**1. Introduction**

Laser-based powder bed fusion of polymers (PBF-LB/P) is a powder-based additive manufacturing process that allows for manufacturing individualized components with a high geometric freedom. To date, PBF-LB/P is predominantly associated with the quasi-isothermal processing of semi-crystalline polymers. Given the continuous heating of the build chamber in quasi-isothermal PBF-LB/P, temperature-induced aging of polymers [1], increased process times due to heating and cooling phases, and the influence of processing times on resulting mechanical properties [2,3] inherently restrict the economic and ecological viability for the cost-efficient manufacturing of polymer components. Considering non-uniform temperature fields occurring in PBF-LB/P, the isothermal assumption merely represents an idealization. With regard to isothermal crystallization kinetics of Polyamide 12, findings by Neugebauer et al. [4] indicate the occurrence of isothermal crystallization during isothermal PBF of polymers. Using a process-integrated approach, Drummer et al. [5] determined the time-dependent occurrence of isothermal crystallization in PBF-LB/P, proposing the possibility of novel process strategies for limiting the isothermal processing zone to the powder bed surface. Considering a time- and temperature-dependency of the isothermal crystallization process, findings by Soldner et al. [6] indicate a non-uniform, geometry-dependent isothermal crystallization [7]. Findings derived by Shen et al. [8] based on a numerical approach indicate a correlation of the

**Citation:** Schlicht, S.; Greiner, S.; Drummer, D. Low Temperature Powder Bed Fusion of Polymers by Means of Fractal Quasi-Simultaneous Exposure Strategies. *Polymers* **2022**, *14*, 1428. https://doi.org/10.3390/ polym14071428

Academic Editors: Swee Leong Sing and Wai Yee Yeong

Received: 21 February 2022 Accepted: 28 March 2022 Published: 31 March 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

underlying cooling rate and resulting residual stress, thus affecting part distortion. Consequently, controlled isothermal crystallization is omnipresent in PBF-LB/P, being influenced by time-, temperature-, and geometry-dependent effects. However, even considering the occurrence of isothermal crystallization in PBF-LB/P, the non-isothermal processing below the crystallization onset temperature of semi-crystalline polymers remains restricted due to the occurrence of stress-induced distortion, specifically curling and warping [9]. Induced by the inhomogeneous crystallization of the polymer melt, curling considerably reduces the process stability, leading to process interruptions. Crystallization-induced shrinkage is inherently bound to the processing of semi-crystalline polymers, thus constituting the requirement of novel strategies for controlling the crystallization kinetics for promoting a uniform, controlled crystallization of each layer. Therefore, the non-uniform crystallization of the applied materials constitutes an inherent challenge for novel processing strategies, focusing on the exposure-induced process optimization.

#### **2. State of the Art**

#### *2.1. Kinetics of Isothermal and Non-Isothermal Crystallization of Polymers*

Quasi-isothermal processing composes the state of the art in laser-based powder bed fusion of polymers. The isothermal assumption in PBF-LB/P implies the predominant occurrence of isothermal crystallization during the build process. A basic modelling of isothermal crystallization processes can be derived using the Avrami equation. Considering non-constant cooling rates, occurring in laser sintering of polymers in quasi-isothermal as well in non-isothermal processing, the Nakamura model, proposed by Nakamura et al. [10, 11] allows for considering non-isothermal crystallization. The macroscopic degree of crystallization, *α*, can be expressed in dependence of the Nakamura kinetics crystallization function *K*(*T*), being closely related to the Avrami function *k*(*T*).

$$\mathfrak{a} = \mathbf{1} - \exp\left[-\left(\int\_0^t \mathbf{K}(t)\mathbf{d}t\right)^n\right] \tag{1}$$

The underlying relation of the Nakamura crystallization rate *K*(*t*) and the Avrami crystallization rate *k*(*t*) can be expressed using a temperature-dependent, dimensionless parameter *n*.

$$K(t) := k(t)^{\frac{1}{n}} \tag{2}$$

Ziabicki [12] described an empirical exponential relation of the crystallization half time *t*1/2, the growth constant *K*<sup>0</sup> and the nucleation rate constant *K<sup>g</sup>* by applying the Lauritzen– Hoffman theory, with the activation energy for polymer diffusion *U\** = 6270 J mol−<sup>1</sup> , the universal gas constant *R*, the temperature value *T*<sup>∞</sup> = *T<sup>g</sup>* − 30 K, indicating a ceased viscous flow, and the equilibrium melting temperature *T* 0 *<sup>m</sup>*, displayed in Equation (3).

$$\frac{1}{t\_{1/2}} = K\_0 \exp\left[-\frac{\mathcal{U}^\*}{R(T - T\_\infty)}\right] \exp\left[\frac{K\_\mathcal{S}(T + T\_m^0)}{2T^2(T\_m^0 - T)}\right] \tag{3}$$

Zhao et al. [13] applied the empirical relation on Polyamide 12, used in laser-based powder bed fusion, by fitting the parameters *K*<sup>0</sup> and *K<sup>g</sup>* based on experimental data obtained from differential scanning calorimetry. The resulting crystallization half times, presented by Zhao et al., exhibit a satisfactory accordance of experimentally obtained and modelled values for sufficiently high cooling rates. Consequently, a reduced processing temperature *T* is correlated with considerably reduced crystallization half times. Considering isothermal crystallization of quenched Polyamide 12 at varying ambient temperatures, Paolucci et al. [14] modelled the temperature-dependent formation of varying crystalline phase compositions of Polyamide 12. Unpressurized crystallization at temperatures exceeding 100 ◦C predominantly leads to the formation of the α-phase [14], implying the formation of an identical crystalline phase within a wide thermal processing window. However, with regard to described crystallization kinetics, a dependency of the applied cooling rate and morphological properties needs to be considered a major influence in low temperature PBF-LB/P.

#### *2.2. Low Temperature Laser-Based Processing of Polymers*

In contrast to powder bed fusion of metal alloys, PBF of polymers is characterized by the avoidance of support structures. With regard to the application of support structures, different approaches for transferring metal-based concepts on the non-isothermal processing of polymers [15] have been described. Resulting material morphologies exhibit significantly reduced spherulite sizes, correlated with an increased elongation at break while exhibiting an insufficient porosity level [16]. In contrast to the application of support structures, the avoidance of curling and warping by means of laser-based preheating has been proposed by Laumer et al. [17], applying simultaneous laser beam irradiation for the isothermal manufacturing of multi-material components. Investigations conducted by Chatham et al. [18] for the manufacturing of polyphenylene sulfide at reduced powder bed temperatures exhibit significantly increased levels of porosity, thus limiting the applicability of produced parts. Consequently, to date, the non-isothermal PBF of polymers is inherently limited with regard to the requirement of support structures and the emergence of insufficient part properties.

#### *2.3. Influence of Exposure Strategies on Superficial Temperature Fields and Part Properties*

The application of a variety of exposure strategies has been described for both metalbased and polymer-based PBF. The interaction of an exposed geometry and the applied exposure strategy on resulting temperature fields is described extensively in recent literature. Exhibiting a reduced thermal penetration depth, increased exposure speeds are correlated with increased superficial maximum temperature values [19,20]. In addition, resulting superficial temperature fields show a dependence on both the applied exposure speed and the underlying scan vector length [19], resulting from a thermal superposition of subsequently exposed scan vectors [20]. Jain et al. [2] describe a correlation of geometry-induced, varying return times of the laser beam and varying mechanical properties of samples manufactured using Polyamide 12, implying an influence of superficial temperature fields on resulting mechanical part properties. Exceeding the monitoring of exposure-induced temperature fields, Greiner et al. (2021) [21] observed an interdependence of applied exposure parameters and the underlying geometry on post-exposure temperature fields, leading to a varying morphological structure of fabricated parts.

Segmented exposure strategies, widely applied in laser-based PBF of metal alloys (PBF-LB/M), exhibit reduced residual stresses, described for the application of steel [22–24] and nickel alloys [25]. Zou et al. (2020) [22] describe a significant influence of the exposure sequence and orientation of distinct segments on resulting residual stress, emphasizing structural advantages of non-linear sequencing compared to linear sequencing. Considering varying exposure patterns of distinct segments, further complex interdependencies of the applied sequence of exposed scan vectors, the scan vector length and the sequence of exposed sub-segments are described. With regard to the inherent geometry-dependence of linear exposure patterns, non-linear exposure patterns are gaining increased attention both in metal- and polymer-based PBF. The application of non-linear exposure patterns was initially described by Yang et al. [26], applying the space-filling, fractal Hilbert curve for the sintering of polymer-bound ceramic particles. Further research on fractal exposure patterns was conducted by Ma et al. [27] and Catchpole-Smith et al. [28], describing reduced stressinduced distortion and the reduced occurrence of heat-induced cracks in PBF of nickel alloys. Greiner et al. [29] described the application of the fractal, space-filling Peano curve for the PBF of Polyamide 12, leading to geometry-invariant temperature fields promoted by the scale-invariant structure of the applied exposure pattern. Therefore, the application of linear exposure patterns is considerably influenced by the exposed cross-section, leading to a reduced reproducibility of part properties. In contrast, applying segmented, fractal

exposure strategies promote the formation of uniform, geometry-invariant temperature fields that could be exploited for low temperature PBF-LB/P. Consequently, the formation of crystallization-induced residual stress and resulting part deformations exhibits a dependence of applied exposure strategies, hence implying the requirement of novel exposure strategies to overcome existing limitations of quasi-isothermal PBF of polymers. mote the formation of uniform, geometry-invariant temperature fields that could be exploited for low temperature PBF-LB/P. Consequently, the formation of crystallization-induced residual stress and resulting part deformations exhibits a dependence of applied exposure strategies, hence implying the requirement of novel exposure strategies to overcome existing limitations of quasi-isothermal PBF of polymers.

occurrence of heat-induced cracks in PBF of nickel alloys. Greiner et al. [29] described the application of the fractal, space-filling Peano curve for the PBF of Polyamide 12, leading to geometry-invariant temperature fields promoted by the scale-invariant structure of the applied exposure pattern. Therefore, the application of linear exposure patterns is considerably influenced by the exposed cross-section, leading to a reduced reproducibility of part properties. In contrast, applying segmented, fractal exposure strategies pro-

#### **3. Methodological Approach for Low Temperature PBF 3. Methodological Approach for Low Temperature PBF**

*Polymers* **2022**, *14*, x FOR PEER REVIEW 4 of 17

The approach presented in this paper focusses on significantly lowering the build chamber temperature while limiting warping and curling of manufactured parts by means of fractal, quasi-simultaneous laser exposure. In contrast to quasi-isothermal PBF, low temperature PBF, as proposed in this paper, relies on the immediate crystallization of distinct exposed segments, considering a material-specific processing windows below the crystallization peak, displayed in Figure 1. The approach presented in this paper focusses on significantly lowering the build chamber temperature while limiting warping and curling of manufactured parts by means of fractal, quasi-simultaneous laser exposure. In contrast to quasi-isothermal PBF, low temperature PBF, as proposed in this paper, relies on the immediate crystallization of distinct exposed segments, considering a material-specific processing windows below the crystallization peak, displayed in Figure 1.

**Figure 1.** Schematic illustration of process-dependent thermal processing windows. **Figure 1.** Schematic illustration of process-dependent thermal processing windows.

Resulting implications for process temperature control include significantly reduced pre-heating times and the immediate removal of manufactured parts subsequent to the build process. Reduced pre-heating times are obtained considering reduced requirements of the thermal homogeneity in contrast to quasi-isothermal processing, thus limiting the required homogenous thermal field to the thickness of the manufactured layer. Resulting process times of non-isothermal and quasi-isothermal processing, respectively, are displayed in Figure 2. Resulting implications for process temperature control include significantly reduced pre-heating times and the immediate removal of manufactured parts subsequent to the build process. Reduced pre-heating times are obtained considering reduced requirements of the thermal homogeneity in contrast to quasi-isothermal processing, thus limiting the required homogenous thermal field to the thickness of the manufactured layer. Resulting process times of non-isothermal and quasi-isothermal processing, respectively, are displayed in Figure 2. *Polymers* **2022**, *14*, x FOR PEER REVIEW 5 of 17

**Figure 2.** Schematic time-dependent temperature variation for applying quasi-isothermal and nonisothermal processing. **Figure 2.** Schematic time-dependent temperature variation for applying quasi-isothermal and nonisothermal processing.

the application of fractal scan paths [26–29] and quasi-simultaneous exposure of distinct segments. Fractal scan path generation applied within the present paper is based on spacefilling, self-avoiding and self-similar curves, commonly referred to as "FASS curves" [30], specifically on the Peano curve [31]. The implementation of fractal, quasi-simultaneous exposure strategies is based on functional recursive programming, conducted in Python 3.8. Resulting exposure strategies are transferred using the Common Layer Interface (CLI) format to allow for the integration of complex exposure strategies into commercially available machinery. Exposure paths, applied for the non-isothermal processing of polymers, include discrete fractal sub-segments that are exposed using fractal sequencing. The resulting exposure strategy of an exemplary square cross-section is displayed in Figure 3.

To allow for the non-isothermal processing of semi-crystalline polymers, restricting

**Figure 3.** Schematic depiction of the applied fractal exposure pattern.

Figure 4.

x

dhatch

y

z

Quasi-simultaneous exposure of varying geometries is based on the repetitive, consecutive exposure of distinct fractal patterns. Each sub-segment constitutes a closed loop, allowing for an uninterrupted quasi-simultaneous exposure, schematically displayed in

Segment size

To allow for the non-isothermal processing of semi-crystalline polymers, restricting the distortion of exposed segments is essential. Based on previous research on the field of segmented exposure strategies [23,25], segmented exposure strategies are combined with the application of fractal scan paths [26–29] and quasi-simultaneous exposure of distinct segments. Fractal scan path generation applied within the present paper is based on spacefilling, self-avoiding and self-similar curves, commonly referred to as "FASS curves" [30], specifically on the Peano curve [31]. The implementation of fractal, quasi-simultaneous exposure strategies is based on functional recursive programming, conducted in Python 3.8. Resulting exposure strategies are transferred using the Common Layer Interface (CLI) format to allow for the integration of complex exposure strategies into commercially available machinery. Exposure paths, applied for the non-isothermal processing of polymers, include discrete fractal sub-segments that are exposed using fractal sequencing. The resulting exposure strategy of an exemplary square cross-section is displayed in Figure 3. To allow for the non-isothermal processing of semi-crystalline polymers, restricting the distortion of exposed segments is essential. Based on previous research on the field of segmented exposure strategies [23,25], segmented exposure strategies are combined with the application of fractal scan paths [26–29] and quasi-simultaneous exposure of distinct segments. Fractal scan path generation applied within the present paper is based on spacefilling, self-avoiding and self-similar curves, commonly referred to as "FASS curves" [30], specifically on the Peano curve [31]. The implementation of fractal, quasi-simultaneous exposure strategies is based on functional recursive programming, conducted in Python 3.8. Resulting exposure strategies are transferred using the Common Layer Interface (CLI) format to allow for the integration of complex exposure strategies into commercially available machinery. Exposure paths, applied for the non-isothermal processing of polymers, include discrete fractal sub-segments that are exposed using fractal sequencing. The resulting exposure strategy of an exemplary square cross-section is displayed in Figure 3.

**Figure 2.** Schematic time-dependent temperature variation for applying quasi-isothermal and non-

Part removal, isothermal

isothermal processing.

Non-isothermal build process

Build chamber

TB2

TB1

TRT

temperature

*Polymers* **2022**, *14*, x FOR PEER REVIEW 5 of 17

processing Isothermal build process

tPreheating < 600 s tCooling < 60 s

Part removal, nonisothermal processing

Process time

**Figure 3.** Schematic depiction of the applied fractal exposure pattern. **Figure 3.** Schematic depiction of the applied fractal exposure pattern.

Quasi-simultaneous exposure of varying geometries is based on the repetitive, consecutive exposure of distinct fractal patterns. Each sub-segment constitutes a closed loop, allowing for an uninterrupted quasi-simultaneous exposure, schematically displayed in Quasi-simultaneous exposure of varying geometries is based on the repetitive, consecutive exposure of distinct fractal patterns. Each sub-segment constitutes a closed loop, allowing for an uninterrupted quasi-simultaneous exposure, schematically displayed in Figure 4. *Polymers* **2022**, *14*, x FOR PEER REVIEW 6 of 17

Figure 4.

**Figure 4.** Fractal, quasi-simultaneous exposure pattern of a distinct sub-segment based on the Peano curve. **Figure 4.** Fractal, quasi-simultaneous exposure pattern of a distinct sub-segment based on the Peano curve.

Quasi-simultaneous exposure is correlated with a significantly increased layer time due to the repetitive exposure of distinct segments, leading to an increase in the layer time Quasi-simultaneous exposure is correlated with a significantly increased layer time due to the repetitive exposure of distinct segments, leading to an increase in the layer

equivalent to the additional number of exposure steps compared to single exposure. Dis-

quence to reduce geometry-induced influences and interferences on resulting temperature fields. Therefore, fractal patterns are applied on sub-segment level and for determining the sequence of consecutive segments. With regard to the scale-invariance of fractal space-filling curves, the sequence of consecutively scanned segments is determined by the

structure of the Peano curve, schematically displayed in Figure 5.

**Figure 5.** Schematic illustration of the fractal exposure sequence of distinct segments.

responding to the specific part contour.

**4. Materials and Methods**  *4.1. Experimental Set-Up* 

x

y

z

Exposure patterns of complex cross-sections are generated applying the algorithm proposed by Yang et al. [26], thus implicitly generating a closed, space-filling curve cor-

All experimental work is conducted using a freely configurable SLS research system, thus allowing for the integration of complex space-filling scan paths, prepared by means curve.

z

y

x

time equivalent to the additional number of exposure steps compared to single exposure. Distinct, repetitively exposed sub-segments are sequenced by applying a fractal exposure sequence to reduce geometry-induced influences and interferences on resulting temperature fields. Therefore, fractal patterns are applied on sub-segment level and for determining the sequence of consecutive segments. With regard to the scale-invariance of fractal space-filling curves, the sequence of consecutively scanned segments is determined by the structure of the Peano curve, schematically displayed in Figure 5. equivalent to the additional number of exposure steps compared to single exposure. Distinct, repetitively exposed sub-segments are sequenced by applying a fractal exposure sequence to reduce geometry-induced influences and interferences on resulting temperature fields. Therefore, fractal patterns are applied on sub-segment level and for determining the sequence of consecutive segments. With regard to the scale-invariance of fractal space-filling curves, the sequence of consecutively scanned segments is determined by the structure of the Peano curve, schematically displayed in Figure 5.

**Figure 4.** Fractal, quasi-simultaneous exposure pattern of a distinct sub-segment based on the Peano

Quasi-simultaneous exposure is correlated with a significantly increased layer time due to the repetitive exposure of distinct segments, leading to an increase in the layer time

*Polymers* **2022**, *14*, x FOR PEER REVIEW 6 of 17

**Figure 5.** Schematic illustration of the fractal exposure sequence of distinct segments. **Figure 5.** Schematic illustration of the fractal exposure sequence of distinct segments.

Exposure patterns of complex cross-sections are generated applying the algorithm proposed by Yang et al. [26], thus implicitly generating a closed, space-filling curve cor-Exposure patterns of complex cross-sections are generated applying the algorithm proposed by Yang et al. [26], thus implicitly generating a closed, space-filling curve corresponding to the specific part contour.
