Anisotropic frontal polymerization in a model resin–copper composite

Frontal polymerization (FP) that involves a self-propagating exothermic reaction wave triggered by local thermal stimuli was first investigated in the 1970s^1,2 and keeps attracting attention from the research community.^3–6 It has been realized with multiple polymer chemistries, including epoxy curing,^7–10 free-radical polymerization,^11,12 and ring-opening metathesis polymerization,^13,14 leading to broad applications, such as cure-on-demand adhesives,¹⁵ deep-eutectic solvents,¹⁶ and hydrogels.^17,18 FP has been proposed as a rapid, efficient, and environmentally friendly approach to fabricate polymeric parts and composites¹⁹ compared to the traditional methodologies based on bulk polymerization achieved with complex pressure and temperature loading cycles, which are time- and energy-consuming.²⁰ 

In FP, the characteristic features of front velocity and front temperature are critical to the properties and morphology of the fabricated polymeric part^13,21–23 and the efficiency of the FP-based manufacturing.²⁴ There are multiple internal and external factors that can influence the front velocity and the front temperature by leveraging the competition between the reaction and diffusion of the system,²⁵ such as the composition of the resin,^26,27 the cure kinetics,^28–30 the pre-cure state,³¹ fluid motion,³² operating temperature,^33,34 pressure,^4,35 gravity,³⁶ and the introduction of a second phase with energy absorption^29,37,38 or thermal conduction functionalities.^39,40 Previous studies have reported that thermally conductive elements aligned with the FP can enhance the front velocity via heat exchange.³³ However, the anisotropic thermal properties associated with systems composed of aligned conductive elements embedded in a resin can lead to orientation-dependent reaction–diffusion mechanisms and FP characteristic features, which have not been systematically investigated.

In the present work, we study the FP of 1,6-hexanediol diacrylate (HDDA) with embedded parallel copper strips, which form a thermally anisotropic system. Experiments and multiphysics numerical analyses demonstrate that front propagation is orientation-dependent due to the different front–metal strip interaction mechanisms. The front shape and propagation speed depend on the thickness of the copper strips. In order to understand the front–metal strip interaction mechanisms, a parametric study on the system size is carried out numerically for front propagation in parallel and perpendicular to the aligned copper strips. These results suggest that the front velocity in both directions increases as the system size decreases and approaches the analytical prediction for homogenized systems. Based on that study of the effect of the system size, a two-dimensional (2D) homogenized model for frontal polymerization in an aligned metal–polymer system is proposed.

This paper is organized as follows: Secs. II and III, respectively, introduce the experimental and computational methods. Section IV first discusses FP in resin–copper anisotropic systems in both experiments and numerical simulations, with emphasis on the effect of the copper strip thickness. In order to demonstrate the orientation-dependent reaction–diffusion mechanisms, we then perform a parametric study of the effect of the strip thickness and the system size for 1D FP in directions parallel and perpendicular to the copper strips, based on which a two-dimensional 2D homogenized model for frontal polymerization in an aligned metal–polymer system is proposed.

The monomer used in the experiments was 1,6-hexanediol diacrylate (HDDA), which was purchased from Allnex (Alpharetta, GA). The initiator used was 1,1,-bis(tert-butylperoxy)-3,3,5-trimethylcyclohexane, referred to as Luperox 231$®$⁠, purchased from Arkema (Pasadena, TX). The two filler materials used in these experiments were Polygloss 90$®$ and Aerosil 200 purchased from KaMin LLC (Macon, GA) and Aerosil$®$ (Mobile, AL), respectively. Aerosil 200 is a fumed silica with a surface area of $200±25$ (⁠$m2/g$⁠) measured by the Brunauer, Emmett, and Teller (BET) method. The copper sheets with a purity of 99.9$%$ were purchased from Chudeng LTD (Changzhou, China), and the impurities are mainly composed of zinc and tin. All materials were used as received.

To prepare the formulation, all materials were weighed to proper amounts and mixed manually until a homogeneous mixture was obtained. The amounts used for each material are reported in parts per hundred resin (PHR), with 1 PHR being the equivalent of 1 part by weight additive or initiator per 100 parts by weight of resin. The resin in all experiments presented hereafter is HDDA. After testing multiple formulations, the best performing filler loading was determined to be 82.5 PHR kaolin and 6 PHR fumed silica in a 10 PHR HDDA/Luperox solution. This mixture was used for all experiments.

Copper was incorporated into the FP systems by taking solid metal sheets and cutting them to the desired dimension of 5 cm $×$ 1 cm $×$ $dCu$⁠, where $dCu$ is the thickness of the sheets. The strips were then embedded into the resin-filler slabs. The correct positions of copper strips were measured with a standard ruler and ensured with the ruler notch when embedded into the resin. There were multiple thicknesses of metal strips applied in the experiments: 0.1, 0.5, 1, 2, and 3 mm.

For each experiment, a constant amount of the formulation was shaped into a slab using a wooden mold with dimensions of $10×5×1cm3$⁠. To initiate the frontal polymerization, an electric soldering iron was heated, and contact was made between the hot soldering iron and one edge of the formulation. This application of heat caused the decomposition of the initiator and the propagation of a polymer front. To track the velocity of the propagating front, a video camera was used to film the reaction with a cm scale in frame. Green phosphorescent powder was dusted over the surface of the formulation to aid in tracking the front, as shown in photos and videos of the experiments. The videos were then analyzed by recording the front stop time at the copper strip in seconds. In addition, the videos were also used to track the front position as a function of time to obtain a velocity in mm/s. Temperatures were recorded for some experiments to compare front temperatures of systems with varying metal strip thicknesses. A Type-K thermocouple was embedded in the middle of the formulation channel and was used in conjunction with a Go!Link USB sensor interface (Vernier) to record the temperature into a laptop through the LoggerLite software (Vernier). All experiments were run in triplicate for statistical analysis.

Figure 1(a) depicts the problem configuration in a 2D domain with the size chosen to be the same as the in-plane dimensions of the experimental setup, i.e., $lx=10$ cm and $ly=5$ cm. The numerical domain comprised two sub-domains corresponding to the HDDA-filler formulation (denoted by the subscript r) and metal strips (denoted by the subscript m). A high temperature was applied to a circular region in the HDDA sub-domain with a radius of 1 mm (marked by the yellow circle) to initiate the FP. Four copper strips were separated by $1cm$⁠, and the outer ones were 1 cm from the top and bottom boundaries. The FP in the HDDA was described by the following reaction–diffusion partial differential equations (PDEs) in terms of the temperature $T$ (in K) and degree of cure $α$ (unitless),

where $κr$ (in $Wm−1K−1$⁠), $Cp,r$ (in $Jkg−1K−1$⁠), $ρr$ (in $kgm−3$⁠), and $Hr$ (in $Jkg−1$⁠), respectively, stand for the thermal conductivity, the heat capacity, the density, and the heat of reaction of the resin. The variable $t$ (in $s$⁠) denotes time. The second relationship is a cure kinetics expression approximating the chemical front based on the $n$th-order model, which was adopted to reproduce free-radical FP in previous works.^24,33 Here, $A$ (in $s−1$⁠) and $E$ in (⁠$Jmol−1$⁠), respectively, denote the pre-exponential factor and the activation energy of the Arrhenius relationship, while $R(=8.314Jmol−1K−1$⁠) is the ideal gas constant.

The transient thermal conduction in the metal domain is described by

The material properties of HDDA resin-filler formulation and copper strips and the cure kinetics parameters adopted in this work to approximate the polymerization are summarized in Tables I and II, respectively.

These governing partial differential equations are accompanied by the following initial and boundary conditions:

where $T0$ (⁠$20°C$⁠), $α0$ (0.01), $Ttrig$ (⁠$250°C$⁠), and $ttrig$ (5 s) are the initial temperature, the initial degree of cure, the triggering temperature, and the triggering time, respectively. Adiabatic boundary conditions are applied to all boundaries. As indicated in Fig. 1(a), only a quarter of the domain was simulated due to the symmetry of the problem. A four-node rectangular mesh with a typical size of 0.2 mm was adopted, and the unadapted numerical system generally contains $∼$30 000 elements. The time step was set as 0.01 s.

The Multiphysics Object-Oriented Simulation Environment (MOOSE),⁴² an open source C++ finite element solver with adjustable mesh adaptivity needed to capture the sharp $T$ and $α$ gradients in the vicinity of the polymerizing front, is adopted in this study. A mesh adaptivity h-level of 3 was applied with respect to both $T$ and $α$⁠. Hereafter, the position of the polymerizing front is identified by the location where $α$ equals 0.5.

The cure kinetics parameters (Table II) and thermal properties (Table I) of the resin-filler formulation lead to a front velocity of 1.67 mm/s and a front temperature of 228.3 $°$C, which are consistent with the experimental measurements of 1.70 mm/s and 225 $°$C.

Figure 1(b) presents several experimental optical images and numerical snapshots of FP in the resin–metal strip system with the thickness of the strips $dCu$ equal to 0.1 mm. In experiments, the location of the polymerization front was identified by the green phosphorescent powder. At first, the front propagates along the direction of copper strips starting at the trigger location (yellow circle) when $t=10$ s and is initially confined between the inner copper strips. At $t=20$ s, the front has crossed the inner strips but is delayed again at the outer strips, while the propagation along with the copper strips proceeds. At $t=35$ s, the front passes the outer strips and reaches the wooden mold. During the propagation, the front maintains an oval-like shape, and the “spikes” observed on the front close to copper strips are caused by the heat exchange between the copper strips and the resin.²⁴ The simulation snapshots presented in Fig. 1(b) show the temperature $T$ and degree-of-cure $α$ fields at $t=10$⁠, 19, and 26 s. The numerically simulated front profiles are highly consistent with those observed in the experiment.

Figure 1(c) compares the front propagation in multiple directions, where $xf$ is the front displacement with respect to the trigger point and $θFP$ is the angle between the front propagation direction and the $x$ axis. In the beginning, $xf$ is identical in all directions, indicating a circular front shape. Deviations in $xf$ occur when the upward moving front (⁠$θFP=90°$⁠, red curve) first reaches the inner copper strip located 5 mm from the trigger point (lower red dashed line), leading to a plateau in the $xf$–$t$ curve indicative of a delay in the front propagation [Fig. 1(b)]. Similar plateaus are also observed with $θFP=60°$⁠, $45°$⁠, and $30°$ in sequence. At the end of the plateau, the FP on the other side of the strip is initiated, and the front is delayed again at the outer copper strip at 15 mm from the trigger point (upper red dashed line). In the direction of the metal strips (⁠$θFP=0°$⁠, black curve), the front propagates in an uninterrupted way and the increasing slope of the $xf$–$t$ curve points to an acceleration of the front.

Figures 1(d) and 1(e), respectively, present the evolutions of temperature $T$ and degree-of-cure $α$ profiles along with the directions $θFP=0°$ and $90°$⁠. In the direction $θFP=0°$ [Fig. 1(d)], the resin temperature (solid lines) right at the front (where $α=0.5$⁠) gradually increases and exceeds the theoretical front temperature in the neat resin at the prescribed initial temperature (⁠$T0=20°C$⁠) $Tf=228°C$⁠.³⁹ This observation can be understood by considering the evolution of the temperature profiles of the inner copper wire (dashed lines). The temperature is higher than the resin temperature ahead of the front due to a higher thermal diffusivity (Table I), and the warmer copper wire heats up the HDDA, which also enhances the front velocity.

In the direction $θFP=90°$ [Fig. 1(e)], the reaction heat diffuses into the copper strip (marked by the thin red rectangular region), leading to more spread-out $T$ and $α$ profiles than $θFP=0°$⁠. The thermal diffusion from the resin to the metal increases the copper temperature and initiates the front propagation on the other side of the strip as the copper temperature reaches $∼140°C$ (see Fig. S1 in the supplementary material) between $t=10$ and 12 s. Discussion about the effects of thickness and thermal properties of the metal strip on the propagation of the polymerization front perpendicular to the metal strips can be found in Figs. S2 and S3 of the supplementary material.

Figure 2(a) presents the numerical snapshots of $T$ and $α$ fields with the copper strip thickness $dCu=0.0$ (without copper strips), 0.2, 0.6, and 1.0 mm. Figure 2(b) shows the corresponding front trajectories in the parallel (⁠$xf,∥$⁠, $θFP=0°$⁠) and perpendicular (⁠$xf,⊥$⁠, $θFP=90°$⁠) directions. As expected, in the absence of copper strips (⁠$dCu=0.0$ mm), the problem is isotropic and the front adopts a circular shape. Both $xf,∥$ and $xf,⊥$ increase linearly with time $t$⁠, indicating a constant front velocity. When $dCu=0.2$ mm (black curves), the front propagation in the parallel direction is accelerated while the FP in the perpendicular direction is delayed, as alluded to Figs. 1(c) and 1(d). The orientation-dependent FP leads to an oval front shape. As the Cu strip thickness increases from 0.4 to 1 mm, the increasing delay in the upward motion of the front associated with the higher heat absorption by the metal strips leads to a different shape of the front at the time at which the front reaches the outer copper strips. The horizontal “hourglass” shape obtained for the thicker metal strips is also observed experimentally [Fig. 2(c)].

Figure 2(d) presents a comparison between experimental measurements and numerical predictions of the time $tFP$ needed to cure the entire rectangular specimen. $tFP$ is normalized by $tFP,0$⁠, the FP duration in the absence of copper strips, to highlight the effect of copper strip thickness $dCu$⁠. $tFP,0$ is 34.35 and 42.68 s in numerical simulations and experiments, respectively. The higher value of $tFP,0$ in experiments than that in simulations is caused by the heat loss to the surroundings, which is not considered in the numerical analyses. $tFP$ increases with a higher $dCu$⁠, primarily due to the slower front propagation in the perpendicular direction [$xf,⊥$ in Fig. 2(b)]. The experimental measurements are consistent with the numerical results.

Given the anisotropic front propagation properties presented in Secs. IV A and IV B, we investigate separately the one-dimensional (1D) propagation of the polymerization in the parallel and perpendicular directions to quantify the effect of copper thickness and system size.

The inset of Fig. 3(a) depicts the composite system for the case of front propagation perpendicular to the embedded metal strip. The system of interest consists of four repeating units with a length $dunit$ and a copper strip thickness $dCu$⁠. Four repeating units are sufficient for the front to reach a quasi-steady-state propagation. The volume fraction of the copper strips is $ϕ=dCu/dunit$⁠. The curves in Fig. 3(a) represent evolution of the front location $xf$ obtained for a unit size $dunit=10$ mm and for three values of the volume fraction $ϕ$⁠: 1$%$ (⁠$dCu=0.1$ mm), 5$%$ (⁠$dCu=0.5$ mm), and 10$%$ (⁠$dCu=1$ mm). As apparent in Fig. 3(a), the step-like features of the $xf$–$t$ curves are increasingly pronounced as $ϕ$ increases. The average front velocity $vf,⊥$ is evaluated as the average slope of the $xf$–$t$ curve.

The results of a parametric study for $dunit$ and $ϕ$ are presented in Fig. 3(b). For all $dunit$ values, $vf,⊥$ monotonously decreases with the increase of $ϕ$⁠. For a given value of $ϕ$⁠, $vf,⊥$ is higher in a smaller unit, as indicated by the black arrow. As the unit size $dunit$ is reduced to 2 mm, $dCu$ values corresponding to $ϕ<10%$ are less or equal to 0.2 mm, smaller than the thermal front width $Lθ$ of 0.79 mm (see Fig. S4 in the supplementary material). For very small values of $dCu$⁠, the front can get through the copper strips with negligible delay. In this case, $vf,⊥$ can be roughly approximated by a homogenized 1D model,

where the overbars denote the homogenized material properties according to the extended Rayleigh model⁴³ (dark blue dots) or the simple series model (light blue dots),

The Rayleigh model corresponds to the fiber-reinforced composites, and the corresponding $vf,⊥$ values are higher than those predicted by the series model, which represents through-thickness metal strip–resin composites and agrees better with the results of the discrete model. $vf,⊥$ can also be predicted with an analytical expression obtained from an asymptotic approximation of the thermal solution in the immediate vicinity of the front,⁴⁴ 

where $T^(ϵ)=T0+(1−α0−ϵ)ρrHr(1−ϕ)ρCp¯$ and $ϵ$ is between 0 and $1−α0$⁠.

The inset of Fig. 3(c) depicts the 1D numerical systems of FP in parallel to the copper strips. The $xf$–$t$ curves illustrate the front propagation with different $ϕ$ values. The front propagation is accelerated due to the heat exchange as indicated by the increasing slope, which converges to a value defined as the average front velocity $vf,∥$⁠.

Figure 3(d) presents $vf,∥$ as functions of $ϕ$ with multiple $dunit$⁠. The non-monotonic trends of $vf,∥$ vs $ϕ$ suggest a system-size-dependent competition between thermal diffusion and available heat of reaction.²⁴ However, for a given value of $ϕ$⁠, the results show an increase in $vf,∥$ with a decrease in $dunit$⁠, as was observed for $vf,⊥$ [Figs. 3(a) and 3(b)]. When $dunit=2$ mm, $vf,∥$ values obtained with the discrete model are close to those provided by the homogenized model [Eq. (4), with $κ⊥¯$ replaced by $κ∥¯$], where $κ∥¯$ can be determined by the parallel model,

The $vf,∥$ values yielded from the homogenized model can also be accurately predicted by the analytical expression in Eq. (6) with “$κ⊥¯$” replaced by “$κ∥¯$⁠.” The divergence observed for $ϕ>6%$ between the results obtained with the discrete model for $dunit=2$ mm and those provided with the homogenized model can be explained by the underestimation of the front temperature associated with the homogenized model [Figs. 4(a) and S5 in the supplementary material].

Based on the agreement between discrete and homogenized models presented in Sec. IV C for the front propagation parallel and perpendicular to the metal strips, we propose the following 2D homogenized anisotropic reaction–diffusion model:

where $κij¯$ is the thermal conductivity matrix of the anisotropic system. For the case when the metallic fibers aligned with the $x$-direction, $κij¯$ can be written as

In order to validate the model described by Eqs. (8) and (9), we compare the FP reproduced by this homogenized model and that simulated with the discrete model. Schematics of problem configurations are shown in the insets of Fig. 4(a). The size of the domain for both discrete and homogenized models was 10 (in $x$⁠,“$∥$”) $×5cm2$ (in $y$⁠,“$⊥$”). In the discrete model, 24 copper strips separated by 2 mm were embedded with the resin. The copper strip thickness $dCu$ was 20, 40, 80, and 120 $μm$⁠, respectively, leading to $ϕ=1%$⁠, $2%$⁠, $4%$⁠, and $6%$⁠. In the homogenized model, $κ∥$ was described by the series model in Eq. (5) to be consistent with the discrete model, while $κ⊥$ was determined with the parallel model in Eq. (7).

Figure 4(a) compares the snapshots of the temperature $T$ distribution at $t=10$ s in the discrete and homogenized models, showing great similarity for all values of the volume fraction. As $ϕ$ increases, the oval front shape is “stretched” in the parallel direction due to the variation in thermal anisotropy, which is captured by both models. Figure 4(b) presents the front location at various time $t$⁠, and the agreement between discrete model curves (D) and homogenized model curves (H) further validates the model described in Eq. (8). The front velocity and the temperature predicted by the discrete and homogenized models match better given a lower $ϕ$⁠.

The front shape in the anisotropic FP system at a time $t$ can be described by the instantaneous front displacement in the parallel and perpendicular directions, which can be regarded as the major and minor axes of the oval front. As demonstrated in Fig. 4(c), the ratio of front displacement in parallel and perpendicular direction $xf,∥/xf,⊥$ values at $t=10$ s predicted by both discrete (blue dots) and homogenized (red dots) models increases with $ϕ$⁠, which corresponds to the “stretched” oval-like shape in Fig. 4(a). This trend of $xf,∥/xf,⊥$ with $ϕ$ is highly consistent with the ratio of velocity in the parallel and perpendicular directions $vf,∥~/vf,⊥~$ analytically predicted by Eq. (6). Discussion about the front temperature $Tf$ predicted by the 2D homogenized model can be found in Fig. S5 of the supplementary material.

We have investigated frontal polymerization in thermally anisotropic systems composed of resin and copper strips with combined experimental, numerical, and analytical approaches. Both experiments and multiphysics numerical analyses demonstrate that the front propagation in thermally anisotropic systems is orientation-dependent, and the front shape depends on the thickness of the copper strips. The front propagation is decoupled into 1D front propagation in the directions parallel and perpendicular to the metal strips to understand the front–metal strip interaction mechanisms. The parametric study on the copper strip thickness and the unit size has revealed that the front velocity in both directions can be enhanced in smaller unit sizes and strip thickness. The front velocity can be approximated by 1D homogenized models as the strip thickness drops below the thermal front width, based on which a 2D homogenized model of FP in anisotropic systems is proposed and validated. The present work provides fundamental understandings of FP in thermally anisotropic systems and sheds light on the rapid, energy-efficient manufacturing of polymer composites enhanced by thermally conductive elements.

See the supplementary material for the details of perpendicular strip FP, bulk polymerization analyses, thermal and chemical front widths, and front temperature in the homogenized models.

This work was supported by the U.S. Air Force Office of Scientific Research through Award No. FA9550-20-1-0194 as part of the Center of Excellence in Self-healing and Morphogenic Manufacturing. The authors also acknowledge the support of the National Science Foundation under Grant No. 1933932 through the GOALI: Manufacturing USA: Energy Efficient Processing of Thermosetting Polymers and Composites. This work was also supported by the NSF EPSCoR-Louisiana Materials Design Alliance (LAMDA) program (Grant No. OIA-1946231) and the Smart Polymer REU (No. CHE1660009).

The authors declare no conflict of interests.

Y.G., S.L., and J.-Y.K. contributed equally to this work.

The data that support the findings of this study are available from the corresponding author upon reasonable request.