1. Introduction

High-temperature dust removal, catalytic conversion, impact energy absorption—these scenarios place seemingly contradictory demands on materials: sufficient mechanical integrity to serve as a load-bearing body, yet three-dimensionally interconnected pore channels to allow fluid or energy to pass through. Over the past decades, engineers have mainly relied on two materials in this gap: porous metals and porous ceramics ^{[1, 2]}. The former, dominated by stainless steels and titanium alloys, exhibit good ductility, weldability, and insensitivity to thermal shock, but in oxidizing atmospheres above $400{}^{\circ }\mathrm{C}$ their surfaces are rapidly eroded and their creep resistance falls short ^[3]. The latter, honeycomb or foam ceramics represented by $\mathrm{SiC}$ and cordierite, can push service temperatures above $1000{}^{\circ }\mathrm{C}$ with excellent chemical stability, yet intrinsic brittleness remains a thorn—a single unexpected thermal shock can trigger catastrophic crack propagation ^[4]. These two systems occupy opposite ends of the performance spectrum, leaving a significant gap in between.

This gap provides an entry point for intermetallic compounds ^[5]. The chemical bonding in intermetallics is hybrid: metallic bonding contributes dislocation mobility, electrical and thermal conductivity, while covalent bonding raises the melting point, stiffness, and oxidation resistance. When iron and silicon, two extremely common elements, combine in a specific proportion, an ${\mathrm{Fe}}_3\mathrm{Si}$ phase with ${\mathrm{DO}}_3$ ordered structure is formed. Its yield strength anomalously rises rather than dropping near $600{}^{\circ }\mathrm{C}$ ^[6], and silicon atoms preferentially diffuse to the surface in oxidizing environments, spontaneously forming a continuous and dense $\mathrm{SiO}{\phantom{A}}_2$ passivation layer ^[7]. More practically, the interdiffusion rates of $\mathrm{Fe}$ and $\mathrm{Si}$ at high temperature are highly asymmetric—$\mathrm{Si}$ diffuses into $\mathrm{Fe}$ much faster than $\mathrm{Fe}$ diffuses into $\mathrm{Si}$. This mass-flow imbalance naturally generates a large number of vacancies during the phase transformation; upon coalescence, these vacancies leave three-dimensionally interconnected micropores ^[8]. In other words, framework construction and pore generation are accomplished in a single sintering step, without the need for additional pore-forming agents or polymeric template burnout. This mechanism is simple, controllable, and uses inexpensive raw materials, quickly attracting several research groups working on high-temperature filtration and catalyst supports ^[9].

From an application perspective, the most outstanding feature of $\mathrm{Fe}\text{--}\mathrm{Si}$ porous materials is that they combine several usually incompatible attributes: decent mechanical strength (compressive strength typically in the range of $5\text{--}20\mathrm{MPa}$), acceptable thermal shock resistance, significantly better corrosion resistance than stainless steel in sulfur-containing flue gas or acidic condensate environments, and high-throughput permeability from the interconnected open-pore structure ^[10]. Their fracture toughness, although certainly inferior to pure metals, is still a notch above cordierite or $\mathrm{SiC}$ ceramics, favoring progressive damage rather than sudden brittle fracture in service ^[11]. Collectively, these characteristics leave $\mathrm{Fe}\text{--}\mathrm{Si}$ porous bodies with almost no direct competitor in oxidizing-corrosive environments at $600\text{--}800{}^{\circ }\mathrm{C}$.

Looking back at the accumulation over the past two decades, the evolution of preparation technologies can be roughly divided into several lines. The earliest-trodden path was powder reactive sintering—mixing $\mathrm{Fe}$ and $\mathrm{Si}$ powders, cold pressing, and vacuum sintering to complete compound synthesis and pore generation in one step ^[12]. This route requires low-end equipment and easily interfaces with existing powder metallurgy production lines, and remains the closest to mass production. Its limitation is that the randomness of particle stacking leads to a wide pore-size distribution, making it difficult to finely design pore morphology. To address this, researchers have introduced two distinct technical approaches. One is dealloying using amorphous ribbons as precursors: an $\mathrm{Fe}\text{--}\mathrm{Si}$ alloy is melt-spun into an amorphous state, partially crystallized by heat treatment, and then the $\mathrm{Fe}$-rich phase is selectively dissolved in an acid solution, leaving a silicide skeleton with pores down to tens of nanometers and specific surface areas much higher than powder products ^[13]. The other route is more direct—sacrificial templating: organic sponges, $\mathrm{NaCl}$ particles, or carbon powders are blended into the powder compact, then burnt out or dissolved during sintering to form oriented or hierarchical pores in situ ^[14]. Since the mid-2010s, the maturation of additive manufacturing techniques such as laser powder bed fusion has begun driving the field from "making blocks" to "printing components," with complex-shaped parts possessing gradient porosity and intricate internal channels now directly formable from digital models ^[15].

In terms of literature trends, the development pace over the past twenty years falls roughly into three stages: before 2010, annual publications were scarce, with work concentrated in a few groups in Germany and Japan, focusing on basic sintering processes and phase characterization ^[16]; from 2010 to 2018, a steady growth phase saw research shift toward quantitative pore-structure-property correlations, with China's output markedly increasing ^[17]; since 2019, the growth rate has further accelerated, with additive manufacturing, nanoporous materials, and hierarchical pore design becoming high-frequency keywords ^[18]. This trajectory indicates that $\mathrm{Fe}\text{--}\mathrm{Si}$ porous materials are advancing from "laboratory materials" toward "engineering candidate materials," though the overall technology readiness level is still climbing.

Problems are also evident. Some contradictions are inherited from other porous materials, e.g., once porosity exceeds about $60\mathrm{\%}$, compressive strength decays to levels unacceptable for engineering—the effective cross-sectional area of the load-bearing framework shrinks, and stress concentration is unavoidable ^[19]. Others are specific to the $\mathrm{Fe}\text{--}\mathrm{Si}$ system: the thermal non-uniformity during powder reactive sintering is particularly pronounced when scaling up; small samples can achieve excellent homogeneity, but when scaling to pilot-scale components, the dispersion of microstructure and pore size across different regions becomes uncontrollable ^[20]. Furthermore, the vast majority of existing mechanical, corrosion-resistance, and permeability data were obtained under single-load or single-environment conditions. In real service, temperature, corrosive media, fluid impingement, and mechanical vibration often act simultaneously, and publicly available data on long-term behavior under such multi-factor coupling are extremely scarce ^[21]. Without such data, engineering design must rely on conservative safety factors, diluting the material's performance advantages.

This paper does not intend to be an exhaustive compendium. Over a thousand entries already exist in literature databases, and repeating them is meaningless. Instead, I aim to extract a coherent logic from published results: how did this material evolve from a block of sintered $\mathrm{Fe}\text{--}\mathrm{Si}$ alloy to one whose pore structure can be tailored for specific applications? What choices were right, and which joints remain unconnected? The subsequent sections are organized as follows: Section 2 dissects the principle differences and process boundaries of four mainstream pore-forming methods; Section 3 discusses structure-property relationships among microstructure, phase constitution, and mechanical, thermal, and permeation properties; Section 4 consolidates several embryonic engineering applications such as high-temperature filtration, damping dissipation, thermal management, and surface protection; Section 5 summarizes the key scientific and technological bottlenecks hindering industrialization; Section 6 provides directional outlooks from perspectives of multi-scale structural design, multi-component composite, and data-driven process development.

2. Main preparation methods and optimization of Fe-Si porous materials

Turning iron and silicon into a porous body essentially accomplishes one thing: while consolidating into a shape, a portion of space is "emptied." But it is precisely the mechanism of this "emptying" that fundamentally distinguishes different technical routes. Currently, four relatively systematic pore-forming approaches exist: reactive sintering relies on vacancies generated by differential atomic diffusion rates; dealloying depends on the huge disparity in electrochemical stability between two phases; sacrificial templating uses a placeholder-removal logic for anti-structure design; and additive manufacturing directly treats pores as part of the printing path. These four sets of logic correspond to four completely different process philosophies, and their respective applicability boundaries are thus defined.

2.1. Powder reactive sintering

Reactive sintering is the most classical and industrially-ready route for $\mathrm{Fe}\text{--}\mathrm{Si}$ porous materials. Its core principle exploits a quite unique intrinsic feature of the $\mathrm{Fe}\text{--}\mathrm{Si}$ system: the diffusion rate of $\mathrm{Si}$ in $\mathrm{Fe}$ is much higher than that of $\mathrm{Fe}$ in $\mathrm{Si}$. When $\mathrm{Fe}$ and $\mathrm{Si}$ powders are mixed, cold-pressed, and heated in vacuum to about $1000\text{--}1200{}^{\circ }\mathrm{C}$, $\mathrm{Si}$ atoms preferentially migrate into the interior of $\mathrm{Fe}$ particles. This asymmetric mass flow disrupts the local material balance, and a large number of vacancies are injected into the diffusion front. When the vacancy concentration becomes supersaturated, they nucleate, grow, and coalesce, leaving pores near the original $\mathrm{Si}$ particle locations. Meanwhile, $\mathrm{Fe}$ and $\mathrm{Si}$ react to form a series of intermetallic compounds—typically the $\mathrm{FeSi}$ phase appears first, then progressively transforms into ${\mathrm{Fe}}_3\mathrm{Si}$ as $\mathrm{Si}$ continues to diffuse, ultimately yielding a framework dominated by the ${\mathrm{DO}}_3$ ordered structure. Pore formation and phase formation are two manifestations of the same process, requiring no external pore-forming agent ^{[8, 12]}.

This route has long held mainstream status because of its process simplicity. Pretreatment of $\mathrm{Fe}$ and $\mathrm{Si}$ powders, mixing, cold pressing, and vacuum sintering—the entire flow is highly compatible with conventional powder metallurgy lines, with low equipment barriers and controllable raw material costs. By adjusting three core parameters—silicon content, powder particle size, and sintering schedule—the final porosity and pore-size distribution can be regulated within a certain range. Table 1 summarizes typical process windows and the corresponding pore-structure characteristics.

Table 1: Typical process parameters and pore-structure characteristics of reactive-sintered Fe-Si porous materials ^{[12, 20]}.

Table 2: Typical mechanical and permeation properties of reactive-sintered Fe-Si porous materials ^{[10, 12, 51, 71]}.

Typical porosity ranges from $35\mathrm{\%}$ to $55\mathrm{\%}$, average pore size from a few microns to thirty-forty microns, and compressive strength can reach $8\text{--}15\mathrm{MPa}$—sufficient to meet the mechanical requirements of most filter supports. Specific performance data vary with process conditions; Table 2 lists representative ranges reported in the literature.

However, the "spontaneity" of pore generation, while offering simplicity, also imposes unavoidable process boundaries. Since pore morphology is entirely determined by the particle packing state and the vacancy coalescence driven by diffusion, any factor affecting powder mixing uniformity and local thermal field is faithfully recorded in the pore structure. Batch-to-batch fluctuations in powder size, spatial differences in green compact density, and temperature non-uniformity within the sintering furnace—all controllable on small laboratory samples—can, once scaled to tubular or plate-shaped components with outer diameters exceeding $200\mathrm{mm}$, cause obvious radial gradients in pore size and phase distribution. Another more fundamental limitation lies in the designability of the pore structure: pore morphology produced by reactive sintering is restricted by diffusion paths and particle geometry, making it difficult to escape the fixed paradigm of "quasi-spherical open pores with wide distribution." When porosity exceeds about $60\mathrm{\%}$, the effective load-bearing area of the framework drops sharply, and strength decays faster than a power-law relation, causing the mechanical reliability of high-porosity products to plummet.

Table 3: Structural comparison of Fe-Si porous bodies from dealloying and reactive sintering ^{[13, 22, 23, 24, 25]}.

2.2. Dealloying method

If reactive sintering "burns out pores," dealloying follows the opposite strategy—first cast a dense multi-phase precursor, then chemically corrode away one phase selectively, leaving a porous skeleton. In the $\mathrm{Fe}\text{--}\mathrm{Si}$ system, this idea usually starts with melt-spinning to produce amorphous ribbons. Because the glass-forming ability of pure binary $\mathrm{Fe}\text{--}\mathrm{Si}$ is limited, small amounts of $\mathrm{B}$ or $\mathrm{P}$ are often added to enhance amorphous stability. The amorphous precursor is partially crystallized by controlled annealing, forming $\alpha \text{-}\mathrm{Fe}$ solid-solution grains dispersed in a residual amorphous matrix; subsequent acid treatment (e.g., dilute $\mathrm{H}{\phantom{A}}_2\mathrm{SO}{\phantom{A}}_4$ or $\mathrm{HCl}$) selectively dissolves the $\alpha \text{-}\mathrm{Fe}$ phase, while the $\mathrm{Si}$-rich phase—containing ${\mathrm{Fe}}_3\mathrm{Si}$ or similar intermetallics—is retained, forming a skeleton with three-dimensionally interconnected nanopores ^{[13, 22, 23]}.

The most attractive feature of dealloying is that it can break through the lower pore-size ceiling of reactive sintering. Reactive-sintered pores rarely go below the micron level, whereas dealloying can stably produce porous bodies with average pore sizes in the $10\text{--}50\mathrm{nm}$ range and specific surface areas exceeding $20{\mathrm{m}}^2/\mathrm{g}$—attributes highly valued in catalysis, sensing, and supercapacitor electrodes. Ligament size and pore size can be jointly tuned via precursor composition and annealing schedule, providing fairly high structural reproducibility under the logic of densification-followed-by-corrosion. Table 3 compares the key structural features of dealloying and reactive sintering methods.

However, the process boundaries of this route are equally sharp. First is the natural limitation in forming dimensions—melt-spinning yields ribbons only tens of micrometers thick, and chemical corrosion can hardly penetrate deep into a bulk body without breaking ligament continuity, thus dealloying ties itself from the outset to products in the form of films or powders. Efficient low-damage integration technologies to assemble these nanoporous ribbons into macroscopic components are still lacking. Secondly, the corrosion process itself is a precise balance between time and structural integrity: insufficient corrosion leaves un-interconnected pores, while over-corrosion thins ligaments and causes brittle framework collapse. Residual $\mathrm{B}$ or $\mathrm{P}$ impurities are difficult to wash out thoroughly, and systematic data on whether they accelerate framework degradation under high-temperature oxidation are absent. Cost is also a practical issue—from amorphous ribbon fabrication to fine corrosion control, the efficiency and batch capacity of the entire chain still lag behind reactive sintering by orders of magnitude.

2.3. Templating method

Although reactive sintering and dealloying differ in principle, they share a common point: the final pore structure is the result of "spontaneous" evolution within the material, with the human role mainly setting boundary conditions. Sacrificial templating, by contrast, reclaims design authority—placeholders are pre-embedded in the powder compact, and after the compact builds adequate framework strength, the placeholders are removed by pyrolysis, dissolution, or vaporization, leaving cavities that are nearly faithful reverse replicas ^{[14, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]}.

The material and geometry of the template directly determine the macro-configuration of the pores. Different templates vary significantly in applicable pore size, porosity, and removal method; Table 4 lists key parameters of several mainstream template systems. The greatest advantage of templating is that it decouples porosity and pore size from the kinetics of sintering densification, so that high porosity no longer inevitably means strength collapse—as long as the skeleton undergoes reasonable secondary sintering densification after template removal, the pore walls themselves can be made sufficiently strong.

Table 4: Common placeholder systems used in templating and their process characteristics ^{[14, 31, 32, 33, 34, 35, 36, 37, 38]}.

Nonetheless, the placeholder-removal logic introduces a new set of operational complications. One arises from the template-removal process: if organic templates are heated too rapidly during pyrolysis, the escape of decomposition gases creates internal pressure, causing wall cracking or even macroscopic part fracture. $\mathrm{NaCl}$ templates require water-washing removal, and if the skeletal pre-sintering is insufficient, water washing can cause local collapse. Another complication is residue contamination: incomplete carbonization of organic foams leaves behind thin glassy carbon layers; impurity cations in salt may enter the matrix at high temperature and degrade corrosion resistance. Moreover, although templating can precisely control the macroscopic geometry of pores, its ability to tune microscopic surface morphology is limited—the internal surfaces of pore walls are usually relatively smooth, with specific surface areas far lower than dealloyed products. These boundaries make templating particularly suitable for scenarios where flow pressure drop and volume filtration are the primary design goals (e.g., high-temperature gas filtration), but inadequate where nanoscale surface activity is required.

2.4. Additive manufacturing method

Additive manufacturing entered the field of porous intermetallics most recently, yet it brings a fundamental paradigm shift in manufacturing. The three foregoing methods, whether by diffusion, corrosion, or placeholders, all produce a homogeneous or statistically homogeneous porous body; additive manufacturing—especially laser powder bed fusion (LPBF)—allows engineers to actively define, with a resolution on the order of a hundred micrometers, where solid, where cavity, and where gradient transition layers exist in three-dimensional space ^{[15, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]}.

The pore-generating mechanism of additive manufacturing can be roughly split into two levels. At the macroscopic level it is topology design: CAD models construct lattice structures (e.g., body-centered cubic, diamond, triply periodic minimal surfaces), and the laser melts the powder layer by layer along the defined path, faithfully reproducing the designed cross-section from a digital model into physical reality. At the microscopic level, the process parameters themselves can also become pore-forming tools: under reduced laser power or increased scan speed, gaps of insufficient fusion and local un-melted powder zones appear between melt tracks; these powder-scale pores superimpose on the macroscopic lattice pores, creating a hierarchical pore structure. For the $\mathrm{Fe}\text{--}\mathrm{Si}$ system, pre-alloyed $\mathrm{Fe}\text{--}\mathrm{Si}$ powders or elemental powder blends are typically used for full-melting printing; the sphericity and flowability of the powder have a significant impact on forming quality. Table 5 lists typical process windows and key influencing factors for LPBF-formed $\mathrm{Fe}\text{--}\mathrm{Si}$ porous bodies.

Table 5: Key process parameters and influences for LPBF-formed Fe-Si porous bodies ^{[41, 42, 43, 44, 45, 46]}.

Additive manufacturing has opened two entirely new doors for this field. The first door is gradient porosity design: a single component can simultaneously have dense load-bearing regions, large-pore low-flow-resistance regions, and fine-pore high-surface-area regions, with smooth transitions among them—something previously achievable only by multi-piece bonding or mechanical assembly. The second door is the integrated forming of complex internal channel networks: for catalytic reactors, compact heat exchangers, and other components requiring internal network-like flow channels, LPBF has become almost the only feasible manufacturing solution.

The price is naturally steep. Cost is the most immediate barrier—whether the specialized powder feedstock for intermetallics or the hourly depreciation of LPBF equipment and inert gas consumption, all dictate that it is temporarily suited only for high-value-added, small-batch, or custom applications. Technical boundaries run deeper: $\mathrm{Fe}\text{--}\mathrm{Si}$ exhibits severe intrinsic brittleness at room temperature, and the steep thermal gradients and repeated thermal cycling during printing can easily induce micro-cracks. Even with optimized parameters, crack density can only be reduced to a certain critical level but rarely eliminated entirely. Furthermore, the laser spot size (usually $50\text{--}100\mu \mathrm{m}$) and powder layer thickness ($20\text{--}50\mu \mathrm{m}$) jointly define a lower resolution limit, which means sub-micron or nanoscale pores cannot be obtained by direct printing but must rely on post-treatment or coupling with dealloying. Surface roughness is also a weakness of additively manufactured porous parts—un-melted powder particles adhering to pore walls both increase flow resistance and can become crack initiation sites under fatigue loading.

2.5. Method comparison and process selection criteria

Viewing the four methods together, two common threads run through them. The first thread is the control precision of pore size and specific surface area: from the wide-distribution micron pores of reactive sintering, to the definable macropores of sacrificial templating, to the precisely controlled nanopores of dealloying, and finally to the cross-scale design of additive manufacturing—designability increases stepwise, but so does cost almost proportionally. The second thread is the shift from "self-organization" to "imposed organization": reactive sintering and dealloying rely heavily on the material's own physicochemical properties to generate pores, with process control indirect and statistical; sacrificial templating and additive manufacturing treat pores as geometric information written proactively, with the process reaching into the specific position and shape of each pore.

These two threads, combined with real industrial constraints, form a rough but effective selection framework. Table 6 provides a comprehensive multi-dimensional comparison of the four methods, which can serve as a preliminary reference for process selection ^{[9, 12, 13, 14, 15]}.

Table 6: Comprehensive comparison of the four pore-forming methods ^{[9, 12, 13, 14, 15]}.

At the same time, it should be recognized that these four routes are not mutually exclusive. Some of the most interesting recent progress occurs precisely at their intersections: using additive manufacturing to print a precursor containing templates, then removing the template to create hierarchical pores; fabricating a macroporous support by reactive sintering and then generating a nanoporous layer on the pore walls by dealloying; or introducing partial reactive sintering into a sacrificial templating route to improve wall densification. These hybrid strategies demonstrate that the so-called "boundaries" of the four methods are not theoretical endpoints but contour lines drawn by current process capabilities. The structure-property relationships discussed in the next chapter are exactly what different pore structures and phase constitutions determine within these contour lines.

3. Microstructure and properties: Mechanical-functional coupling across multi-scale structures

If Chapter 2 addressed "how to make it," this chapter confronts "what is the thing made like" and "why is it like this." The complexity of porous materials lies in the fact that their macroscopic behavior is never a monotonic function of a single parameter—porosity, pore-size distribution, pore morphology, ligament cross-section, grain size, phase constitution, degree of order, defect distribution—these elements intertwine to jointly determine mechanical load-bearing capacity, fluid transport efficiency, and environmental tolerance. Decomposing these coupling relationships is both the epistemological core for understanding $\mathrm{Fe}\text{--}\mathrm{Si}$ porous materials and the methodological basis for guiding process optimization.

3.1. Pore-structure parameter system and characterization methods

Before entering the discussion of properties, it is necessary to establish a clear parameter language. The pore structure of a porous body is not a vague "porous state" but is defined by several independent yet interrelated geometric parameters. The most fundamental level consists of three scalar parameters: porosity, average pore size, and specific surface area. Porosity defines "how much void," average pore size answers "how large the voids are," and specific surface area answers "how much interface exists." There is no simple functional relationship among the three—two materials with the same porosity can be composed of a few large pores or a multitude of small pores, and their average pore size and specific surface area can differ by more than two orders of magnitude.

The second level is distribution parameters. The pore-size distribution curve—whether obtained by mercury intrusion porosimetry, bubble-point method, or gas adsorption-desorption isotherms—carries far more structural information than the average pore size. A narrow, sharp distribution implies structural homogeneity, with the responses of various locations to fluid or stress tending to synchronize; a broad, multi-modal distribution hints at heterogeneity, with possible preferential flow channels or stress concentration zones. For filtration applications, the upper tail of the pore-size distribution is often more critical than the average value—the largest $5\mathrm{\%}$ of pores determine the lower limit of interception precision. Similarly, pore morphology must be quantified: pore shape factors (e.g., roundness, aspect ratio), open porosity (the fraction of interconnected pores in total pore volume), pore connectivity (described by coordination number—the average number of neighboring pores connected to a given pore), and pore orientation distribution (isotropic versus layered directional arrangement).

The third level is the microstructural parameters of the ligament/pore wall. The porous skeleton is not merely "the walls between pores"; it is itself a microstructural entity. Ligament cross-section size, the grain morphology inside the ligament (equiaxed or columnar), phase constitution and degree of order, grain-boundary and phase-boundary characteristics, and the distribution of micro-cracks and micro-pores—these second-tier structural features determine the intrinsic mechanical response and chemical stability of the skeleton itself. For the $\mathrm{Fe}\text{--}\mathrm{Si}$ system, whether the ligament interior is a single-phase ${\mathrm{DO}}_3$ ordered solid solution or mixed with $\mathrm{FeSi}$ second phase, whether the grain size is nanometer or micrometer scale, and whether silicon oxides or borides segregate at grain boundaries, all leave clear fingerprints on macroscopic properties.

This three-level parameter system—macroscopic pore geometry, mesoscopic pore morphology, and microscopic skeleton organization—together constitutes the bridge connecting processing and performance. Any single performance index is rarely determined by a single parameter; the interplay and trade-offs among parameters are the core difficulty in porous material design.

3.2. Mechanical behavior: From uniaxial compression to service reliability

Mechanical performance is the first hurdle for porous intermetallics moving from laboratory to engineering application. Compared with dense metals, the mechanical response of porous bodies exhibits three notable characteristics: severe strength attenuation, nonlinear elastic modulus, and brittle fracture tendency. All three are closely related to pore structure, but in different ways.

3.2.1. Strength-porosity relationship and its deviation

That the strength of porous materials decreases with increasing porosity is expected. The real engineering question is: how fast does it drop, and does it follow a predictable law?

The classical Gibson-Ashby model provides a concise power-law framework: for open-cell foams, yield strength scales with relative density to the power $1.5$; for closed-cell foams, the exponent lies between $1$ and $2$. A large body of experimental data shows that reactive-sintered $\mathrm{Fe}\text{--}\mathrm{Si}$ porous materials roughly follow this trend—when porosity increases from $30\mathrm{\%}$ to $55\mathrm{\%}$, compressive strength decreases from about $25\mathrm{MPa}$ to about $8\mathrm{MPa}$, with a fitted exponent between $1.5$ and $2.0$ ^{[1, 51]}. This seemingly well-behaved decay curve, however, exhibits a clear inflection when porosity exceeds about $60\mathrm{\%}$: strength drops abruptly, with actual values systematically lower than Gibson-Ashby predictions. This implies that in the high-porosity region, some additional weakening mechanism is activated.

Researchers gradually realized that the root of this deviation lies in the geometric percolation transition of pores from "isolated" to "connective." In the low-porosity regime, pores approximate spherical defects dispersed in a continuous matrix, the stress concentration effect of each pore is relatively independent, and equivalent-medium models can reasonably describe it. When porosity exceeds a critical value (about $55\mathrm{\%}\text{--}60\mathrm{\%}$ for the $\mathrm{Fe}\text{--}\mathrm{Si}$ system), pores begin to interconnect over large distances, ligaments transform from "thick walls" into "slender columns," and the loading mode shifts from predominantly compressive body stress to framework deformation involving significant bending components. At this point, the weakest ligaments—exactly those located at the upper tail of the pore-size distribution with the smallest cross-sectional area—become the "mechanical fuse points" of the entire skeleton. The overall strength is no longer determined by the average ligament size but by the extreme-value statistical distribution of the weakest links. This is a physical picture that the homogenization framework of Gibson-Ashby cannot capture.

Table 7: Behavior differences across porosity ranges.

3.2.2. Nonlinear decrease of elastic modulus

The attenuation of elastic modulus with porosity is more severe than that of strength—a general rule for porous bodies. To understand this, one must distinguish two different measurement concepts: the engineering elastic modulus is the secant slope of the stress-strain curve in the low-strain segment, but for porous bodies containing micro-cracks, the curve may not be strictly linear even within a $0.1\mathrm{\%}$ strain range. Micro-cracks already close or open under very small loads, causing loading-unloading paths to not coincide and the modulus to vary with the number of loading cycles.

For $\mathrm{Fe}\text{--}\mathrm{Si}$ porous materials, the elastic modulus is usually assessed by ultrasonic measurement of longitudinal wave velocity, to avoid pseudo-nonlinearity from micro-crack closure during mechanical loading. Sonic-measured modulus data indicate that as porosity increases from $30\mathrm{\%}$ to $50\mathrm{\%}$, the dynamic elastic modulus drops rapidly from about $200\mathrm{GPa}$ for dense ${\mathrm{Fe}}_3\mathrm{Si}$ to $60\text{--}90\mathrm{GPa}$—a decrease ($\sim 60\mathrm{\%}\text{--}70\mathrm{\%}$) far greater than the corresponding strength decrease ($\sim 40\mathrm{\%}\text{--}50\mathrm{\%}$) ^[63]. This accords with the general intuition in solid mechanics: pores not only reduce the load-bearing area but also significantly reduce structural stiffness, but the degree of stiffness loss depends on pore shape—flattened pores along the loading direction drastically reduce stiffness, while transversely elongated pores do the opposite. This also explains why the equiaxed pores produced by sacrificial templating generally provide a higher elastic modulus at the same porosity than the irregular pores from reactive sintering.

Table 8: Comparison of elastic modulus and strength attenuation in Fe-Si porous materials.

3.2.3. Fracture toughness and inherent brittleness

If strength and elastic modulus still leave acceptable engineering margins for $\mathrm{Fe}\text{--}\mathrm{Si}$ porous bodies, fracture toughness is an inherently deficient indicator. At room temperature, dense ${\mathrm{Fe}}_3\mathrm{Si}$ is itself a brittle intermetallic, with a bulk fracture toughness $K_{\mathrm{IC}}$ typically only $2\text{--}5\mathrm{MPa}\cdot {\mathrm{m}}^{1/2}$ ^[57]. The introduction of pores only makes matters worse.

The fracture behavior of porous bodies must consider brittleness superposition at two scales. At the macroscopic scale, pores are stress concentrators and crack nuclei. For open-cell structures, ligament cross-sections experience significant bending stresses, and the extremely low fracture work of $\mathrm{Fe}\text{--}\mathrm{Si}$ means that a ligament will undergo cleavage fracture after very little plastic deformation. Once a ligament breaks, the load redistributes to adjacent ligaments, causing a chain-reaction collapse—macroscopically exhibiting a completely brittle failure mode, with the stress-strain curve dropping vertically to zero after the peak. At the microscopic scale, cleavage planes in ${\mathrm{Fe}}_3\mathrm{Si}$ grains (the $\{110\}$ families of the ${\mathrm{DO}}_3$ ordered structure are low-index cleavage planes) provide obstacle-free paths for crack extension. When grains are pinned by silicon oxides or boride precipitates, the crack may be forced to deflect, but the energy consumed by deflection is extremely limited and far from producing a macroscopic toughening effect ^{[11, 53]}.

This dual-scale brittleness means that the mechanical design of $\mathrm{Fe}\text{--}\mathrm{Si}$ porous materials cannot follow the plastic energy-absorption logic of metallic porous materials. Its failure gives almost no warning, with the transition from "intact" to "shattered" extremely abrupt under critical load. Therefore, conservative allowable-stress design criteria should be adopted for load-bearing $\mathrm{Fe}\text{--}\mathrm{Si}$ porous components, supplemented by nondestructive testing methods (such as acoustic emission monitoring) to capture early signals of micro-crack propagation ^[58].

Table 9: Fracture toughness characteristics and brittle failure mechanisms of Fe-Si porous materials.

3.2.4. Behavior under thermal stress fields

In high-temperature service environments—exactly the most anticipated application scenario for $\mathrm{Fe}\text{--}\mathrm{Si}$ porous materials—thermal stress becomes an unavoidable issue. A filter tube undergoes temperature changes of tens to hundreds of degrees during start-up and shut-down; due to the lag in heat conduction inside the skeleton, spatial gradients arise, and gradients mean thermal stress.

The $\mathrm{Fe}\text{--}\mathrm{Si}$ system possesses two thermodynamically advantageous characteristics: one, its coefficient of thermal expansion (CTE) is $(10\text{--}14)\times {10}^{-6}{\mathrm{K}}^{-1}$, close to common stainless steels, reducing thermal mismatch stress with metallic support structures ^[64]; two, ${\mathrm{Fe}}_3\mathrm{Si}$ maintains the ${\mathrm{DO}}_3$ ordered structure from room temperature to about $800\mathrm{K}$, with no solid-state phase transformation, hence no additional stress from phase-change volume effects. However, the porous structure itself amplifies the damaging potential of thermal stress. The stress concentration factor around pores increases with the sharpness of the pore morphology; the long-axis tips of irregular pores may experience stresses several times the macroscopic average. Repeated thermal cycling leads to thermal fatigue—micro-cracks initiate from these stress-concentration sites, slowly propagate, until some critical ligaments lose their load-bearing capacity.

Thermal shock experiments—rapidly plunging a sample taken from a furnace into cold water—simulate an extreme condition. For reactive-sintered $\mathrm{Fe}\text{--}\mathrm{Si}$ porous bodies, when the thermal shock temperature difference $\mathrm{\Delta }T$ is about $300\text{--}500{}^{\circ }\mathrm{C}$, strength retention can exceed $70\mathrm{\%}$; when $\mathrm{\Delta }T$ exceeds $600{}^{\circ }\mathrm{C}$, penetrating cracking usually occurs ^[62]. The key material parameters influencing thermal shock resistance are the combination of thermal conductivity, elastic modulus, fracture strength, and CTE—ideally high thermal conductivity (reducing gradients), low modulus and high strength (withstanding larger strain without fracture), and low CTE (reducing strain amplitude). Unfortunately, for a given chemical system, trade-offs often exist among these parameters; therefore, improving thermal shock resistance requires multi-parameter synergistic optimization rather than conventional single-variable adjustment.

Table 10: High-temperature thermal stress characteristics and thermal shock resistance of Fe-Si porous materials.

3.3. Fluid transport performance: Permeability and filtration characteristics

In the vast majority of applications, porous materials need to interact with fluids—gaseous or liquid media flowing through pore channels, either being filtered, distributed, or participating in mass transfer and reaction. The essence of fluid transport performance is how fast the porous body allows media to pass under a given driving force; filtration performance concerns what particle size it can intercept and at what pressure drop cost.

3.3.1. Darcy permeability and its deviation

At sufficiently low flow velocities, flow in porous media obeys Darcy's law: volumetric flow rate is proportional to the pressure-drop gradient, with the proportionality factor being permeability $\kappa$ divided by fluid viscosity $\mu$. Permeability is an inherent geometric property of the porous body, independent of fluid type (provided no chemical reaction occurs between fluid and skeleton). In the Darcy flow regime, permeability $\kappa$ is jointly determined by porosity $\epsilon$, pore size $d$, and tortuosity $\tau$; the classic Kozeny-Carman relation writes $\kappa \propto {\epsilon }^3/[(1-\epsilon {)}^2\cdot S^2]$, where $S$ is specific surface area ^[71]. The core message of this relation is that permeability is extremely sensitive to porosity—a $20\mathrm{\%}$ increase in porosity can multiply permeability several times; at the same porosity, structures with coarser average pore size and lower specific surface area have higher permeability.

Experimentally measured permeability of reactive-sintered $\mathrm{Fe}\text{--}\mathrm{Si}$ porous bodies typically falls in the range ${10}^{-13}$ to ${10}^{-11}{\mathrm{m}}^2$, comparable to porous stainless-steel filter media, but lower than the typical levels of honeycomb ceramics ^{[71, 72]}. This is related to the relatively low porosity (usually not exceeding $55\mathrm{\%}$) and the wide pore-size distribution of $\mathrm{Fe}\text{--}\mathrm{Si}$ porous bodies—the fine-pore tail significantly raises specific surface area, thereby compressing permeability.

When the flow velocity increases and the Reynolds number exceeds the laminar critical value, inertial effects become noticeable, and Darcy's law must be supplemented by a Forchheimer correction term: $\text{pressure-drop gradient}=(\mu /\kappa )v+\beta \rho v^2$, where $\beta$ is the inertial resistance coefficient and $\rho$ is fluid density. For gaseous media, this non-Darcy regime is often non-negligible—high-temperature gases have increased viscosity and decreased density, so the superficial velocity (mass flux converted) can easily enter the inertia-dominated range. This means that when designing high-temperature gas filtration systems, selecting membranes based solely on Darcy permeability is insufficient; the additional pressure drop contributed by the inertial term must be accounted for, to avoid the actual operating point deviating from design expectations ^[73].

Table 11: Permeability and flow characteristics of Fe-Si porous materials.

3.3.2. Trade-off between filtration efficiency and pressure drop

Filtration is one of the oldest and most widespread functions of porous materials. For depth filtration—where particles are captured inside pore channels rather than accumulating on the surface—filtration efficiency depends on two mechanisms: size sieving and inertial impaction/diffusive capture. Size sieving directly intercepts particles larger than the pore-throat size, a hard threshold determined by the upper tail of the pore-size distribution. Inertial impaction and diffusive capture act on sub-micron and nano-particles, respectively, and are jointly regulated by flow velocity and pore size.

A practical advantage of $\mathrm{Fe}\text{--}\mathrm{Si}$ porous bodies is the natural chemical activity of their skeleton surfaces—$\mathrm{Fe}$ atoms in ${\mathrm{Fe}}_3\mathrm{Si}$ form a thin iron-oxide layer under high-temperature oxidizing atmospheres, which has a certain chemisorption capacity for some gas-phase components and fine particles, thus providing an auxiliary capture mechanism beyond pure physical interception ^{[72, 76]}. For sulfur- or chlorine-containing high-temperature gases—such as coal-gasification syngas—whether the $\mathrm{Fe}\text{--}\mathrm{Si}$ skeleton undergoes excessive sulfidation or chlorination directly affects the long-term stability of filtration efficiency. Preliminary laboratory data indicate that under conditions of $\mathrm{H}{\phantom{A}}_2\mathrm{S}$ concentration below several hundred ppm and temperature $500\text{--}600{}^{\circ }\mathrm{C}$, the sulfide layer formed on the $\mathrm{Fe}\text{--}\mathrm{Si}$ surface has controllable thickness, and permeability and filtration efficiency do not significantly degrade in the short term ^{[81, 82]}. However, long-term exposure data exceeding hundreds of hours are still very scarce—a critical knowledge gap impeding the entry of $\mathrm{Fe}\text{--}\mathrm{Si}$ porous materials into coal-chemical and biomass-gasification sectors.

Filtration design and permeability are always in conflict. Pursuing higher filtration precision—i.e., intercepting smaller particles—requires reducing pore size, especially trimming the upper tail of the pore-size distribution. But reducing pore size inevitably lowers permeability, meaning that at the same flux, a larger filtration area or higher driving pressure difference is needed. A high driving pressure difference not only increases energy consumption but also imposes greater fluid-drag force on the weak pore walls, potentially triggering micro-crack propagation and filtration-precision degradation over long-term operation. This conflict is amplified in the $\mathrm{Fe}\text{--}\mathrm{Si}$ system by its inherent brittleness—operators dare not apply overly high pressure difference to avoid framework fracture, yet permeability is insufficient to deliver the required flux at low pressure difference. An industrial compromise is to use an asymmetric membrane structure: a fine-pore $\mathrm{Fe}\text{--}\mathrm{Si}$-based membrane is coated (by dip-coating or spraying) onto a macroporous support produced by reactive sintering; the support provides mechanical strength and low flow resistance, while the membrane layer provides filtration precision ^{[111, 112]}. The success of this scheme depends on the membrane-support bonding strength and interfacial stability during long-term service, which is also one of the currently active research directions.

Table 12: Filtration characteristics, advantages, and engineering conflicts of Fe-Si porous materials.

3.4. Thermophysical properties: Heat transfer behavior from steady-state to transient

As materials intended for high-temperature applications, the thermophysical properties of $\mathrm{Fe}\text{--}\mathrm{Si}$ porous bodies not only determine their own service behavior but also indirectly affect structural reliability via thermal-mechanical coupling paths. The two core parameters are thermal conductivity and coefficient of thermal expansion—they control the establishment of steady-state temperature fields and the magnitude of thermal stress, respectively.

3.4.1. Structural sensitivity of effective thermal conductivity

The effective thermal conductivity of a porous body is a function of the skeleton-phase thermal conductivity, pore-phase thermal conductivity, porosity, and pore-structure topology. For porous metals, the intrinsic skeleton conductivity is much higher than that of the gas in the pores (air $\sim 0.026\mathrm{W}/(\mathrm{m}\cdot \mathrm{K})$, while dense ${\mathrm{Fe}}_3\mathrm{Si}$ is $\sim 10\text{--}20\mathrm{W}/(\mathrm{m}\cdot \mathrm{K})$), so heat is mainly conducted along ligaments, with pores acting as thermal barriers. Effective thermal conductivity decreases with increasing porosity, but the rate of decrease depends heavily on pore shape and connectivity. Closed-cell structures usually fall between Hashin-Shtrikman upper and lower bounds; open-cell structures, due to thinner ligament cross-sections and longer heat-flow paths, show more pronounced attenuation ^[63].

Existing experimental data indicate that the effective thermal conductivity of reactive-sintered $\mathrm{Fe}\text{--}\mathrm{Si}$ porous bodies at room temperature is approximately $2\text{--}8\mathrm{W}/(\mathrm{m}\cdot \mathrm{K})$, roughly linearly decreasing with porosity. This range is lower than that of dense stainless steel ($\sim 15\mathrm{W}/(\mathrm{m}\cdot \mathrm{K})$), but several times higher than ordinary ceramic foams (e.g., $\mathrm{Al}{\phantom{A}}_2\mathrm{O}{\phantom{A}}_3$ foam, usually $0.5\text{--}3\mathrm{W}/(\mathrm{m}\cdot \mathrm{K})$), reflecting the intermediate thermal conductivity of intermetallics "between metals and ceramics." For scenarios requiring both high-temperature filtration and heat exchange (e.g., filter tubes in fluidized-bed reactors that contact hot gas on one side and a cooler surface on the other), this moderate thermal conductivity is an appropriate design compromise—it is neither too high (avoiding excessive conductive heat loss through the tube) nor too low (avoiding building up excessively large temperature gradients within the tube wall) ^{[63, 68]}.

Table 13: Effective thermal conductivity characteristics of Fe-Si porous materials.