Equilibrium-gated pattern formation: How molecular dissociation thermodynamics drive emergent behavior in dissipative polymeric systems

Significance

Materials with spatially varying properties offer unique advantages, such as enhanced toughness and adaptability, over uniform materials. Traditionally achieved via additive manufacturing, this study presents an alternative approach, harnessing a multistep chemical reaction mechanism to spontaneously generate such patterns during polymerization. We identify a key feature: equilibrium-gated pattern formation. This feature drives polymerization by locally activating dormant initiators, regulating their supply to subsequent reaction steps. Small shifts in chemical equilibrium modulate this supply, destabilizing the system’s thermochemical balance. Together, these factors create nontrivial, emergent behavior that shapes the material’s spatial organization. Integrating computational and experimental approaches, we provide a framework for designing synthetic materials with self-organizing properties, opening pathways for customizable, high-performance polymeric materials.

Abstract

Emergent patterns in biological systems arise through dissipative processes that balance reaction and transport phenomena, producing highly functional properties from self-regulating mechanisms. Synthetic fabrication, by contrast, often relies on user-controlled, multistep methods that lack the self-organizing capabilities of natural systems. Inspired by nature, we sought chemical systems that integrate strongly coupled reaction and transport phenomena, identifying frontal ring-opening metathesis polymerization (FROMP) as a method capable of creating diverse forms and functions through reactive processing. By employing discrete molecular initiators, FROMP allows precise control of key reaction steps—inhibition, initiation, and propagation. Using an integrated computational and experimental framework, we uncover how near-equilibrium inhibition dynamics, coupled with far-from-equilibrium reaction kinetics, drive pattern formation in frontally polymerized synthetic materials. We propose the concept of equilibrium-gated pattern formation, demonstrating how initiator chemistry can be tuned to achieve programmable macroscale properties. Our study reveals a surprising insight: Emergent behavior in FROMP systems arises from the inhibition-dominated regime of resin composition, expanding prior observations that such behavior is confined to a narrow compositional space near the boundary between front quenching and uniform front propagation. We identify a broader compositional window, far from the quenching regime, where emergent behavior reliably manifests. This expanded design space significantly enhances the operational flexibility of reactive systems and their capacity for self-organization. These insights provide a roadmap for designing bioinspired materials with self-organizing capabilities, unlocking possibilities in synthetic manufacturing.

As disordered as nature may appear, countless natural phenomena adhere to consistent patterns and regularity (e.g., phyllotaxy, mackerel sky, sand ripples) (1). Across nature, complex organizational patterns integral to biological structure and function spontaneously emerge through dissipative, far-from-equilibrium reaction-transport processes at multiple hierarchical levels, from molecular pathways to tissues and organs (2, 3). Competing reaction-transport phenomena at the local molecular scale give rise to symmetry-breaking events conducive to emergent behavior at the larger organismal level (4). This process imparts numerous functionalities for biological survival, including camouflage [e.g., skin markings (5, 6)], mating [e.g. peacock’s colorful tail (7)], sensing [e.g., skin ridges and striations (8)], and predation [e.g., multiscale organization in stomatopods appendage for improved fracture energy (9)], Fig. 1A.

Fig. 1.

Emergent behavior arising from dissipative reaction-transport phenomena across natural and synthetic systems. (A) Examples from biology, physics, and engineering: (i) Skin markings of a zebra, which contribute to thermoregulation and camouflage (Deep Dream AI-generated image). (ii) Oscillatory patterns in a Belousov–Zhabotinsky reaction, demonstrating potential for smart material applications (Image credit: Futurum/Robert E. Klein). (iii) Transmission electron microscopy (TEM) image of a silver dendrite morphology, highlighting growth dynamics. [Reproduced with permission from Fang et al. (10), Copyright 2013, American Chemical Society]. (iv) Phase-separation in a LiFePO₄ battery electrode, illustrating Lithium-rich regions in white. [Reproduced with permission from Cogswell and Bazant (11), Copyright (2012) American Chemical Society]. (B) Synthetic patterns observed in FROMP systems: (i) Stable front propagation in a liquid monomer resin (light pink), transitioning to a solid polymer (gold). (ii) Schematic representation of a spinning front destabilized by reaction-transport feedback, forming distinct soft amorphous and semicrystalline domains. (C) Illustration of enhanced mechanical properties (e.g., toughness) arising from spontaneous patterning in FROMP systems; see ref. 12.

Contrary to the self-regulating reaction-transport feedback loops that drive emergent behavior in nonequilibrium natural systems, hierarchical organization in synthetic materials typically relies on user-controlled manufacturing strategies for automation and standardization (13). These conventional methods often involve complex, multistep processes. Inspired by biological patterning, we envision an alternative route to creating complex synthetic materials with programmable properties by exploiting dissipative reaction-transport phenomena and symmetry-breaking events. Elegant demonstrations of nonequilibrium reaction-transport phenomena span across physics, material science, and engineering, including spinodal decomposition in electrode materials for energy storage (11), self-oscillating polymeric gels by virtue of the Belousov–Zhabotinsky reaction for smart materials (14), eutectic alloy solidification (15), and dendritic growth (10, 16), Fig. 1A.

Less explored, thermally mediated reaction-transport phenomena in synthetic materials have emerged as promising candidates for achieving autonomous patterning in structural materials. These systems leverage thermal diffusion, which surpasses mass transport by orders of magnitude to drive self-sustaining reactions (12, 17). Among these, FROMP stands out as a versatile approach for reactive processing. This exothermic, self-sustaining reaction is initiated by an external stimulus—thermal, chemical, or photo—which ignites a localized reaction front (18). A key feature of FROMP is its reliance on well-defined initiator complexes, whose chemistry can be intentionally manipulated to precisely control each step of the olefin metathesis cycle: inhibition, initiation, and propagation. The reaction front propagates through the liquid monomer resin, transforming it into a solid polymer as heat is efficiently transported into the unreacted monomer phase, governed by the delicate balance of reaction kinetics, exothermicity, and thermal diffusion, Fig. 1B.

Provided the balance between heat generation, reaction kinetics, and thermal transport is maintained, the reaction front propagates steadily and uniformly (19, 20). However, disruptions to this delicate balance—such as heat loss or variations in the resin chemistry—destabilize the reaction-transport feedback loop, leading to unsteady propagation characterized by pulsating, spinning, or aperiodic dynamics, Fig. 1B, (12, 21, 22). These nonuniform fronts generate localized high-temperature regions, which modulate the polymer microstructure during polymerization, resulting in the formation of alternating soft amorphous and hard semicrystalline domains with varying stiffness and improved toughness, Fig. 1C. We have previously leveraged this behavior to spatially program the polymer microstructure on the millimeter scale, highlighting the potential of FROMP for the autonomous fabrication of synthetic materials with spatially varying properties (12, 23, 24). Despite these advances, a mechanistic understanding of how chemical composition influences the reaction-transport dynamics remains elusive.

Previous works to describe the emergence of nonuniform front dynamics in FROMP systems have primarily relied on linear stability analysis, incorporating concepts from combustion physics, such as the Zeldovich number (21, 22, 25–29). While these approaches provide valuable insights, they do not account for the mechanistic details of FROMP, which include an inhibition, initiation, and propagation step (30, 31). Importantly, experimental studies have demonstrated that changes in the resin chemistry—such as increasing the concentration of the phosphite inhibitor or modifying ligands in the ruthenium initiator—can significantly influence the propensity for emergent behavior to manifest (12). These observations suggest that the underlying chemical details play a critical role in the emergence of spontaneous patterns. Here, we aim to establish a mechanism-based framework that relates the reaction kinetics to nonuniform front propagation, providing a deeper understanding of how chemical composition drives emergent behavior in FROMP systems.

To address these limitations, we integrate a computational and experimental framework, employing a mechanism-based FROMP model—grounded in the principles of reaction kinetics—to uncover the chemical origins of emergent behavior in frontally polymerized synthetic materials. Central to this work is the concept of equilibrium-gated pattern formation, which arises from the temperature-regulated ligand dissociation dynamics. Although this step operates near equilibrium, we demonstrate how the dynamic interplay between the temperature-regulated inhibitory ligand dissociation and the far-from-equilibrium olefin metathesis reactions influences the balance of reaction kinetics, exothermicity, and heat transport in the resin system. By modulating the supply of the active ruthenium initiator to the olefin metathesis cycle, these localized inhibition dynamics are shown to disrupt the uniform front propagation, giving rise to emergent behavior.

Our findings establish a framework for controlling macroscale emergent behavior in FROMP systems through molecular design of initiators with tunable activity. This work highlights the vast potential of unexplored initiators, paving the way for designing materials with programmable spatial properties through FROMP. By efficiently screening the chemical design space, this insight unlocks possibilities for advancing the self-organizational capabilities of synthetic materials with enhanced engineering properties.

Results

Model Overview.

We present here a brief overview of the mechanism-based thermochemical model to describe the multistep FROMP kinetics. We refer the reader to our prior work for a systematic description of the model formulation (32). The model is grounded in the conventional principles of reaction kinetics and establishes a mechanism-based description of the FROMP kinetics through a three-step sequence (33, 34), which includes an:

1. Inhibition step, which gates the reactivity of the dormant 16-electron ruthenium initiator, (II) by thermally activated dissociation of the coordinated phosphorus-based ligand, (PR₃). This process leads to the formation of an active 14-electron initiator complex, (AI), prior to entry in the olefin metathesis cycle, Fig. 2(1). We assume that the equilibrium between (II) and (AI) is rapidly established and adjusts locally with temperature changes, enabling a dynamic balance between inhibition and activation (i.e., near-equilibrium dynamics). We hypothesize that this temperature-dependent shift in the equilibrium state plays a critical role in the emergent behavior of the system, highlighting the importance of near-equilibrium dynamics. Thermodynamically, the inhibition equilibrium is characterized by both a positive enthalpy ($\mathrm{\Delta }{H}^{\text{o}}>0$) and positive entropy ($\mathrm{\Delta }{S}^{\text{o}}>0$) of ligand dissociation.^* This thermodynamic profile gives rise to a threshold temperature above which dissociation becomes favorable, as the opposing temperature-coupled enthalpic effects are outweighed by the favorable entropic contributions. The resulting sigmoidal production of active species (Fig. 3D) reflects this thermally responsive gating behavior, where the reaction is effectively regulated by the local equilibrium constant, $K_{\text{eq}}(T)$, derived from both enthalpic and entropic contributions through the van’t Hoff equation,

   $$K_{\text{eq}}=\text{exp}(-\frac{\mathrm{\Delta }{H}^o}{\mathit{RT}}+\frac{\mathrm{\Delta }{S}^o}{R})$$ [1]

   Starting with a ([II₀], [AI₀], [${\text{PR}}_3^0$]) resin composition, it can be shown that the temperature-coupled production of the active ruthenium initiator, [AI⁺], during the inhibition equilibrium evolves as a function of the reactants through the following relationship:

   $$\begin{array}{cc}\hfill [\mathbf{A}{\mathbf{I}}^{+}]& =-\frac{[\mathbf{A}{\mathbf{I}}_0]+[\mathbf{P}{\mathbf{R}}_3^0]+K_{\text{eq}}}{2}+...\hfill \\ \hfill & ...+\frac{1}{2}\sqrt{\begin{array}{c}([\mathbf{A}{\mathbf{I}}_0]+[\mathbf{P}{\mathbf{R}}_3^0]+K_{\text{eq}}{)}^2\hfill \\ -4([\mathbf{A}{\mathbf{I}}_0][\mathbf{P}{\mathbf{R}}_3^0]-K_{\text{eq}}[\mathbf{I}{\mathbf{I}}_0]).\hfill \end{array}}\hfill \end{array}$$ [2]

   The generated active ruthenium complex subsequently enters the olefin metathesis cycle, affecting the kinetics of the ensuing initiation and propagation steps.
2. Initiation step, which involves the 14-electron active ruthenium initiator, (AI), coordinating an olefinic monomer substrate to first form a four-coordinate ruthenium-olefin adduct, (B). The ruthenium-olefin adduct undergoes initiation by [2+2] cycloaddition and subsequently cycloreversion, Fig. 2(2), resulting in the formation of a ruthenium alkylidene with a single ring-opened monomer attachment, (C). This process is accompanied by ring strain relaxation, contributing to the heat release. The rate of formation of the ruthenium alkylidene, (C), computes to

   $$\frac{d[\mathbf{C}]}{\mathit{dt}}=k_i^{\mathit{eff}}[\mathbf{A}\mathbf{I}](1-\alpha ),$$ [3]

   with $k_i^{\mathit{eff}}$ = $A_i·\text{exp}(-\frac{E_a^i}{\mathit{RT}})$, an effective temperature coupled initiation rate coefficient described by an Arrhenius-type kinetics. Here, $A_i$, represents an initiation pre-exponential factor in units of [s⁻¹], and $E_a^i$, an initiation activation energy in units of [J/mol]. For nomenclature convenience, we additionally introduce the degree of cure, $\alpha$ = $\frac{[\mathbf{M}]}{[{\mathbf{M}}_0]}$ $\in$ [0,1], with [M₀] denoting the initial concentration of the olefinic monomer in the resin system, and [M], the concentration of the olefinic monomer converted through polymerization. As such, a state of $\alpha$ = 0 represents the uncured liquid monomer resin, while $\alpha$ = 1, a state of complete conversion of the liquid resin into a solid polymer.
3. Propagation step, which involves the ruthenium alkylidene, (C), sequentially reacting with $n$-olefinic monomer units in an irreversible chain growth polymerization process, similar in mechanism to the initiation step. This process leads to the formation of a solid polymer, Fig. 2(3). By virtue of the law of mass action and accounting for the sequential coordination of the olefinic monomers to (C), the rate at which the monomer units convert into a solid poly-olefin is computed as

   $$[{\mathbf{M}}_0]\frac{d\alpha }{\mathit{dt}}=k_p^{\mathit{eff}}[\mathbf{C}](1-\alpha ),$$ [4]

   with $k_p^{\mathit{eff}}$ = $A_p·\text{exp}(-\frac{E_a^p}{\mathit{RT}})$, an effective temperature- coupled propagation rate coefficient, similar in form to the earlier description of $k_i^{\mathit{eff}}$.

Fig. 2.

Schematic representation of a mechanism-based model for FROMP, rooted in reaction kinetics and consisting of three key steps (1) Inhibition equilibrium—Thermal activation of dormant initiators through the dissociation of inhibitory ligands. (2) Initiation—Coordination and activation of olefinic monomers, triggering the olefin metathesis cycle. (3) Propagation—Sequential chain growth polymerization, which predominantly contributes to the exothermic heat release, driving the self-sustaining reaction front.

Fig. 3.

(A) Schematic illustrating how increasing the inhibitor concentration affects the emergence of spontaneous patterns in FROMP systems, as observed by Paul et al. (12). (B) Temperature-dependent production of the active ruthenium complex as a function of the starting dormant initiator concentration for different inhibitor loadings in 10,000:1:x resin formulations. Higher inhibitor concentrations reduce the amount of active ruthenium complex generated at a given temperature. (C) Schematic illustration of controlled patterning through the design of Grubbs’ chemical initiators with varying activity, inspired by experimental observations from Paul et al. (12) on the effect of the R¹—alkylidene ligand. (D) Temperature-coupled production of the active ruthenium complex as a function of the starting dormant initiator concentration, with variation in the inhibition equilibrium constant, $K_{\text{eq}}$, for a 10,000:1:1 resin formulation. Lower values of $K_{\text{eq}}$ (Left to Right) strengthen inhibition, reducing the amount of the active ruthenium complex formed at a given temperature. (E) Map of simulation results showing the polymerization front velocity for a 10,000:1:1 resin formulation as a function of the standard entropy of the ligand dissociation-association equilibrium ($\mathrm{\Delta }{S}^o$). Results illustrate the transition from stable uniform front propagation to emergent behavior as inhibition strengthens. (F) Framework for controlling the far-from-equilibrium emergent behavior in dissipative FROMP systems by modulation of the localized near-equilibrium inhibition dynamics through the design and screening of initiators (e.g., M204 and M207) with tailored activity. Initiator designs are schematically represented by distinct fingerprints, with ligand groups (i.e., L, X, R¹, and PR₃) color-coded to mimic Fig. 2 (1) such that different substituents of the same group are marked by their geometric shape.

Last, the governing equation for the temperature evolution with heat release during FROMP of the liquid monomer resin is described by the classical heat balance equation,

$$\kappa {\mathrm{\nabla }}^{2}T+\rho H_r\frac{\partial \alpha }{\partial t}=\rho C_p\frac{\partial T}{\partial t}.$$ [5]

Here, $\kappa$ [$\frac{\text{W}}{\text{m}·\text{K}}$], C_p [$\frac{\text{J}}{\text{kg}·\text{K}}$], $\rho$ [$\frac{\text{kg}}{{\text{m}}^3}$] denote the thermal conductivity, specific heat capacity, and density of the resin, while $H_r$ [$\frac{\text{J}}{\text{kg}}$], the total enthalpy of the polymerization reaction.

Jointly, Eqs. 1–5 describe the mechanism-based FROMP model for a total of four solution variables ([AI⁺], [C], $\alpha$, $T$). We note that the FROMP reaction steps are thermochemically coupled in a feedback loop, with transient heat release during the olefin metathesis playing a central role in maintaining the system far from equilibrium.^† As a last remark, we note that any effects associated with molecular diffusion are neglected in this framework, as the reaction front propagates primarily due to thermal (heat conduction) rather than mass diffusion. We refer the reader to Materials and Methods for a description of the model implementation.

On the Origin of Emergent Behavior.

State-of-the-art conventional FROMP models utilized to study emergent behavior in dissipative polymerization systems are phenomenological in nature and constructed on a single propagation step (23, 28). As extensively discussed in ref. 32, these models do not capture the critical role of the resin chemistry on the macroscopic FROMP observables (e.g., front velocity). Recent experiments by Paul et al. (12) on the frontal polymerization of a 1, 5 - cyclooctadiene (COD) monomer with different Grubbs’ initiators and a tributyl phosphite inhibitor, P(OBu)₃ (TBP), provide a systematic study on the role of resin chemistry on the polymerization front dynamics. Particularly, subtle changes in the resin chemistry,—either by increasing the concentration of the phosphite inhibitor, Fig. 3A, or replacing individual ligands (e.g., R¹– alkylidene group, Fig. 3C) in the Grubbs’ initiator,—were observed to destabilize the front, resulting in spinning front dynamics conducive to spontaneous patterning. Owing to their chemistry-blind nature, phenomenological FROMP models are unable to rationalize these observations, pointing to the need for a unifying framework.

We employ the mechanism-based model to inform such rationale and establish a chemically grounded understanding of the origin of emergent behavior in FROMP-ed polymers. For the sake of anchoring the model to an actual FROMP system, we focus on a COD monomer resin formulation, Fig. 2, in which a Grubbs’ second-generation initiator (e.g., M204, Fig. 3F) and a TBP inhibitor are dissolved for controlled bulk reactivity.^‡ Doing so enables us to establish a baseline for probing the effect of variations in the resin chemistry on the polymerization front dynamics.^§

The relevant physicochemical parameters for our COD/M204/TBP resin system are tabulated in Table 1. The mechanism-based nature of the framework allows virtually all the physicochemical parameters to be found from the literature, either experimentally or from ab-initio computations.

Table 1.

Physicochemical parameters for numerical FROMP simulations of emergent behavior in a COD:M204:TBP resin system

Parameter	Value	Source
$\kappa$	0.133 W/(m$·$K)	(23)
$\rho$	882 kg/m3	(23)
$C_p$	1838 J/(kg$·$K)	(23)
$H_r$	219 kJ/kg	(23)
$\mathrm{\Delta }{H}^o$	26.1 kCal/mol	(35, 36)
$\mathrm{\Delta }{S}^o$	51 Cal/(mol$·$ K)	(36)
$E_i^{\mathit{eff}}$	19.12 kCal/mol	(37)
$A_i^{\mathit{eff}}$	6.0 $·{10}^{12}{\text{s}}^{-1}$	Calibrated
$E_p^{\mathit{eff}}$	17.7 kCal/mol	(38)
$A_p^{\mathit{eff}}$	1.8 $·{10}^{14}{\text{s}}^{-1}$	Calibrated

To align the model with observations (12), we first inspect the role of increasing the resin inhibitor concentration on FROMP kinetics as illustrated in Fig. 3A. In line with Eq. 2, the inhibitor concentration enters the inhibition equilibrium step and regulates the temperature-coupled amount of the active 14-electron ruthenium species, [AI⁺], that can be generated during dissociation. Starting with a ([II₀], [${\text{PR}}_3^0$]) resin formulation, such effect can be quantified by modeling the production of the active ruthenium complex, [AI⁺], as a function of the starting dormant initiator concentration, [II₀].

Fig. 3B illustrates such an evolution for a 10,000:1:x monomer/initiator/inhibitor resin formulation for four different TBP inhibitor concentrations, while maintaining the initiator concentration fixed. Noticeably, an increase in the TBP inhibitor loading retards the activation of the dormant ruthenium initiator (i.e., stronger inhibition), disrupting the supply of the latter to the olefin metathesis cycle. This retardation is demonstrated by a rightward shift in the curves, with higher temperatures required to achieve a particular active initiator supply. By restricting the supply of the active initiator to the olefin metathesis cycle, the increase in inhibitor concentration establishes a localized bottleneck, which disrupts the regulatory feedback mechanism between the olefin metathesis reaction kinetics, exothermicity, and heat transport into the unreacted monomer. We hypothesize that it is precisely the creation of an inhibition-dominated regime which—by regulating the supply of the active initiator to the olefin metathesis cycle—controls emergent behavior in dissipative frontally polymerized systems. This hypothesis builds on the seminal work of Sanford et al. (30), in which the dissociation of the phosphorus-based ligand for N-heterocyclic ruthenium alkylidenes was identified as a rate-limiting step in the reaction sequence of ring-opening olefin metathesis.

To further test the hypothesis, we computationally explore a separate scenario in which the inhibition strength is modulated by virtue of the equilibrium constant, $K_{\text{eq}}$. From conventional reaction thermodynamics, modulations in the equilibrium constant, $K_{\text{eq}}$, regulate the reaction extent by shifting the reaction quotient either toward the products or the reactants. Experimentally, modulations in the equilibrium constant at a fixed temperature are achieved by modifying the structure of ruthenium initiators with different ligand substituents, Fig. 3C. As discussed by Sanford et al. (30), ancillary ligands (e.g., phosphine, alkylidene, halide groups) alter the steric and electronic structure of ruthenium initiators (Discussion–Summary and Outlook), modulating their dissociation thermodynamics as defined in Fig. 2(1).

In line with Eq. 1 and for illustrative proof of concept, we systematically adjust $K_{\text{eq}}$ by varying the standard entropy of the phosphine ligand dissociation, $\mathrm{\Delta }{S}^{\text{o}}$, over published literature ranges (36, 39), while preserving the standard enthalpy of dissociation, $\mathrm{\Delta }{H}^{\text{o}}$, fixed to the value reported in Table 1. For completeness, we emphasize that both $\mathrm{\Delta }{H}^{\text{o}}$ and $\mathrm{\Delta }{S}^{\text{o}}$ contribute to the temperature-coupled equilibrium constant, $K_{\text{eq}}$. We explored both scenarios: holding $\mathrm{\Delta }{H}^{\text{o}}$ fixed while varying $\mathrm{\Delta }{S}^{\text{o}}$, and vice versa (SI Appendix, Fig. S3).

Similar to the inhibitor loading investigation, we quantify the effect of $K_{\text{eq}}$ on the inhibition equilibrium by modeling the temperature-regulated production of the active ruthenium complex, [AI⁺], as a function of the starting dormant initiator loading, [II₀], Fig. 3D. Akin to the effect of increase in the inhibitor concentration, a decrease in $K_{\text{eq}}$ by virtue of diminishing favorable entropy change (Left to Right) strengthens inhibition, making it less favorable for the phosphine ligand to dissociate. As such, $K_{\text{eq}}$ introduces a regulatory knob, Fig. 3D, which—by subtle modifications to the ruthenium initiator—modulates the propensity for the active initiator complex to readily form prior to entry in the olefin metathesis cycle.

In line with our hypothesis, we subsequently assess the effect of strengthening inhibition by virtue of $K_{\text{eq}}$ (i.e., ruthenium initiator design) on the propensity for emergent behavior to arise. We do so by computationally mapping the 1D polymerization front velocity, V_f, as a function of $\mathrm{\Delta }{S}^{\text{o}}$ for a 10,000:1:1 resin formulation and determine the mode of front propagation for each combination (i.e., stable planar vs. pulsating). Fig. 3E illustrates the simulated results at two different initial resin temperatures, $T_0$ = 15 ^°C and $T_0$ = 23 ^°C, with combinations resulting in stable planar fronts shown in red, while those resulting in pulsating fronts conducive to desirable emergent behavior in green.^¶ In line with our hypothesis, strengthening inhibition (i.e., decreasing $\mathrm{\Delta }{S}^{\text{o}}$) modulates the supply of the active initiator to the olefin metathesis cycle, affecting in turn the nonequilibrium kinetics of the subsequent initiation and propagation steps. This cascade effect results in a decreasing front velocity, Fig. 3E. As the ability to readily dissociate the phosphorus-based ligand diminishes, gradual transition to an inhibition-dominated regime ensues. This transition disrupts the balance between thermal diffusion and the olefin metathesis reaction kinetics, inducing a shift from stable uniform to nonlinear front dynamics conducive to emergent behavior. These results are consistent with the experimental observations by Paul et al. (12) who, by replacement of the R¹– alkylidene ligand in Grubbs’ initiators (e.g., M202, M204, and M207), demonstrated a pronounced change in the propensity of the resin system to spontaneously pattern.

More importantly, these findings demonstrate how localized near-equilibrium inhibition dynamics couple with the far-from-equilibrium reaction kinetics to produce nontrivial emergent behavior in dissipative polymerization systems. On the basis of these findings, we propose the concept of equilibrium-gated emergent behavior in frontally polymerized systems. This proposition puts forward a framework for control over the macroscopic far-from-equilibrium emergent behavior in FROMP-ed polymers by modulation of the localized near-equilibrium inhibition dynamics through design of initiators of varying activity (i.e., dissociation thermodynamics, $\mathrm{\Delta }{\text{H}}^{\text{o}}$ and $\mathrm{\Delta }{\text{S}}^{\text{o}}$), Fig. 3F. Given the large library of initiators and inhibitors capable of frontally polymerizing, this insight advances a framework for rapid molecular screening—in combination with density functional theory—of the chemical design space (i.e., L, X, R¹, and PR₃ ligands) to identify optimal formulations for architecting emergent behavior in synthetic polymers, Fig. 3F. Such insights can significantly accelerate the experimental design and development protocols, currently reliant on iterative time and material costly processes. Achieving this can uncover opportunities for programmable synthetic materials.

Mapping Composition to Emergent Behavior.

We present here a detailed study of our prototypical COD/M204/TBP resin system through a synergistic computational and experimental approach. Through such effort, we assess the ability of our mechanism-based FROMP model to map variations in the resin chemistry and processing conditions to emergent behavior in frontally polymerized systems. Such studies are crucial for establishing an integrated, closed-loop communication between computational models and experiments, enabling accelerated design.

Using the physicochemical properties in Table 1, we perform 1D simulations to compute macroscopic FROMP observables pertinent to variations in the front velocity with change in the resin formulation. We experimentally sweep through a large range of monomer-to-initiator loading equivalents, [1,000 to 50,000], while maintaining the initiator-to-inhibitor loading ratio fixed. We refer the reader to Materials and Methods for a description of the experimental methods.

Fig. 4A illustrates the comparison between the simulated and experimental front velocities for a X:1:1 resin system at room temperature. As described in refs. 32 and 39, a decrease in the monomer-to-initiator loading ratio increases both the concentration of the dormant initiator (and effectively the amount of the active initiator that can be generated for entry in the olefin metathesis cycle) and that of the inhibitor in the initial resin formulation. As we sweep from 50,000 to 10,000 monomer-to-initiator equivalents (Right to Left), the continuous increase in the amount of the active initiator that can be generated at high temperatures for entry in the olefin metathesis cycle dominates the macroscopic FROMP behavior. As a result, an increase in front velocity is initially observed. The increase in front velocity is also evident from Fig. 4B, which illustrates the simulated time evolution of the 1D front location for the different resin formulations.

Fig. 4.

(A) Comparison of the experimental and simulated front velocities for a X:1:1 COD/M204/TBP resin system as a function of the monomer-to-initiator loading ratio. (B) Simulated time evolution of the 1D polymerization front location with varying monomer-to-initiator loading ratio, illustrating the differences in front progression dynamics. (C) Experimental map of the initial resin temperature ($T_0$) with variation in the monomer-to-initiator loading ratio for a X:1:1 resin formulation. Resin formulations resulting in stable planar fronts are indicated in red, while those leading to spontaneous patterns are shown in green.

As the monomer-to-initiator loading ratio is lowered past 10,000 parts, the front velocity plateaus, Fig. 4A. This result points to the emergence of an inhibition-dominated regime attributed to the further increase in the inhibitor concentration with decrease in the monomer-to-initiator loading. The additional increase in inhibitor concentration retards the activation of the dormant ruthenium initiator, disrupting the supply of the latter to the subsequent FROMP reaction steps. Such disruption establishes a localized inhibition-dominated regime, which dictates the macroscopic FROMP behavior and prevents the further increase in front velocity. A similar behavior is evident from Fig. 4B, in which the simulated time evolution of the front location for [1,000 to 5,000] monomer-to-initiator equivalents is nearly coincident. Interestingly, as the monomer-to-initiator loading ratio decreases from 50,000 to 1,000 parts (Bottom to Top curve), a transition from stable uniform to nonlinear front dynamics conducive to emergent behavior is numerically observed. This simulation result bears a two-fold significance:

• i)First, it supports our hypothesis that the occurrence of a localized inhibition-dominated regime, which disrupts the nonequilibrium feedback loop between the FROMP reaction kinetics, exothermicity, and heat transport into the unreacted monomer, controls the emergent behavior in dissipative polymerization systems at the macroscale.
• ii)Second, it expands upon prior observations that emergent behavior in FROMP systems necessarily arises closer to front quenching (12). These observations strictly hold true for resin formulations with a fixed monomer-to-initiator equivalence, while the inhibitor loading gradually increases, Fig. 3A. This insight has important ramifications from a design standpoint and unlocks capabilities for architecting frontally polymerized synthetic materials with programmable spatial properties, while preserving reasonable front velocities for curing.

To further test the validity of i) and ii), we experimentally map the initial resin temperature window, $T_0$, for emergent behavior in a X:1:1 resin formulation, Fig. 4C. Noticeably, as the monomer-to-initiator loading ratio decreases from 10,000 to 1,000 parts (i.e., increasing inhibitor concentration), the initial resin temperature window for spontaneous patterning significantly widens. Isolating on an initial resin temperature of choice and moving leftward along the monomer-to-initiator equivalence axis (also the direction of increasing front speeds), an increased propensity for emergent behavior is observed. We refer the reader to SI Appendix, Fig. S1 for experimental images of the fully cured COD/M204/TBP resins collected at 24-h postcure for different initial resin temperatures. On the effect of increasing the initiator-to-inhibitor equivalence, we experimentally map emergent behavior for a select few initial resin temperatures for X:1:2 resin formulations. SI Appendix, Fig. S4 illustrates the expansion in the initial resin temperature for emergent behavior to reliably manifest compared to X:1:1 formulations, showcasing the critical role of increasing the inhibitor concentration on the propensity of the system to spontaneously pattern. These results corroborate our 1D simulation findings, Fig. 4B, and illustrate how mechanistically grounded computational models—in contrast to composition-blind phenomenological FROMP models—are able to inform experimental design protocols, departing from iterative, inefficient approaches for the identification of resin conditions conducive to emergent behavior.

To better illustrate the dynamics of emergent behavior in a COD/M204/TBP resin system, we qualitatively replicate the map in Fig. 4C by performing 2D simulations. Fig. 5A illustrates contours of the point-wise maximum temperature acquired in the resin system at the conclusion of the FROMP simulations. Localized high-temperature regions serve as nucleation sites for emergent behavior in frontally polymerized systems by virtue of symmetry-breaking events. As such, their tracking provides an easy approach to quickly assess resin conditions (e.g., resin formulation, initial resin temperature) conducive to emergent behavior. Noticeably, the simulated FROMP results demonstrate a widening in the initial resin temperature window for patterning as the monomer-to-initiator equivalence decreases from 10,000 to 2,500 parts (Left to Right). Particularly, pronounced patterns (i.e., distinct high-temperature marks) emerge downward of an initial resin temperature, $T_0$ = 10 ^°C for a 10,000:1:1 resin formulation, 18 ^°C for a 5,000:1:1, and 23 ^°C for a 2,500:1:1, qualitatively reproducing the experimental findings in Fig. 4C. Moreover, these results expand the state-of-the-art computational capabilities, demonstrating the ability to computationally map emergent behavior in frontally polymerized systems over a wide range of monomer loadings and resin processing conditions. We refer the reader to SI Appendix, Fig. S5 for additional FROMP simulations in which the stable planar-propagating resin formulations from Fig. 5A are supplemented with higher inhibitor loadings, resulting in emerging spontaneous patterns as the ability to readily dissociate the inhibitory phosphorus-based ligand diminishes.

Fig. 5.

(A) Contours of the maximum temperature extracted at the end of the FROMP simulations for X:1:1 COD/M204/TBP resin formulations. Legend boundaries are prescribed in the [$T_0$, $T_{\text{max}}$ + 40 ^°C] range, where $T_{\text{max}}$, is the maximum adiabatic front temperature analytically calculated as $T_{\text{max}}=T_0+\frac{H_r}{C_p}(1-{\alpha }_0$). A starting degree of cure, ${\alpha }_0=0.01$, is used across all simulations (Finite Element Analysis). Localized high-temperature regions act as nucleation sites for emergent behavior in FROMP systems through symmetry-breaking events, providing a convenient method for assessing resin conditions conducive to emergent behavior. From Left to Right, the propensity for emergent behavior increases as the monomer-to-initiator equivalence decreases from 10,000 to 2,500 parts. (B) Experimental images of the fully cured COD/M204/TBP resin system in a rectangular mold for an initial resin temperature T₀ = 18 ^°C. Images are acquired 24 h postcure and illustrate the downward-propagating front dynamics through the distinct formation of interconnected soft amorphous and hard semicrystalline domains. Numerical simulations and experiments are in good qualitative agreement, demonstrating the influence of variations in the resin chemistry (e.g., monomer-to-initiator loading) and processing conditions (e.g., initial resin temperature, $T_0$) on emergent behavior. *Experimental images enhanced for better visualization.

We additionally compare the $T_{\text{max}}$ profiles from numerical simulations to the experimental images of the polymer final product acquired 24 h postcure to allow for the distinct visualization of patterned soft amorphous and hard semicrystalline domains arising from modulations in the polymer molecular backbone [e.g., cis–trans ratio (23)] by localized high temperatures. To better visualize the front dynamics, experimental images are acquired for resin formulations frontally polymerized in a rectangular mold, similar to the experimental setup by Paul et al. (12) and described in detail in Materials and Methods. Fig. 5B illustrates such a comparison for [2,500 to 10,000]:1:1 resin formulations, demonstrating the consistent formation of front traces with highly regular spin modes that transverse back and forth across the resin domain in both numerical simulations and experiments. As discussed in ref. 12, such emergent behavior gives rise to materials with spatially varying properties (e.g., stiffness, toughness) of broad engineering appeal.

The dynamics of emergent behavior in frontally polymerized resin systems can be precisely visualized by tracking the front propagation (i.e., degree of cure, $\alpha$) with time. Fig. 6 (Top row) illustrates a zoomed-in view of the front dynamics for a representative 5,000:1:1 COD/M204/TBP FROMP simulation and compares the numerical results to the experimentally acquired imagery (Bottom row) from Paul et al. (12). Similar to Fig. 5B, one observes the distinct formation of spin mode dynamics transversing back and forth across the resin domain in good qualitative agreement with experiments and published literature. We refer the reader to SI Appendix, Fig. S6 for more complex multihead spin mode dynamics. A formal discussion on the effect of reactor geometry on the resulting frontal polymerization spin modes is accessible in the work of Pojman et al. (27).

Fig. 6.

Simulated (Top) versus experimental (Bottom) spin mode front dynamics during the frontal polymerization of a COD/M204/TBP resin system, demonstrating the capacity of the mechanism-based FROMP model to qualitatively reproduce the dynamics of emergent behavior in dissipative FROMP systems. Experimental images reproduced with permission from Paul et al. (12).

Toward synergistic computational and experimental efforts for accelerated design of architected polymers, these results demonstrate the utility of our mechanism-based model for rapid exploration of the chemical design space, linking thermochemical processes grounded in reaction kinetics at the molecular scale to the complex emergent behavior at the macroscale polymer architecture.

Beyond M204–A Demonstration.

To further illustrate the ability to control emergent behavior in FROMP systems by design of the ruthenium initiator, we perform 2D simulations of the previous COD resin system in combination with a distinct ruthenium initiator species, MX,^# whose ability to readily dissociate the inhibitory phosphorus-based ligand is diminished. To achieve this, we preserve all the parameters to our mechanism-based model unchanged and prescribe a lower $\mathrm{\Delta }{S}^{\text{o}}$ from 51 to 47.6 Cal/(mol$·$K) in line with our discussion from On the Origin of Emergent Behavior. Fig. 7A illustrates contours of the maximum temperature, $T_{\text{max}}$, acquired throughout the polymerization process at the conclusion of the simulations for [2,500 to 10,000]:1:1 COD/MX/TBP resin formulations with variation in the initial resin temperature, $T_0$. By comparing Figs. 5A and 7A, a widening in the initial resin temperature for emergent behavior across all monomer-to-initiator resin equivalents is observed. Contrary to M204, pronounced patterns (i.e., distinct high-temperature traces) are numerically acquired for initial resin temperatures as high as $T_0$ = 23 ^°C for a 10,000:1:1 resin formulation, $T_0$ = 28 ^°C for a 5,000:1:1 resin formulation, and $T_0$ = 33 ^°C for a 2,500:1:1 resin formulation. This behavior is expected owing to the more inhibitory nature of the MX ruthenium initiator compared to M204. Moreover, it illustrates how material design capabilities by virtue of the resin processing conditions can be significantly expanded through careful design of the initiator chemistry and pertinent dissociation thermodynamics.

Fig. 7.

(A) Contours of the maximum temperature extracted at the end of the FROMP simulations for X:1:1 COD/MX/TBP resin formulations. Legend boundaries are prescribed in the [$T_0$, $T_{\text{max}}$ + 40 ^°C] range. The ruthenium initiator species, MX, reproduces the emergent behavior observed in COD/M207/TBP resin systems. (B) Experimental map of the initial resin temperature ($T_0$) as a function of the monomer-to-initiator loading ratio for a X:1:1 COD/M207/TBP resin formulation. Formulations resulting in emergent behavior are indicated by green squares, while those resulting in stable planar fronts are shown in red. The stronger inhibitory nature of the M207 initiator leads to a significant expansion of the initial resin temperature range supporting emergent behavior, marked in light green. Maximum initial resin temperatures for COD/M204/TBP resin formulations beyond which spontaneous patterns cease are represented by shaded green circles.

Experimentally, it can be demonstrated that the MX ruthenium initiator qualitatively reproduces the emergent behavior of a COD/M207/TBP resin system.^‖ Fig. 7B illustrates the experimental map of emergent behavior as a function of the initial resin temperature for different monomer-to-initiator loading equivalents of a X:1:1 COD/M207/TBP resin system. Here, M207 formulations conducive to emergent behavior are illustrated as green squares, while those resulting in stable planar fronts in red. We refer the reader to SI Appendix, Fig. S2 for experimental images of the fully cured COD/M207/TBP resins collected at 24-h postcure for different initial resin temperatures. Marked in light-green color, a significant widening in the original M204 envelope for emergent behavior (shaded region) is experimentally observed as the phenyl, Ph-ligand, of the alkylidene R¹-group for an M204 initiator is substituted for a 3-methyl-2-butenylidene ligand in M207. Concretely, distinct spontaneous patterns manifest for initial resin temperatures upward of $T_0$ = 25 ^°C for a 10,000 and 5,000:1:1 resin formulation, and $T_0$ = 30 ^°C for a 2,500:1:1 resin formulation. These results are in good qualitative agreement with numerical simulations, Fig. 7A, and corroborate similar reports in the literature by Paul et al. (12) on the role of the alkylidene ligand on the propensity for emergent behavior to manifest in FROMP systems.

Discussion–Summary and Outlook

Patterns are ubiquitous in nature, from the molecular arrangements in crystals and snowflakes, to the complex morphogenic growth for structural development in biological systems and the formation of galaxies. Once obscure, the governing laws that dictate emergent behavior across numerous physics and chemo-biological systems are now better understood by the principles of nonequilibrium thermodynamics and symmetry-breaking events governed by the competing reaction-transport phenomena. Coupled in a feedback loop, dissipative reaction-transport processes regulate the consumption and replenishment of key ingredients at the smaller material scale to dictate in turn the autocatalytic system response at the macroscale (4). The self-regulating nature of emergent behavior across physics, developmental biology, and engineering systems has created avenues for design of materials with self-organizational properties.

A distinct avenue, frontal polymerization by virtue of ring-opening metathesis and precise chemistry control has emerged as a promising approach for manufacturing synthetic polymers with programmable spatial properties (e.g., stiffness, toughness). However, the ability to harness emergent behavior through FROMP requires a fundamental understanding of the inherent link between the resin thermochemistry, reaction-transport processes, and the final polymer product. State-of-the-art FROMP models are phenomenological in nature and constructed on a single propagation step. As a result, they fail to account for the complex multistep FROMP dynamics, particularly the critical influence of resin chemistry—such as variations in resin formulation or changes to initiator chemistry—on macroscopic observables like front velocity and front behavior (e.g., planar uniform vs. nonuniform dynamics).

This work presented an integrated computational and experimental framework, employing a mechanism-based FROMP model to uncover the underlying physicochemical principles governing emergent behavior in synthetic polymers. The concept of equilibrium-gated pattern formation in FROMP systems was introduced, elucidating how localized near-equilibrium inhibition dynamics—associated with the activation of a dormant initiator—couple with the far-from-equilibrium metathesis reactions to dictate nontrivial emergent behavior at the macroscale. At the foundation of this concept is the occurrence of a localized inhibition-dominated regime which, by thermally modulating the supply of the active initiator to the olefin metathesis cycle, disrupts the fine balance between the FROMP kinetics, exothermicity, and heat transport into the unreacted monomer. Emergence of such a regime by variations in the resin chemistry, either by increase in the inhibitor concentration or tuning the initiator, was demonstrated in good agreement with both in-house experiments and literature (12).

On the basis of these findings, a roadmap for modifications to the initiator to achieve programmable macroscale properties is proposed. Thermodynamically, temperature-regulated inhibition dynamics associated with the ligand dissociation in N-heterocyclic ruthenium alkylidenes depend on both the molecular environment and the ligand nature, see ref. 30. Subtle influences could arise from:

1. Electronic Properties. Electronic properties of ligand substituents around the ruthenium center play a pivotal role in governing the dissociation equilibrium by modulating the metal-ligand bond strength. Electron-donating ligands, such as the N-heterocyclic carbenes (NHCs) and certain alkylidene substituents (i.e., alkyl), destabilize the metal center through electron density donation. This destabilization often increases the propensity for ligand dissociation, leading to larger equilibrium constants ($K_{\text{eq}}$). Conversely, less electron-donating ligands (i.e., certain halides) stabilize the metal-ligand bond, hindering dissociation. For example, small electronically neutral groups (e.g., H) are less effective at labilizing the phosphine ligand. Similarly, the choice of halide ligand (-X) impacts dissociation, with larger, more electron-donating halides or alkoxides promoting dissociation via steric and electronic effects. Together, these variations in electronic properties fine-tune the dissociation equilibrium, subtly shaping the enthalpic and entropic contributions that govern the catalytic system’s thermochemical behavior.

2. Steric Environment. Large bulky ligands induce steric strain around the metal center, weakening the metal-ligand bond and making dissociation less endothermic. Nevertheless, a tightly bound ligand is often highly constrained in the bound state, whereas weaker binding permits looser ligand conformations even prior to dissociation. This raises the question of whether sterically congested ligands experience a smaller entropy change upon dissociation, potentially compensating for their reduced endothermicity. The dual role of steric effects—modifying both the enthalpy and entropy of dissociation—highlights their complex thermodynamic impact.

3. Inhibitory Ligand Flexibility and Size. Rigid ligands impose a fixed spatial arrangement around the metal center, reducing the extent of favorable entropy gain upon dissociation by restricting conformational freedom. In contrast, flexible ligands or large bulky substituents experience a significant increase in conformational freedom upon dissociation. This unleashing of conformational degrees of freedom (i.e., translational and rotational motion), contributes to a greater entropy gain for large ligands compared to smaller flexible ones. Smaller ligands, though typically more flexible when bound, provide a smaller contribution to the overall entropy gain upon dissociation, as much of their conformational freedom is already realized in the bound state.

4. Medium Effects (e.g., viscosity). As polymerization progresses, the environment undergoes significant changes, particularly in viscosity. The increasing viscosity of the reaction medium hinders the molecular motion of the dissociated ligand. This restriction curbs entropic gains that would otherwise arise from dissociation in a less viscous environment. These effects highlight the dynamic interplay between the evolving polymer matrix and the thermodynamics of ligand dissociation during the course of the reaction.

Acting individually or in combination, the aforementioned factors subtly modulate the thermodynamics of ligand dissociation, which govern the inhibition dynamics. As such, a thorough examination of the thermodynamics of ligand dissociation by screening of distinct ligand substituents with varying steric and electronic properties can provide valuable insights into how these subtle molecular influences manifest in the macroscopic system response. Given the large library of unexplored initiator and inhibitor chemistries capable of frontally polymerizing, this insight unlocks avenues for design of materials with programmable spatial properties by precise modulation of the local inhibition dynamics and transport properties at the molecular level.

When combined with density functional theory and machine learning for rapid material screening, the proposed roadmap would enable integrated computational platforms that connect molecular-level kinetic and thermodynamic analysis with continuum-level simulations. These platforms can predict emergent behavior based on resin chemistry and processing conditions, facilitating informed and efficient material design (e.g., inverse design). Efforts of the nature proposed are at their inception, yet steadily receiving more traction (40). Notable advancements in this direction involve the successful deployment of virtual databases for molecular description of large repositories of organophosphorus ligands [i.e., Kraken (41)]. Building on these initiatives, we are actively working toward an integrated, multiscale computational platform that links ab-initio molecular screening with continuum-level inputs to enable rapid, computation-driven materials design (SI Appendix, Density Functional Theory (DFT) Computations). Achieving this can unlock opportunities for design and manufacturing of architected synthetic polymers with superior self-organizing and adaptive capabilities over the more conventional additive manufacturing platforms.

Materials and Methods

Finite Element Analysis.

The fully coupled system of Eqs. 1–5 is numerically solved using the finite element method (FEM) through the development of both a 1-D and 2-D staggered solver discretized with continuous first-order Lagrange elements using the open-source FEniCS computing platform (42). All material properties serving as input to the FEM are reported in Table 1. To numerically solve for the concentration degrees of freedom, ([C$(x,y,t)$], $\alpha (x,y,t)$), an explicit Euler scheme with a sufficiently small time discretization for numerical accuracy is utilized. Upon casting Eq. 5 into a linear variational problem, the partial differential equation governing heat diffusion is implicitly solved for the temperature field, $T(x,y,t)$, using an iterative conjugate-gradient Krylov solver.

A key challenge associated with the FROMP modeling is the need to capture the sharp gradients in temperature and degree of cure present in the moving front. The ability to resolve such sharp gradients requires a highly refined spatial discretization of the simulation domain. On this note, a uniform mesh with a sufficiently small element size, $h_e$ = 2 $\mathrm{\mu }$m, for a 2-D simulation domain of length, $l$ = 10 mm and width, $w$ = 2 mm, is employed for our meshing needs. The fully coupled system of equations is supplemented with the following initial conditions, $\alpha (x,y,0)=0.01$, [C$(x,y,0)$] = 0, $T(x,y,0)=T_0$, for a starting ([M₀], [II₀], [${\text{PR}}_3^0$]) monomer/initiator/inhibitor resin composition to systematically replicate the resin chemistry and processing conditions from experiments, see SI Appendix.

To initiate FROMP, a trigger temperature, T_trig = 160 °C, is applied for a short period of time, $t\in [0,t_{\text{trig}}]$, at the left edge ($x=0,y\in [0,w]$). A trigger time of 5 s is applied for 1D simulations, and 8s for 2D ones. Past t = t_trig, the left boundary is insulated. Adiabatic conditions are imposed at the remaining boundaries throughout the simulation, motivated by our intent to isolate on the chemical origins of emergent behavior in dissipative frontally polymerized systems, decoupled from secondary effects associated with heat losses. We refer the reader to SI Appendix, Fig. S7 for additional numerical simulations accounting for boundary heat losses to demonstrate their effect on emergent behavior. All data visualization and postprocessing is performed through ParaView, an open source postprocessing visualization engine.

Experiments—Materials.

1,5-Cyclooctadiene and tributylphosphite, P(OBu)₃ (TBP) were purchased from TCI Chemicals. 5-ethylidene-2-norbornene (ENB), Grubbs 2nd generation initiator (M204), and Grubbs initiator M207 (M207) were purchased from Millipore Sigma. All materials were used as received unless otherwise stated.

Frontal Polymerization.

COD monomer was filtered through a plug of basic alumina before use to remove the stabilizer and any oxidized species. 97:3%-volume COD:ENB mixtures were prepared in batch by measuring 97 mL COD in a graduated cylinder and adding 3 mL ENB via a syringe. For briefness, we describe here the methodology for the synthesis of a 2,500:1:1 COD-ENB:M204:TBP resin formulation. Tabulated monomer/initiator/inhibitor quantities for synthesizing the remaining resin formulations in this work are delegated to SI Appendix, Tables S2 and S3, following the same herein procedure.

To prepare a 2,500:1:1 COD-ENB:M204:TBP resin formulation, 2.0 $\mathrm{\mu }$L (0.0074 mmol) of TBP was added via a 10 $\mathrm{\mu }$L microsyringe to a vial containing 2.00 g (18.4 mmol) of premixed 97:3%-volume COD:ENB. The monomer and inhibitor mixture was added to a vial containing 6.25 mg (0.0074 mmol) M204. The resulting mixture was sonicated in an ultrasonic water bath kept below 18 ^°C for three minutes or until all the initiator had dissolved. The homogeneous resin was used immediately or stored at 0 ^°C before being transferred to a test tube or mold for FROMP experiments.

Front Speed Measurements.

Approximately 2 g of the synthesized resin was pipetted into a 10 mm diameter glass test tube. Front speed measurements were performed in triplicate, mixing resin separately for each trial. For formulations where mixing 2 g of resin required measuring less than 0.75 mg of initiator (i.e., 40,000:1:1), a single batch of 6 g of FROMP resin was mixed and used for all three trials. The initial resin temperature, $T_0$, was measured using a handheld Omega HH806AW thermocouple reader equipped with a K-type thermocouple. The test tube was cooled or heated as necessary inside a temperature-controlled oven until a T₀ = 23.0 °C ± 0.5 °C readout was recorded.

FROMP was initiated by running a 4 A direct current through a section of a 24 gage Kanthal A-1 ground resistance wire (TEMco, SKU$\#$RW0250), dipped at the top surface of the liquid resin, $\sim$1 cm in depth. The current was increased to 5 A if a front failed to form and shut off as soon as a front was observed. Videos of the downward-propagating polymerization front were captured at 25 FPS using a Canon EOS R5 camera, equipped with a Canon EF-100 mm 1:2.8 macrolens or iPhone 15. Front speed measurements were determined from video recordings using either the open-source physics software package Tracker® or a custom-built Python algorithm based on the OpenCV library. Specimens were subsequently allowed to cool to room temperature, left to crystallize overnight, and examined the following day to identify traces of pattern formation. Images were captured using an Epson V500 Photo flatbed scanner.

Rectangular Specimen Preparation.

Rectangular molds were constructed by clamping a U-shaped 1/4 inch thick high-temperature silicone rubber gasket (McMaster-Carr, #1460N26) between a pair of 1 mm thick standard glass slides, spanning 10 mm in width (VWR, #48300-026) (12). Approximately 3 mL of chilled FROMP resin was pipetted into the mold inside a custom-built environmental chamber preheated to an ambient temperature 5 °C warmer than the target initial resin temperature, $T_0$, to minimize the in-plane heat losses during FROMP. The initial resin temperature was measured until the target temperature rounded to the closest whole degree was reached. FROMP was initiated by running 4 A of direct current through a section of a 24 gage Kanthal A-1 resistance wire, dipped just into the top $\sim$1 cm of the resin. The power supply was shut off once the front had propagated midway down the resin. Fully polymerized samples were allowed to cool to room temperature, left to polymerize overnight, and inspected the following day for pattern formation. Images were captured using an Epson V500 Photo flatbed scanner.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.