Additive Manufacturing of Biomimetic Near-Zero CTE Optical Structures

Abstract

Super invar, with its near-zero coefficient of thermal expansion (CTE), has a great potential to be used in the design and fabrication of high-precision optical structures, such as optical mirror substrates. In order to reduce the weight and maintain the strength of the mirror substrate, several biomimetic lattice designs were investigated in this paper. The static modeling provides a systematic study on different types of biomimetic mirror substrates. The impact of structure parameters, such as the wall thickness, lattice unit length, height of the lattice structure, and the thickness of the side plate, are also studied. It turns out that the three-layer lattice-structured composite mirror substrate has the best performance. With AM techniques, three-layer gyroid optical structures, which are not possible to fabricate with conventional manufacturing technology, were designed and printed with our in-house-built AM machine. The stiffness test of the gyroid specimens was in good agreement with the modeling results. The gyroid structure shows about a 20% improvement over the honeycomb structure. The gyroid design reduces the equivalent density to 1.8 g/cm³ and has an order-of-magnitude improvement on the thermal deformation, while maintaining a comparable strength with that of beryllium.

1. Introduction

Biological materials are ingeniously designed and optimized tools that are employed by nature for organisms to survive and thrive within challenging environments. They represent the elegant strategies that fulfill a variety of not only mechanical but also functional needs (energy absorption, fracture toughness, and stiffness), as they are generally simple in composition but efficient in performance. By utilizing exquisite structures instead of chemical complexity, biological materials surpass their synthetic counterparts in many properties and functions. Lattice structures, formed by a repeated arrangement of cells (honeycomb, gyroid, bone, nacre, sandwich, etc.), are commonly used to tailor physical and mechanical properties [1,2,3,4,5]. However, studies on biomimetic structures are mainly focused on energy and biomedical engineering, and the materials used are limited to hydrogels, polymers, graphene, and their derivatives [6,7].

Super invar alloy consists of about 32 weight percent nickel, roughly 5 percent cobalt, balance iron, and trace amounts of other metals and minerals like copper, aluminum, and manganese. Super invar has minimal thermal expansion at room temperature. The optical structures made by super invar can maintain a high accuracy and dimensional thermal stability. As shown in Figure 1, super invar’s CTE (1 μ/K) is only one-sixth of the beryllium thermal expansion coefficients (6.7 μ/K) [8,9,10]. However, super invar has a relatively high density (8.15 g/cm³), which makes it less attractive for applications requiring light-weight components.

Additive manufacturing (AM), such as Laser Powder Bed Fusion (L-PBF), is the right technology to make the structured mirror to reduce the weight, thermal deformation, and stress. Compared with conventional manufacturing methods, many complex structures can be easily printed [11,12,13,14]. The AM technique enables us to further improve optical structures such as mirror substrates, optical breadboards, optical stages, brackets, etc., so to obtain a higher strength, a lower equivalent density, or weight.

In this paper, a systematic investigation is undertaken, from design to manufacturing, on super invar optical structures in consideration of the thermal expansion coefficients, elastic modulus, and equivalent density. Section 2 details the design of super invar optical mirror substrates by taking advantage of AM techniques and biomimetic structures such as honeycombs and gyroids. Section 3 describes the AM of super invar-structured mirror supports. Section 4 discusses the conclusions and perspectives.

2. Design and Modeling of Super Invar Optical Mirror Substrates

2.1. Static Analysis of Optical Mirror Substrates

A small-size three-dimensional (3D) model of an optical mirror substrate was built with SolidWorks 2016×64 Edition, based on the James Webb Space Telescope, as shown in Figure 2 [15,16,17]. The dimension of the 3D model was set at 10 inches. As shown in Figure 3, one side of the mirror substrate is a solid plate and the other side of the mirror substrate is a honeycomb support structure. The model was used to study the deformations of different structures under gravity force loads. To calculate the plate deformation under the gravity force, the gravity acceleration g was assumed to be 9.8 m/s² (on the Earth). As shown in Figure 4, the center of the struts is fixed. The gravity is perpendicular to the surface of the mirror substrate. The material of the model is super invar. The mechanical properties of super invar are listed in

Table 1. Mechanical property of super invar.

Super Invar
Elastic Modulus	145,000	N/mm2
Poisson’s Ratio	0.23	N/A
Mass Density	8150	kg/m3
Tensile Strength	483	N/mm2
Yield Strength	276	N/mm2
Thermal Conductivity	10	W/(m·K)
Specific Heat	500	J/(kg·K)

.

As shown in

Table 2. Convergence study of the static model.

	Element Size	Total Nodes	Total Elements	Displacement Under Gravity
	mm	-	-	µm
#1	5	103,579	52,205	0.85
#2	3	145,295	73,771	0.86
#3	2	440,636	222,433	0.89
#4	1	2,411,576	1,400,054	0.89

, the mesh size of the model was varied to investigate the convergence of the static simulation. The results converged when the element size was reduced to 1 mm. Therefore, in this paper, the element size of the static model is set to 1 mm.

In Figure 4, the maximum displacement D1 of the plate deformation of the mirror due to the gravity is defined and used to evaluate the performance of the mirror substrate.

2.2. Design of Honeycomb Optical Mirror Substrate

In order to find the structure with a low equivalent density and to keep the gravity displacement at a minimum, the honeycomb structure was used. The honeycomb structure provides sufficient rigidity for ultra-high-precision optics and reduces the weight of the mirror substrates. As shown in Figure 5, to optimize the design parameters of the honeycomb structure, six honeycomb structures are analyzed. The gravity displacement and equivalent density of different designs of the mirror are compared. This provides a baseline to further build up multi-layer composite structures. The dimension parameters are defined in Figure 6 and

Table 3. The dimensions and the gravity displacements of the plates.

	L	d3	d2	Gravity Displacement	Total Mass	Equivalent Density	Displacement × Density
	mm	mm	mm	μm	g	kg/m3	μkg/m2
Case 1	25.4	2	10	1.452	868	1726.2	2506
Case 2	12.7	2	10	1.556	1609	3199.7	4979
Case 3	6.35	2	10	1.265	2709	5387.3	6815
Case 4	25.4	1	10	1.444	450	894.9	1292
Case 5	12.7	1	10	1.611	868	1726.2	2781
Case 6	6.35	1	10	1.634	1609	3199.7	5228
Solid Plate	-	-	10	0.9926	3415.2	8149.5	8089

.

The gravity displacement and the equivalent density are also listed in Table 3. As the gravity displacement and the equivalent density all should be small, the product of the gravity displacement and the equivalent density is used to compare the performance of the structures, and are listed in Table 3. The smaller the value, the better the performance of the structure should be. In Case 4, the displacement is relatively low and the plate has the minimum mass. The product of the gravity displacement and the equivalent density is also the smallest. Case 4 was therefore chosen to be the honeycomb structure for the following simulations.

2.2.1. Composite Plate—Two Layers

To obtain both a high strength and low equivalent density, the honeycomb and the solid plate were combined together to form a two-layer composite plate, which is shown in Figure 7. The dimension parameters of the two-layer composite plate are defined in Figure 8 and Table 3.

In

Table 4. Gravity displacement of the two-layer composite plates.

Honeycomb Thickness	Solid Plate Thickness	Total Thickness	Gravity Displacement	Total Mass	Equivalent Density	Displacement × Density
mm	mm	mm	μm	g	Kg/m3	μkg/m2
0	12	12	0.6997	4098	8149.5	5702
1	11	12	0.7788	3802	7560.9	5888
3	9	12	1.017	3208	6379.6	6488
6	6	12	1.385	2319	4611.7	6387
9	3	12	1.155	1429	2841.8	3282
10	2	12	0.9497	1133	2253.1	2140
11	1	12	0.7368	836	1662.5	1225

, the gravity displacement of the two-layer composite plate reaches the maximum point when the thickness of the honeycomb structure equals the thickness of the solid plate. When the honeycomb thickness is 10 mm and the ratio of the honeycomb thickness to the whole thickness is larger than 0.8, the deformation of the two-layer composite plate’s gravity displacement is less than 1 µm, as shown in Figure 9.

2.2.2. Composite Plate—Three Layers

A three-layer composite plate is shown in Figure 10. The three-layer composite plate has one honeycomb structure in the middle, sandwiched by two solid plates. The modeling results are summarized in Table 4.

As shown in

Table 5. Gravity displacement of the three-layer composite plates.

Solid Plate Thickness—d1	Honeycomb Thickness—d2	Solid Plate Thickness—d1	Total Thickness	Gravity Displacement	Total Mass	Equivalent Density	Displacement × Density
mm	mm	mm	mm	μm	g	kg/m3	μkg/m2
1	10	1	12	0.4315	1133	2253.1	972
1	8	1	10	0.5923	1043	2489.0	1474
1	6	1	8	0.8979	953	2842.8	2552
1	4	1	6	1.592	863	3432.4	5464
1	2	1	4	3.9	773	4611.7	17,985

, the three-layer composite plate has a better performance. When the honeycomb thickness is 10 mm and the plate thickness is 12 mm, the gravity displacement is only 0.4315 µm, which is even smaller than that of the solid plate. The product of the gravity displacement and the equivalent density is the smallest. The three-layer composite plate has a better performance when compared with the two-layer design of the same height.

2.2.3. Composite Plate—Five Layers

For the five-layer plate, shown in Figure 11 and

Table 6. Gravity displacement of the five-layer composite plate.

Solid Plate Thickness1	Honeycomb Thickness1	Solid Plate Thickness2	Honeycomb Thickness2	Solid Plate Thickness3	Total Thickness	Gravity Displacement	Total Mass	Equivalent Density	Displacement × Density
mm	mm	mm	mm	mm	mm	μm	g	Kg/m3	μkg/m2
1	4.5	1	4.5	1	12	0.5207	1429.4	2841.8	1480

, the gravity displacement of the plate with the thickness of 12 mm is 0.5207 µm, which is bigger for that of the three-layer plate. Moreover, the equivalent density of the five-layer composite plate is also much higher. So, the five-layer design is not as good as the three-layer design.

A comparison summary of various honeycomb designs is given in Figure 12 and

Table 7. Comparison of different plate designs.

	Total Thickness	Gravity Deformation	Total Mass	Equivalent Density	Displacement × Density
	mm	μm	g	kg/m3	μkg/m2
One layer	12	0.6997	4098	8150	5703
Two layers	12(11-1)	0.7368	836	1663	1225
Three layers	12(1-10-1)	0.4315	1133	2253	972
Five Layers	12(1-4.5-1-4.5-1)	0.5207	1429	2843	1480

. The three-layer composite plate obviously has the best performance. The gravity deformation is about 60% of the solid plate and the equivalent density of the three-layer design is about one-quarter of the solid plate. When compared with the two-layer design, the product of the displacement and the density of the three-layer design was reduced by 20%. The three-layer composite plate is difficult to fabricate with conventional manufacturing techniques. However, by taking advantage of the AM technique, the three-layer design can be fabricated.

The three-layer design is further improved by adjusting the heights. As shown in

Table 8. Simulation results of the plate with different dimensions.

10 Inches	Total Height	Gravity Deformation	Mass	Equivalent Volume	Equivalent Density	Displacement × Density
	d1-d2-d1
	mm	μm	g	mm3	kg/m3	μkg/m2
Three layers	12(1-10-1)	0.4315	1133	512,028	2253	972
Three layers—A	22(1-20-1)	0.1588	1546	892,760	1732	275
Three layers—B	42(1-40-1)	0.0735	2431	1,704,360	1426	104
Solid beryllium	24	0.02385	1860	1,005,704	1850	44

, the height of the honeycomb structure can be increased to reduce the deformation and the equivalent density of the plate. When the height of the plate is 22 mm, the equivalent density is 1.732 g/cm³, which is smaller than beryllium. The product of the gravity deformation and the equivalent density is only 275.

In order to obtain the powder out of the hollow plate after the AM treatment, one side of the plate was designed to have small holes, as shown in Figure 13. The radius of the hole is 2 mm. These holes can affect the strength of the plate. The gravity displacement is shown in

Table 9. Gravity displacement of the three-layer composite plate with holes on one side.

	Solid Plate Thickness—1	Honeycomb Thickness	Solid Plate Thickness—2	Total Thickness	Gravity Deformation	Total Mass
	mm	mm	mm	mm	μm	g
Three layers	1	10	1	12	0.4315	1133
Three layers—one surface with holes	1	10	1	12	0.4366	1126

. With the holes on one side of the plate, the gravity displacement is almost the same. The change is only about 1.2%, which is acceptable.

2.3. Design of Gyroid Optical Mirror Substrate

Figure 14 shows a flow chart to design a triply periodic minimal surface (TPMS, including gyroid) optic parts that PolarOnxy has developed by combining commercial software (Solidworks 2016, nTop 2024, and ANSYS 2023 R1) and our internally developed algorithm. This tool enables us to optimize the optical mirror substrate. The details of using these software tools to design the TPMS components have been described in references [18,19], and we will not repeat them here [18,19].

Figure 15 shows a schematic design of a 10-inch three-layer sandwiched gyroid mirror substrate using nTopology software. This type of structure includes a 10 mm gyroid core structure layer sandwiched by two 1 mm thick solid layers. The thicknesses of these three layers must be optimized to meet the specifications of the size, temperature, and stiffness of the mirror substrate. Figure 16 shows a modeling example in comparison to the gravity deformations between two gyroid structures using super invar.

Table 10. Performance comparison of the 10-inch gyroid mirror substrates.

Materials	10 inches	Lattice Length	Total Height d1-d2-d1	Wall Thickness d3	Gravity Deformation	Total Mass	Equivalent Density
		mm	mm	mm	μm	g	g/cm3
Beryllium	Three layers Gyroid infill	30	12(1-10-1)	1	0.4315	313	0.511
Beryllium	Three layers Gyroid infill	40	12(1-10-1)	1	0.3758	272	0.476
Super Invar	Three layers Gyroid infill	30	12(1-10-1)	1	0.472	950	1.89
Super Invar	Three layers Gyroid infill	40	12(1-10-1)	1	0.485	904	1.79

gives a performance comparison for the following two materials: beryllium and super invar. It shows that the equivalent density is reduced by five times compared with the solid version, while keeping a comparable deformation. Further optimization on the gyroid structure may lead to a weight reduction by over 10 times.

Table 11. Thermal deformations at various temperature variation ranges.

Maximum Thermal Deformation of 10-Inch Plate over ΔT		10 °C	20 °C	30 °C
Super Invar—Gyroid	μm	0.08525	0.9237	2.644
Beryllium—Gyroid	μm	42.6	85.8	129.1
Beryllium—Solid	μm	44.2	83.44	124.5

shows the thermal-induced deformation modeling results for 10-inch gyroid mirror substrates using Be and super invar. It shows that super invar has obvious advantages (two orders of magnitude lower deformation) in a temperature-varying environment.

2.4. Comparison Between Honeycomb and Gyroid Structures

Taking the three-layer composite plate as an example for comparison between the honeycomb and gyroid structures, the composite plate (4 × 10 × 1 inches) was infilled with a gyroid lattice or the honeycomb structure, as shown in Table 11. When the equivalent density was kept the same, the stiffness of the two plates was modeled and compared, while the upper surface was set at an evenly distributed pressure of 10⁵ Pa, as shown in Figure 17.

The equivalent density of the plate with different infill structures was 1.8 g/cm³. When the plate was infilled with the honeycomb structure, the displacement was 29.93 µm. With the same wall thickness and equivalent density, the displacement of the plate infilled with the gyroid structure was about 26.22 µm. The displacement was decreased by 12.4% when using the gyroid lattice (Case 1). When the gyroid lattice wall thickness was doubled to 0.56 mm, the material was moved from the surface plate to the middle lattice structure to keep the same density. The displacement of the plate was further reduced by another 10%, as indicated in Case 2 in

Table 12. Simulation results of different designs.

	Dimensions	Lattice Type	Lattice unit Length	Lattice Wall Thickness	Surface Wall Thickness	Displacement	Equivalent Density	Frequency
	Inch		mm	mm	mm	µm	kg/m3	Hz
1	4 × 10 × 1	Gyroid	28 × 28 × 28	0.28	2	26.22	1833	357
2	4 × 10 × 1	Gyroid	28 × 28 × 28	0.56	1.765	23.54	1833	350
3	4 × 10 × 1	Gyroid	14 × 14 × 14	0.28	1.765	24.27	1833	342
4	4 × 10 × 1	Honeycomb	-	0.26	2	29.93	1811	346

, while the natural frequencies of the plate with one fixed side wall are almost the same. Further reducing the wall thickness and lattice unit length (Case 3) did not bring better results.

3. Additive Manufacturing of Super Invar Structured Parts

Our in-house-built L-PBF AM machine was able to print super invar samples with lengths of 3 inches. The AM machine was well described in other publications and will not be repeated here [10,12].

3.1. Test of Coefficient of Thermal Expansion (CTE)

Two groups of 2-inch rectangular samples were printed with super invar powder (American Element, Los Angeles, CA, USA) in the AM machine, identified as group 20221011 (horizontal X) and 20221012 (horizontal Y), and were used to test the coefficient of thermal expansion (CTE). Figure 18 shows the printed CTE samples with the orientation marked on the printing substrates. For each group, there were four samples divided into the following two comparison groups: two without the HIP treatment (A1 and A2) and two with the HIP treatment (A3 and A4). The conditions of the HIP treatment are 1100 °C and 100 MPa for 4 h. All of the samples were tested by a third party (TPRL, Inc. 3080 Kent Avenue, West Lafayette, IN, 47906, USA) for the thermal expansion testing from −50 °C to 150 °C. A dual push-rod dilatometer (Theta Dilatronics II) was used to measure the linear thermal expansion from 100 to 1800 K, following the testing procedure of ASTM standard E228 [20]. The differential expansion between the sample and a known standard reference material was measured as a function of the temperature. The expansion of the sample was computed from this differential expansion and the expansion of the standard. The measurements were made under computer control and linear expansion was calculated at preselected temperature intervals.

The mean CTE at a temperature T was calculated by dividing the change in the length in the sample by the change in the temperature, as follows:

$${\alpha }_{mean}={\displaystyle \frac{\frac{L_T-L_0}{L_0}}{T-T_0}}$$

where T₀ is the reference temperature (normally 20 °C or 70 °F).

Figure 19 shows the test results of the CTE with a different orientation. A similar CTE result obtained for the X and Y orientations indicates an isotropic property. This is very important for optical structures.

Figure 20 shows the complete test results of the CTE both with and without the HIP treatment. Both groups of CTE curves were similar, which indicates an excellent repeatability control. With the HIP treatment, zero CTE is achieved for all four samples (Figure 20). This gives another way to fine-tune the CTE with heat treatment.

3.2. AM on Hollow Structures

Printing on top of hollow structures without extra supporting structures underneath is difficult. Using the super invar powder, we tried to print the one-sixth part of the mirror substrate, as shown in Figure 13, with dimensions of 1 inch, 1.5 inches, and 2 inches, so to understand the largest span that a top plate can be printed on without adding extra supporting structures underneath. After printing, the parts were cut in half with an electrical discharge machine (EDM) to check the quality. All three prints were good, as shown in Figure 21.

Table 13. Test results of the surface roughness and relative density of the printed mirror substrates.

Serial Number	Surface Roughness (µm)	Relative Density % (Microscopic Image)
1 inch	20.5	98.2%
1.5 inches	22.9	96.5%
2 inches	23.4	96.3%

lists the testing parameters on the surface roughness and relative density, which were very good.

3.3. AM of Three-Layer Honeycomb Mirror Substrate

A complete 3 inch mirror substrate sample of 25 mm in height was printed, as shown in Figure 22. The laser AM parameters are listed in

Table 14. AM parameters for printing the three-layer mirror substrate.

	Layer Height	Hatch Mode	Hatch Space	Laser Power	Laser Scan Speed
	mm	-	mm	W	mm/s
Mirror plate	0.05	Lines	0.06	120	90

. The bottom solid plate is 3 mm while the top is 2 mm. The height of the honeycomb is 20 mm. There are very few defects visible on the sample. The equivalent density was measured as 1.9 g/cm³ by using the Archimedes method. The relative density measured by microscopic imaging was 98.7% and the surface roughness was 17.3 µm.

3.4. AM of Gyroid Mirror Substrate

Different types of 3-inch optical mirror substrates with different wall thicknesses were designed. The design with a 0.52 mm wall thickness achieved the best stiffness of the plate. Figure 23 shows the design. The thickness of the bottom plate was set at 5 mm. The thickness of the side wall was 1 mm.

The AM parameters in Table 12 were used for printing the optical gyroid structures. Figure 24 shows the sample after printing. The sample was cut out of the substrate by an EDM. After polishing, the polished surface was used to characterize the quality of the AM process with a microscope. Figure 25 shows the polished solid side of the optical mirror substrate. A relative density as high as 99% was achieved.

3.5. Stiffness Test of Gyroid Structures

Figure 26 shows three additively manufactured specimens with super invar powders. The design dimensions of the specimens are listed in

Table 15. Comparison of the test results with the simulation results.

Specimens	Lattice Size	Mean Lattice Wall Thickness	Side Wall Thickness	Equivalent Density	Stiffness-Test	Stiffness/Density Test	Stiffness Simulation	Stiffness/Density Simulation
	mm	mm	mm	kg/m3	N/mm		N/mm
#1	14 × 14 × 14	0.607	2.2	3402	7767	2.28	7812	2.30
#2	9 × 9 × 9	0.524	2.1	3637	10,256	2.82	9596	2.64
#3	14 × 14 × 14	0.79	1.9	3436	9302	2.71	8620	2.51

. Specimen #3 was designed by moving some material from the side wall to the gyroid lattice structure. As shown in Figure 27, the three-point bend testing (stiffness) of the super invar gyroid samples using a Tinius Olsen 10,000 lbs capacity Benchtop load frame. The testing procedures were in accordance with ATS QA Manual Rev. 23. The supporting span was 40 mm and the force was applied in the middle of the specimens.

As shown in Figure 28, when the displacement increased from 0 to 0.5 mm, the corresponding force was increased slowly. That may have been caused by the small particles on the surface. After the small particles on the surface were flattened by the force, the whole specimen began to deform. The slopes of the curves were then calculated by using the data after 0.5 mm of displacement. Table 15 lists the test results. The simulation results are in good agreement with the test data, and the difference is within 7%. When comparing specimen #1 with specimen #2, specimen #2 has a higher stiffness-to-density ratio. This indicates that a smaller lattice size and a thinner wall thickness provide a better design for the composite plate. However, the minimum wall thickness is limited by the AM machine; thus, the limit for the lattice density or size was set for a specific AM machine.

Furthermore, moving some material from the side wall to the lattice structure may increase the material utilization efficiency. This is confirmed by specimen #3, which had a better performance than specimen #1, despite both having the same lattice size.

4. Summary and Conclusions

In this paper, super invar was used to design a variety of biomimetic lattice optical structures by taking advantage of the unique AM features. The CTEs of the additive manufactured super invar test coupons showed zero CTE for different printing orientations, which is critical for optical structures requiring a high precision over a wide temperature range. Honeycomb and gyroid structures were used to investigate the performance of optical mirror substrates. The impact of the lattice parameters, such as the wall thickness, lattice unit length, height of the honeycomb or gyroid structure, and the thickness of the side plate, were systematically studied. It turns out that the three-layer composite plate has the best performance. The performance of the three-layer composite plate was further optimized to design and fabricate both the honeycomb and gyroid optical mirror substrates. Some important conclusions are listed as follows:

(a)

Super invar has obvious advantages (two-orders-of-magnitude lower deformation) in temperature-varying environments.

(b)

The smaller lattice size and thinner wall thickness provide a better design for the composite plate.

(c)

Moving some material from the side wall to the lattice structure may increase the material utilization efficiency.

(d)

The design with the 0.52 mm wall thickness achieved the best stiffness of the plate.

(e)

Gyroid structures showed over 20% better performance than the honeycomb structures.

(f)

As high as a 99% relative density was achieved for the biomimetic lattice structures.

(g)

It is feasible to print on top of hollow structures with a 2-inch span without extra supporting structures underneath.

(h)

There was a good agreement between the simulation and experiment in the stiffness test.