Additively manufactured metallic TPMS lattice structures: design strategies, fabrication, multifunctional properties, and applications

Abstract

The utilization of the triply periodic minimal surface (TPMS) has emerged as a highly effective approach for designing lattice structures due to its smooth surfaces, intricately connected lattice structures, and precisely controlled geometric properties. However, the full potential of metallic TPMS lattice structures remains underexplored due to three primary challenges: (i) the computational complexity involved in multi-scale design optimization, (ii) manufacturing limitations in achieving defect-free lattice architectures, and (iii) an incomplete understanding of the process-structure-property relationships necessary for multifunctional applications. To address this gap, a comprehensive analysis encompassing design, manufacturing, properties, and applications is essential. This paper presents such a study of metallic TPMS lattice structures. First, we introduce geometric design methods for generating digital TPMS models tailored to specific requirements. Building on this, we summarize additive manufacturing techniques employed in fabricating these structures, highlighting how additive manufacturing mitigates structural design constraints and enables high-quality, accurate fabrication. Subsequently, we elaborate on the diverse properties and multidisciplinary applications of additively manufactured metallic TPMS lattice structures. Finally, by synthesizing recent advancements in this field, we discuss existing limitations and offer perspectives on future research directions.

Introduction

Metals have appealing properties such as high strength and ductility, high machinability, high melting point, required chemical and dimensional stability, and excellent thermal/electric conductivity¹. Therefore, they are used in various industries, including the military, transportation, building, energy, biomedical, and others. In the past, bulk metal materials were often used, and the vast majority of structures were solid structures². The porosity of solid structures can thus be taken to be zero. Internal porosity is even regarded as a fault as a result of poor manufacturing quality. However, many unique lattice structures exist in nature, including bones, corals, honeycombs, wood, and so on^3,4. Natural lattice structures were discovered to have low density, high specific strength, and versatility^5,6,7. Lattice structures composed of metals and their alloys (as well as other types of materials) have garnered significant attention and development in recent years, inspired by these natural forms.

In a range of fields, including transportation, aircraft, athletics, and biomedicine, metallic lattice structures provide the benefits of decreased emissions, increased surface area, and superior mechanical and other properties^8,9,10,11. Lattice structures have traditionally been made using investment casting^12,13, wire-woven methods^14,15, direct foaming^16,17, powder metallurgy with space holder^18,19, and other techniques^{9,20,21,22,23,24,25,26,27}. Traditional manufacturing technologies have obvious limitations in creating complicated planned structures and/or the potential of architectures. In summary, lattice structures produced by current manufacturing techniques have two significant drawbacks: (i) random structure and (ii) constrained utilization architecture. Random lattice structures exhibit unstable performances because the walls’ pores and thickness are difficult to precisely regulate, which is undesirable in practical settings. For another, the availability of certain types of designs is constrained by the fact that non-random lattice structures made using normal manufacturing techniques generally need extra assembly or bonding stages, which complicate the production process and even make producing lattice structures challenging. Since these shortcomings significantly restrict the creation and use of metallic lattice structures, it is anticipated that non-random designs will be produced using a method that is more user-friendly, efficient, and structure-flexible.

Innovative additive manufacturing introduces fresh approaches to fabricating lattice structures²⁸. With additive manufacturing processes becoming more advanced, especially in metal additive technology, integrating lattice structures using additive methods presents an increasingly mature technology for lightweight design. Most additive manufacturing lattice structures consist of simple geometric units (such as columns, spheres, and squares) formed via Boolean operations. These geometric models, resulting from Boolean operations, exhibit uneven transitions at the unit connections, often featuring sharp corners and turns. Consequently, stress concentrations arise at these junctures when the structure undergoes stress, potentially leading to premature failure during extended use. Additionally, within the additive manufacturing process, straight rods in structures, particularly long, thin horizontal ones, tend to collapse due to inadequate support, resulting in formation failures^29,30,31. Given the mechanical and fabrication limitations of these lattice structures, researchers are now exploring innovative lattice designs. One noteworthy example is the triply periodic minimal surface (TPMS) lattice structure, which is the primary focus of this review. A systematic literature search was conducted using Web of Science (website: http://www.webofscience.com) as the primary academic database, employing keywords such as “triply periodic minimal surface,” “additive manufacturing,” and “metallic lattice structure.” To ensure relevance to recent advancements, the search was limited to peer-reviewed articles published between 2015 and 2025. Subsequently, titles and abstracts were reviewed to exclude studies falling outside the scope of this review, with priority given to high-impact journals and pioneering works. The selected papers were then categorized based on key themes, including the design, manufacturing, mechanical properties, thermal properties, permeability, and multi-physical applications of metallic TPMS lattice structures. Comparative analysis, such as TPMS lattices versus traditional lattices, was performed to identify the strengths, limitations, and potential future directions of current research. Reflecting increasing interest, the volume of scientific research concerning metallic TPMS lattice structures and their applications has grown substantially over the past decade. This trend is illustrated in Fig. 1, which shows the cumulative number of TPMS-related publications from 2015 to 2025.

Fig. 1: Cumulative number of publications on metallic TPMS lattice structures from 2015 to 2025.

Data is surveyed from Web of Science (website: http://www.webofscience.com).

Minimal surfaces are those with minimal area and zero mean curvature, adhering to specific constraints that prevent self-intersection. These surfaces are periodic along all three coordinate axes, and a periodic arrangement of minimal surfaces in space forms a TPMS lattice structure³². The unique properties of TPMS, such as adjustability in terms of range, curvature, and period, make it a versatile option^33,34. Additionally, TPMS functions enable complex calculations like Boolean, modulation, and convolution³⁵. Consequently, TPMS lattice structures offer two key advantages compared to other designs. Firstly, mathematical functions can precisely define the entire structure. Adjusting function parameters directly modifies fundamental properties like porosity or volume-specific surface areas. Secondly, unlike traditional lattice structures, TPMS’s continuous surface design inherently offers superior topological optimization and self-support during the additive manufacturing process. First, the continuous nature of TPMS eliminates abrupt geometric transitions and sharp discontinuities, common sources of stress concentration in discretized or voxel-based topologically optimized structures. By maintaining smooth curvature and uniform material distribution, governed by mathematical functions such as the level-set equation, TPMS avoids localized stress peaks, aligning with topology optimization’s theoretical objective of maximizing stiffness-to-weight ratios while ensuring mechanical robustness. Second, TPMS are mathematically derived from energy-minimizing principles, such as minimal surface area for given boundary conditions, mirroring the goals of topology optimization frameworks aimed at maximizing performance under constraints. Their intrinsic periodicity and symmetry facilitate seamless scalability while avoiding artificial anisotropy often introduced by voxel-based optimization methods. Third, unlike conventional topology optimization, which often prioritizes a single objective, TPMS-based designs can simultaneously optimize multi-physical properties due to their high surface-area-to-volume ratios and interconnected porosity. This continuous surface enables concurrent enhancements in properties such as permeability (for fluid flow) and thermal/mechanical efficiency, a critical advantage for applications requiring multifunctional performance. Finally, the smooth, self-supporting geometry of TPMS reduces manufacturing complexity in additive manufacturing by minimizing the need for post-optimization smoothing or support structures. Consequently, the as-designed topology closely matches the as-manufactured structure, which is a persistent challenge in traditional topology optimization workflows. The array of advantages has led to a surge in scientific interest surrounding TPMS. In contrast to conventional porous material research, TPMS necessitates a multidisciplinary approach spanning computer graphics, manufacturing science, mechanics, thermology, biology, and chemistry, as depicted in Fig. 2.

Fig. 2

Overview of this review of metallic TPMS lattice structures includes: design, additive manufacturing methods, properties, and applications.

Although TPMS is an intriguing investigation hotspot in a variety of domains, its benefits are underutilized. Currently, most studies concentrate solely on the performance or application of a single discipline. The current research trajectory is disorganized and scattered. Interdisciplinary research on metallic TPMS lattice structures is required to expand the use of metallic TPMS lattices. TPMS, for example, can be created using computer-aided design (CAD) algorithms with complex forms and gradient porosity. Most TPMSs, whether thermal or chemical, are still conventional TPMS lattice structures with consistent porosity and simple forms. Furthermore, manufacturing procedures must be enhanced as an intermediate step between the design and use of metallic TPMS lattice systems. Current additive manufacturing technologies have major hurdles when dealing with complex metal TPMS lattice systems with outward forms and interior architecture. While new manufacturing techniques are being presented, the computational efficiency and precision of the path-planning process must be improved even more. To increase production quality, more manufacturing restrictions should be incorporated during the design phase. Based on the above information, a review of TPMS design, production, performance, and application will be presented. Figure 2 depicts the organization of this work. This paper introduces the concept and characteristics of the TPMS lattice structure. After that, the geometric design of TPMS and the precise additive manufacturing process are discussed. The existing properties of the metallic TPMS lattice structure are reviewed, and the applications of the metallic TPMS lattice structure in different fields are described. The limitations, challenges, and future opportunities of additive manufacturing metallic TPMS lattice structures are also discussed based on the findings of the review. This review will provide insights into the additive manufacturing of metallic TPMS lattice structures and provide a wealth of information on the design, manufacture, and performance of TPMS lattice structures for their various applications.

Design of metallic TPMS lattice structures

The formulation of design strategies yields the essential three-dimensional (3D) models, constituting the foundational core of TPMS lattice structures. In practice, geometric attributes such as porosity and volume-specific surface areas wield a pronounced influence on performance outcomes. Thus, the geometric design establishes a bedrock for overseeing performance dynamics across diverse disciplines. In contradistinction to conventional foam lattice constructions, TPMS lends itself to the creation of intricate elements, mirroring the symmetrical elegance of lattice frameworks. The TPMS design methods used in this study include gradient TPMS, heterogeneous TPMS, TPMS configurations with complex external contours, and topology-optimized TPMS.

Gradient TPMS lattice structures

Three unique features distinguish TPMS from other lattice structures. Firstly, TPMS is characterized by its implicit surface representation, describable through algebraic equations, often abbreviated as f (x, y, z) = C, where C remains constant, making it an iso-surface. Secondly, TPMS exhibits three distinct oscillation modes, with easily adjustable distribution ranges and periods via function parameters. Last but not least, TPMS is recognized as a minimal surface, signifying zero mean curvature. Its smooth contours bear a resemblance to natural elements like leaves and soap bubbles³⁶. In addition, TPMS structures are found in natural systems such as biological cubic membranes³⁷, butterfly wing scales³⁸, biomineralized skeleton of the knobby starfish³⁹, and the exoskeletons of weevils⁴⁰, as shown in Fig. 3.

Fig. 3: Natural TPMS-like lattice structures.

a Characteristic projection of mitochondrial tomography reconstruction in ameba³⁷. b Comparison of an artificial Gyroid structure with the natural wings of a butterfly³⁸. c SEM image of biomineralized skeleton of the knobby starfish³⁹. d Partial enlarged SEM image of weevil exoskeleton and schematic diagram of single diamond structure⁴⁰.

TPMS can change various parameters, such as periodicity and relative density, to adjust its mechanical properties. In Eq. (1), where n_i is the number of cell repetitions in the x, y, or z directions and L_i is the absolute size of the structure in that direction, periodicities are determined by the ω values (ω_x, ω_y, and ω_z)⁴¹. Schwarz discovered the first example of TPMS in 1865⁴⁰. Following that, in 1883, Neovius found a new sort of TPMS⁴⁰. Schwarz and Neovius discovered five different varieties of TPMS: Schwarz primitive, Schwarz Diamond, Schwarz Hexagonal, Schwarz crossing layers of parallels, and Neovius surface. Schoen then described the Gyroid-surface, which is the most well-known type of TPMS, in 1970⁴². More research has been done on a triply periodic minimal surface, such as Gyroid, Schwarz Primitive, Schwarz Diamond, and Neovius⁴³. As a result, as indicated in Table 1, common TPMS units can be described^41,44. Due to the implicit feature, the aforementioned straightforward implicit functions are required for both geometries and performances. Most existing TPMS, like parametric surfaces in the CAD realm, require discretization as mesh models for display or additive manufacturing. TPMS appears to divide the space into two halves, f (x, y, z) < C and f (x, y, z) > C. It should be noted that TPMS is a type of surface with no wall thickness. It takes another materialization procedure to create TPMS lattice structures. By directly offsetting the TPMS surfaces with a constant wall thickness, the sheet TPMS lattice structures will be created. Network TPMS lattice structures are the name given to the two components that are separated by TPMS.

$${\omega }_{i}=2\pi \frac{{n}_{i}}{{L}_{i}}(\mathrm{with}\,i=x,y,z)$$

(1)

Table 1 Mathematical expressions of different TPMS units

The volume ratio between two separated segments of a network TPMS lattice structure is solely determined by the curvature parameter (C). Consequently, adjusting the period parameter (ω) and C independently can yield graded or non-uniform TPMS lattice structures. In the context of sheet TPMS, it becomes imperative to account for wall thickness. Remarkably, sheet TPMS, exhibiting diverse combinations of C and wall thickness, could achieve comparable relative densities. In prior work, the parameters C and ω are created with graded values to produce a graded surface⁴⁵. Although the surface is substantially varied by varying the period parameter ω values, the continuity and smoothness of TPMS can still be maintained, as demonstrated in Fig. 4a. In Fig. 4b, relative densities can be directly changed by changing C⁴⁶. Furthermore, the graded sheet TPMS lattice structures with varying offset wall thicknesses can be created, as shown in Fig. 4c⁴⁷.

Fig. 4: Graded TPMS lattice structures.

a Adjusting ω to generate graded TPMS lattice structures⁴⁵. b Adjusting C to generate graded network TPMS lattice structures⁴⁶. c Adjusting wall thickness to generate graded sheet TPMS lattice structures⁴⁷.

In general, nature often features graded lattice structures. An illustrative instance lies in the gradual shift of porosities within human bones from cancellous to cortical^48,49. As a result, mimicking natural designs with graded lattice materials is an effective strategy. It should be noted that graded lattice structures can also be constructed using traditional lattice or foam architectures. When weighed against these alternatives, graded TPMS lattice structures exhibit dual advantages. To begin with, the graded porosities can be carefully adjusted using TPMS capabilities. For instance, the C can be defined as a coordinate-related function. Figure 4 depicts a linear change in the porosities of graded TPMS. Notably, TPMS readily facilitates the creation of more intricate graded lattice structures, featuring non-linear porosity variations. Secondly, owing to its function-driven attributes, TPMS ensures inner surfaces retain smoothness, optimal connectivity, and continuous integrity. Alterations in curvature, period, and wall thickness can all be flexibly applied. To generate graded lattice structures, TPMS can allow more design freedom.

Heterogeneous TPMS lattice structures

Most applied engineering constructions are made up of a variety of materials to fully use the advantages of different materials. Similarly, while no TPMS unit is ideal and possesses all advantages, the evaluation of TPMS lattice structures in engineering commonly employs relative density, porosity, and volume-specific surface areas as fundamental performance indices. Table 1 illustrates that a single TPMS unit cannot simultaneously achieve optimal performance in all aspects. This has led to the development of heterogeneous TPMS lattice structures to address increasingly complex application requirements. It is important to note that the constituent material of heterogeneous TPMS can be the same. These TPMS units are treated as distinct components in the creation of heterogeneous porous structures. Fortunately, each TPMS component can be described mathematically, allowing the selection of appropriate TPMS units in various locations to achieve specific goals. For instance, the Gyroid and Primitive structures can be effectively combined, as depicted in Fig. 5a⁵⁰, while retaining the smooth surfaces characteristic of heterogeneous TPMS lattice structures. The unit weight can be adjusted following linear or sigmoidal principles during transitions between regions. This design freedom enables the selection of any TPMS unit in specific locations to cater to unique applications, as illustrated in Fig. 5b⁵¹. Figure 5c highlights the benefits of heterogeneous TPMS and presents comparisons between heterogeneous and graded TPMS, emphasizing the variation in porosities among different regions of natural structures⁵². The graded Primitive structures can mimic natural architectures by creating porous structures with similar porosity distributions. However, further consideration is required to replicate complex functions comprehensively. Notably, although Primitive structures exhibit higher porosity and permeability, Diamond structures offer higher Young’s modulus at the same relative density⁵². As a result, the strategic utilization of different units is necessary in various areas.

Fig. 5: Heterogeneous TPMS lattice structures.

a Heterogeneous TPMS lattice structures from Primitive to Gyroid lattice structures⁵⁰. b A hybrid disc/cylinder-shaped scaffold model of Gyroid-surface and Diamond-surface with different units in different regions⁵¹. c Comparison of graded and heterogeneous TPMS lattice structures⁵².

The internal surfaces of heterogeneous TPMS remain smooth, akin to the benefits seen in graded TPMS. The transition zones, however, may be dramatically altered due to topology changes. To improve continuity, reasonable transition regulations are required. Compared to graded TPMS, heterogeneous TPMS provides more design options.

TPMS lattice structures with complex external shapes

Previous discussions have explored a range of methods for constructing interior lattice characteristics to fulfill diverse criteria. While many of these constructions feature simple geometries like cubes or spheres, practical applications often demand porous materials with complex freeform surface structures. For instance, tissue engineering scaffolds or implants must closely mimic defective regions. However, traditional CAD algorithms for lattice structures have primarily focused on interior pores, neglecting collaboration with form design techniques. Conventional approaches involve four steps to create a lattice scaffold with bone morphology. Firstly, a 3D model is reconstructed based on bone characteristics. Secondly, the envelope region is determined by computing the maximum and minimum diameters in three dimensions. Thirdly, within the envelope region, porous structures like TPMS are formed. Finally, Boolean operations are utilized to determine the intersections between the 3D bone model and lattice structures within the envelope region. By following these steps, one can achieve optimal lattice structures with essential internal architectures and desired exterior shapes. However, this method has three significant drawbacks. First of all, the efficiency and accuracy of 3D Boolean operations prove unreliable when constructing intricate lattice structures. Multiple modeling errors might arise post-mesh tailoring and rearrangement computations. Additionally, lattice structures located outside the target area but within the envelope region minimally affect the outcome. Lastly, managing the relative positions of exterior forms and internal pores, particularly in non-uniform lattice systems, poses a challenge. Slight variations can result in entirely different pore distributions.

The scalar field describes distances between the target model and porous structures in the envelope region, enhancing Boolean operation efficiency⁵³. Figure 6a depicts the design results of TPMS lattice structures with complex external shapes. However, creating the scalar field takes time, without eliminating inherent Boolean operation drawbacks. The inherent disadvantages of Boolean operations are not eliminated. Yoo et al.⁵⁴ immediately mapped TPMS units to the intended 3D models, inspired by the finite element approach. Voxelization was applied to the original target model. Following that, each TPMS unit is then transferred from parametric to spatial using shape functions, as shown in Fig. 6b. This technique fully avoids the traditional Boolean operation. TPMS structure porosities are easily modified by changing voxelization density. Nevertheless, the TPMS unit shapes altered during mapping, possibly losing initial smooth features. Furthermore, both voxelization and mapping occur after obtaining the target 3D model. Adjusting the target model requires repeating all prior stages, which is time-consuming in applications like medical implants. Figure 6c illustrates solid T-spline use for TPMS porous structures with freeform exteriors⁵⁵. T-spline is a useful technique for creating freeform surfaces. The parametric space, as a type of parametric surface, is regular and easily split. As a result, the solid T-spline can be easily separated into multiple cubes for TPMS extraction. Furthermore, the control points can be used to save TPMS parameters to create graded or heterogeneous TPMS porous structures. The local refinement of T-spline fine-tunes local geometry or porosity features, considerably enhancing design freedom. As a result, this strategy can considerably boost design freedom. Similar approaches based on B-spline were also introduced⁵⁶. However, the overall efficiency of the process is not as high as predicted. The two-dimensional (2D) design technique is a superior alternative for lattice structures where design efficiency is more critical than iterative design requirements⁵⁷. The external shapes will be sliced as stacked sections, as seen in Fig. 6d. 2D TPMS contours extracted within external form mesh layers. Layered TPMS regions with freeform shapes obtained as layered areas after offsetting and 2D Boolean operations, directly manufacturable via additive manufacturing.

Fig. 6: TPMS lattice structures with complex external shapes.

a CAD model of the same vertebra with a Gyroid-type porous architecture by Boolean⁵³. b The Tibia scaffold is composed of a Primitive-surface by shape function mapping method⁵⁴. c TPMS lattice structures are designed based on solid T-splines⁵⁵. d 2D design strategy for TPMS lattice structures with freeform shapes⁵⁷.

Topology-optimized TPMS lattice structures

While the aforementioned methods allow for generating TPMS lattice structures with varying porosity distributions, selecting appropriate porosity values for different regions remains crucial. As a porous structure, TPMS lattice structure has a significantly lower mass than a solid structure under the same envelope volume. While porosity inherently reduces mechanical strength, solid structures often possess excessive performance capabilities. Therefore, replacing solid structures with porous ones is an effective lightweighting strategy, leading to substantial savings in materials, energy, and manufacturing time. It is important to acknowledge the inherent trade-off between mechanical performance and structural weight. The design of 3D-printed lightweight structures focuses on balancing material usage and physical properties, which involves strategically designing porosity to achieve an optimal performance. Topology optimization stands as a potent methodology within the realm of structural optimization, enabling the derivation of optimal material density distributions contingent upon the defined design domain, applied loads, and imposed constraints. In a solid material, each calculation element has a density of either 0 or 1. To mitigate the presence of elements exhibiting intermediate densities, the solid isotropic material with penalization (SIMP) method was developed. In contrast to solid materials, TPMS lattice structures generated using implicit equations can be mapped to density fields obtained from topology optimization. The relative density of these TPMS lattice structures can be adjusted continuously from 0 to 1. Consequently, the integration of topology optimization with TPMS lattice structures has become an increasingly prevalent and promising area of research.

Wang et al.⁵⁸ propose a performance-driven design strategy for multiscale gyroid lattice structures, wherein a topologically optimized density distribution is mapped onto the unit cell dimensions (Fig. 7a). Applying this multiscale optimization method, they constructed a three-point bending beam that exhibited an approximate 9.6% increase in bending strength and a roughly 46.8% improvement in bending displacement when compared to a uniform beam. Panesar et al.⁵⁹ conducted a detailed comparison of topology designs employing different lattice structures, as shown in Fig. 7b. Given the alignment of performance requirements and manufacturing constraints, a gradient TPMS generated by mapping relative density onto the grayscale density solution derived from topology optimization emerges as a particularly desirable choice. This approach effectively reconciles the need to satisfy performance requisites with the limitations imposed by manufacturing processes. Li et al.⁶⁰ optimized the relative density distributions of TPMS structures for heat-sink applications. Their findings demonstrated that graded TPMS lattice structures, within the same volume ratio, could reduce the maximum temperature from 66.1 to 57.4 °C compared to uniform TPMS structures. This highlights the potential for improved heat transfer performance through density grading, as further illustrated in Fig. 7c, which demonstrates that a broader density range can amplify heat transfer performance. In summary, topology optimization offers a valuable approach for designing TPMS lattice structures by tailoring relative density distributions to meet diverse, multidisciplinary performance targets. However, the iterative nature of topology optimization can be computationally expensive, making the development of more efficient methods a key area of ongoing research. Addressing these challenges is crucial for generating TPMS models with optimized geometries and performance characteristics suitable for specific applications.

Fig. 7: Topology optimization of TPMS lattice structures.

a Example of multiscale three-point bending beam generation based on topology optimization⁵⁸. b Comparisons of different TPMS lattice structures design strategies⁵⁹. c Heat-sink topology optimization⁶⁰.

Additive manufacturing methods of metallic TPMS lattice structures

The manufacturing quality significantly impacts the efficiency of TPMS lattice structures. The porous characteristics and complicated topology of the TPMS lattice structures provide significant obstacles to the accuracy and effectiveness of existing manufacturing techniques, even though they may be simply manufactured by additive manufacturing layer by layer. In this part, related work on metallic TPMS lattice structures and additive manufacturing techniques are examined.

Mainstream methods of additive manufacturing for metallic TPMS lattice structures

Recent decades have seen the development of several additive manufacturing technologies, some of which are suitable for producing metals and alloys. These metal additive manufacturing technologies include powder bed fusion (PBF)^61,62, direct energy deposition (DED)^63,64, fused deposition modeling (FDM)^65,66, and laminated object manufacturing (LOM)^67,68, direct ink writing (DIW)^69,70,71, binder jetting^72,73, joule print^74,75, liquid metal additive manufacturing^76,77,78, electrochemical additive manufacturing (ECAM)^79,80,81, and cold spray additive manufacturing (CSAM)^82,83,84. Figure 8 shows the various metal additive manufacturing schematics. However, certain additive manufacturing technologies (e.g., LOM⁶⁸ and CSAM⁸⁵) are not capable of producing precision metal lattice structures. Until now, metallic structures have been produced mainly by PBF, DIW, and binder jetting. Due to the complex topology and complex porous design of metallic TPMS lattice structures, they are mainly produced by PBF technology. PBF can be further subdivided into laser powder bed fusion (L-PBF), electron beam powder bed fusion (EB-PBF), and large-area pulsed laser powder bed fusion (L-APBF), depending on its energy source⁸⁶. Metallic TPMS lattice structures can also be fabricated using material extrusion additive manufacturing (MEAM) and binder jetting techniques.

Fig. 8: Schematic illustrating several metal additive manufacturing techniques.

a powder bed fusion², b direct energy deposition², c fuse deposition modeling², d laminated object manufacturing², e direct ink writing⁶⁹, f binder jetting⁷², g joule print⁷⁴, h liquid metal additive manufacturing⁷⁶, i electrochemical additive manufacturing (ECAM)⁷⁹, and j cold spray additive manufacturing (CSAM)⁸².

Laser powder bed fusion

L-PBF technology is the mainstream manufacturing technology for the 3D printing of metallic TPMS lattice structures. Direct metal laser sintering (DMLS), selective laser sintering (SLS), and selection laser melting (SLM) are a few different types of L-PBF^87,88. For SLS and SLM, all procedures are carried out in a chamber with a protective environment (often argon or nitrogen) to stop oxidation during manufacturing^89,90. In the chamber, the powder layer is first deposited and leveled. The powder is next focused and scanned by a laser beam, and the CAD structure is placed on the slice or layer to induce a quick fusion. The metallic melt then cools down to solidify by the pre-designed CAD model. The appropriate slice of the CAD model and the chosen scanning approach determine the laser beam’s scan path for each layer. Due to their high precision, L-PBF technologies are extremely beneficial for creating metallic TPMS lattice structures with incredibly intricate geometries.

It’s important to highlight that processing parameters, including layer thickness (t), laser power (P), scanning speed (v), hatching space (h), scanning strategy, spot size (d), and stage preheating temperature, play a crucial role in determining the final quality of the manufactured parts^90,91. A balanced combination of these factors should be carefully considered to achieve optimal component performance. Yang et al.⁹² employed μLPBF to fabricate four TPMS lattice structures from Zr-based metallic glass powder. The μLPBF system, unlike conventional LPBF machines, uses a 25 μm laser spot size to achieve high printing precision⁹³. They successfully manufactured unit cell sizes ranging from 1 to 2.5 mm, and the 3D-printed TPMS lattices exhibited high accuracy and excellent manufacturing quality. The minimum unit cell size achievable for TPMS lattice structures through L-PBF largely depends on the laser focus diameter and the size of the metal powder particles. Utilizing finer metallic powder and a laser with a smaller focus diameter can enable the creation of smaller unit cells with thinner struts. However, it’s important to note the presence of bonded particles on the surfaces, contributing to surface roughness, as depicted in Fig. 9a. Through meticulous planning and slicing by Yavari et al.⁹⁴, laser paths were strategically designed to prevent overlapping contours at intervals of 77 μm, resulting in dense walls closely resembling the design file (Fig. 9b). The lattice structures created using additive manufacturing closely mirrored the morphological attributes of the CAD design. Al-Ketan et al.⁹⁵ employed gas-atomized maraging steel fine powder in SLS to fabricate TPMS lattice structures. The step effect associated with layer-by-layer printing is evident in Fig. 9c, albeit less pronounced for TPMS structures. The SEM image indicates favorable printability. Furthermore, the final quality of TPMS samples is also influenced by the inherent part shape. Fan et al.⁹⁶ utilized Ti6Al4V powders to craft graded TPMS lattice structures via SLM. Notably, particles partially melted on the surfaces are observable in Fig. 9d. Thinner walls in graded structures might offer less support compared to those with uniform wall thickness. Inaccuracies decreased along the gradient direction, leading to larger thickness deviations. Ma et al.⁹⁷ compared SLM-fabricated TPMS with the intended models through CT scans. As shown in Fig. 9e, the pore size values of all constructed TPMS structures were smaller than those of the planned models. This investigation revealed that higher design porosity correlated with reduced manufacturing errors and increased manufacturing stability. The observed inaccuracies ranged between 46 and 80 μm. While L-PBF is capable of producing metallic TPMS lattice structures, defects such as porosity, incomplete melting, and cracks can be present. These defects lower part density and subsequently impact performance. Hence, optimizing machining parameters and geometries can significantly enhance the performance of TPMS lattice structures. However, due to the numerous processing parameters and the intricate topology of TPMS, optimization remains a challenging endeavor. Ongoing research continues to delve into refining the fabrication process^46,98,99.

Fig. 9: Metallic TPMS lattice structures fabricated by L-PBF.

a SEM images of the external surface of 3D-printed TPMS lattices reveal excellent printing accuracy and surface quality, exhibiting minimal adherence of unmelted powder particles. The scale bar represents 500 um⁹². b Laser path optimized TPMS lattice structure⁹⁴. c SEM images of the TPMS lattice structures by SLS⁹⁵. d Graded metallic TPMS lattice structures fabrication⁹⁶. e Comparison of preparation and design of metallic TPMS lattice structures⁹⁷.

Material extrusion additive manufacturing

While PBF dominates current research on metallic TPMS lattice structures, MEAM and binder jetting offer attractive, cost-effective alternatives. In MEAM, the initial step involves producing a feedstock composed of metal powder and a binder. Following printing, the binder is removed via debinding and sintering, resulting in the final metallic part¹⁰⁰. Due to the extrusion process, MEAM is better suited for shelled parts with defined infill structures¹⁰¹. This approach reduces material usage and printing time, while enabling the creation of components with tailored effective properties, significantly enhancing lightweight design potential¹⁰². Research on MEAM indicates that printing parameters such as build orientation, infill type, infill density, and layer thickness influence the final part quality and mechanical performance¹⁰³.

Rosnitschek et al.¹⁰⁴ utilized a Markforged Inc. device to fabricate Inconel 625 parts with a gyroid infill structure using MEAM. Their study aimed to understand the influence of the infill structure on dimensional accuracy and effective tensile properties to assess its potential for lightweight engineering. The results showed that neither the printing direction nor the infill structure significantly affected geometric deviations or sintering shrinkage, with deviations ranging from 1.04% (width) to 3.41% (height), yielding an overall average deviation of 2.23%. However, analysis of fracture surfaces of gyroid flatwise specimens revealed that the gyroid lattice structure could not withstand the debinding and/or sintering processes. The infill pattern collapsed, with all infill material accumulating at the bottom of the sample, rendering the gyroid flatwise specimen unusable in its current state (as illustrated in Fig. 10a). This indicates a need to increase the infill density when using gyroid lattice structures to achieve successful prints. The infill structure proved beneficial within the elastic range, where a 40% weight reduction only resulted in a 20% decrease in the specimen’s mechanical properties.

Fig. 10: Metallic gyroid structures fabricated by MEAM and binder jetting.

a Comparison of the collapsed filling structure of the gyroid flatwise specimen with the successfully sintered filling structure of the gyroid upright specimen, along with dimensional deviations between the sintered samples and the CAD models¹⁰⁴. b SEM images of the sintered gyroid lattice structure and a comparison between the μCT reconstructed model and the CAD model¹⁰⁶.

Binder jetting

In binder jetting, metal powder or sand is selectively joined layer by layer using a binder, a polymeric liquid, necessitating post-processing, such as sintering or high isostatic pressing, to densify the printed part. Unlike laser/electron beam melting methods, binder jetted parts avoid thermal stresses or deformation caused by large thermal gradients¹⁰⁵. Advanced binder jetting printers, such as ExOne’s X1160Pro, achieve printing speeds exceeding 10 L/h, significantly faster than the 0.3 L/h rate of LPBF printers¹⁰⁶.

Mostafaei et al.¹⁰⁵ demonstrated binder jetting’s capability in fabricating intricate structures by producing complex-shaped, high-precision dentures. These features position binder jetting as a transformative technology for expanding the adoption of TPMS lattice structures in aerospace and biomedical applications. Xie et al.¹⁰⁶ pioneered the fabrication of gyroid lattice structures with varying relative densities using Ti6Al4V alloy via binder jetting. The dimensions of the binder jetting additively manufactured gyroid lattice structures in the X, Y, and Z directions deviated from the designed model by less than 0.2 mm, while the relative density deviated by less than 3%. The relative density deviations were mainly attributed to excessive binder penetration at the model edges. SEM images (Fig. 10b) revealed that the gyroid lattice structure surface consisted of powder particles tightly connected through sintering necks, resulting in surface roughness. Furthermore, a comparison between μCT scan reconstructions and CAD designs indicated that the actual wall thickness slightly exceeded the designed wall thickness. Binder penetration at the boundaries was identified as a major factor affecting dimensional accuracy and boundary uniformity, which cannot be corrected by simply adjusting the scaling factor of the print file. It’s expected that industrial-grade binder jetting printers can produce finer geometries and smoother surfaces by increasing nozzle resolution and reducing ejected droplet size¹⁰⁷.

Post-processing

The as-built condition of TPMS lattice structures might not be suitable for direct applications. Specifically, the internal defects existing in the lattice structures are undesired as they increase the level of variations during the measurement of mechanical properties and decrease the mechanical strength¹⁰⁸. Furthermore, the microstructure of additive manufacturing is generally sub-optimal as the directional cooling may cause anisotropy, and fast cooling rates may form microstructural phases with inferior properties. Consequently, post-processing methods are employed, including heat treatment, and surface modification.

Heat treatment

Rapid cooling during the L-PBF process introduces substantial internal thermal stresses. Heat treatment is frequently employed to enhance microstructures and minimize these stresses. Yan et al.³² were the first to report the effect of heat treatment on TPMS-based Ti6Al4V structures manufactured using the SLM technique. The as-built samples exhibit excellent columnar grains of α’ martensitic with a width of 100–300 nm. After heating at 680 °C for 4 h and then later sandblasting, α’ martensite was converted to a mixture of α and β with a width of 500–800 nm being the dominant phase (Fig. 11a). The heat treatment removes the partially melted particles and hence decreases the surface roughness. Due to heat treatment, the grain size increases, and the presence of the β phase decreases the micro-hardness of the samples from (4.01 ± 0.34) GPa to 3.71 ± 0.65 GPa. Sun et al.¹⁰⁹ used heat treatment to enhance the compressive deformation mechanism of AlSi10Mg produced via SLM. Heat treatment increased grain size, oriented grains towards <100>, and shifted the fracture mechanism of TPMS lattice structures from brittle to ductile (Fig. 11b). Heat treatment can also increase the energy absorption capacity of the lattice structures. Al-Si10-Mg lattice structures were observed to significantly improve energy absorbing capacity at the expense of its reduced strength¹¹⁰. Maskery et al.¹¹¹ investigated the effect of heat treatment on the double G-TPMS lattice structure of Al-Si10-Mg. The heat treatment decreases the ultimate tensile strength of the structure by 12%, at the same time significantly enhancing its ductility. In contrast, Al-Ketan et al.¹¹² investigated the effect of heat treatment on the TPMS lattice structures of maraging steel. After the heat treatment, the toughness of the samples was extensively reduced. The decrease in toughness and increase in strength was due to the nickel-rich intermetallic phase formation during the heat treatment.

Fig. 11: Comparison of as-built and heat-treated TPMS lattice structures.

a Microstructure of the struts in both as-built and heat-treated TPMS lattice structures³². b SEM images of the fractured surfaces from both the as-built and heat-treated samples¹⁰⁹.

Surface modification

Additively manufactured metallic components may display rough surface finishes, notably lattice structures from L-PBF with partially melted particles¹¹³. The mechanical properties of metallic lattice systems, notably fatigue strength, would decrease as a result of the increasing surface roughness of such linked particles. It has been suggested to use techniques including blasting^113,114, chemical etching¹¹⁵, and electrochemical polishing¹¹⁵ to remove the adhered particles. To further improve the surface roughness of the metallic TPMS lattice structure, Yang et al.¹¹⁶ utilized sandblasting to mitigate step effects and particle influence on 316L TPMS lattice structure surfaces. The sandblasting causes the work hardening due to the local plastic deformation on the structure’s surface and the improvement in surface finish as shown in Fig. 12a. The improvement in mechanical properties was observed, Young’s modulus of sandblasted samples was increased from 1097.87 ± 20.86 MPa to 1163.69 ± 23.33 MPa, and yield strength increased from 19.15 ± 0.45 MPa to 20.95 ± 0.35 MPa, while the increment of 26.18 ± 0.33 MPa to 27.65 ± 0.38 MPa was observed in plateau stress. Additionally, the fatigue properties were also improved, fatigue strength was increased from 9.1 to 11.7 MPa at 2 × 10⁶ cycles, and the endurance limit was increased from 0.35 to 0.45. Shivank A. Tyagi et al.¹¹⁷ used blast finished and electro-polished to obtain the surface finish and integrity required for very fine pore size (100–200 μm) SS 17-4PH-based TPMS lattice structures as shown in Fig. 12b. Surface roughness is significantly reduced by up to 85% for electro-polishing and up to 50% for blast finishing, even inside complex surfaces. Electrochemical polishing may diminish strut diameter, weakening lattice structures, and necessitating a balance between surface roughness and structural strength, as highlighted by Pyka et al.¹¹⁵. To prevent strut damage and material loss, Soro et al.¹¹⁸ proposed intermittent etching to maintain structural integrity while improving surface roughness. There are other methods of surface modification, for instance, shot peening^119,120, laser peening¹²⁰, water jet peening¹²⁰, low plasticity burnishing¹²¹, and others. These methods are not used for any TPMS structures and need to be explored.

Fig. 12: Comparison of different surface modifications.

a SEM images compare as-built 316L TPMS lattice structure samples with sandblasted ones, highlighting improvements in mechanical properties¹¹⁶ and b blasted finished versus electro-polished for SS 17-4PH based TPMS lattice structure improvement¹¹⁷.

Properties of metallic TPMS lattice structures fabricated by additive manufacturing

Recent years have witnessed significant attention directed towards enhancing the performance of metallic TPMS lattice structures. The objective of combining geometric design and additive manufacturing is to emulate the impressive attributes of natural lattice structures. This section analyzes mechanical, thermal, or permeability properties according to different disciplines.

Mechanical properties

Extensive research has primarily centered on assessing the mechanical attributes of metallic TPMS lattice structures, outweighing investigations in other domains. Whether deployed as scaffolds, energy absorbers, or heat exchangers, the fundamental mechanical performance of TPMS lattice structures is pivotal to ensuring their stability and reliability¹²². Geometric freedom in additive manufacturing opens a plethora of opportunities for part optimization and performance gains across industries.

The compression deformation behavior of lattice structures may be broken down into three distinct stages (Fig. 13): linear elastic deformation, plastic deformation, and densification^29,123. During elastic deformation, the material reaction is linear elastic with a modulus corresponding to the elastic modulus of the structural material. However, Ashby et al. advocate measuring the unloading modulus for the most accurate estimate of the elastic modulus of metallic lattice structures since it better captures the structure’s performance²⁹. When the elastic limit is reached, plastic deformation occurs, with unit lattice structures yielding or buckling. In bending-dominated structures, deformation proceeds with an essentially constant stress, known as the plateau stress, but in stretch-dominated structures, the stress necessary for further deformation oscillates. When unit lattice structure components deform sufficiently to make contact with other components, preventing further deformation, the densification strain is reached, and densification begins as stress rises sharply. Strut-based lattice structures, such as body-centered cubic (BCC), face-centered cubic (FCC), octet, Weaire-Phelan, Kelvin and honeycomb configurations, deform primarily through tensile action under mechanical load. This efficient load transfer along the trusses results in higher stiffness and initial collapse strength. Conversely, sheet-based structures, like TPMS structures, deform primarily through bending. This behavior allows the material to absorb energy effectively before densification, giving these structures high compliance.

Fig. 13

General compressive behavior of stretch and bending-dominated lattice structures during elastic deformation, plastic deformation, and densification^29,123.

Maskery et al. investigated the compressive failure modes shown in Fig. 14a¹¹¹. where the strength was continually lost and regained when each layer crumbled and was squeezed into the one below. After densifying each layer, the structure got stronger, recovering up to 90% of its initial crushing strength before reaching 50% strain. Yang et al.¹²⁴ first described the graded TPMS’s capacity to absorb energy. The failure mechanism is significantly influenced by the cell size. According to Fig. 14b, as cell size increases, the capacity to absorb energy diminishes. The majority of prior research has been on compressive loading of metal lattices, but tensile characteristics have received less attention despite their importance in the envisioned applications. This is due to the difficulty in fabricating tensile lattice specimens devoid of stress concentration locations. One disadvantage of traditional designs of tensile samples consisting of solid-lattice-solid stacks is their tendency to fail prematurely at the solid-lattice interface due to an abrupt change in density. Furthermore, layer-by-layer production leads to non-uniform residual stress distribution and manufacturing flaws at the solid-lattice interface. Bharath et al.¹²⁵ examined various lattice structures (Diamond, Gyroid, Fischer, IWP, and Primitive) under compression and tension, revealing topological dependencies in elastic modulus, yield strength, and energy absorption values (Fig. 14c). Importantly, the Fisher structure displayed superior properties compared to sheet diamond lattices in both compression and tension. As shown in Table 2, numerous metallic TPMS lattice architectures have been created and studied.

Fig. 14: Mechanical behaviors.

a Compression failure modes and energy absorption of metallic TPMS lattice structures¹¹¹. b Compression behavior of different metallic TPMS lattice sizes¹²⁴. c Comparison of the topological dependence of the mechanical properties of different TPMS lattice structures in tension and compression¹²⁵. d Comparison of fatigue behavior of TPMS and octahedron¹²⁶.

Table 2 Various metallic TPMS lattice structures are produced by different additive manufacturing technologies, and their mechanical properties (obtained from compressive tests)

To further elucidate the mechanical performance of metallic TPMS lattice structures, a comparative analysis was conducted using a range of lattice structures, and the relationships between relative elastic modulus and relative strength as functions of relative density were plotted (Fig. 15). The dataset reveals that lattice topology significantly influences stiffness efficiency. Specifically, bending-dominated structures, such as BCC and cubic lattices, exhibit rapid stiffness degradation with increasing porosity due to localized bending at the intersections of the struts. In contrast, stretch-dominated structures, including octahedral trusses and diamond lattices, demonstrate superior stiffness retention at lower relative densities. Notably, TPMS structures, such as Gyroid and Primitive lattices, outperform traditional strut-based designs due to their continuous and smooth surface topologies. This geometric characteristic facilitates enhanced stress redistribution and minimizes localized stress concentrations, consequently improving mechanical efficiency. For instance, 316L stainless steel TPMS lattice structures exhibit a relative modulus of 0.02, significantly exceeding that of conventional bending-dominated structures. Furthermore, the Ashby chart depicting the relationship between relative density and yield strength highlights a significant disparity between bending- and stretch-dominated structures (Fig. 15b), confirming the substantial strength loss experienced by bending-dominated structures at high porosities. While BCC and cubic lattices have been extensively studied, their poor strength retention at low densities limits their effectiveness in structural applications. Conversely, stretching-dominated structures, TPMS lattice structures offer a more optimal balance, maintaining strength while minimizing stress concentration.

Fig. 15: Relationship between relative elasticity modulus and relative strength as a function of relative density for a wide array of architected lattice structures.

a Relative elasticity modulus and b relative strength depicted against the relative density. The theoretical limits on stiffness and strength are marked by black and blue lines, representing their upper and lower bounds based on scaling principles¹⁶⁷.

Many high-value technological applications rely on the fatigue performance of L-PBF metallic lattice structures. Both biomedical and aeronautical components, for example, are subject to strict cyclic loading limitations²⁹. Fatigue failure in additively manufactured lattice structures, similar to that observed in other porous metallic materials, progresses through three distinct stages. Stage I is characterized by an initial creep-like deformation driven by localized stress concentrations, leading to microstructural rearrangements and subsequent crack nucleation; this early stage is critical in determining the overall fatigue resistance of the structure. As cyclic loading continues, Stage II begins with crack initiation and is followed by crack propagation, as accumulated cyclic strain causes gradual material degradation. Finally, Stage III involves crack coalescence and a rapid decrease in stiffness, culminating in structural failure, as illustrated in Fig. 16. The mechanical characteristics of the bulk material, the lattice’s relative density, and unit cell topology have the biggest effects on the fatigue properties of lattice structures. Speirs et al.¹²⁶ investigated the compressive fatigue behavior of three different lattices (octahedral, cellular gyroid and sheet gyroid) of SLM nitinol scaffolds, as shown in Fig. 14d. When TPMS and the traditional octahedral beam lattice structure were evaluated at the same relative density, it was discovered that TPMS had superior static mechanical properties. The lattice structure with a higher relative density can resist greater loads and more cycles than the one with a lower relative density. Experimental studies have consistently demonstrated the superior fatigue resistance of metallic TPMS lattice structures, particularly gyroid, primitive, and I-WP structures, when compared to conventional strut-based designs. This enhancement is primarily attributed to the smooth surface morphology and continuous, curvilinear load paths characteristic of TPMS lattice structures, which effectively minimize stress concentrations and, consequently, delay fatigue-crack initiation. Conversely, strut-based topologies, such as BCC and octet-truss lattices, are prone to early failure due to stress amplification at their nodal intersections. Moreover, fatigue performance is significantly influenced by loading conditions. While strut-based lattices like octet-truss structures may exhibit acceptable endurance under uniaxial compression, TPMS lattice structures generally demonstrate greater reliability under multiaxial or shear-dominant loading scenarios. This improved performance under complex loading stems from their inherent geometric isotropy and stress-delocalizing capabilities. These observed differences highlight the critical need for systematic comparative analyses of fatigue behavior across diverse lattice types, geometric configurations, and porosity levels, as will be detailed in the subsequent dataset presentation and discussion. Table 3 compares the fatigue strength of different metallic lattice structures produced by different additive manufacturing technologies, which gives a comparison among the unit cells, porosity, and fabrication methods.

Fig. 16

The total strain evolution of metallic lattice structures under cyclic loading typically exhibits a three-stage fatigue damage progression^167,168.

Table 3 Fatigue properties of different metallic lattice structures produced by additive manufacturing. The data were obtained from compression-compression fatigue tests

TPMS designs consistently exhibit superior fatigue performance in both low-cycle and high-cycle fatigue regimes. This is attributed to their continuous surface geometry, which minimizes stress concentration and promotes uniform stress distribution. Conversely, strut-based architectures typically experience earlier fatigue failure due to geometric discontinuities and localized strain accumulation at nodal connections. While the mechanical properties of existing TPMS geometries have been extensively studied, the tensile fatigue behavior of metallic TPMS lattice structures remains underexplored. Different TPMS cell types offer unique advantages, requiring informed choices for specific applications and balancing interdisciplinary needs. The changing needs of multifunctional applications require careful tradeoffs.

Thermal properties

Managing excess heat generated during operation poses a significant challenge for computer microchips as well as large-scale machinery and equipment. Over time, metallic lattice structures have gained extensive use as effective heat exchangers. Complex structures with high porosity substantially augment internal heat exchange regions. This advantage has led to the prevalent use of parameterized lattice structures or stochastic metal foams in modern heat exchangers. Theoretically, TPMS structures, characterized by linked and smooth pores, offer superior heat exchange material options. The previously mentioned adjustable geometries and porosity characteristics provide invaluable benefits for further enhancing heat transfer efficiency. Despite the clear advantages, TPMS heat exchangers remain less common in practical engineering applications compared to conventional lattice counterparts. Nevertheless, limited existing research has delved into and illustrated the advantages of TPMS’s thermal performance.

Kaur et al.¹²⁷ explored TPMS’s flow and heat transfer properties, using the commercially available open-cell foam known as the TKD structure for performance evaluation. Figure 17a illustrates the heat transfer distribution of the Primitive, Gyroid, and TKD structures with the same porosity, viewed from different perspectives. Comparatively, the Gyroid exhibits an average heat transfer coefficient 1.07 times higher than that of TKD. Furthermore, Gyroid structures can dissipate heat twice as rapidly as TKD structures at the same temperature differential. Liu et al.¹²⁸ investigated the average surface temperature, thermal resistance, and heat transfer coefficient of four different TPMS lattice structure heat sinks made of AlSi7Mg using L-PBF. Under forced convection, homogeneous TPMS exhibit varying heat transfer performance, with the Primitive structure demonstrating superior comprehensive performance, while the Gyroid structure excels under natural convection conditions (Fig. 17b). Notably, the P-Quadratic II structure outperforms both homogeneous and gradient TPMS designs in comprehensive heat transfer under forced convection and natural thermal conductivity conditions. Specifically, the average convection heat transfer coefficient of the P-Quadratic II structure is 13.44–19.78% higher than that of the homogeneous Primitive structure. Qureshi et al.¹²⁹ harnessed TPMS as heat transmission structures in phase transition materials for thermal energy storage. The TPMS lattice structures are integrated into phase change materials, causing them to transition from solid to liquid upon heating. Figure 17c shows that at 60 seconds, with a consistent 90% porosity, the TPMS lattice structure exhibits a higher liquid fraction compared to the Kelvin structure. Regarding pure thermal conduction, the IWP structure performs better than the Gyroid, Primitive, and Kelvin structures. In natural convection, the Primitive structure surpasses the Gyroid, IWP, and Kelvin structures. Al-Ketan et al.¹³⁰ also investigated the heat transmission capabilities of graded TPMS. The analysis considered temperature contours and velocity streamlines of uniform and graded diamonds from different angles, as shown in Fig. 17d. Both uniform and graded structures possess identical porosities, with uniform TPMS showcasing 22% larger surface areas than graded ones. The experimental findings highlight that the porosity grading structure induces a 27.6% pressure drop, while the decrease in convective heat transfer is limited to 15.7%. This emphasizes the influence of porosity distribution design on heat transmission performance control. The performance of the heat transmission may be effectively controlled by choosing the right porosity distribution design.

Fig. 17: Thermal properties of metallic TPMS lattice structures.

a Heat transfer distribution of Primitive, Gyroid, and TKD¹²⁷. b The heat-sink experiment of homogeneous TPMS heat sinks under natural convection conditions¹²⁸. c The liquid fraction at 60 s after heat transfer by different TPMS lattice structures¹²⁹. d Temperature contours and velocity vectors of the uniform and graded Diamond¹³⁰.

To conclude, TPMS structures stand out as exceptional choices for heat transfer applications due to their smooth porous architectures and high-volume-specific surface areas. In certain specialized applications, TPMS’s thermal performance even surpasses that of traditional porous structures like foams or lattices. The thermal performances of TPMS can be effectively regulated by its unit type, relative density, and structural characteristics, analogous to mechanical properties. However, current research remains inadequate for more intricate applications. There’s a dearth of research on the thermal performance of graded, heterogeneous, and multiscale TPMS. Subsequent studies should prioritize these interrelated concerns for a more comprehensive understanding.

Permeability properties

As previously mentioned, TPMS structures exhibit a high degree of interconnection and possess non-tortuous pores that lend themselves well to mass transfer applications. This advantageous characteristic makes TPMS lattice materials suitable for various engineering purposes, including porous electrodes for fuel cells and batteries, porous filters, and scaffolds for tissue engineering. Permeability, a vital metric for assessing fluid flow efficiency, hinges on several porous characteristics: pore size, tortuosity, porosity, and interconnectivity. Recent investigations have delved into these properties, establishing a theoretical foundation for engineering applications.

Ma et al.¹³¹ explored the mass transfer capabilities of Gyroid structures while creating scaffolds mimicking bone structures. The Gyroid’s inherent properties lead to minimal turbulence. Figure 18a illustrates the mass transfer traits of bone-mimicking Gyroid scaffolds, depicting fluid pressure and permeability within these structures. Adjusting pore diameter can augment permeability, and the permeability of gyroid scaffolds can mirror that of actual bones. Experimental results underscore that TPMS structures can achieve notable permeability levels while maintaining lower porosity. However, it’s important to note that elevating porosity to enhance permeability might compromise TPMS’s mechanical performance. A balance must be struck between mechanical attributes and permeability to align with diverse application needs^45,132,133. Asbai-Ghoudan et al.¹³⁴ developed an analytical model incorporating intended architecture, pore size, and porosity to predict TPMS permeability. The projected and observed permeabilities differ by less than 5%. In Fig. 18b, the correlation between permeability (k) and pore size is depicted for Fisher-Koch S, Gyroid, and Schwarz Primitive structures, along with linear fits of permeability coefficients based on porosity (ϕ) for each architecture. The Schwarz Primitive structure consistently boasts the highest permeability. However, the Gyroid structure exhibits superior permeability for structures with 50% porosity. TPMS structures demonstrating enhanced permeability at the same relative density could emerge as the preferred choice across various applications.

Fig. 18: Permeability properties of metallic TPMS lattice structures.

a The mass transfer characteristics of the bone-mimicking scaffolds of Gyroid structure¹³¹. b The Correlation between the permeability k and the pore size for Fisher-Koch S, Gyroid, and Schwarz Primitive and linear fitting of the coefficients of the permeability curves according to porosity ϕ for each architecture¹³⁴.

In general, TPMS permeability research has gained prominence in recent years. TPMS constructions undeniably contribute to heightened permeability^{135,136,137,138}. In contexts like tissue engineering scaffolds or bone implants that necessitate mass transfer, precise control over permeability is pivotal. Yet, accurately estimating and assessing TPMS permeability remains a challenge. Much of the recent research has centered on the standard cube-shaped TPMS, much like the focus on other performance aspects. While some endeavors have explored the permeability of graded TPMS, further investigation is warranted to unravel the mechanisms governing porosity distribution’s influence, heterogeneous TPMS, and multiscale TPMS.

Multiple applications of metallic TPMS lattice structures

Optimal TPMS products find diverse applications through geometry design, precise additive manufacturing, and tailored properties. Recently, the significance of TPMS applications has surged, garnering substantial research focus. The topology-dependent multifunctionality of TPMS lattice structures has become a focal point of inquiry. With their sleek surfaces and intricate interconnected pores, TPMS structures seamlessly integrate across various domains. This section underscores the robust potential of metallic TPMS lattice structures by examining their representative applications within mechanical, thermal, biological, and chemical realms.

Functional components

The fundamental mechanical attributes of metallic TPMS lattice structures have undergone thorough scrutiny. These structures can serve as energy or impact absorbers, capitalizing on their capacity for lightweight, stress distribution, and deformation management^139,140. Remarkably, TPMS’s weight-to-volume ratio surpasses that of solid constructs. In engineering practice, TPMS structures stand poised for direct integration as functional components. Alkebsi et al.¹⁴¹ harnessed TPMS lattice structures as turbine blades, optimizing their porosity distribution through topology optimization techniques. This innovation led to superior lightweight characteristics, enhanced stress management, and mitigated deformation, surpassing previous models. Wang et al.¹⁴² pioneered the use of TPMS structures to create soft robot joints. Experimental outcomes reveal that the Primitive surface excels as a choice for flexure hinges. This choice elevates compliance and the compliance ratio significantly compared to conventional leaf flexure hinges. Beyond these instances, TPMS structures offer an array of captivating mechanical applications, attesting to their versatility and promising potential across a broad spectrum of engineering contexts.

Heat exchangers

TPMS’s exceptional heat transfer attributes stem from its high-volume-specific surface areas. Consequently, its primary thermal application lies in the domain of heat exchangers. Attarzadeh et al.¹⁴³ explored diamond surfaces for heat exchange purposes, yielding promising experimental results that suggest heightened potential for efficient heat transmission between heat sources and moving air. Exploiting smaller wall thicknesses could further amplify the heat exchanger’s thermal prowess. The supercritical Brayton cycle, based on carbon dioxide (CO₂), is widely used in engineering. Nonetheless, modern heat exchangers’ cycle efficiency and heat transfer capabilities often fall short of desired expectations. Li et al.¹⁴⁴ attempted to utilize TPMS structures inspired by biological systems as heat exchangers aimed to address this shortfall. Gyroid and Diamond structures, boasting smooth surfaces, exhibit notably reduced flow separation compared to printed circuit heat exchangers. Gyroid structures notably foster turbulent kinetic energy production, beneficial for efficient heat transfer. Additionally, TPMS demonstrates a significantly elevated heat transfer rate compared to printed circuit heat exchangers. Impressively, TPMS also boasts a higher heat transfer coefficient, ranging from 16% to 100%. However, it’s worth noting that despite these advancements, research and applications of TPMS in the thermal domain are still in their infancy compared to other fields. The majority of TPMS lattice structure heat exchangers in use today exhibit uniform porosities and conventional configurations. A captivating challenge that necessitates further exploration pertains to enhancing the heat transfer capabilities of TPMS lattice structure heat exchangers within compact volumes.

Biological implants and scaffolds

In the biological field, TPMS structures have recently gained considerable traction. Remarkably, TPMS’s geometry and topology mirror those found in natural structures, endowing it with distinct advantages for biological applications. Medical implants and scaffolds for tissue engineering stand out as two prevalent applications for TPMS lattice materials. Unlike traditional lattice designs, TPMS exhibits sleek surfaces conducive to cell attachment and growth. Furthermore, its intricate interconnected porous structures and specialized surface regions with significant volumes facilitate the efficient transport of nutrients and waste. Current biological engineering harnesses TPMS structures effectively, capitalizing on their exceptional biological performance and the previously mentioned tunable mechanical and mass transport capabilities.

Li et al.¹⁴⁵ explored early osteointegration of TPMS Ti6Al4V scaffolds, observing their capacity to stimulate bone ingrowth. Stable contact between implants and surrounding bone tissues was achieved just 5 weeks post-implantation. Micro-CT scans underscored pervasive bone growth throughout TPMS porosity structures. Hsieh et al.¹⁴⁶ juxtaposed lattice and TPMS scaffolds to underscore distinctions among various porous structures. Hsieh et al. juxtaposed lattice and TPMS scaffolds to underscore distinctions among various porous structures. TPMS scaffolds exhibited superior bone development compared to the octet-truss lattice. Additionally, TPMS scaffolds exhibited lower fatigue failure rates than lattice structures. Barba et al.¹⁴⁷ compiled recommended TPMS pore diameters for different bone phases. The variable porosity of TPMS enables construction with different hole diameters to facilitate bone colonization or vascularization. IWP and Neovius structures proved effective in establishing suitable osseointegration zones at larger unit cells. Alabort et al.¹⁴⁸ confirmed the feasibility of utilizing 3D printing to apply TPMS as metallic bones, with TPMS structures exhibiting excellent osteointegration potential. Possessing appropriate stiffness and strength, TPMS can emulate both cortical and trabecular bones, positioning it as a formidable contender for various biological applications. In summation, TPMS structures hold promise across a broad spectrum of biological applications. By crafting exceptional TPMS implants or scaffolds with pertinent geometries and attributes, the demands of the human biological environment can be effectively met.

Catalysts and reactors

TPMS’s interconnected structures and expansive specific surface areas offer utility across diverse applications. In the chemical realm, for instance, the amplification of chemical contact areas can notably elevate the efficiency and quality of chemical reactions. As a result, TPMS structures hold significant promise as catalysts or reactors, an avenue that recent research has intriguingly explored.

In the present landscape of chemistry and energy, hydrogen energy is a crucial subject. The methanol steam reforming process stands as an efficient hydrogen production method. Yet, the catalyst support profoundly impacts the catalytic reaction rate and conversion rate within micro-reforming reactors. Addressing this, Lei et al.¹⁴⁹ utilized TPMS structures as catalytic supports, leading to enhanced hydrogen generation performance compared to commercial alternatives. Notably, the controllable macro-to-micro scale of TPMS empowers the manipulation of flow fields and reaction rates within the methanol steam reforming process. In the context of CO₂ methanation, Baena-Moreno et al.¹⁵⁰ devised TPMS microreactors, contrasting their performance against honeycomb solutions to elucidate TPMS’s advantages. Enhanced mass and heat diffusion processes translate into a noteworthy 14% enhancement in CO₂ conversion under optimal conditions. Expanding on TPMS structures, Sun and colleagues developed solar vapor-generating devices¹⁵¹. These apparatuses deliver exceptional evaporation capabilities with substantially lower photothermal component loading compared to similar devices. It’s pertinent to note that the majority of TPMS lattice structures currently in use adhere to uniform lattice designs. The performance of gradient or heterogeneous TPMS lattice structures necessitates heightened research attention in the future, as these structures hold promising prospects across various applications.

Limitations and challenges

While metallic TPMS lattice structures have shown remarkable functional flexibility, their complex geometries have traditionally posed significant challenges for conventional manufacturing methods. Undoubtedly, additive manufacturing technologies have ushered in a new era for designing and fabricating intricate metallic TPMS lattice structures. However, it’s important to recognize that additive manufacturing technologies, while groundbreaking, are not without limitations and challenges when it comes to producing metallic TPMS lattice structures across various industries. It is crucial to consider these limitations and challenges within the realm of additive manufacturing technologies and lattice structures, as shown in Fig. 19.

• (1) During the geometric design phase, the design freedom afforded by current methods remains insufficient to meet the demands of complex applications. This limitation is particularly pronounced in the design of complex TPMS lattice structures, where the majority of existing CAD algorithms rely on Boolean operations. While effective, these operations incur significantly higher computational time and space consumption compared to the creation of standard solid models. Consequently, the computational processes involved in generating intricate TPMS structures are susceptible to interruption due to resource exhaustion.

• (2) Despite the advantages of additive manufacturing, the resulting part quality often requires improvement. While adjusting process parameters and implementing post-processing techniques can mitigate defects, challenges remain, particularly in the context of metal TPMS lattice structures. For instance, in L-PBF, the presence of partially or completely unmelted powder adhering to the structure’s surface increases surface roughness, necessitating post-processing. However, these post-processing steps must carefully balance surface finish and structural integrity. Even μLPBF, which offers improved precision, still faces the issue of powder adherence on TPMS lattice surfaces. Critically, these adhered particles can act as crack initiation sites during component deformation. MEAM offers a cost-effective and easy method for producing TPMS lattice structures, but it yields low precision and is highly sensitive to processing parameters. While binder jetting offers advantages in low printing costs and high printing efficiency, binder penetration at the boundaries significantly affects dimensional accuracy and surface finish. This issue cannot be corrected solely by adjusting the scaling factor of the print file, thus limiting the production of fine geometries and smooth surfaces.

• (3) Although CAD algorithms and additive manufacturing techniques can generate different geometric features, most performance control studies are still based on uniform metal TPMS lattice structures. Only a few mechanical properties studies are based on graded TPMS. While compression testing is frequently employed in mechanical deformation studies of additively manufactured metallic TPMS lattices due to its relative simplicity, tensile fatigue testing offers a valuable, complementary approach. Specifically, tensile fatigue testing can provide critical insights into the structural deformation behavior of diverse metallic TPMS lattice materials, thereby enhancing our understanding beyond that gained solely from compression-based analyses. The mechanical testing under tensile loading can present a useful understanding of the structural deformation of different TPMS lattice materials. More research is needed on the heat and mass transfer properties of metallic TPMS lattice structures, especially heterogeneous TPMS and TPMS structures with complex external shapes.

• (4) Topology optimization is a complex process. If the design model exceeds the machinability of the device, the metal TPMS lattice structure may not be manufactured with additive manufacturing technology. However, iterative optimization is time-consuming, and more efficient topology optimization methods remain a focus of research. Furthermore, two-dimensional density results lack the precision needed for real-world engineering problems. In practice, additional constraints further increase computational cost. These issues require further investigation to generate ideal TPMS models with appropriate geometries and performance.

• (5) Designing and manufacturing metallic TPMS lattice structures requires a multidisciplinary applications, encompassing materials science, mechanics, finite element analysis, and heat and mass transfer. Critically, their multifunctional performance requires further investigation. Balancing competing requirements poses a significant challenge. Therefore, realizing metallic TPMS lattice structures with comprehensive applications demands multidisciplinary collaboration to construct structures with optimal porosity distributions using appropriate TPMS unit cells.

• (6) The absence of dedicated standards for metal TPMS lattice structures leads to significant data variation. This lack of standardized assessment, in comparison to other lattice types, severely limits objective comparison across studies. From a design perspective, unified parametrization is needed, necessitating ISO/ASTM guidelines based on TPMS lattice structure topology classification and rating. From a manufacturing perspective, the printing process window can be determined based on TPMS geometry and machine calibration protocols, given the absence of specific process parameters. Finally, inconsistent testing methodologies, including non-standardized procedures, missing surface quality metrics, and disparate porosity analysis techniques, contribute to characterization challenges.

Fig. 19

The limitations and challenges of metallic TPMS lattice structures encompass geometric design, printing defects, complex TPMS, topology optimization, comprehensive applications, and specific standard.

Conclusions

This review systematically summarizes recent research findings concerning the design, manufacturing, property, and applications of metallic TPMS lattice structures. Specifically, TPMS design methodologies represent a multiscale design strategy tailored to complex actual complex requirements, encompassing features such as gradient porosity, heterogeneous characteristics, complex freeform external shapes, and topology optimization. However, current methods in the geometric design phase still exhibit limitations in design freedom, hindering their ability to meet the demands of complex applications. Therefore, more efficient and adaptive design strategies are needed to construct ideal metallic TPMS lattice structures. Subsequently, advanced manufacturing techniques are essential to translate CAD models into functional physical objects. Traditional manufacturing methods often struggle with the intricate geometries of TPMS structures, rendering them difficult to produce via milling or turning. The rapid advancement of additive manufacturing technologies has thus been crucial in facilitating the development and application of metallic TPMS lattice structures. The three-dimensional porous features can be realized through the layer-by-layer accumulation of two-dimensional slices. Metallic TPMS lattice structures can be manufactured with high precision across various scales using different 3D printing techniques, such as L-PBF, MEAM, and binder jetting, depending on the specific manufacturing requirements and materials. Nevertheless, appropriate post-processing techniques remain necessary to address issues such as residual stress, surface roughness, and mechanical anisotropy. In terms of performance, multifunctional TPMS lattice structures, capable of simultaneously withstanding mechanical loads and enhancing heat and mass transfer properties, represent a critical frontier in future research. This multifunctionality can be controlled through both geometric design approaches and additive manufacturing techniques. In summary, with optimized geometries, precise manufacturing quality, and reliable performance, metallic TPMS lattice structures hold significant promise for successful application across diverse fields in the future.