Electric Nanosecond Pulsing Assisted Sintering as a Transformative Paradigm for Non-Equilibrium Processing of Powder Materials

Abstract

Sintering has traditionally been governed by slow, near-equilibrium thermal pathways in which thermal excitation and structural evolution are inseparably coupled. Here we introduce electric nanosecond pulsing (ENP)-assisted sintering as a kinetically constrained, interface-dominated consolidation processing paradigm in which transition phenomena are triggered within a narrowly defined non-equilibrium activation domain. By temporally compressing energy delivery to durations shorter than thermally driven diffusion and mechanical relaxation, ENP drives powder systems into a distinctly non-equilibrium state characterized by extreme spatial localization of Joule heating at particle interfaces. Using pressure-assisted sintering and furnace-free ultra-fast sintering as model systems, we demonstrate that ENP enables selective interfacial activation, transient melting confined to particle contacts, and accelerated neck growth within microsecond processing windows. Continuum electro-thermo-mechanical simulations, complemented by microscale current-constriction analysis, reveal that geometric amplification of current density at powder contacts generates localized temperature spikes while the volume remains comparatively unaffected. Beyond a processing advance, ENP reframes sintering as an interface-dominated, far-from-equilibrium phenomenon. This paradigm provides access to the innovative sintering pathways for materials that are inaccessible to conventional thermal or field-assisted routes. By shifting control from bulk temperature to interfacial electric transients, ENP establishes a new framework for non-equilibrium materials manufacturing.

Graphical abstract

Keywords

Electric nanosecond pulsing (ENP) / Non-equilibrium materials processing / Interface-dominated sintering / Current constriction effects / Localized Joule heating

Highlight

	● Electric nanosecond pulsing establishes a new non-equilibrium sintering paradigm.
	● Rapid pulses enable temporally and spatially decoupled electro-thermal response.
	● Extreme current constriction induces localized Joule heating at particle interface.
	● Selective interfacial bonding is achieved without bulk densification or grain growth.
	● Furnace-free ultra-fast sintering of ceramics is achieved via intrinsic Joule heating.

1 Introduction

Powder sintering is a cornerstone of materials manufacturing, enabling the consolidation of particulate matter into dense or architected solids with tailored microstructures and properties [1–5]. Over the past decades, field-assisted sintering techniques (FAST), also referred to as spark plasma sintering (SPS) or electric current-assisted sintering, have emerged as powerful alternatives to conventional furnace-based sintering [6–9]. By applying an external electric field and current during consolidation, FAST enables accelerated densification, reduced processing temperatures, and improved microstructural control compared to purely thermal routes [10–12]. These advantages have made FAST a widely adopted platform for processing various structural and functional materials [13–15]. Despite its success, FAST remains fundamentally constrained by its second-scale timescale and moderate electric field strength, where densification is still largely governed by near-equilibrium thermal transfer and stress-assisted mechanisms. In conventional FAST, the applied current predominantly acts as a volumetric Joule heating source, leading to rapid yet largely homogeneous temperature elevation within the compact [16,17]. As a result, the attainable processing window of traditional FAST for driving strongly non-equilibrium phenomena, such as ultra-fast mass transport or transient metastable states, remains limited [18,19]. This motivates the exploration of new sintering paradigms capable of accessing extreme temporal and field regimes beyond those of FAST.

Recent advances in electric nanosecond pulsing (ENP) challenge this long-standing view and point toward a qualitatively different regime of powder processing. A defining conceptual shift introduced by ENP-assisted sintering is the re-localization of control from the bulk to interfaces. In contrast to conventional FAST, ENP-assisted sintering employs ultrashort electrical pulses in nanosecond timescale with ultrahigh electric field strengths, delivering energy into powder compacts on timescales far shorter than thermal dissipation or mechanical relaxation [20]. This extreme combination of high field intensity and ultra-fast energy deposition fundamentally alters how electric, thermal, and structural responses are coupled during sintering. Rather than inducing global thermal equilibrium, ENP drives the system into a highly non-equilibrium processing regime, where local responses dominate over bulk averaging effects [21,22]. The ultra-fast delivery of energy in ENP-assisted sintering gives rise to localized Joule heating, concentrated at interparticle constrictions where current density is intrinsically amplified. Because of the micro- to nanosecond pulse duration, such localized temperature excursions can reach extreme values while the overall bulk temperature remains comparatively low [23]. This spatial and temporal decoupling of heating from the macroscopic thermal field enables selective activation of interparticle interfaces without promoting excessive grain growth or global densification [24]. Consequently, ENP-assisted sintering opens a pathway to engineer microstructures that are inaccessible via conventional FAST or furnace sintering. Beyond localized thermal effects, ENP introduces a suite of non-thermal electric field-driven phenomena that further distinguishes it from traditional sintering methods. The intense transient electric fields can modify defect configurations [25], enhance diffusion kinetics [26], alter recrystallization kinetics [27], promote dislocation motion [28], and promote oxide removal [29–31] across particle contacts and grain boundaries. As a result, ENP can promote rapid neck formation, interface restructuring, and transient bonding states that deviate markedly from equilibrium thermodynamics. Such effects are particularly pronounced in powder systems, where the high density of interfaces provides fertile ground for electric field and matter interactions.

From a broader perspective, ENP-assisted sintering reframes powder consolidation as a problem of driving matter far from equilibrium with precision and selectivity. Rather than minimizing gradients and waiting for equilibrium to be established, this approach exploits extreme transients to selectively activate specific length scales, particularly at particle contacts or grain boundaries. In doing so, ENP enables microstructural outcomes that reconcile seemingly contradictory objectives, such as strong interparticle bonding without global densification, or rapid consolidation without excessive grain growth. These attributes make ENP especially attractive for advanced applications requiring non-equilibrium microstructures, including architected porous alloys, 3D-printed materials, and defect engineering in functional materials. In this paper, ENP-assisted sintering challenges the traditional paradigm that sintering must proceed through slow, near-equilibrium thermal pathways. Instead, it establishes a new framework in which interfaces, rather than bulk phases, become the primary site of control, and in which extreme electric and temporal conditions are harnessed to engineer matter far from equilibrium. As understanding of the underlying electro-thermal-microstructural coupling deepens, ENP-assisted sintering is poised to exert a profound impact on next-generation materials processing and manufacturing science.

2 Materials and Methods

Monosized, highly spherical 316L stainless steel powders (d = 50 μm, T_m = 1425 °C) produced by plasma rotating electrode processing (PREP) were used as the raw material for the innovative ENP-assisted sintering process (Fig. 1). The use of this well-defined powder system provides an ideal experimental platform for directly observing particle bonding and interfacial microstructure under ENP processing. The concept of ENP-assisted sintering originates from field-assisted sintering technologies, particularly spark plasma sintering. This recently developed ENP technology is characterized by the quasi-instantaneous generation of consecutive electric pulses with ultra-high intensity (about 10¹⁰–10¹¹ A/m²) and ultra-short duration (< 1 μs). These features enable rapid Joule heating and strong electric-field effects, establishing a powerful and practical paradigm for eliciting distinctive neck-growth kinetics and interfacial phenomena in powder materials under electric fields.

The sample tooling configuration proposed for ENP-assisted sintering is inspired by conventional SPS setups, which have been extensively demonstrated to be effective under low-voltage, high-current conditions. In the present work, this configuration is extended to operate under high-voltage, high-current regimes. Figure 2 presents a schematic of the ENP-assisted sintering configuration, along with the electropulsing profile applied to the as-received 316L stainless steel powders. An alumina ceramic tube was mounted on the ENP system as an electrically insulating mold. To facilitate powder loading, a secondary alumina tube with an inner diameter of 0.5 mm and a length of 4 mm was inserted into the primary tube. The 316L stainless steel powders were loaded into this secondary tube, with molybdenum (Mo) bars (2 mm in length and 0.5 mm in diameter) placed at both ends to serve as punches. Although the powders were initially in a loose state, a relative density of approximately 55% was achieved before sintering. The small powder sample size minimizes skin effects associated with electric current flow while ensuring a high current density during ENP-assisted sintering. Moreover, the high electrical conductivity (5.2 × 10⁻⁸ Ω·m) and high melting point (about 2900 K) of molybdenum make it an excellent material choice for the punches. Copper (Cu) rods were used as spacers to connect the sample with the electrodes in the ENP device. An axial load of approximately 10 MPa was applied to promote the densification of powder materials during ENP-assisted sintering by allowing axial displacement while maintaining stable contact with the powder materials.

In addition, to demonstrate the capability of achieving ultra-fast sintering using ENP technology, Y-stabilized ZrO₂ (YSZ) powders (Tosoh Corporation, Japan) were used as the starting material. A cylindrical green compact with a diameter of 2 mm and a height of 4 mm was prepared by uniaxial pressing under a load of 2.5 kN. The compacted ZrO₂ sample was then wrapped with a thin Mo foil of 20 μm thickness. The resulting ZrO₂@Mo assembly was placed inside the primary ceramic tube of the ENP device and subjected to one thousand consecutive pulses with a duration of 1 μs at a frequency of 1 kHz.

In high-voltage discharging technology, contact quality and alignment are among the most important yet challenging issues in the design and operation of the tooling system. Accordingly, effective electrical insulation was ensured on both sides of the setup to guarantee safe operation. All contact surfaces between the Cu rods and Mo punches were prepared to be flat and smooth, and a thin Cu foil was inserted at the interface to improve electrical contact. To prevent misalignment caused by external loading, spring support was applied to the upper Cu rod to maintain the force direction along the central axis of the configuration. The electric pulsing parameters were programmed into the ENP device. In this study, multiple pulses with a duration of 1 μs were applied. The newly developed ENP technology is capable of delivering ultra-short electric pulses with durations as short as a few hundred nanoseconds; however, a duration of 1 μs was selected here for demonstration purposes. The capacitor in the ENP device was charged to 600 V. Upon triggering the solid-state switch, the stored electrical energy was instantaneously released and discharged into the powder sample. The entire pulsing process was completed on the micro- to nanosecond timescale. The experimentally measured output current density in the sample is shown in Figure 2.

3 Results

3.1 Interfacial bonding in 316L stainless steel powders via ENP-assisted sintering

The exploration of ENP-assisted sintering begins with a fundamental question: can electrical energy be delivered in a temporally and spatially resolved manner sufficient to selectively activate interfacial bonding in loose powder assemblies, without triggering volume heating? In this ultra-fast sintering paradigm, two ENP pulses with a duration of 1 μs, a frequency of 100 kHz, and a current density of about 10¹⁰ A/m² were applied to 316L stainless steel powders. Please note that this is the nominal current density calculated by the measured current across the nominal cross-sectional area of the sample. The electric current is measured through a high-voltage differential probe on the constant resistors (about 95 mΩ) which are in series with the sample. The localized current density at powder contacts cannot be experimentally measured and will instead be estimated using finite element method (FEM) simulations in a later section. Prior to ENP treatment, the high electrical resistance of the loosely packed powder compact (about 150 kΩ) was obtained by measuring the whole sample fixture, reflecting discrete mechanical contacts and poor electrical percolation. Strikingly, after ENP processing, the resistance dropped sharply to about 30 mΩ when the neck with a metallurgical contact was formed. This indicates a dynamic resistance collapse associated with the formation of an electrically continuous network of newly established interfacial bonds in powder sample, as well as the dielectric breakdown resulting from the removal of surface oxides on the powders. Figure 3 reveals that, following ENP-assisted sintering, the powders are no longer loosely packed; instead, an architected porous 316L stainless steel structure with robust interparticle bonding is obtained. Importantly, enlarged views confirm that after only two 1 μs pulses, no obvious densification or macroscopic shrinkage occurs. The powder morphology remains essentially intact, yet discrete particle contacts transform into metallurgically bonded junctions. This decoupling of bond formation from bulk densification represents a conceptual departure from conventional sintering strategies, including spark plasma sintering and flash sintering, where densification and bonding are intrinsically coupled through sustained thermal exposure. Here, intense and highly localized Joule heating at powder contact points drives transient melting and rapid metallurgical fusion, while the powder interiors remain comparatively unaffected.

To probe the mechanistic origin of this interfacial transformation, individual spherical 316L powders were subjected to a single 1 μs pulse at about 10¹⁰ A/m². As shown in Figure 4, localized melting at powder contacts leads to the formation of fine grains bridging the microscale particles. These fine grains are signatures of rapid solidification and recrystallization of a transient liquid phase generated under non-equilibrium electrical excitation. A single pulse produces a limited population of such fine grains, whereas an additional pulse increases their number, demonstrating that ENP enables incremental and programmable interfacial structural evolution. The ability to engineer nano-scale features precisely at particle junctions, while preserving the original powder morphology, introduces a new degree of freedom in architecting hierarchical microstructures. The resulting porous dual-modal material, characterized by microscale particles interconnected through nano-scale interfacial regions, embodies a harmonic structural motif anticipated to synergistically enhance specific strength and ductility. The dramatic reduction in electrical resistance, from kilo-ohm to milli-ohm levels, further underscores that ENP-assisted sintering fundamentally reconfigures the electrical and metallurgical topology of the powder compact. Once initial interfacial bonds are established, the resistance contrast between particle contacts and particle interiors becomes negligible. Consequently, subsequent pulses no longer generate pronounced localized Joule heating; instead, energy deposition transitions from interfacial concentration to more homogeneous volume heating.

In essence, ENP operates as a transient interfacial activator: it first exploits electrical heterogeneity to trigger localized bonding, then naturally shifts toward bulk heating once percolation is achieved. Such adaptive energy localization is absent in conventional thermal sintering frameworks. Collectively, these observations position ENP-assisted sintering not merely as an incremental processing modification, but as a transformative paradigm for powder consolidation. By temporally compressing energy delivery to the micro-nano second regime and spatially confining thermal effects to particle interfaces, ENP enables the formation of mechanically robust green bodies with architected dual-modal structures, prior to any significant densification. This capability opens a pathway in powder metallurgy. To preserve the interfacial fine grains and harmonic architecture during subsequent consolidation, rapid thermal routes such as spark plasma sintering (SPS) or ultra-fast high-temperature sintering (UHS) may be employed. In this context, ENP-assisted sintering provides a foundational interfacial design platform, redefining the sequence and physics of bonding in particulate materials and establishing a new regime of non-equilibrium, interface-dominated materials processing.

3.2 Electric discharging sintering of 316L stainless steel via ENP-assisted sintering

Beyond the formation of interfacial bonding, achieving densification (or sintering neck growth) of powder materials is a key objective in ENP-assisted sintering technology. Fundamentally, sintering is a diffusion-driven, time-dependent dynamic process. To enable sufficient heat transfer and mass diffusion, twenty consecutive pulses with a duration of 1 μs each were applied at a frequency of 10 Hz and a current density of approximately 1 × 10¹⁰ A/m² to the 316L stainless steel powders. Figure 5A presents the morphology of spherical 316L stainless steel powders after consolidation by ENP-assisted sintering. Evidently, the result is superior to that obtained under the previous processing condition, where the interfacial bonding and fine grain formation were observed. In the present case, a measurable degree of shrinkage is achieved, although full densification has not yet been realized. Enlarged views of the 316L stainless steel samples clearly reveal pronounced sintering neck formation and growth. As shown in Figure 5B and 5C, the spherical powders appear to have softened and deformed under the applied load (about 10 MPa). The magnified image provides a representative example of sintering neck growth driven by localized Joule heating at the particle contact regions (Fig. 5D). Notably, the application of 20 consecutive pulses at 10 Hz corresponds to an extremely short total processing time of only two seconds. Such ultra-rapid sintering neck formation and growth, indeed, can be attributed to localized Joule heating and accelerated diffusion near the interparticle contact regions. In contrast to traditional furnace sintering, the incremental contribution of electric field effects in this high-intensity ENP process warrants further systematic investigation.

To further increase the processing time without inducing overheating, 316L stainless steel powders were subjected to ENP-assisted sintering using 1000 pulses with a duration of 1 μs at a low frequency of 1 Hz. Under this condition, the global temperature rise of the powder volume is minimized, while the contribution of electric field effects during discharge can be considered. Figure 5E shows the morphology of spherical 316L stainless steel powder samples after this ENP-assisted sintering condition. Although no significant densification is observed, sintering neck formation between particles is evident. It should be noted that achieving substantial particle densification under electric current without significant Joule heating may require pulsing times on the order of hours or even days [32], which would diminish the advantages of ENP-assisted sintering as an electrically driven ultra-fast non-equilibrium metallurgical technology. Therefore, the exploration of ENP-assisted sintering was redirected toward processing under high electric intensity, high pulsing frequency and a limited number of pulses, enabling rapid temperature buildup through Joule heating. Specifically, ten consecutive 1 μs-duration pulses at a frequency of 100 kHz and a current density of 1.5 × 10¹⁰ A/m² were applied to the 316L stainless steel powder sample. An external load of 200 g (about 10.2 MPa) was applied to maintain stable electrical contact during densification. Figure 5F presents the morphology of the consolidated powder sample after this ENP-assisted sintering. Enhanced sintering neck growth under high-intensity, quasi-instantaneous electric pulsing is clearly observed. Compared to the previous ultra-fast sintering condition, this sample exhibits a higher degree of densification or shrinkage, although full densification is still not achieved. Figure 5G shows a detailed view of the powder sample along the electric current direction, while Figure 5H presents the view perpendicular to the current direction. As discussed earlier, sintering neck formation and growth in ENP-assisted sintering are associated with softening and localized melting at powder contacts. Evidently, well-developed sintering necks are formed and show a preferential alignment along the current direction.

Notably, the total processing time in this ENP condition is only 100 μs, highlighting the extreme non-equilibrium nature of the process. The ultra-rapid formation and growth of sintering necks cannot be solely explained by conventional thermally activated diffusion. Instead, they likely arise from a coupled thermo-electric transport process in which localized Joule heating and intense electric fields act synergistically. Under such high-intensity pulsing, particle contacts experience concentrated electric fields and current crowding, leading to localized temperature spikes and enhanced atomic mobility. In addition to thermal activation, field-driven mass transport mechanisms, particularly electromigration, may contribute significantly to accelerated neck growth and preferential alignment along the current direction. The observed anisotropic neck morphology therefore suggests that electric field effects are not merely auxiliary but play an active and directional role in microstructural evolution. Unlike conventional FAST or furnace sintering, where thermally driven diffusion dominates and electric effects are difficult to decouple, ENP-assisted sintering operates in a regime where extreme electric intensity and ultra-fast temporal scales enable direct interrogation of electrically driven transport phenomena. In this sense, ENP-assisted sintering provides not only a processing technique but also a unique experimental platform for probing non-equilibrium interfacial kinetics under coupled electro-thermal conditions.

3.3 Furnace-free ultrafast sintering of zirconia via ENP-assisted sintering

The distinctive attributes of ENP processing lie in its capacity for rapid Joule heating and pronounced electric-field effects, enabling ultra-fast microstructural modification at narrowed spatial and temporal scales. Extending these advantages to sintering, ENP provides a compelling pathway toward ultra-fast sintering, such as flash sintering (FS), by synergistically leveraging both thermal and field-driven mechanisms. Conventional flash sintering is typically a two-stage process: the compact is pre-heated in a furnace to a temperature near or above the flash onset, after which an electric field is applied to trigger rapid densification. While effective, this approach relies on external thermal infrastructure and relatively slow furnace heating. In contrast, ENP-assisted ultra-fast sintering offers the preliminary demonstration of a one-stage, furnace-free configuration in which heat generation and field activation are intrinsically coupled within the tooling architecture. Figure 6A presents the schematic configuration for ENP-assisted ultra-fast sintering. Compacted zirconia (ZrO₂) powders were encapsulated by a thin molybdenum (Mo) foil. Upon application of high-intensity electric pulses, the conductive Mo coating is rapidly heated by Joule dissipation, acting as a localized heat initiator. Through heat conduction, the ZrO₂ compact is driven above its flash onset temperature, at which point the current pathway transitions into the ceramic body and activates rapid densification. This design imposes a critical requirement on the surrounding coating: it must exhibit high electrical conductivity at low temperature to concentrate initial current flow, but its effective conductivity must decrease at elevated temperature so that the electric field can penetrate and trigger flash in the ceramic. Graphite powder represents an alternative coating candidate; it heats readily by Joule effect and becomes electrically insulated upon rapid oxidation around 600–700 °C, which coincides with the flash onset temperature of zirconia.

The overarching objective of this tooling strategy is to realize ultra-rapid, furnace-free sintering without any pre-heating step. An external pressure of about 3 MPa was applied through the upper Cu punch to reduce contact resistance, and a thin Cu foil was inserted between the punch and the powder compact to further enhance electrical and thermal contact stability. For experimental validation, a 20 μm-thick Mo foil was used to prepare the ZrO₂@Mo powder assembly (Fig. 6B). The inset shows the physical appearance of the prepared sample, where the Mo foil tightly encapsulates the compacted ZrO₂. In the furnace-free ENP-assisted ultra-fast sintering experiment, one thousand consecutive pulses with a duration of 1 μs at a frequency of 1 kHz were applied. The measured peak current was approximately 5200 A, corresponding to an electric field of about 500 V/cm. Under such extreme electrical input, substantial Joule heating was generated in the thin Mo foil within an ultrashort time, rapidly elevating the local temperature and subsequently initiating ultra-fast sintering in the ZrO₂. As shown in Figure 6B, ZrO₂ powders adhered to the Mo foil, appearing darkened and partially melted, followed by recrystallization. This observation indicates that localized temperatures were sufficiently high to induce significant mass transport and densification. However, the application of one thousand high-frequency pulses proved excessive. The melting of the Mo foil implies that the temperature exceeded 2623 °C (the melting point of Mo). Given the extremely short processing duration (about 1 s), heat diffusion throughout the compact was insufficient, leading to pronounced thermal gradients. Consequently, a highly inhomogeneous microstructure developed in the ZrO₂ after ENP-assisted sintering (Fig. 6C).

Further characterization of the sintered ZrO₂ is presented in Figure 7. The as-received powders consist of nanoscale and agglomerated particles (Fig. 7A). After furnace-free ENP-assisted ultra-fast sintering, clear spatial heterogeneity is observed. At the center of the sample, morphology closely resembles that of the raw powder, with negligible shrinkage (Fig. 7B). The ultra-short thermal exposure limited heat penetration into the bulk, preventing the center region from reaching the flash onset temperature. In contrast, the sub-surface region experienced substantial densification and grain growth (Fig. 7C). Although full densification was not achieved, these encouraging results convincingly demonstrate the feasibility of furnace-free ENP-assisted ultra-fast sintering of ZrO₂ and probably a broad range of ceramic materials. It is reasonable to expect that strategic optimization of the tooling architecture and pulsing protocol will enable the simultaneous realization of high electric field intensities and efficient thermal transport, thus promoting spatially uniform and ultra-fast responses.

From a broader perspective, ENP-assisted ultra-fast sintering represents a transformative paradigm in field-assisted densification. Unlike conventional sintering, in which thermal preheating and electrical activation are temporally separated, ENP intrinsically integrates ultra-fast Joule heating, intense electric fields, and transient non-equilibrium conditions within microsecond timescales. Such a unique processing regime enables localized and programmable energy delivery that minimizes global thermal exposure, promotes non-equilibrium activation of mass transport that may effectively reduce the apparent sintering temperature, and provides spatiotemporal control over current pathways, allowing a deliberate transition from a metallic initiator to the ceramic compact. With optimized pulsing parameters and improved thermal management to prevent excessive overheating and thermal gradients, furnace-free ENP-assisted ultra-fast sintering holds significant promise for scalable and energy-efficient ceramic processing. Beyond zirconia, this concept may open new frontiers for the ultra-fast densification of refractory and functional ceramic materials under precisely controlled non-equilibrium conditions.

4 Discussion

4.1 Understanding macroscopic non-equilibrium behavior in ENP-assisted sintering

The ultra-fast and highly localized characteristics of ENP-assisted sintering render real-time, in-situ monitoring during the process practically infeasible; consequently, only the initial and final states can be experimentally captured. This underscores the necessity of establishing a comprehensive numerical framework capable of capturing both macroscopic and microscopic behaviors throughout the ENP-assisted sintering process of powder materials. Continuum sintering theory has achieved considerable success in modeling densification kinetics and microstructural evolution under various sintering conditions [33,34]. The main constitutive relationship in this framework connects the external load, represented by the components of the stress tensor {\sigma }_{ij} (externally applied stress) to the strain rate tensor’s components {\dot{\epsilon }}_{ij} through the nonlinear-viscous relation:

(1){\sigma }_{ij}=\frac{\sigma \left(W\right)}{W}\left[\phi {\dot{\epsilon }}_{ij}+\left(\psi -\frac{1}{3}\phi \right)\dot{e}{\delta }_{ij}\right]+P_L{\delta }_{ij}

where W represents the equivalent strain rate, and \sigma \left(W\right) is the equivalent stress; \phi and \psi are respectively the normalized shear and bulk modules; {\delta }_{ij} is the Kronecker delta ({\delta }_{ij}=1 if i=j and {\delta }_{ij}=0 if i\ne j); \dot{e} represents the volume change rate of the porous body, or the first invariant of the strain rate tensor; P_L is the effective sintering stress. The normalized shear and bulk viscosity moduli, \phi and \psi, along with the effective sintering stress P_L, are expressed in terms of porosity \theta within the framework of the modified Skorokhod-Olevsky model [35]:

(2)P_L=\frac{3\alpha {\left(1-\theta \right)}^2}{r_g}

(3)\phi ={\left(1-\theta \right)}^2

(4)\psi =\frac{2}{3}\frac{{\left(1-\theta \right)}^{11.35}}{{\theta }^{0.49}}

where \alpha stands for the surface energy of the material and r_g for the average particle radius.

In the case of ENP-assisted sintering with externally applied stress, the porous powder materials are considered as power law creep. The equivalent stress, which governs the material’s constitutive behavior, can be expressed as

(5)\sigma \left(W\right)=A{W}^m

Considering the semi-empirical relationship for diffusion creep in the sintering process, the structure-dependent material constant A can be expressed as [36]:

(6)A=\tilde{A}{T}^m{\left(\frac{G}{G_0}\right)}^{2}exp\left(\frac{m\mathrm{\Delta }{H}_{SD}}{RT}\right)

here \tilde{A} is the mechanism-dependent material constant, T is the temperature, m is the strain-rate sensitivity exponent, G is the grain size, G_0 is the initial grain size, \mathrm{\Delta }{H}_{SD} is the activation energy for self-diffusion, and R is the gas constant. The equivalent strain rate is:

(7)W=\frac{1}{\sqrt{1-\theta }}\sqrt{\phi {\dot{\gamma }}^2+\psi {\dot{e}}^2}

here \dot{\gamma } is the shape change rate and \dot{e} is the volume change rate. In Cartesian coordinate system, volume change rate and shape change rate can be written as [33,37]:

(8)\dot{e}={\dot{\epsilon }}_{11}+{\dot{\epsilon }}_{22}+{\dot{\epsilon }}_{33}

(9)\dot{\gamma }=\sqrt{2\left({\dot{\epsilon }}_{12}^2+{\dot{\epsilon }}_{13}^2+{\dot{\epsilon }}_{23}^2\right)+\frac{2}{3}\left({\dot{\epsilon }}_{11}^2+{\dot{\epsilon }}_{22}^2+{\dot{\epsilon }}_{33}^2\right)-\frac{2}{3}\left({\dot{\epsilon }}_{11}{\dot{\epsilon }}_{22}+{\dot{\epsilon }}_{11}{\dot{\epsilon }}_{33}+{\dot{\epsilon }}_{22}{\dot{\epsilon }}_{33}\right)}

A piecewise Arrhenius-type function, previously proposed, was used to describe the viscosity parameter {\eta }_0, introducing a defined transition temperature to differentiate the material’s behavior across two distinct thermal regimes [38]:

(10)

(11)T_T=\frac{Q_{c,2}-Q_{c,1}}{R\ln \left({\displaystyle \frac{A_1}{A_2}}\right)}

Using 316L stainless steel as the exemplary material, the transition temperature determined from the experimentally derived viscosity was found to be close to the delta-ferrite transformation temperature predicted by thermodynamic calculations [38]. Moreover, this model reproduced the densification behavior observed in dilatometry experiments across both sides of the transformation temperature, accurately reflecting the increase in shrinkage rate. These constants were previously obtained from binder jetted 316L stainless steels produced with free sintering at heating rates between 2 °C/min and 15 °C/min [38]. The values of the constants applied in viscosity parameter {\eta }_0 are provided in Table 1. The overall response of the porous viscous material depends on the porosity \theta, and its densification can therefore be quantified through the evolution of porosity. The volume change rate of the porous material can be linked to the evolution of porosity through the mass conservation equation:

(12)\dot{e}={\dot{\epsilon }}_{11}+{\dot{\epsilon }}_{22}+{\dot{\epsilon }}_{33}=\frac{\dot{\theta }}{\left(1-\theta \right)}

The grain growth with porosity influence is modeled by the following equation:

(13)\dot{G}=\frac{dG}{dt}=\frac{k_0}{3G^2}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{Q_G}{RT}\right){\left(\frac{{\theta }_c}{{\theta }_c+\theta }\right)}^{\frac{3}{2}}

Here G is the grain size, k_0 is the pre-exponent term (2.965 × 10⁶ μm³/s) and Q_G is the activation energy for grain growth (164.8 kJ/mol) for 316L stainless steel, R is gas constant (8.314 J/[mol·K]), T is the temperature, and{\theta }_c is the critical porosity (0.0514), and t is the time.

In the simulation of ENP-assisted sintering process, the temperature rise in the powder sample is primarily caused by heat conduction from localized Joule heating in the powder sample. The electro-thermal coupling by Joule heating is generally governed by:

(14)\mathrm{\nabla }\cdot \stackrel{\rightharpoonup }{J}=\mathrm{\nabla }\cdot \left({\sigma }_{elec}\stackrel{\rightharpoonup }{E}\right)=0

(15)\mathrm{\nabla }\cdot \left(-\kappa \mathrm{\nabla }T\right)+\rho C_p\frac{dT}{dt}=\mathit{J}\mathit{E}

Here \stackrel{\rightharpoonup }{J} is current density, \stackrel{\rightharpoonup }{E} is electric field strength, {\sigma }_{elec} is electric conductivity, \kappa is thermal conductivity, \rho is density, and C_p is thermal capacity. Natural cooling at the outer surface of the fixture was modeled as a convective boundary condition, which is governed by:

(16)q_{conv}=h_{conv}\left(T_{air}-T\right)

Here natural convection heat transfer coefficient h_{conv} of 8 W/(m²·K), and ambient temperature T_{air} of 293 K is applied in this simulation. The upper Cu electrode was defined as a boundary current source, while the lower Cu electrode was grounded. The current profile was determined by electric current pulse curve experimentally measured in ENP-assisted sintering of 316L stainless steel powders. The electro-thermal property data of materials used in this simulation are as shown in Table 2.

Figure 8 presents the continuum electro-thermo-mechanical simulation of the ENP-assisted sintering process under 1 μs-duration pulses at 100 kHz with a current density of 1.5 × 10¹⁰ A/m², focusing on the transient macroscopic response of the powder compact. Figure 8A shows the temporal evolution of the macroscopic temperature rise during a 12 μs pulsing window. Owing to the extremely high current density, rapid Joule heating is triggered within microseconds. However, despite this sharp temperature increase, the total duration remains far shorter than the characteristic timescale required for significant diffusion-controlled densification or grain growth. This is quantitatively confirmed in Figure 8B, where the simulated porosity and grain size evolution reveal negligible macroscopic densification and essentially no grain coarsening during most of the pulsing period. The powder compact experiences temperature elevation but remains structurally frozen in a quasi-instantaneous non-equilibrium state. Only when the pulsing time approaches about 9.5 μs does a critical transition occur. The accumulated Joule heat drives the global temperature toward an overheating regime, resulting in incipient melting and rapid densification.

It demonstrates that ENP-assisted sintering operates in a regime where energy delivery is temporally compressed to such an extent that the thermal field can rise without immediately activating long-term diffusion or densification. In contrast to conventional FAST/SPS, where seconds temporal level heating intrinsically combines temperature rise with densification, ENP-assisted sintering decouples temperature from structural evolution until a critical energy accumulation is reached. Figure 8C further illustrates the spatial distribution of current density when the macroscopic temperature approaches the melting point. Under this condition, the current density distribution is relatively uniform throughout the compact. This reflects the fact that once electrical percolation is established and resistive contrast diminishes, the powder compact behaves macroscopically as a quasi-homogeneous conductor. However, when overheating occurs (Fig. 8D), localized melting and densification alter the electrical topology. The current distribution becomes increasingly non-uniform. This feedback between electrical conductivity evolution and local thermal response creates a nonlinear electro-thermal instability that accelerates melting once the critical threshold is crossed. Taken together, the distinctive characteristics of ENP-assisted sintering indicate that it is governed not by slow thermal equilibration, but by a sharp transition between a non-equilibrium effect and a runaway overheating regime. Within the micro-nano second domain, ENP-assisted sintering provides a unique processing window where substantial energy can be injected without triggering global densification or grain growth. This feature is inaccessible to conventional field-assisted sintering technologies and represents the foundation of selective, non-equilibrium capability of ENP processing.

4.2 Understanding microscopic localized behavior in ENP-assisted sintering

While the macroscopic simulation clarifies the global electro-thermal response, it does not capture the intrinsic localization phenomena occurring at particle boundaries. To resolve this, microscale simulations were performed on a 3 × 3 array of 50 μm 316L particles subjected to a 100 ns pulsing duration (Fig. 9A). Because the particles in this microscale model are treated as fully dense solids, densification analysis based on porosity evolution is not applicable. Instead, deformation and mass transport within the particles are described using a Nabarro-Herring creep mechanism governed by volume diffusion. An external axial pressure of about 10 MPa is applied. In this model, the sintering stress is defined as P_{L0}=3\alpha /r_g, representing the normal tensile stress acting on the particle surface due to surface tension. Figure 9B clearly shows intense current localization at particle boundaries. The current density is highly anisotropic and strongly dependent on boundary orientation relative to the applied electric field. Particle interfaces perpendicular to the current direction experience extreme current crowding, whereas those parallel to the current flow remain comparatively unaffected. Most strikingly, the current density at contact constrictions can reach values up to two orders of magnitude higher than within the particle interiors. This amplification arises from geometric constriction effects and electrical discontinuity at particle boundaries. Unlike macroscale uniform heating, the ENP pulse intrinsically concentrates electrical energy at interparticle boundaries. The consequence of this current amplification is illustrated in Figure 9C. Owing to the intense localized Joule heating, the boundary temperature rapidly exceeds the melting point within approximately 67 ns, whereas the particle volume exhibits negligible temperature rise. This phenomenon reflects an extreme spatial and temporal decoupling of the thermal response at nanosecond time scale, in which melting is confined to the boundary while the bulk remains thermally inert, and the overall temperature of the assembly stays low. Such behavior is not achievable in conventional furnace sintering or traditional SPS, as thermally driven relaxation process acts to homogenize temperature gradients over a significantly longer timescale.

Moreover, the sharp thermal gradients that develop at particle contacts give rise to pronounced thermally induced stress concentrations. Taking the diffusion creep flow caused by externally applied pressure into account as well, the micro-scale simulations predict local stress levels exceeding 200 MPa in the boundary regions. Under such ultra-high current densities in ENP-assisted sintering, these highly localized stress fields can coincide with intense current concentration at the interfaces, thereby amplifying the role of non-thermal electric field effects and making them an integral component of the overall mass transport process. Thus, it is demonstrated that ENP-assisted sintering is intrinsically an interface-dominated process at microscopic scale. Rather than heating the powder assembly uniformly, the ENP electric pulse selectively activates particle boundaries through geometric electric current constriction, localized Joule heating, extreme thermal gradients, and the resulting stress concentrations, with possible contributions from field effects under electron flow. This highly localized interfacial activation provides a mechanistic explanation for experimental observations, where metallurgical bonding and fine grain formation occur in the absence of significant macroscopic densification.

When considered together, a coherent multi-scale mechanistic framework has been established for ENP-assisted sintering. At the macroscale, ENP compresses energy delivery into powder materials on the microsecond timescale, creating a transient non-equilibrium window before bulk densification or long-range diffusion can meaningfully proceed. Within this brief window, the microscale phenomena dominate. Geometric current crowding at particle boundaries generates extreme current density, which in turn produces localized Joule heating and intense stress concentrations that enable selective interfacial bonding. The temporal decoupling between microsecond processing time and characteristic diffusion time, as well as the spatial decoupling between interface and volume, constitute the fundamental advantage of ENP-assisted sintering. Rather than homogenizing temperature and passively waiting for equilibrium diffusion to govern microstructural evolution, ENP actively exploits electrical heterogeneity and geometric constriction to target particle interfaces with precision. In doing so, it reframes sintering as an interface-activated, far-from-equilibrium process, offering a fundamentally different paradigm for powder consolidation.

5 Conclusions

This work establishes ENP-assisted sintering as a new non-equilibrium paradigm for powder materials processing. Unlike furnace sintering or conventional field-assisted sintering technologies, where thermal diffusion governs structural evolution over seconds to minutes, ENP-assisted sintering operates as a temporally and spatially concentrated energy delivery process in which densification is triggered within a narrowly defined non-equilibrium interface-activated domain within micro- to nanosecond windows. This extreme regime transiently resides in a metastable state in which thermal excitation and structural evolution are decoupled. At the microscopic level, geometric current constriction at particle contacts amplifies current density by orders of magnitude, triggering highly localized Joule heating and steep thermal gradients. These effects drive transient melting, fine grain formation, and accelerated neck growth, while the powder volume remains comparatively cool. At the macroscopic level, ENP-assisted sintering creates a transient non-equilibrium window in which substantial energy can be injected without immediately activating diffusion-controlled densification or grain coarsening. This dual-scale mechanism explains the central capability of ENP: selective interface activation without mandatory bulk densification. Such decoupling unlocks microstructural combinations previously considered contradictory, including strong interparticle bonding without shrinkage, rapid consolidation with suppressed grain growth, and programmable evolution of metastable interfacial states. Ideally, in porous metals, this can enable architected dual-modal structures composed of microscale load-bearing particles bridged by nanoscale bonded interfaces. In ceramics, it can enable furnace-free ultra-fast sintering through intrinsic coupling of ultra-fast Joule heating and electric-field activation. More broadly, ENP-assisted sintering shifts the conceptual foundation of sintering. Rather than minimizing gradients and waiting for equilibrium diffusion to proceed, this approach exploits extreme electrical transients to deliberately drive mass transport far from equilibrium. Interfaces, not bulk phases, become the primary locus of control in powders. The governing physics transitions from thermally activated diffusion to coupled electro-thermal transport under ultra-fast excitation.

Looking forward, several research directions emerge. Quantitative characterization of electrically driven mass transport, particularly electromigration and defect engineering, remains essential to effectively distinguish thermal from non-thermal contributions. Advanced diagnostics capable of resolving interfacial dynamics at micro-nano second time scale would further illuminate transient sintering behaviors. Optimization of tooling architectures and pulse protocols will be critical for achieving spatially uniform consolidation while preserving non-equilibrium advantages. Extending ENP-assisted sintering to architected porous materials, additively manufactured feedstocks, and defect-engineered functional materials may open transformative pathways for energy-efficient and spatially selective manufacturing. In essence, ENP-assisted sintering offers a new route for powder consolidation as an interface-activated, far-from-equilibrium process governed by extreme spatial and temporal conditions. By relocating control from bulk heating to interfacial transients, it establishes a new scientific and technological framework for next-generation materials processing.

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