Undercooled liquid metal particles as gateway to low-temperature processing and non-equilibrium dynamics

Summary

The continuous need to overcome thermal and miniaturization challenges in the fabrication of integrated circuits calls for novel alternative pathways to device and interconnect manufacturing. Efforts to achieve this goal, however, have been challenging due to limitations of the traditional equilibrium-based (Gibbsian) view of metal processing. This perspective demonstrates the application of metastable-metal processing as a platform for low-temperature and heatless solders with their subsequent effects on interconnect formation. In the process of realizing the solder, new understanding of non-equilibrium thermodynamics has emerged primarily around the solidification of deeply undercooled metals. This has led to resolution of the Kauzmann paradox (entropy catastrophe) through expansion of associated phase space to capture the contribution of tensions that drive deep undercooling.

Graphical abstract

Thermodynamics; Manufacturing; Materials science; Electronic materials

Introduction

The nature of metals processing is often limited by our ability to shape and join them. Metal processing often requires liquefaction at high temperatures followed by immediate solidification. This significant energetic requirement persists from traditional metal casting,¹ soldering/interconnect fabrication² to additive manufacturing³ technologies, which are currently limited by our current view of equilibrium phase diagrams. From these limitations, ambient liquid metals (primarily gallium) emerge as a highly attractive alternative for those who desire metallic properties with high formability without the associated melting energy tax, leading to a massive upsurge of liquid metals research in the past decade (in fields such as electronics, catalysis, energy storage, robotics, and much more).^4,5,6,7 With the current rate of advancements in electronic devices and internet of things (IoT), novel materials and integrated chip design pathways are becoming a necessity to surmount the ever-increasing energy and miniaturization demand in electronics manufacturing due to the looming limits of Moore’s law.^8,9 Current challenges in fulfilling this need arise from the slow advancement in materials discovery and methodologies utilized (high processing temperature of typical lead-free solder joints, T > 240°C),² which has since inspired arrays of solutions for various component bonding.^4,10,11,12 The International Electronics Manufacturing Initiative (iNEMI) board assembly technical roadmap predicts that low temperature soldering (LTS) usage will increase significantly by 2027, by which the need for this technology will exponentially increases.¹³ Low-temperature solders have become increasingly attractive to avoid dynamic warpage, heat-driven defects, and mechanical flaws under traditional soldering process. The LTS are also vital in high-density packaging, especially in small ball-grid arrays (BGAs) and thin printed circuit boards (PCBs), especially with increasing complexity, use of hybrid materials, and device scaling. Additionally, the development of low-temperature metal processing (removal of thermal requirements) will further the advancements of flexible-, wearable- and bio-electronics.^{14,15,16,17,18} In this perspective, we explore the advancements in stabilization and impact of undercooled metal particles^19,20 that circumvent challenges of high-temperature processing while increasing efficiency of soldering and quality of resulting joints (Figure 1A).

Figure 1.

Low-temperature metals processing facilitated by frustrated undercooled metastable state

(A) Schematic of the modified energy landscape through surface engineering of undercooled liquid metal core-shell (ULMCS) particles highlighting an alternative pathway beyond high-temperature processing.

(B) The stress generated by the core-shell architecture frustrates the undercooled liquid and increases the energy barrier to solidification, resulting in deep undercooling and freezing point offset.

(C) Schematic of differences in diffusion mechanism between high and low heat intermetallic formation during soldering.

(D) Schematic of expanded phase landscape due to presence of $⟨{\hat{T}}_{\delta }⟩$ inducing divergence away from the traditional view of equilibrium thermodynamic processes.

Undercooling is a common phenomenon in metal alloys; however, it is often thought to be transient since the system inherently relaxes to equilibrium.^20,21,22,23 Thermodynamically, the undercooled state is metastable and out-of-equilibrium. These factors limit current utilization and adoption of undercooled materials as a viable, processable, and functional system. Recent progress, however, has shown that nucleation in these systems can be frustrated by isolating the bulk within a micron-sized core-shell architecture, where an in situ formed passivating layer suppresses heterogeneous nucleation.^24,25,26 Concomitant speciation and segregation presents organized immobilized surface layers. We and others,^{24,25,26,27,28} have shown that this surface speciation and organization can perturb homogeneous nucleation. This concept requires divergence away from the equilibrium view of metallic systems by utilizing thermodynamic frustration of the nucleation process, maintaining a non-equilibrium steady state through surface-driven stresses and forcing the system to a metastable state far from equilibrium, leading to deep suppression of freezing point (Figure 1B). The stabilization of these systems is achieved through surface engineering with a core-shell architecture (Figure 1B). The principal driving force in preventing nucleation comes from gradient in chemical potential emanating from the self-organized passivating oxide layer (typically 0.7–5 nm)^{24,29,30,31,32} and the bridging ordered interface underneath it. Kinetics and equal a priori probabilities leads to a sharp concentration gradient, hence, a chemical potential gradient (Δμ) across the thin surface layer(s). For brevity, this sharp energy gradient is captured as a surface stress tensor, $⟨{\hat{T}}_{\delta }⟩$, which represents non-PV work (δw’) and acts as a counterbalance to the dissipation of free energy by the system (ΔG = ΔH-TΔS+δw’). This surface energy-derived stress tensor has been defined as:

$$⟨{\hat{T}}_{\delta }⟩=\int {\oint }_1^n\left({\sum }_i^m\frac{A_H}{24\pi D_0^2}\right)dAdz$$ (Equation 1)

where n is the number of atoms in the surface layer, i and m are the number of interfaces and metal elements present. A_H is Hamaker’s constant, D₀ is the interatomic distance between the surface atoms, and the integral is taken across the whole surface area (A) along the surface thickness (z). Maximization of this surface stress can be achieved through felicitous choice of composition or balancing enthalpic and entropic changes in the system.^26,28 This force balance results in stable undercooled metal particles that can be synthesized in high yields and stored for extended periods while maintaining metastability.²⁴ Production of stable undercooled metal particles enables new applications of known alloys while supporting alternative metal processing pathways. Compared to other stable liquid-phase alloys, undercooled liquids enable benefits from liquid-like processability whilst also offering solid-state performance and resilience upon activation. Solidification occurs with concomitant loss of surface plasticity, thus reduced corrosion compared to analogous liquid metals.

Further, consideration of interconnect formation under low-temperature conditions necessitates evaluation of coupled heat and mass diffusion balance for this non-equilibrium process. Transport (flow) and intermetallic formation between solder alloy and the electronic contacts play a crucial role in determining the quality of the joints. Simultaneous heat and mass transfer that occur during the reflow process can be defined by the Soret-Dufour effect,^33,34,35 where the dissipation function (Ψ) is described as:

$$\Psi =-J_S·\nabla T-\sum _{i=1}^{n-1}{j}_i·{\nabla }_T\left({\mu }_i^{\prime }-{\mu }_n^{\prime }\right)$$ (Equation 2)

where J_S is the entropy flow (related to heat flow, J_q, as $J_S=\raisebox{1ex}{$J_q^{″}$}\!\left/ \!\raisebox{-1ex}{$T$}\right.$), ∇T is the temperature gradient, $\sum _{i=1}^{n-1}{j}_i$ is total mass flux, with ${\nabla }_T\left({\mu }_i^{\prime }-{\mu }_n^{\prime }\right)$ being the temperature-dependent chemical potential gradient for n components. Despite the dependency of ∇T on both the former (thermal/entropy) and latter (compositional) components of this function, external factors that may perturb dissipation behavior are not currently being considered through this relationship. Lower ∇T imply slow diffusion, hence limited mass transport during intermetallic formation; however, in undercooled solders, other sources of energy (e.g., stored and released δw’ due to $⟨{\hat{T}}_{\delta }⟩$) can compensate for the transfer deficit(s) leading to equivalent mass across the joint without requiring the same degree of ∇T (Figure 1C).

This perspective explores the impact of temperature in soldering/intermetallic formation and how high energy requirements in interconnect formation can be mitigated using undercooled liquid metals. We explored situations in which the diffusion flux balance can be optimized to achieve optimal bonding conditions without requiring significant external thermal energy input, enabling lower reflow temperatures for interconnect formations by ∼100°C. Further, we explore the effect of metastability on how an undercooled system navigates alternative relaxation pathways, expanding our current viewpoint of the 2-dimensional phase space for opportunities in higher dimensional landscapes (Figure 1D).

Influence of temperature on diffusion and intermetallic formation

Understanding how metal processing temperature can be lowered requires an ability to tune when solidification occurs. Undercooled liquid metal core-shell (ULMCS) particles emerge as a perfect platform to demonstrate and study this phenomenon due to their ability to possess and maintain a controlled high degree of undercooling. Whilst a high degree of undercooling is typically achieved through complex experimental setups (e.g., containerless solidification, flux-assisted, levitating droplet, etc.),^19,36,37,38 ULMCS particles are commonly synthesized using the shearing liquid into complex particles (SLICE) method.³⁹ This methodology is a simple, scalable derivation of the droplet emulsion technique, enabling micro-to nano-particle formation by exploiting complex force balances (shear, gravity, drag, centrifugal, and buoyancy) to trigger droplet formation through the Rayleigh-Plateau limit.³⁹ Shearing of metallic melts under an organic acid environment leads to simultaneous etching of the surface of the metal, as well as binding of the organic ligands (conjugate basis of the acid) on the passivating oxide layer. The formed shell creates a protective outer organic layer which prevents further coalescence and Ostwald ripening while also enabling and maintaining the influence of $⟨{\hat{T}}_{\delta }⟩$ as described in the introduction.^32,39,40,41 The versatility of this method enables ULMCS synthesis using alloy systems of various melting temperatures with different organic shell combinations to tune inductive effects, $⟨{\hat{T}}_{\delta }⟩$, and ultimately the degree of undercooling. To demonstrate the applicability of ULMCS, commercially available solder alloy, SAC 305 (Sn - 3.0 Ag - 0.5 Cu), is used to highlight influence of temperature on intermetallic formation and viability of this technology under commercial standards. Core-shell SAC305 particles can be undercooled to temperatures below 100 °C ($\raisebox{1ex}{$\Delta T$}\!\left/ \!\raisebox{-1ex}{$T_m$}\right.=0.25$, Figure 2A), enabling much lower reflow temperatures in lieu of the traditional 240 °C.^24,28,42 These particles have shown long-term resilience and stability, maintaining their metastable undercooled state for >3 years under ambient storage as previously reported.^24,26 This stability, however, is highly dependent on the integrity of the surface oxide, in which oxidation through prolonged storage, aggressive etching, thermal cycling, or high-temperature conditions can lead to reduction in $\raisebox{1ex}{$\Delta T$}\!\left/ \!\raisebox{-1ex}{$T_m$}\right.$ or loss of undercooling altogether when the surface oxide is breached.²⁴ Evolution of the surface oxide perturbs $⟨{\hat{T}}_{\delta }⟩$ and δw’ which dominates the counterbalance of enthalpy-entropy contribution to free energy; hence, over time, relaxation of the system will inherently lead the system back to equilibrium conditions.²⁴

Figure 2.

Undercooling of commercial SAC305 (Sn - 3.0 Ag - 0.5 Cu) solder and Field’s metal (51 In – 32.5 Bi – 16.5 Sn) for low heat and heatless soldering and intermetallic formation

(A) Differential scanning calorimetry (DSC) trace of undercooled SAC305 particle (insert: scanning electron microscopy, SEM image of undercooled core-shell SAC305 particles).

(B) Meniscus neck formation of both high-temperature processed commercial SAC305 solder and a core-shell SAC305 particle joined between two copper plates (insert: zoom-in of IMC).

(C) Energy dispersive spectroscopy (EDS) map of undercooled SAC305-copper plate joint.

(D) Concentration profile of SAC305 IMC as indicated on Figure 2C.

(E) SEM images of joints created using Field’s metal at 25°C and 100°C respectively.

(F) Average EDS line profile displaying IMC at each respective temperature condition.

(G) Changes in copper ratio at the IMC intersection with increasing reflow temperature.

(H) Changes in IMC thickness with increasing reflow temperature.

All error bars represent the standard deviation.

A key aspect that determines good joint formation in solders is wettability and flowability as the metal liquid phase forms and bonds with a solid. Here, the meniscus that formed on the edges of the joint (so-called solder fillet) between two copper plates indicates proper wetting, flow, and coalescence comparable to commercial solder processed at higher temperatures (240°C) despite being processed at temperatures much lower than its melting point (Figure 2B). Another important metric to determine the reliability of the solder joint is the formed intermetallic compound(s) (IMC) between the solder and the substrate. The thickness of the IMC typically ranges around 2-4 μm for a long-lasting solder joint (thicker joints will cause intermetallic diffusion, and a thinner interface indicates weak interaction and joining). The formation of a high-quality interconnect requires a strong interface between the solder alloy and the substrate, which necessitates interdiffusion across the surface (Figure 2C). Low-temperature joining of these solders displays an interface profile akin to that of commercial solders, albeit with different structural features (Figure 2D). On average, the low-temperature joint displays a similar IMC formation behavior to that of a commercial joint with identical composition distribution across the interface (Table 1), suggesting that the commercial reflow temperature requirements can be further optimized to improve efficiency in electronics manufacturing.

Table 1.

Core-shell SAC intermetallic depth profile

Distance (μm)	Sn, avg. (at%)	Cu, avg. (at%)	Ag, avg. (at%)	Estimate phase specification
0	0.50	99.38	0.11	Cu
0.81	27.20	72.66	0.11	Cu3Sn
2.42	43.94	55.82	0.24	Cu6Sn5
4.40	57.43	41.98	0.56	Cu5Sn6
4.97	94.90	1.45	3.51	Sn

For ambient (heat-free) soldering, Field’s metal (In-Bi-Sn, 51-32.5-16.5 wt %, respectively) ULMCS has been previously utilized due to its ability to undercool much lower than ambient room temperature T_f = −50 °C).^24,28,42 A quick glimpse into the cross-section of joints formed with and without thermal input (∇T) using Field’s metal reveals similar features with identical diffusion profiles (Figures 2E and 2F). This similarity highlights that interface IMC formation between distinct metallic structures can be attained through alternate sources of work (e.g., the relaxed $⟨{\hat{T}}_{\delta }⟩$ and mechanical force), which bypasses the need for heat during interfacial IMC formation. The energy released by the thermodynamic stress, which promotes and stabilizes the undercooled state, likely translates to high reaction affinity, A_j (for reaction _j) given that;

$$A_j=-{\sum }_{i=1}^n{M}_i{\mu }_i{v}_{ij}$$ (Equation 3)

where M_i = molecular mass of species i, μ_i = chemical potential of species i, v_ij = specific stoichiometric coefficient of species i in reaction j. This suggests a coupled relation between the undercooling mechanism (composition and stoichiometry) and the resulting reaction affinity during formation of the IMC, which in turn is related to energy/entropy dissipation through the phenomenological balance relations (Equation 2). Given the divergent nature in mass, and cohesive energy dependency of diffusion, an estimated gradient in composition (Table 1) is expected across the IMC that should also relate to the total energy dissipated by the system. Without detailed analysis through high-resolution electron diffraction, however, the crystallinity or metastable nature of these approximated species cannot be confirmed. Slight differences in the total dissipated energy at higher temperatures (>100 °C) or a divergence in reaction affinity to Cu for In and Sn are expected at higher temperatures due to changes in atomic mobility and associate reaction activation barrier (Figure 2G). Despite the compositional differences, however, the net width of the IMC layer remains relatively constant (∼1.8 ± 0.3 μm), irrespective of the processing temperature and in agreement with expected values (Figure 2H). These results highlight that lowering temperature requirements from soldering may have minimal impact on formed IMCs. Given that the ULMCS contains metal in the liquid form, the release of liquid metal from the core-shell architecture and concomitantly the stored energy (under minimum ∇T) still enables flow, wetting, and interdiffusion needed to form IMC between metals.

Other phenomena that differ when thermal demand is lowered from the solidification of metals are the nature of flow and densification during soldering. The ULMCS undergo a limited densification rate (compensated by high viscosity with cooling) that is dependent on the strain propagation as described by Scherer’s constitutive model for load-driven viscous sintering^17,43:

$${\dot{\epsilon }}_x={\dot{\epsilon }}_f+\frac{1}{E}\left[{\sigma }_x+v\left({\sigma }_y+{\sigma }_z\right)\right]$$ (Equation 4)

where ${\dot{\epsilon }}_x$ and ${\dot{\epsilon }}_f$ is the strain rate in the x-direction and free strain respectively, E is the elastic modulus, v is Poisson’s ratio, and σ is the applied stress in each Cartesian direction. As such, there are high limitations in the thickness and mass diffusion balance that must be achieved due to non-uniformity of stress and the need to accommodate the removal of thermal stimuli in the system (Figure 3A). This thermal-limited densification process can lead to partial/incomplete solidification, leading to defects and joint weaknesses under high current density conditions.¹⁷ This mechanically driven liquefaction and densification mechanism (as opposed to thermal densification) also brings an important argument on how cross-sectional analysis on solder joints is typically performed. Soldered samples often undergo rigorous sectioning and polishing prior to being imaged and analyzed, given that a smooth flat surface is needed (Figures 3B and 3C). It can be seen, however, that an unpolished, wire-cut sample still displays similar IMC interface thickness (Figure 3D) despite the highly deformed surface (Figure 3B). Given the high penetration depth of EDS analysis (∼3 μm), most line scans can capture concentrations deep below the surface. A more accurate surface profiling (up to ∼10 nm deep), without rigorous extraneous work, can better be performed through methods like angle-resolved X-ray photoelectron spectroscopy (ARXPS). Whilst polished images provide a more pristine and cleaner surface (Figures 3B and 3C), the amount of mechanical work performed on the surface of the sample can lead to alteration in the IMC due to added external energy provided to the system, therefore enabling further diffusion or surface smudging (comparing Figure 2F, ∼1.5 μm thickness, and 3D, ∼1 μm thickness). This argument can be valid for other preparation methods which involve energy inputs beyond mechanical work such as; Electrical Discharge Machining (EDM) wire cutting, focused ion beam (FIB), etching (chemical), and laser sectioning (photonic) where the presence of dissimilar force gradients/thermodynamic potential (∇ϕ) can perturb atomic level structural configurations, especially for highly mobile atoms in soft solder alloys. Herein, we suggest that special care and attention need to be taken when processing and analyzing these soft materials, hence the need to deploy at least two complementary processing methods and compare results.

Figure 3.

Further analysis on heat-free solders

(A) SEM-EDS map of heat-free Field’s metal joint on copper.

(B) SEM image of wire-cut cross-section of heat-free joint displaying a large amount of mass transfer due to mechanical stress.

(C) SEM image of low-speed diamond saw cut and polished cross-section of heat-free joint displaying significantly cleaner surface, however, with much more mechanical work performed.

(D) EDS average line scan of wire-cut joint IMC.

Error bars represent the standard deviation.

Technological impact of undercooled liquid metal particles

Low-temperature solders have been incorporated into standard electronics processing, showing the ability to integrate with current state-of-the-art manufacturing methodology (Figures 4A and 4B).^44,45,46 This demonstration utilizes a viscous formulation of the undercooled particles to create a solder paste such that the particles can be deposited using a manual stencil printing method directly into a BGA (BGA169) chip. This effort was done to minimize the reflow temperature of SAC305 solders and curtail defects and warpage due to thermal stress during cooling and mismatch in coefficient of thermal expansion (CTE).⁴⁷ Utilization of core-shell SAC305 solders minimizes this thermal gradient whilst maintaining flowability and IMC formation. Thermal and mechanical stability tests have also been performed on joints created using these solders, which can withstand extended thermal cycling and multiple mechanical drop tests.^44,45,46 The ability to tune and lower the processing temperature of solder alloys without compromising on joint quality or mechanical stability enables wider integration capabilities for metallic interconnects on otherwise incompatible substrates or devices. The physical attributes of these soft, polydisperse, micron-sized particles render them highly adaptive, therefore, enabling incorporation into ink formulations and composites. This feature enables printability and formability into complex patterns and structures using methods such as screen-printing, direct write, stencil printing, and direct deposition (Figure 4C).^17,18,48,49

Figure 4.

Technological and scientific impact of undercooled liquid metal core-shell particles

(A) Cross-section of SAC305 solder joints from undercooled microcapsules processed at 180°C.

(B) SEM images of solder joints formed on BGA (Insert: zoom in on formed intermetallic with average IMC thickness of 2.5 μm).

(C) Patterned conductive trace printed using a 2D plotter.

(D–F) Screen-printed undercooled core-shell particles on non-traditional complex surfaces (e.g., commercial Jello, live rose flower, and cow brain, respectively).

(G) Ashby plot of resistivity vs. density displaying the wide range of ULMCS properties.

(H) Schematic of phase diagram expansion due to the presence of an added thermodynamic tensor.

(I) Relative entropy changes during the melting and freezing cycle of Sn-Ag-Cu core-shell particles displaying asymmetric entropy generation. Figures (A and B) are reprinted with permission from the author of the original article, Tevis and Paramanik.⁴⁴ Figures (C–E and G) are reprinted or adapted with permission from Martin et al.¹⁷ Copyright 2019 Wiley Publishing. Figure (F) is reprinted or adapted with permission from Martin et al.¹⁸ Copyright 2020 Wiley Publishing. Figures (H and I) are reprinted or adapted with permission from Martin et al.²⁸ under CC BY-NC-ND license. Copyright 2025 Wiley Publishing.

Furthermore, the polydisperse, non-Hertzian contact nature of these particles facilitates self-filtration that enabling mechanical bonding to imprintable, irregular, or textured surfaces (Figures 4D–4F). Mechanical bonding is especially useful for bio-electronic applications where metals cannot directly chemically bond onto low surface energy biological surfaces.¹⁸ The surface-dependency of the metastable state facilitates various activation methods ranging from mechanical, chemical, and photonic stimuli,^17,50 which creates variable density and conductivity that is otherwise not available through traditional pathways (Figure 4G). Given the difference in solidification behavior compared to traditional processes, the microstructure and mechanical behavior of these formed structures are inherently different from their equilibrium-processed solid state, leading to a brittle nature and limited stability under continuous exposure to stress.¹⁷ With an appropriate flux, however, the oxide shell can be dissolved and washed off, reducing the brittleness and defect density. Whilst this technology has been picked up by a commercial solder manufacturer, there are still other barriers that limit larger-scale utilization of ULMCS in everyday electronics (e.g., mass production, batch stability, process windows, and market readiness). Despite the lower energy requirements needed to functionalize these solders, the differences in solidification profile may necessitate procedural changes in electronics assembly lines, that have been standardized for most devices, hence requiring a general overhaul in the industrial scale where unfamiliarity with the technology may underwhelm the benefits. From these demonstrations, however, overcoming the necessity to maneuver the equilibrium phase landscape can create alternate processing pathways and opportunities by removing higher thermal energy dependency in metal joining, enabling ease of composite, mixed-material integration, and expansion of the processing-structure-property relationship in metals.^17,48

Emerging scientific impact from undercooled liquid metal particles

Beyond enabling non-traditional metals processing pathways, the presence of an asymmetric stress gradient in these core-shell particles presents a perfect platform to further our understanding of non-equilibrium phenomena and associated phase landscape. This is especially important when investigating second-order (i.e., glass transition, T_g)⁵¹ or more complex higher-order phase transitions.⁵² The idea of the vitrification limit and the ideal glass transition based on the Vogel-Fulcher-Tamman (VFT)^53,54 relation is highly limited to assumptions of the predecessor phase’s behavior (undercooled liquids). More recent investigation on metastable liquid phase change behavior by us and others, however, reveals that preferential order, collective motion, and interfaces of these systems render them more complex than a simple transient state.^27,28,55,56 Observation of non-equilibrium, metastable phase transitions in materials with degenerate energy landscapes are often limited by the observable timescale accessible to investigate these events (e.g., the ideal glass transition or Kauzmann’s paradox).^57,58,59 Existence of a tunable, engineered thermodynamic driving force in the form of $⟨{\hat{T}}_{\delta }⟩$, however, expands this non-equilibrium window (transition window, δT = 3°C for typical bulk alloy vs. ∼23°C for ULMCS) by perturbing the entropy dissipation behavior and slowing down the kinetics of phase transition beyond traditional limits.²⁸ ULMCS’ ability to achieve stable, controllable deep undercooling allows for investigation of the intermediate phase spaces that are otherwise transient and inaccessible (Figure 4H).²⁸ This capability can be utilized to navigate through the complex energy landscape of metastable materials, where slow controlled evolution of the surface (tuning $⟨{\hat{T}}_{\delta }⟩$)^32,50 can be used as a knob to alter phase transition temperatures and rates. Such control over phase transition behavior led to the realization of the Kauzmann temperature (T_k) across large temperature windows and at varying timescale (Figure 4I). This expansion in δT scales inversely with the configuration entropy ($\raisebox{1ex}{$\delta T$}\!\left/ \!\raisebox{-1ex}{$T_{\text{trs}}$}\right.\sim \raisebox{1ex}{$k_B$}\!\left/ \!\raisebox{-1ex}{$\Delta S_c$}\right.$, where T_trs is the initial transition temperature, k_B is Boltzmann constant and ΔS_c is configurational entropy) where: (1) a sharp transition in heat capacity (c_p) is only possible for large differences in S_c between the liquid and solid and (2) a discontinuity in c_p only exists at a thermodynamic limit. Existence of $⟨{\hat{T}}_{\delta }⟩$ alters the entropy production in the system ($\sigma =\sum _{i,j}^m{L}_{ij}{X_{i}X}_j$ where L_ij is the reciprocal matrix and X_i are associated force vectors)^60,61 due to imbalance in the reciprocal forces acting on the system (flux in ≠ flux out, L_ij≠L_ji). This leads to alteration in the configurational and relaxation dynamics of the undercooled liquid as temperature reduces,⁵⁵ hence the need to represent a divergence from a 2-dimensional landscape due to the influence of $⟨{\hat{T}}_{\delta }⟩$ (Figure 4H). Further, in the case of $⟨{\hat{T}}_{\delta }⟩$ stabilized undercooled metals, these expansion exist despite changes in cooling rates, ascertaining that this is not a kinetic limited behavior.²⁸ The resolution of the Kauzmann paradox (and associated entropy catastrophe) through the expansion of phase landscape by accounting for the associated thermodynamic stressor that drives the undercooling is realized.²⁸ We inferred that the Kauzmann paradox existed not due to limited experimental timescales; rather, it is due to limited perspective and dimensionality in how we perceive the phase landscape, which is often traditionally represented in two-dimensional. With the advancements of modern non-equilibrium thermodynamics theories (with renormalization group,⁶² thermodynamics speed limits,⁶³ information theory,⁶⁴ etc.), however, we see that many, unconsidered internal and external thermodynamic potentials possess deeper influence in how complex phases are formed. Mapping out these potentials, however, require increased dimensionality (to 3D and beyond),⁶⁵ hence further emphasizing the importance of expansion in our view and understanding of the phase landscape. This ultimately raises the question whether there are undiscovered opportunities in the phase landscape that have yet to be conquered and if we are currently being blindsided by our limited perspective.

Conclusions and outlook

This perspective explores the past, current, and future potential of undercooled liquid metals as a platform for technological and scientific advancements. The ability to suppress and control phase transitions and the liquid-phase range of metallic systems is a critical aspect in tackling the emergence of soft-/bio-electronics and device miniaturization. Extending the workable range of metallic liquid phase through metastable states reveals unique opportunities and new processing pathways that are otherwise inaccessible through a traditional thermodynamic view. Despite this promising opportunity, however, understanding of this space (e.g., how to access/control them, what phases/properties can potentially be present, structural relaxation mechanism and degree of order present in the metastable liquid) is still highly limited. Whilst steady progress on breaking down the roles of various thermodynamic potentials in driving metastability is made, further awareness of this subject and its applicability is highly needed to advance the science. Studies on the fundamentals of non-equilibrium thermodynamics have not been at the forefront since initial breakthrough in the 1930s by Onsager and Prigogine; however, with increasing complexity of modern material systems, it is becoming increasingly apparent that this subject needs to be revisited and further expanded. This expansion of the materials processing landscape leads to the realization that there are large knowledge gaps, unanswered scientific questions, and undiscovered technological potential that has yet to be conquered. Given the continuous need to innovate, this perspective provides a fresh viewpoint on metals processing, what is available and what is to come; the potential opportunities and future research direction of this technology are summarized herein.

Adaptability and energy efficiency

The reduction of metal’s processing temperature enables adaptation of metallic systems to otherwise incompatible systems. At a larger scale, this reduction in heat results in significantly increased energy efficiency, especially given the volume of ever-increasing electronics production demands. The ability to bypass certain energy barriers in a metal’s life cycle (from mining to manufacturing, including recycling) also enables substitutive processing pathways that enable flexibility and opportunities.

Expansion of the phase landscape

The presence of $⟨{\hat{T}}_{\delta }⟩$ in ULMCS acts as a source of driving force in the non-equilibrium undercooled liquid landscape, leading to controlled entropy dissipation behavior. Tuning this asymmetric tensor enables access to microstates between the liquid-solid states which tap into unique material properties and structures. This expansion of the phase landscape provides alternative pathways for materials design, in which proper understanding, quantification, and control of internal and external stresses that are present in the material system can lead to discovery of new multiphase material systems with unique interfacial-driven effects.

Extension beyond single stimuli processing

Thermal limitations of metals processing can be overcome through different driving forces. Understanding how different perturbations (e.g., mechanical, chemical, electrical, photon, etc) alter the atomic structures of these systems can lead to unique materials processing approaches, especially when these forces are coupled. Detailed understanding of energy/force balance in material systems as well as their preferential reactivity based on atomic characteristics can lead to more efficient pathways for materials process design by utilizing thermodynamically favorable interactions.

Acknowledgments

This work was supported in part by the Center for Complex Particle Systems (COMPASS) through an NSF grant award no. 2243104 and by the North Carolina State University startup funds for MT.

Author contributions

Conceptualization: A.M. and M.T.; methodology: A.M. and J.P.; data curation: A.M. and J.P.; investigation: A.M. and J.P.; supervision: A.M. and M.T.; validation: A.M. and M.T.; visualization: A.M.; writing – original draft: A.M.; writing – review and editing: A.M., J.P., and M.T. All authors have given approval to the final version of the manuscript.

Declaration of interests

M.T. has patent applications and/or registrations related to this work. Key patents include: US12208471B2, Solder materials including supercooled micro-capsules and alloyed particles. US12226831B2, Direct printing and writing using undercooled metallic core-shell particles. US12583059B2, Solder paste on-demand apparatus. US12030126B2, undercooled liquid metallic droplets having a protective shell.

Declaration of generative AI and AI-assisted technologies in the writing process

No generative AI and AI-assisted technologies were used in the writing process of this manuscript.