shape (elongation direction and aspect ratio) and texture of the epitaxial columnar grains all need to be predicted. The grain size is observed to be proportional to the melt pool size [10]. The elongation is along the direction of dendrite growth. The texture depends on the geometric and thermal profile of the melt pool and also the texture of the base metal.
3. Model Formulation 3.1. Melt pool The laser beam is assumed to be a point heat source and part of the laser power is absorbed by the powder layer when the laser beam travels through. When the laser beam reaches the substrate, it will interact with the substrate and generate a temperature field (and melt pool) due to heat conduction. Figure 1 is a schematic of the trailing (solidifying) part of the melt pool formed under the moving laser beam. The quasi-steady state temperature field and melt pool can be modeled using the Rosenthal solution [14]:
𝜂𝑃
𝑇(𝑥0 , 𝑦0 , 𝑧0 ) = 𝑇0 + 2𝜋𝑘 𝑒 −𝜆𝑉𝑥0
𝑒
2 2 −𝜆𝑉√𝑥2 0 +𝑦0 +𝑧0
(1)
√𝑥02 +𝑦02 +𝑧02
where 𝑥0 , 𝑦0 , 𝑧0 are the location of a point in the coordinate system attached to the moving point source, 𝑇0 is the initial temperature of the solid, 𝜂 is the absorption coefficient of laser energy by the substrate, 𝑘 is the thermal conductivity, and 1/2𝜆 is the thermal diffusivity of the material. Using the Rosenthal model, the local thermal variables such as thermal gradient 𝐺 = |𝜵𝑇| and cooling rate 𝑇̇ = 𝜕𝑇/𝜕𝑡 can also be calculated along the boundary of the melt pool. Another important solidification parameter is the solidification rate 𝑅, which is related to the beam velocity and melt pool shape: 𝑅 = 𝑉𝑐𝑜𝑠𝜃
(2)
where 𝜃 is the angle between direction of the moving source and growth direction of the solidifying material. The local solidification parameters not only depend on the process parameters but also vary along the melt pool. At a location (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 ) on the solidifying front of the melt pool, the local solidification mode depends on the local 𝐺 and 𝑅. In Hunt’s model, the CET transition is characterized by a critical 𝐺 𝑛 /𝑅 ratio [8]. If the 𝐺 𝑛 /𝑅 is greater than a critical value, the solidification is fully columnar, and dendrite arms will grow from the solid-liquid interface.
3.2. Local epitaxial columnar dendrite growth: SX and PX base In the case of SLM AM, the solid (base) metal and the liquid metal on either side of the solid-liquid interface have similar chemical composition, hence the same crystal structure. Columnar grains will grow epitaxially from the seed crystal provided by the base metal. The dendrite arm will have the same crystallographic orientation as the seed crystal, and the growing direction will be along the <001> axis of the seed crystal that is most closely aligned with the thermal gradient vector [33]. The SX base (Fig. 2a) can only provide a seed crystal of a fixed orientation identical to the SX base. Hence, the dendrite will have the same crystallographic orientation as the SX base metal. The dendrite growing direction will be along one of the <001> axes (for FCC) of the SX base that has the maximum: 𝑐𝑜𝑠𝜑 = (𝒎ℎ𝑘𝑙 ∙ 𝑮𝑖 )/|𝑮𝑖 |
(3)
where 𝑮𝑖 is the local thermal gradient vector at the location (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 ) and 𝒎ℎ𝑘𝑙 (𝒎100,𝒎010 and 𝒎001 ) are the direction vectors of the three <001> axes of the SX base. The PX base (Fig. 2b) could provide seed crystals of different orientations. Within a small element surface around a point (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 ), we assume initially there are multiple (𝑁𝑠𝑒𝑒𝑑 ) possible seed crystals instead of one seed crystal with a fixed orientation in the case of the SX base. The texture (crystallographic orientation distribution) of a polycrystal metal is usually represented by a collection of a large number of grain orientations. For each possible seed crystal (j=1: 𝑁𝑠𝑒𝑒𝑑 ), the crystallographic orientation (𝒎ℎ𝑘𝑙,𝑗 ) can be determined in a statistical way, i.e., by randomly choosing one orientation out of the collection of
orientations that represent the PX base texture. The growing direction of the final dendrite from the location (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 ) will be along the direction of 𝒎ℎ𝑘𝑙,𝑗 of the seed crystal 𝑗 that has the maximum: 𝑐𝑜𝑠𝜑 = (𝒎ℎ𝑘𝑙,𝑗 ∙ 𝑮𝑖 )/|𝑮𝑖 |
(4)
Naturally, the crystallographic orientation of the final dendrite will be the same as the selected seed crystal 𝑗. One special case is the fine-grain random-texture PX base. If the base metal is a fine-grained randomtextured polycrystal, it is safe to assume that, at each location, there is statistically always a seed crystal that provides one <001> direction that is parallel to the thermal gradient and the dendrite arm from this seed crystal will outgrow all others. The growing direction of the final dendrite arms will always be perpendicular to the boundary (parallel to the thermal gradient vector) along the front. The procedures outlined above should give this result if the number of grain orientations representing the random texture and the number of initial possible seed crystals are both large enough. To summarize, the columnar epitaxial dendrite grows from a location will have the same crystallographic orientation as the seed crystal, and the growing direction will be along one of the <001> axes of the seed crystal that is most closely aligned with the thermal gradient. The seed crystal is fixed with the same orientation as the base in the case of SX base. In the case of the PX base, the final seed crystal can be determined, in a statistical way, by the texture of the base and thermal gradient at the solid-liquid interface.
3.3. Solidification microstructure: SX and PX substrate The epitaxial columnar dendrite growth at a specific location has been discussed. Now the overall solidification microstructure can be analyzed by first considering one solidified section that solidifies from bottom to up as the solidifying front is sweeping through. The overall solidification microstructure depends on the microstructure of the substrate. Figure 3 is a schematic side-by-side comparison of the solidification microstructure on SX and PX substrates with both the longitudinal (i.e. along the laser scan direction) and transverse cross-section view. Fig. 3a and 3b are the schematics of the longitudinal cross-section at 𝑡0 with the heat source moving from left to right. The parts behind the solidifying front of the melt pool are the already solidified microstructures, which will become clear after the analysis of solidified sections in the following discussion. For one solidified section, the substrate serves as the base metal only at the beginning of solidification (defined the time as 𝑡0 in Fig. 3), Later, the already solidified metal will serve as base for the following solidification until the solidification is complete (defined the time as 𝑡1 in Fig. 3). At the beginning of solidification (𝑡0 ) on a SX substrate (Fig. 3c), the dendrite arms with the same crystallographic orientation as the substrate will grow in one of the <001> axis directions. Depending on the local thermal gradient direction and the substrate orientation, the dendrite growth from each location could be along the original dendrite arm direction in the SX (oriented) or not (mis-oriented) [31, 32]. Such oriented/mis-oriented dendrites will keep growing until the occurrence of CET (𝑡𝐶𝐸𝑇 , if it occurs) and the rest of solidification will be complete in the equiaxed mode. The columnar region of the section solidified
from a SX substrate is still a SX with the same crystallographic orientation but different dendrite growth directions at different locations. At the beginning of solidification (𝑡0 ) on a PX substrate (Fig. 3d), at different locations of the bottom solidifying boundary, dendrite arms with different crystallographic orientation will nucleate and compete with neighboring dendrites. Some of the dendrites will outgrow others, reach the final trunk diameter, and keep growing into individual elongated grains until the occurrence of CET (𝑡𝐶𝐸𝑇 ) or the completion of solidification. In the columnar region, the solidified section is now a polycrystal. If the crystallographic orientations of the elongated grains are known, the texture of this solidified section will be determined.
3.4. Prediction of the solidification texture on PX substrates The texture of one solidified section can be obtained by a numerical procedure. The number of columnar grains in the solidified section is related to the cooling rate and also proportional to the section area (melt pool size). Assuming the number of grains in one section is 𝑁𝑔𝑟𝑎𝑖𝑛 , and then the solidifying boundary can be discretized into 𝑁𝑔𝑟𝑎𝑖𝑛 small segments. The average thermal gradient at each segment can be obtained using the melt pool model outlined in section 3.1. The crystallographic orientation of the dendrite from each segment, which will grow into an elongated grain, can be determined by the procedure outlined in section 3.2. The resulting overall texture of one solidified section can be obtained by adding all the determined grain orientations into a collection. Now, for the texture of one solidified section, the total number in the collection of grain orientations is 𝑁𝑔𝑟𝑎𝑖𝑛 . The crystallographic orientation of each grain in one solidified section will have one <001> axis being deterministic, i.e., closely aligning (even parallel) to the thermal gradient. The other two axes are somewhat probabilistic, i.e., only loosely restrained by the orientation distribution (texture) of the substrate. Since a statistical step of selecting the possible seed crystal is involved, numerically repeating the texture estimation for the solidified section could give different results each time regarding two <001> axes of the solidified grains (again, one <001> will be determined), which physically represents a different solidified section along the track. The procedure can be repeated to obtain and collect the texture of a number (𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 ) of solidified sections to represent the texture of a track, then the total number in the collection of grain orientations is 𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 × 𝑁𝑔𝑟𝑎𝑖𝑛 . In reality, one layer is built by many short segments of tracks. Since we are assuming the melt pool and solidification being quasi-steady, segments of tracks of the same scanning direction in one layer could be considered belonging to one track. Hence, the aforementioned texture estimation of a track can represent the texture of a layer. However, if the scan direction in one layer is bidirectional, then the forth and back scan direction will generally give different results. In such a case, the layer is considered consisting of two tracks and the texture of both tracks need to be estimated and collected. If we use 𝑁𝑡𝑟𝑎𝑐𝑘 to denote the type of track (𝑁𝑡𝑟𝑎𝑐𝑘 = 1 for unidirectional and 𝑁𝑡𝑟𝑎𝑐𝑘 = 2 for bidirectional), for the texture of one layer, the total number in the collection of grain orientations is 𝑁𝑡𝑟𝑎𝑐𝑘 × 𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 × 𝑁𝑔𝑟𝑎𝑖𝑛 . For the first layer, the texture of the substrate (building platform) is usually random, and the resulting texture of the first layer will be non-random. This non-random first layer texture will serve as the substrate texture for the second layer and so on. The rotation of scan direction between layers can be accounted for by rotating the substrate (previous layer) texture in the opposite way. Hence, the effect of scan strategy is taken into account. If we repeat the texture estimation to obtain and collect the texture of a
number (𝑁𝑙𝑎𝑦𝑒𝑟 ) of layers to represent the texture of 3D material, the total number in the collection of grain orientations is 𝑁𝑙𝑎𝑦𝑒𝑟 × 𝑁𝑡𝑟𝑎𝑐𝑘 × 𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 × 𝑁𝑔𝑟𝑎𝑖𝑛 .
4. Results and discussion The solidification texture of AlSi10Mg processed by the EOS DMLS system was estimated using the procedure outlined above. First, the Rosenthal solution is used to simulate the melt pool generated during the processing of AlSi10Mg by the EOS DMLS system. The values for the process parameters employed are 𝑃 = 200 W, 𝑉 = 1400 mm/s, 𝑇0 =23 oC [34]. The material properties employed are k = 150.0 W/(m∙K), 1/2𝜆 = 8.418e-5 m²/s, and 𝑇𝑚 = 660 oC [35] and the absorption factor is assumed to be 𝜂 = 0.5. Assuming that the heat source travels from left to right, the results obtained from the Rosenthal solution are illustrated in Fig. 4. The original Rosenthal model was employed to obtain the thermal gradient profile on the solidifying boundary at the beginning of solidification of one solidified section. Now the predicted bottom solidifying boundary is a semi-circle from original Rosenthal model (Fig. 5). The number of grains in one solidified section was assumed to be 𝑁𝑔𝑟𝑎𝑖𝑛 = 20 and hence the bottom solidifying boundary was then discretized into 20 small segments (Fig. 5). The number of 𝑁𝑔𝑟𝑎𝑖𝑛 = 20 is based on the experimental results of Ti-6Al4V processed by EBM [36], in which the average beta grain width was plotted against the average experimentally measured effective melt pool and approximately 20 beta grains were seen across the melt pool width. The average thermal gradient at each segment was obtained using the Rosenthal model. In predicting the texture of the first layer, the initial platform serves as the substrate and its texture is represented using a collection of 5,000 randomly distributed grain orientations. In each segment (𝑖 = 1: 20 in Fig. 5), the dendrite (grain) orientation is determined by equation (4) with 𝑁𝑠𝑒𝑒𝑑 = 100. The resulting texture of one solidified section can be obtained by adding all the determined grain orientations into a collection. The procedure was repeated to obtain and collect the texture of 𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 = 1,000 solidified sections to represent the texture of one track. The scanning pattern being simulated here is a unidirectional scanning direction in one layer (𝑁𝑡𝑟𝑎𝑐𝑘 = 1), and hence the texture of a track is that of a layer. Fig. 6a is the estimated texture of the first layer (determine and collect dendrite orientation from segment 𝑖 = 1: 20 in Fig. 5 and repeat 1000 (𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 ) times) growing from the random-textured substrate (platform) in the form of <001> pole figure. At the first glance, one would presume a <001> fiber texture along the scanning direction (SD) which was also concluded by Thijs et al. [34]. However, a different interpretation could be obtained by examining the texture of the collections of grains that grow from one particular location (determine and collect dendrite orientation from one particular segment 𝑖 in Fig. 5 and repeat 1000 (𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 ) times). Fig. 6b, c, d are the estimated textures of the collection of 1,000 grains growing from three different locations (segment 𝑖 = 1, 5, 10 in Fig. 5) in the form of <001> pole figure. For a collection of 1,000 grains growing from one particular location, the texture is a strong <001> fiber along the thermal gradient of this location. The reason for the strong <001> fiber is that the dendrites (grains) grow from one particular location at the bottom solidifying boundary will have one <001> axis being deterministic. One <001> axis will be closely aligning (even parallel) to the thermal gradient while the other two axes are somewhat probabilistic (Section 3.4). Texture of the track/layer (Fig. 6a) can now
be considered as the collection of 20 (𝑁𝑔𝑟𝑎𝑖𝑛 ) fiber texture components instead of one fiber texture component along the SD direction. The scanning pattern being simulated here is that there is no rotation of the scanning direction between layers, hence the resulting texture of the previous layer will be used directly as the substrate texture for the next layer. Fig. 7 is the estimated texture by the aforementioned procedures and its evolution with the number of layers. At the fifth layer (Fig. 7e), the texture reaches a steady state, and the steady state texture has a similar pattern and intensity to that obtained from the experiment [34] (Fig. 7f) using the same process parameter values and scanning pattern. In terms of pattern, the layer texture begins with a seemingly <001> fiber component along SD (which is actually a collection of 20 (𝑁𝑔𝑟𝑎𝑖𝑛 ) fiber texture components) at the first layer (Fig. 7a). With the increase of layers, the directions of fiber components change from evenly distributed to concentrating around a maximum intensities of −45o to +45o to the building direction (BD). In terms of intensity, the maximum MRD (multiples of random density) of the experimental texture is 3.7 (Fig. 7f), while the prediction of the steady-state texture (Fig. 7e) is 2.5. The difference in these values is likely due to the inherent inaccuracy in the original Rosenthal model from the various simplifying assumptions. Since the bottom solidifying boundary predicted by the original Rosenthal solution will always be a semicircle, the variation of process parameters such as P and V will only affect the magnitude of the thermal gradient on each segment but not the direction. According to equation (4), only the direction of the thermal gradient affects the selection of seed crystals. As a result, by using the original Rosenthal solution, the process parameters P and V have no effect on the solidification texture. The texture is only affected by the substrate texture and the scanning pattern. Better agreement between the predicted and experimental texture can be expected if finite element analysis is employed instead, as the solidifying boundary is not fixed as semi-cycle, and the effect P and V can be captured by using FEA. Another advantage of using FEA is to further improve the temperature field prediction, as well as remove the assumptions of semi-infinite boundary conditions, temperaturedependent thermal properties, and no latent heat. This will be explored in the future to determine the accuracy and efficiency of using them to predict the grain texture using the proposed processmicrostructure model.
5. Conclusions In this work, a quantitative method linking the microstructure (texture) and the process parameters (laser power, scanning speed, preheat and scanning patterns) has been developed for the SLM metal AM processes. To achieve the above goal, 1) the Rosenthal model was used to predict the melt pool geometry and thermal profile and 2) a quantitative semi-empirical method to estimate the texture of the epitaxial columnar grains which grow from polycrystal substrates was developed. In the texture estimation, one solidified section was assumed to have a constant number of grains, where each grows from an epitaxial columnar dendrite from the bottom solidifying boundary. The crystallographic orientation of each dendrite was determined using the local average thermal gradient direction of the segment on the bottom solidifying boundary together with a statistical step of selecting seed crystals from the substrate. The texture of one solidified section was obtained by determining and collecting the orientation of each dendrite. The texture of one track was obtained by repeating the
estimation and collection of texture for a number of sections. The texture of a layer was considered the same as the track if the scanning is unidirectional. Finally, the resulting texture of one layer was used as the substrate texture for the next layer and the effect of scanning direction rotation between layers was taken into account. The proposed method was used to estimate the texture and its evolution with the number of layers for EOS-DMLS-processed AlSi10Mg (unidirectional scanning direction in one layer and no rotation of scanning direction between layers). The texture reaches a steady state after five layers, and the steady state texture has similar pattern and intensity to that obtained from the experiment using the same process parameter values and scanning pattern. In the future, a more advanced melt pool model based on the FEM method will be used to obtain more accurate geometry and thermal profile of the melt pool, which should result in a more accurate texture prediction. The improved process-microstructure model with the more advanced melt pool model will be used to predict the texture of SLM metals using scan patterns other than uniaxial scanning direction in one layer and rotation of scanning direction between layers. The findings from these research directions will be reported in future publications. Acknowledgement The financial support from the National Science Foundation (CMMI- 1434077) is gratefully acknowledged.
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Figure 1. A schematic of the trailing (solidifying) part of the melt pool formed under the moving laser beam.
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Figure 2. Epitaxial columnar dendrite arm growth from (a) SX and (b) PX base metals.
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Figure 3. Longitudinal cross-section at beginning of solidification (𝑡0 ) of one section on a (a) SX and (b) PX substrate; and transverse cross-section at beginning of solidification (𝑡0 ) on a (c) SX and (d) PX substrate.
Figure 4. Predicted 3D melt pool size and shape of the EOS DMLS-processed AlSi10Mg employing the Rosenthal solution. The laser is moving from left to right.
Figure 5. Transverse cross-section at beginning of solidification on a PX substrate with the bottom solidifying boundary predicted by original Rosenthal solution.
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Figure 6. (a) First layer texture of columnar grains grow from random-textured substrate (platform) in the form of <001> pole figure. And texture of the collection of 1000 grains grow from three different locations, i.e., segment i = 1(b), i = 5 (c), i = 10 (d) in Fig. 5 in the form of <001> pole figures.
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Figure 7. Texture evolution of solidified microstructure estimated by the proposed method for EOS-DMLS-processed AlSi10Mg: (a) first layer, (b) second layer, (c) third layer, (d) fourth layer, (e) fifth layer, (f) experiment [34] using the same process parameter and scanning pattern.
S2214-8604(16)30074-4 http://dx.doi.org/doi:10.1016/j.addma.2017.05.005 ADDMA 173
To appear in: Received date: Revised date: Accepted date:
19-4-2016 3-5-2017 3-5-2017
Please cite this article as: Jian Liu, Albert C.To, Quantitative texture prediction of epitaxial columnar grains in additive manufacturing using selective laser melting (2010), http://dx.doi.org/10.1016/j.addma.2017.05.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Quantitative texture prediction of epitaxial columnar grains in additive manufacturing using selective laser melting7 Jian Liu, Albert C.To* Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15261 *
Corresponding author ([email protected])
Highlights
A texture prediction method was proposed for epitaxial columnar grains in SLM. The texture prediction method was combined with the melt pool prediction. Process and microstructure were linked quantitatively for the metal SLM AM process. Texture evolution with the number of layers for SLM AlSi10Mg was simulated. The simulated texture showed pattern and intensity similar to experiment results.
Abstract Metal additive manufacturing (AM) such as selective laser melting (SLM) has the powerful capability to produce very different microstructural features, hence different mechanical properties in metals using the same feedstock material but different values of process parameters. However, the relation between processing-microstructure is mostly investigated by experiments, which is expensive and time-consuming since the parameter space is quite large. The lack of a reliable theoretical model of the processingmicrostructure relationship of AM material is preventing AM technology from being widely adopted by the manufacturing community. Hence, the goal of this work is to establish the link between the microstructure (texture) and the process parameters (laser power, scanning speed, preheat and scanning strategy) of a metal SLM process. To achieve the above goal, a quantitative semi-empirical method is proposed to predict the texture of the epitaxial columnar grains grown from polycrystal substrates. Combined with the melt pool prediction by the Rosenthal solution, the processing and microstructure were linked together quantitatively. The proposed method is used to estimate the texture evolution with the number of layers for EOS-DMLS-processed AlSi10Mg (unidirectional scanning direction in one layer and no rotation of scanning direction between layers). The texture reaches a steady state after five layers, and the steady state texture has similar pattern and intensity to that obtained from the experiment using the same process parameter values and scanning strategy.
Keyword: additive manufacturing; selective laser melting; texture; microstructure
1. Introduction Selective laser melting (SLM) is a powder bed fusion metal AM technique that utilizes a moving laser beam to melt the powder particles in each layer selectively, which leaves behind metallic layers joined together to form the desired geometry [1, 2]. Previously, optimization of the process parameters was predominately for the goal of minimizing the porosity [3]. For metallic structural components, the
microstructure such as grain size, shape and texture also significantly influence the properties, including elastic modulus, strength, anisotropy, plastic deformation behavior, fatigue and crack propagation properties. In general, AM has the ability to produce very different microstructural features in the metals deposited using the same feedstock material but different values of process parameters [4-7], which cannot be achieved by traditional metal part fabrication techniques due to their inability to control heat transfer conditions at very small length scales. The possibility of controlling the microstructure of metals using SLM has spurred intense research in predicting microstructure from the given process parameters. This work proposes a new model that takes the key SLM process parameters as input and predicts the grain texture in a quantitative manner. The proposed model builds upon previous works in predicting the columnar-to-equiaxed transition (CET) through melt pool modeling as a function of the process parameters. For instance, Gaumann et al. [8] integrated the modified Rosenthal melt pool model [9] and Hunt’s CET model to obtain processing–microstructure maps which define processing windows for different morphologies, specifically in their case, to ensure columnar mode in the laser repair of single crystal (SX) components. Beuth and co-workers [10] translated the solidification map of Ti–6Al–4V in the 𝐺-𝑅 (thermal gradient-solidification rate) space [11] to a process map in the 𝑃-𝑉 (beam power-velocity) space using finite element analysis to obtain the G and R values. Finite element analysis was performed for the deposition of single beads of Ti-6Al-4V via electron beam (EBM) wire feed AM processes. The local 𝐺 and 𝑅 not only depend on the process parameters, but also vary up to an order of magnitude along the melt pool. The results presented were only for the top of the melt pool, which is a critical location. Dehoff et al. [6] developed electron beam scan strategies to achieve site-specific control of the texture of Inconel 718 based on principles of columnar-to-equiaxed transitions. An analytical model for fusion welding melt pool prediction compiled by Grong [12] was used to guide the choice of process parameters to produce controlled spatial variation of G and R that will generate different solidification grain morphology according to the solidification map of Inconel 718 [13]. Above are some examples in integrating the melt pool prediction together with the CET model. Note the research is limited to a qualitative evaluation of the overall grain morphology by evaluating the predicted 𝐺 𝑛 /𝑅 ratio at a critical point of the melt pool. No model to predict the detailed grain texture was proposed. The effect of process parameters on the texture is the subject of research in this paper. To the best of the authors' knowledge, there has been no research yet to quantitatively model and predict the texture of the solidified metal when the base metal is polycrystalline which is the case in powder-bed SLM. The objective of this work is to develop a quantitative semi-empirical method to estimate the texture of the epitaxial columnar solidified microstructure on a polycrystal (PX) substrate. Combined with the melt pool prediction by the Rosenthal solution, the process parameters and microstructure can be linked together quantitatively. This paper is organized as follows. Section 2 reviews the necessary background for the formulation of the model including the melt pool models, the CET model and the epitaxial columnar growth on single crystal (SX) substrates. After introducing the Rosenthal melt pool model and discussing the local epitaxial columnar dendrite growth from both SX and PX bases, Section 3 presents the proposed method to estimate the texture in detail. Section 4 presents the results and discussion of melt pool profile and the texture evolution with the number of layers, of AlSi10Mg processed by the EOS DMLS system. This is followed by conclusion and future works in Section 5.
2. Background Inherent to the SLM and other high-energy beam based process is the formation of a pool of molten metal (i.e. melt pool) directly below the heat source with melting occurring ahead of the heat source and solidification behind it. The solidified microstructure depends on local solidification conditions at the trailing edge of the melt pool. Hence, the melt pool geometry (i.e. size and shape) and thermal profile (i.e. thermal gradient and cooling rate, etc.) need to be first predicted as a function of the process parameters. Then the microstructure can be predicted using the melt pool profile based on solidification theory. The following provides the background information necessary to discuss formulation of the model. The temperature field around the moving heat source can be predicted by simple analytical models [14] or complex numerical simulations [15-19]. Experimentally, the temperature field can be measured using the IR or near-IR cameras [20, 21]. Once the temperature field is known, the melt pool shape can be obtained as the liquidus isotherm, i.e., the surface at which 𝑇 = 𝑇𝑚 (𝑇𝑚 is the melt temperature of the material and the undercooling of the solidifying front is ignored). The local thermal variables such as thermal gradient 𝐺 = |𝜵𝑇| and cooling rate 𝑇̇ = 𝜕𝑇/𝜕𝑡 can also be calculated along the boundary of the melt pool. SLM is a rather complex process controlled by mass transfer (powder) and by heat and fluid flow in the melt pool. For this reason, experimental measurement of the complex physical phenomena is difficult because the process is both highly transient and localized. Numerical simulations consider these physical phenomena (mass, energy and momentum conservation) and solve the governing equations using numerical techniques within a certain spatial and temporal domain. The finite element thermal analysis [15-17] and coupled flow-thermal modeling and simulation usually performed at the powder scale [18, 19] are two representative groups of numerical simulations. Those complex numerical simulations are usually time-consuming and not suited for applications such as real-time control [22] and the exploration of the entire process parameter space for process optimization [23]. For computational efficiency, analytical solutions or correlation models in simple mathematical form are useful because essential physics are taken into account without the need for numerical simulation. It usually deals with the quasi-steady state solution in the moving coordinate system attached to the heat source. Rosenthal [14] first proposed the analytical solution of the temperature field relative to a point heat source that is moving on a semi-infinite solid substrate. In order to obtain a simple close form solution, that work considered only the conduction of heat along with other simplifications such as 1) temperature-independent thermal properties, 2) no latent heat, beside the assumptions of point source and semi-infinite boundary condition. These solutions can be used to predict the temperature field at a distance far from the heat source, but the prediction of temperature in the vicinity of the heat source is only a rough approximation. For example, the transverse cross-sections of the melt pool predicted by Rosenthal solution will always be a semi-circle, which will have an impact on the prediction of columnar solidification texture. Cline and Anthony [9] modified Rosenthal’s theory to include a two-dimensional (2D) surface Gaussian distributed heat source with a constant distribution parameter and found an analytical solution. Their solution is a significant step to improve the temperature prediction in the near heat source regions [24]. A similar solution has been derived by Eagar and Tsai [25]. Although the analytical solution for 3D double ellipsoidal heat source has also been derived [24], it is more useful for
the deep penetration welding. For simplicity, only the original Rosenthal model will be considered in this work. With the melt pool geometry and thermal profile, and based on the recent solidification theory, the resulting solidified microstructure such as the grain size, shape and texture can be predicted. The important solidification parameters include thermal gradient 𝐺, cooling rate 𝑇̇ and solidification rate 𝑅. In general, the solidification morphology can be planar, cellular, columnar dendritic or equiaxed dendritic, depending on the cooling rate and the 𝐺 𝑛 /𝑅 values at the solid-liquid interface [26]. The CET model considers the competition between (1) the columnar dendrite growth of the seed crystal provided by the partly melted base metal, and (2) the nucleation and growth of equiaxed grains within the undercooled liquid ahead of the tip of the dendrite. The analytical model for the CET model was first developed by Hunt in 1984 [27]. The volume fraction of equiaxed grains was obtained by integrating the growth rate from the time that nucleation starts until the time that the columnar front reaches the equiaxed grains [8]. In Hunt's simplified CET model, different 𝐺 𝑛 /𝑅 ratio corresponds to different volume fraction ∅ of equiaxed grains. Hunt proposed that the structure is fully equiaxed if ∅ > 0.49 (columnar dendrites will be completely blocked), fully columnar if ∅ < 0.0066 and a mixture of columnar and equiaxed if 0.0066 < ∅ < 0.49. Hunt's simplified CET model can be represented as a microstructure/solidification map in the 𝐺-𝑅 space in the form of the two curves of 𝐺 𝑛 /𝑅 which corresponds to ∅ = 0.49 and ∅ = 0.0066, respectively [28]. Such microstructure/solidification maps in the 𝐺-𝑅 space have been experimentally calibrated for many materials such as Ti–6Al–4V [11], Ni-based super alloy Inconel 718 [13] and CMSX-4 [8]. In the case of a single crystal substrate as in the case of laser repair of SX superalloy components, epitaxial columnar growth will result in a SX solidification microstructure, which is preferred. Hence, the CET has been the focus in early studies of laser processing of single crystal components to ensure columnar growth and to avoid equiaxed growth mode [28-30]. Also, since the thermal gradient direction varies along the boundary when base metal orientation is constant, it is possible for the changes of dendrite growth direction. This phenomenon is known as oriented to misoriented transition (OMT) and has also been observed and modeled in the repair of single-crystal turbine parts using the laser-based deposition process [31, 32]. When the substrate is a polycrystal, the grain size, shape (elongation direction and aspect ratio) and texture of the epitaxial columnar grains all need to be predicted. The grain size is observed to be proportional to the melt pool size [10]. The elongation is along the direction of dendrite growth. The texture depends on the geometric and thermal profile of the melt pool and also the texture of the base metal.
3. Model Formulation 3.1. Melt pool The laser beam is assumed to be a point heat source and part of the laser power is absorbed by the powder layer when the laser beam travels through. When the laser beam reaches the substrate, it will interact with the substrate and generate a temperature field (and melt pool) due to heat conduction. Figure 1 is a schematic of the trailing (solidifying) part of the melt pool formed under the moving laser beam. The quasi-steady state temperature field and melt pool can be modeled using the Rosenthal solution [14]:
𝜂𝑃
𝑇(𝑥0 , 𝑦0 , 𝑧0 ) = 𝑇0 + 2𝜋𝑘 𝑒 −𝜆𝑉𝑥0
𝑒
2 2 −𝜆𝑉√𝑥2 0 +𝑦0 +𝑧0
(1)
√𝑥02 +𝑦02 +𝑧02
where 𝑥0 , 𝑦0 , 𝑧0 are the location of a point in the coordinate system attached to the moving point source, 𝑇0 is the initial temperature of the solid, 𝜂 is the absorption coefficient of laser energy by the substrate, 𝑘 is the thermal conductivity, and 1/2𝜆 is the thermal diffusivity of the material. Using the Rosenthal model, the local thermal variables such as thermal gradient 𝐺 = |𝜵𝑇| and cooling rate 𝑇̇ = 𝜕𝑇/𝜕𝑡 can also be calculated along the boundary of the melt pool. Another important solidification parameter is the solidification rate 𝑅, which is related to the beam velocity and melt pool shape: 𝑅 = 𝑉𝑐𝑜𝑠𝜃
(2)
where 𝜃 is the angle between direction of the moving source and growth direction of the solidifying material. The local solidification parameters not only depend on the process parameters but also vary along the melt pool. At a location (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 ) on the solidifying front of the melt pool, the local solidification mode depends on the local 𝐺 and 𝑅. In Hunt’s model, the CET transition is characterized by a critical 𝐺 𝑛 /𝑅 ratio [8]. If the 𝐺 𝑛 /𝑅 is greater than a critical value, the solidification is fully columnar, and dendrite arms will grow from the solid-liquid interface.
3.2. Local epitaxial columnar dendrite growth: SX and PX base In the case of SLM AM, the solid (base) metal and the liquid metal on either side of the solid-liquid interface have similar chemical composition, hence the same crystal structure. Columnar grains will grow epitaxially from the seed crystal provided by the base metal. The dendrite arm will have the same crystallographic orientation as the seed crystal, and the growing direction will be along the <001> axis of the seed crystal that is most closely aligned with the thermal gradient vector [33]. The SX base (Fig. 2a) can only provide a seed crystal of a fixed orientation identical to the SX base. Hence, the dendrite will have the same crystallographic orientation as the SX base metal. The dendrite growing direction will be along one of the <001> axes (for FCC) of the SX base that has the maximum: 𝑐𝑜𝑠𝜑 = (𝒎ℎ𝑘𝑙 ∙ 𝑮𝑖 )/|𝑮𝑖 |
(3)
where 𝑮𝑖 is the local thermal gradient vector at the location (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 ) and 𝒎ℎ𝑘𝑙 (𝒎100,𝒎010 and 𝒎001 ) are the direction vectors of the three <001> axes of the SX base. The PX base (Fig. 2b) could provide seed crystals of different orientations. Within a small element surface around a point (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 ), we assume initially there are multiple (𝑁𝑠𝑒𝑒𝑑 ) possible seed crystals instead of one seed crystal with a fixed orientation in the case of the SX base. The texture (crystallographic orientation distribution) of a polycrystal metal is usually represented by a collection of a large number of grain orientations. For each possible seed crystal (j=1: 𝑁𝑠𝑒𝑒𝑑 ), the crystallographic orientation (𝒎ℎ𝑘𝑙,𝑗 ) can be determined in a statistical way, i.e., by randomly choosing one orientation out of the collection of
orientations that represent the PX base texture. The growing direction of the final dendrite from the location (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 ) will be along the direction of 𝒎ℎ𝑘𝑙,𝑗 of the seed crystal 𝑗 that has the maximum: 𝑐𝑜𝑠𝜑 = (𝒎ℎ𝑘𝑙,𝑗 ∙ 𝑮𝑖 )/|𝑮𝑖 |
(4)
Naturally, the crystallographic orientation of the final dendrite will be the same as the selected seed crystal 𝑗. One special case is the fine-grain random-texture PX base. If the base metal is a fine-grained randomtextured polycrystal, it is safe to assume that, at each location, there is statistically always a seed crystal that provides one <001> direction that is parallel to the thermal gradient and the dendrite arm from this seed crystal will outgrow all others. The growing direction of the final dendrite arms will always be perpendicular to the boundary (parallel to the thermal gradient vector) along the front. The procedures outlined above should give this result if the number of grain orientations representing the random texture and the number of initial possible seed crystals are both large enough. To summarize, the columnar epitaxial dendrite grows from a location will have the same crystallographic orientation as the seed crystal, and the growing direction will be along one of the <001> axes of the seed crystal that is most closely aligned with the thermal gradient. The seed crystal is fixed with the same orientation as the base in the case of SX base. In the case of the PX base, the final seed crystal can be determined, in a statistical way, by the texture of the base and thermal gradient at the solid-liquid interface.
3.3. Solidification microstructure: SX and PX substrate The epitaxial columnar dendrite growth at a specific location has been discussed. Now the overall solidification microstructure can be analyzed by first considering one solidified section that solidifies from bottom to up as the solidifying front is sweeping through. The overall solidification microstructure depends on the microstructure of the substrate. Figure 3 is a schematic side-by-side comparison of the solidification microstructure on SX and PX substrates with both the longitudinal (i.e. along the laser scan direction) and transverse cross-section view. Fig. 3a and 3b are the schematics of the longitudinal cross-section at 𝑡0 with the heat source moving from left to right. The parts behind the solidifying front of the melt pool are the already solidified microstructures, which will become clear after the analysis of solidified sections in the following discussion. For one solidified section, the substrate serves as the base metal only at the beginning of solidification (defined the time as 𝑡0 in Fig. 3), Later, the already solidified metal will serve as base for the following solidification until the solidification is complete (defined the time as 𝑡1 in Fig. 3). At the beginning of solidification (𝑡0 ) on a SX substrate (Fig. 3c), the dendrite arms with the same crystallographic orientation as the substrate will grow in one of the <001> axis directions. Depending on the local thermal gradient direction and the substrate orientation, the dendrite growth from each location could be along the original dendrite arm direction in the SX (oriented) or not (mis-oriented) [31, 32]. Such oriented/mis-oriented dendrites will keep growing until the occurrence of CET (𝑡𝐶𝐸𝑇 , if it occurs) and the rest of solidification will be complete in the equiaxed mode. The columnar region of the section solidified
from a SX substrate is still a SX with the same crystallographic orientation but different dendrite growth directions at different locations. At the beginning of solidification (𝑡0 ) on a PX substrate (Fig. 3d), at different locations of the bottom solidifying boundary, dendrite arms with different crystallographic orientation will nucleate and compete with neighboring dendrites. Some of the dendrites will outgrow others, reach the final trunk diameter, and keep growing into individual elongated grains until the occurrence of CET (𝑡𝐶𝐸𝑇 ) or the completion of solidification. In the columnar region, the solidified section is now a polycrystal. If the crystallographic orientations of the elongated grains are known, the texture of this solidified section will be determined.
3.4. Prediction of the solidification texture on PX substrates The texture of one solidified section can be obtained by a numerical procedure. The number of columnar grains in the solidified section is related to the cooling rate and also proportional to the section area (melt pool size). Assuming the number of grains in one section is 𝑁𝑔𝑟𝑎𝑖𝑛 , and then the solidifying boundary can be discretized into 𝑁𝑔𝑟𝑎𝑖𝑛 small segments. The average thermal gradient at each segment can be obtained using the melt pool model outlined in section 3.1. The crystallographic orientation of the dendrite from each segment, which will grow into an elongated grain, can be determined by the procedure outlined in section 3.2. The resulting overall texture of one solidified section can be obtained by adding all the determined grain orientations into a collection. Now, for the texture of one solidified section, the total number in the collection of grain orientations is 𝑁𝑔𝑟𝑎𝑖𝑛 . The crystallographic orientation of each grain in one solidified section will have one <001> axis being deterministic, i.e., closely aligning (even parallel) to the thermal gradient. The other two axes are somewhat probabilistic, i.e., only loosely restrained by the orientation distribution (texture) of the substrate. Since a statistical step of selecting the possible seed crystal is involved, numerically repeating the texture estimation for the solidified section could give different results each time regarding two <001> axes of the solidified grains (again, one <001> will be determined), which physically represents a different solidified section along the track. The procedure can be repeated to obtain and collect the texture of a number (𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 ) of solidified sections to represent the texture of a track, then the total number in the collection of grain orientations is 𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 × 𝑁𝑔𝑟𝑎𝑖𝑛 . In reality, one layer is built by many short segments of tracks. Since we are assuming the melt pool and solidification being quasi-steady, segments of tracks of the same scanning direction in one layer could be considered belonging to one track. Hence, the aforementioned texture estimation of a track can represent the texture of a layer. However, if the scan direction in one layer is bidirectional, then the forth and back scan direction will generally give different results. In such a case, the layer is considered consisting of two tracks and the texture of both tracks need to be estimated and collected. If we use 𝑁𝑡𝑟𝑎𝑐𝑘 to denote the type of track (𝑁𝑡𝑟𝑎𝑐𝑘 = 1 for unidirectional and 𝑁𝑡𝑟𝑎𝑐𝑘 = 2 for bidirectional), for the texture of one layer, the total number in the collection of grain orientations is 𝑁𝑡𝑟𝑎𝑐𝑘 × 𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 × 𝑁𝑔𝑟𝑎𝑖𝑛 . For the first layer, the texture of the substrate (building platform) is usually random, and the resulting texture of the first layer will be non-random. This non-random first layer texture will serve as the substrate texture for the second layer and so on. The rotation of scan direction between layers can be accounted for by rotating the substrate (previous layer) texture in the opposite way. Hence, the effect of scan strategy is taken into account. If we repeat the texture estimation to obtain and collect the texture of a
number (𝑁𝑙𝑎𝑦𝑒𝑟 ) of layers to represent the texture of 3D material, the total number in the collection of grain orientations is 𝑁𝑙𝑎𝑦𝑒𝑟 × 𝑁𝑡𝑟𝑎𝑐𝑘 × 𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 × 𝑁𝑔𝑟𝑎𝑖𝑛 .
4. Results and discussion The solidification texture of AlSi10Mg processed by the EOS DMLS system was estimated using the procedure outlined above. First, the Rosenthal solution is used to simulate the melt pool generated during the processing of AlSi10Mg by the EOS DMLS system. The values for the process parameters employed are 𝑃 = 200 W, 𝑉 = 1400 mm/s, 𝑇0 =23 oC [34]. The material properties employed are k = 150.0 W/(m∙K), 1/2𝜆 = 8.418e-5 m²/s, and 𝑇𝑚 = 660 oC [35] and the absorption factor is assumed to be 𝜂 = 0.5. Assuming that the heat source travels from left to right, the results obtained from the Rosenthal solution are illustrated in Fig. 4. The original Rosenthal model was employed to obtain the thermal gradient profile on the solidifying boundary at the beginning of solidification of one solidified section. Now the predicted bottom solidifying boundary is a semi-circle from original Rosenthal model (Fig. 5). The number of grains in one solidified section was assumed to be 𝑁𝑔𝑟𝑎𝑖𝑛 = 20 and hence the bottom solidifying boundary was then discretized into 20 small segments (Fig. 5). The number of 𝑁𝑔𝑟𝑎𝑖𝑛 = 20 is based on the experimental results of Ti-6Al4V processed by EBM [36], in which the average beta grain width was plotted against the average experimentally measured effective melt pool and approximately 20 beta grains were seen across the melt pool width. The average thermal gradient at each segment was obtained using the Rosenthal model. In predicting the texture of the first layer, the initial platform serves as the substrate and its texture is represented using a collection of 5,000 randomly distributed grain orientations. In each segment (𝑖 = 1: 20 in Fig. 5), the dendrite (grain) orientation is determined by equation (4) with 𝑁𝑠𝑒𝑒𝑑 = 100. The resulting texture of one solidified section can be obtained by adding all the determined grain orientations into a collection. The procedure was repeated to obtain and collect the texture of 𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 = 1,000 solidified sections to represent the texture of one track. The scanning pattern being simulated here is a unidirectional scanning direction in one layer (𝑁𝑡𝑟𝑎𝑐𝑘 = 1), and hence the texture of a track is that of a layer. Fig. 6a is the estimated texture of the first layer (determine and collect dendrite orientation from segment 𝑖 = 1: 20 in Fig. 5 and repeat 1000 (𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 ) times) growing from the random-textured substrate (platform) in the form of <001> pole figure. At the first glance, one would presume a <001> fiber texture along the scanning direction (SD) which was also concluded by Thijs et al. [34]. However, a different interpretation could be obtained by examining the texture of the collections of grains that grow from one particular location (determine and collect dendrite orientation from one particular segment 𝑖 in Fig. 5 and repeat 1000 (𝑁𝑠𝑒𝑐𝑡𝑖𝑜𝑛 ) times). Fig. 6b, c, d are the estimated textures of the collection of 1,000 grains growing from three different locations (segment 𝑖 = 1, 5, 10 in Fig. 5) in the form of <001> pole figure. For a collection of 1,000 grains growing from one particular location, the texture is a strong <001> fiber along the thermal gradient of this location. The reason for the strong <001> fiber is that the dendrites (grains) grow from one particular location at the bottom solidifying boundary will have one <001> axis being deterministic. One <001> axis will be closely aligning (even parallel) to the thermal gradient while the other two axes are somewhat probabilistic (Section 3.4). Texture of the track/layer (Fig. 6a) can now
be considered as the collection of 20 (𝑁𝑔𝑟𝑎𝑖𝑛 ) fiber texture components instead of one fiber texture component along the SD direction. The scanning pattern being simulated here is that there is no rotation of the scanning direction between layers, hence the resulting texture of the previous layer will be used directly as the substrate texture for the next layer. Fig. 7 is the estimated texture by the aforementioned procedures and its evolution with the number of layers. At the fifth layer (Fig. 7e), the texture reaches a steady state, and the steady state texture has a similar pattern and intensity to that obtained from the experiment [34] (Fig. 7f) using the same process parameter values and scanning pattern. In terms of pattern, the layer texture begins with a seemingly <001> fiber component along SD (which is actually a collection of 20 (𝑁𝑔𝑟𝑎𝑖𝑛 ) fiber texture components) at the first layer (Fig. 7a). With the increase of layers, the directions of fiber components change from evenly distributed to concentrating around a maximum intensities of −45o to +45o to the building direction (BD). In terms of intensity, the maximum MRD (multiples of random density) of the experimental texture is 3.7 (Fig. 7f), while the prediction of the steady-state texture (Fig. 7e) is 2.5. The difference in these values is likely due to the inherent inaccuracy in the original Rosenthal model from the various simplifying assumptions. Since the bottom solidifying boundary predicted by the original Rosenthal solution will always be a semicircle, the variation of process parameters such as P and V will only affect the magnitude of the thermal gradient on each segment but not the direction. According to equation (4), only the direction of the thermal gradient affects the selection of seed crystals. As a result, by using the original Rosenthal solution, the process parameters P and V have no effect on the solidification texture. The texture is only affected by the substrate texture and the scanning pattern. Better agreement between the predicted and experimental texture can be expected if finite element analysis is employed instead, as the solidifying boundary is not fixed as semi-cycle, and the effect P and V can be captured by using FEA. Another advantage of using FEA is to further improve the temperature field prediction, as well as remove the assumptions of semi-infinite boundary conditions, temperaturedependent thermal properties, and no latent heat. This will be explored in the future to determine the accuracy and efficiency of using them to predict the grain texture using the proposed processmicrostructure model.
5. Conclusions In this work, a quantitative method linking the microstructure (texture) and the process parameters (laser power, scanning speed, preheat and scanning patterns) has been developed for the SLM metal AM processes. To achieve the above goal, 1) the Rosenthal model was used to predict the melt pool geometry and thermal profile and 2) a quantitative semi-empirical method to estimate the texture of the epitaxial columnar grains which grow from polycrystal substrates was developed. In the texture estimation, one solidified section was assumed to have a constant number of grains, where each grows from an epitaxial columnar dendrite from the bottom solidifying boundary. The crystallographic orientation of each dendrite was determined using the local average thermal gradient direction of the segment on the bottom solidifying boundary together with a statistical step of selecting seed crystals from the substrate. The texture of one solidified section was obtained by determining and collecting the orientation of each dendrite. The texture of one track was obtained by repeating the
estimation and collection of texture for a number of sections. The texture of a layer was considered the same as the track if the scanning is unidirectional. Finally, the resulting texture of one layer was used as the substrate texture for the next layer and the effect of scanning direction rotation between layers was taken into account. The proposed method was used to estimate the texture and its evolution with the number of layers for EOS-DMLS-processed AlSi10Mg (unidirectional scanning direction in one layer and no rotation of scanning direction between layers). The texture reaches a steady state after five layers, and the steady state texture has similar pattern and intensity to that obtained from the experiment using the same process parameter values and scanning pattern. In the future, a more advanced melt pool model based on the FEM method will be used to obtain more accurate geometry and thermal profile of the melt pool, which should result in a more accurate texture prediction. The improved process-microstructure model with the more advanced melt pool model will be used to predict the texture of SLM metals using scan patterns other than uniaxial scanning direction in one layer and rotation of scanning direction between layers. The findings from these research directions will be reported in future publications. Acknowledgement The financial support from the National Science Foundation (CMMI- 1434077) is gratefully acknowledged.
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Figure 1. A schematic of the trailing (solidifying) part of the melt pool formed under the moving laser beam.
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Figure 2. Epitaxial columnar dendrite arm growth from (a) SX and (b) PX base metals.
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Figure 3. Longitudinal cross-section at beginning of solidification (𝑡0 ) of one section on a (a) SX and (b) PX substrate; and transverse cross-section at beginning of solidification (𝑡0 ) on a (c) SX and (d) PX substrate.
Figure 4. Predicted 3D melt pool size and shape of the EOS DMLS-processed AlSi10Mg employing the Rosenthal solution. The laser is moving from left to right.
Figure 5. Transverse cross-section at beginning of solidification on a PX substrate with the bottom solidifying boundary predicted by original Rosenthal solution.
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Figure 6. (a) First layer texture of columnar grains grow from random-textured substrate (platform) in the form of <001> pole figure. And texture of the collection of 1000 grains grow from three different locations, i.e., segment i = 1(b), i = 5 (c), i = 10 (d) in Fig. 5 in the form of <001> pole figures.
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Figure 7. Texture evolution of solidified microstructure estimated by the proposed method for EOS-DMLS-processed AlSi10Mg: (a) first layer, (b) second layer, (c) third layer, (d) fourth layer, (e) fifth layer, (f) experiment [34] using the same process parameter and scanning pattern.