Structure and Flow-Viscosity of Filled-Polymer-Based 3D Printing Ink: Exploration through Coarse-Grained Molecular Dynamics

Abstract

The addition of nanofiller particles to a polymer matrix has long been known to enhance or modify the composite’s mechanical and rheological properties. However, quantitatively capturing such changes with molecular level simulations remains computationally challenging. Toward that goal, we performed coarse-grained molecular dynamics of a nanocomposite system at a fixed (25 vol %) filler loading under nonspecific, weak polymer–filler interactions representative of a broad class of technologically important materials. We report several interesting results, including: (1) the equilibrium chain-configuration remains Gaussian-like as in an unfilled melt; (2) smaller filler particles display a stronger tendency to cluster; (3) larger fillers act as plasticizers by reducing the entanglement density and accelerating the chain mobility; and (4) fillers enhance the tensile response modulus, with the effect being stronger for larger particles. We also simulate cluster breakup, yielding, and elongational flow under an applied time-linear tensile strain and study the flow viscosity as a function of filler-size and chain-length.

1. Introduction

Polymer nanocomposites (PNCs) represent a class of materials that consist of nanoparticles (NPs) dispersed in a polymer matrix. In recent years, there has been a renewed interest in the field of PNCs predominantly fueled by the potential to alter or enhance material properties, such as electrical conductivity, , thermal conductivity, , and rheological characteristics, − with applications in fields as diverse as ground/air transportation, , smart materials, shock-absorbing supports, athletic equipment, separation processes, drug delivery, catalysis, and food packaging. In particular, the tunable rheological properties of PNCs have made them particularly attractive for additive manufacturing (aka 3D printing), − where precise control over the rheological response of the printer’s “ink” across a range of applied conditions is critical.

Recent experimental advances showcase the rheological flexibility of PNCs where these materials enable extrusion-based additive manufacturing techniques such as FDM (fused deposition modeling) and direct ink writing (DIW). In particular, DIW inks must exhibit shear-thinning behavior so that the material flows readily under stress during extrusion. The ink formulation must also have sufficient yield stress and equilibrium storage modulus to maintain its shape after deposition. − Polydimethylsiloxane (PDMS)-based inks, often reinforced with fumed silica NPs, are widely used due to tunable viscoelastic properties that emerge from microscopic polymer–filler interactions. , Understanding and controlling these rheological properties are essential when seeking to optimize inks for both on-machine performance and the resulting quality of the 3D printed parts.

NPs in the polymer matrix tend to aggregate and affect the overall structure, dynamics, and mechanical response of the PNCs. There have been attempts to investigate the clustering behavior of NPs using techniques such as X-ray/neutron scattering, − electron microscopy, − and atomic force microscopy (AFM). Ramier et al. focused on studying the structure of NP aggregates, while Genix et al. explored polymer conformation in the presence of NP aggregates using scattering techniques. Zare and co-workers studied the formation of silica NP aggregates in polypropylene-based PNC systems using transmission electron microscopy to analyze samples ranging from 2.5 to 15 wt % filler. It was shown that smaller NPs exhibit a higher tendency to aggregate, leading to overall larger cluster sizes with increasing NP content. Further characterization of these aggregates using AFM by Mélé et al. revealed a heterogeneous size distribution. Such aggregates significantly affect the conformation of polymer chains, with techniques like small-angle neutron scattering capable of extracting such information.

The distribution of NPs in a polymer matrix alters the dynamics of both the polymers and NPs, consequently affecting the overall properties of the system. Theorists have developed models , that aim to predict and understand polymer diffusion, and different experimental − and simulation techniques − have been used to measure polymer dynamics in PNCs. The movement of polymer chains in PNCs is more inhibited than in neat polymer melts because the chains must diffuse around the impenetrable NPs. In an experimental study, the polymer diffusion in PNCs composed of entangled polystyrene and silica NPs was measured using elastic recoil detection. As the NP content increased from 0 to 50 vol %, a decrease in polymer diffusion was observed. This reduction in mobility was found to be more pronounced for longer polymer chains. These results are in excellent agreement with the entropic barrier model, , which relates the slowdown in polymer diffusion in the presence of NPs to entropy loss due to reduction in chain conformations. Other techniques have supported and expanded these insights. Nuclear magnetic resonance techniques such as pulsed gradient spin echo have been employed for measuring the translational diffusion of entangled polyethylene in zinc oxide-filled polyethylene PNCs. In another contribution, neutron spin echo experiments on poly­(ethylene-alt-propylene) (PEP) chains with silica NPs showed that this effect is not limited to chains directly touching the NPs; even chains in the bulk experience a collective slowdown. Complementing these experiments, a number of simulation works focused on understanding the dynamics of the polymers closer to the surface of the NP, different NP loading, and NP-polymer interactions have been reported. Another important phenomenon is the diffusion of NPs into aggregate formation, which can significantly affect the bulk properties of PNCs. This has prompted a number of theoretical models, − experiments, , and computational studies ,− on the dynamics of NPs within PNCs. An experimental study by Grabowski and Mukhopadhyay quantitatively measured the dynamics of gold NPs in entangled poly­(n-butyl methacrylate) (PBMA) polymer melts using fluctuation correlation spectroscopy. Their measured diffusion values for small gold NPs were higher than those predicted by the Stokes–Einstein (SE) equation. The SE model was found to be accurate only for NP sizes larger than the average radius of the gyration of the polymer chains.

Through the years, the mechanical properties of PNCs have been well-researched and documented. Studies have consistently shown that the addition of even small quantities of NPs (as low as 5 wt %) to the polymer matrix can lead to a substantial enhancement in the modulus of PNCs. Osman et al. investigated the shear effects in high-density polyethylene filled with calcite, revealing that the formation of agglomerates substantially increases the low shear-rate viscosity. However, at moderate shear rates, these agglomerates slowly break apart and result in shear thinning. Zhu et al. observed similar filler dissociation trends in 1,4-polybutadiene (PBD)/silica-based PNCs under intense oscillatory and step-shear stresses.

Many of the above properties are generically true for a wide class of polymers and do not depend on the atomistic constituent details. Thus, there have been a number of simulation studies on the structure, dynamics, and rheological properties of PNCs where polymers are represented at the bead-spring level, with coarse-grained interactions described by the well-traveled Kremer–Grest (KG) model. Moghimikheirabadi et al. studied the structure and dynamics of KG polymers in the presence of NPs with up to a 60 vol % filler. They performed structure factor analyses for both the polymer melts as a whole and for single chains separately. For the melts, they observed a peak in the low q region whose position remained relatively unchanged with different NP loadings, while individual chains in the presence of NPs behaved as if they were in an ideal melt state under equilibrium conditions. These authors also determined the mean square displacement (MSD) of polymers and NPs separately and found that the mobility of both the polymer chains and the NPs are a strong function of NP volume fraction with a monotonic decrease in mobility with increasing NP loading. Decreased polymer and NP mobility trends were also observed by Riggleman and co-workers under high NP loading, who concluded that segmental dynamics in PNCs strongly depends on the number of NPs adjacent to the monomers. Another study by the same group investigated the stress–strain properties of PNC melts and showed significantly higher yield stress and elastic modulus as compared to the neat NP system. A similar study by Hagita et al. showed that “aggregated” NPs produce higher stress values compared to “dispersed” NPs. A more exhaustive discussion of the literature on coarse-grained modeling for PNC systems has been captured in a number of recent review articles. −

However, despite extensive experimental and computational studies on PNCs, a systematic understanding of such structural, mechanical, and dynamical effects under weak, nonspecific polymer–filler interactions remains limited. Prior simulation efforts have largely focused on strong filler–polymer attractions or varying filler loadings, making it difficult to isolate purely entropic and energetic contributions. Moreover, only a few bead–spring investigations have explored how NP size and chain length jointly dictate clustering, entanglements, and mobility, and even fewer have examined extensional flow and cluster breakup under deformation, which are central to additive manufacturing ink performance.

In this work, we build on insights from prior studies and employ the KG bead–spring model to simulate large-scale systems (∼up to 10.9 million beads per system) to systematically study the dynamics of PNCs within the polymer melt, their clustering behavior, and their macroscopic mechanical and rheological properties as a function of polymer chain length and NP radius. In our simulations, we set the interaction strengths for all beads at 1k _B T, a parameter choice also employed in several other studies , in the literature. This choice of interactions differs from that for other systems, such as ionomers, where ≥1k _B T interaction strengths are employed to model specific chemical functionalities and associations within polymer melts. By maintaining a fixed filler loading of 25 vol % (that is industrially relevant for many applications) and employing uniformly weak interaction potentials while varying NP radius and polymer chain length, the intent of the present work was to isolate entropic effects and establish useful processing–structure–property relationships for filled-polymer inksan aspect not systematically addressed in previous studies.

2. Methods

Equilibrium and nonequilibrium molecular dynamics (MD) simulations were conducted using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) on models of bulk PNC systems represented within periodic simulation cells. The polymer chains were modeled as coarse-grained KG bead springs, while the NPs were modeled as hollow spheres with a dense packing of surface beads, as described below. Structural models of PNCs were created by initial random placement of NPs within an empty simulation box using PACKMOL, as shown in Figure a,b. This was followed by “growing” polymer chains within the simulation cell volume with a biased random walk approach. Initial configurations used an ultralow polymer density ρ ≈ 0.01 and polymer-NP collision detection to ensure zero overlap of polymer beads with the NPs, as illustrated in Figure c. The NPs, which are significantly larger than the beads comprising the polymer strands, were represented by a collection of beads arranged as a hollow sphere and treated as rigid bodies using the “fix rigid” command in LAMMPS, following an approach similar to that used in Moghimikheirabadi, Riggleman, and Karnes studies. More specifically, each NP of radius r _NP consists of (int) 4πr _NP ² beads evenly distributed on the surface of the sphere with one additional particle located at the center of the NP and defined as a different LAMMPS “type” to facilitate analyses. To relax the polymer chains around the NPs and escape any high energy initial configurations, early MD steps implement a soft repulsive interaction potential using the “pair_style soft” command in LAMMPS.

1.

Snapshot of (a) the distribution of NPs within a simulation box with low density. (b) is the zoom image of a NP size r _f = 10σ. (c) is the snapshot following the introduction of polymer chains l = 20 beads per chain into the simulation box in the presence of NPs. (d) is the state after compressing the box to the polymer density of 0.87.

Each NP was designed with a central Lennard-Jones (LJ) site, and the NP surface comprised of approximately 4πr _f ² LJ interaction sites, corresponding to a near-uniform and dense (i.e., impenetrable) surface bead distribution. The NP radius (r _f) was defined by the distance from the center LJ bead to the surface LJ sites. NPs of three different radii, i.e., r _f = 5σ, 10σ, and 20σ, were considered in this work, with the number of surface beads per NP being 314, 1256, and 5024, respectively. Each PNC system was modeled with 110 NPs of uniform size and a fixed NP volume fraction of 25 vol %. Across all systems studied, the resulting net number of beads (polymer + NP beads) ranged from 234,540 to 10,913,140, with the corresponding simulation box lengths (l _x = l _y = l _z ) ranging from 67.4σ to 251.2σ.

We studied polymer chains of different lengths, l = 20, 100, and 200 beads per chain. As in the standard KG model, each polymer chain was represented as a sequence of beads, with each bead physically representing a Kuhn monomer, and consecutive beads connected by a finitely extensible nonlinear elastic spring with an average bond length of approximately b ≈ 0.96σ. Additionally, all beads (polymer beads and NP surface beads) within a cutoff distance of 2.5σ experience a pairwise nonbond interaction of the LJ form (with 1–2, 1–3, and 1–4 nonbond exclusion for intrachain neighbors).

Following standard convention, LJ-reduced units were employed for all quantities. Thus, mass was expressed in units of mass m of each bead, length was expressed in units of LJ parameter σ, energy expressed in units of LJ parameter ε, time expressed in units of $\tau =\sqrt{m{\sigma }^2/\epsilon }$ , temperature expressed in units of k _B T/ε, density expressed in units of m/σ³, and pressure expressed in units of ε/σ³. As mentioned in the Introduction, the purpose of this study was to probe a PNC system with weak, nonspecific interaction between the polymer and NPs. Thus, all nonbond interactions were set equal to 1k _B T, i.e., ε_{polymer–polymer} = ε_{polymer–filler} = ε_{filler–filler} = 1.0 (in reduced units). And to ensure numerical stability all simulations employed a small time-step Δt = 0.001.

We select LJ-reduced units as opposed to SI or experimental units parametrized for a particular polymer system since this approach permits the discussion of general trends in polymer behavior. While conversion from LJ to experimental values is not straightforward, selection of values for σ, ε and m, based on a polymer of interest, may facilitate this conversion for the purpose of discussion, but we remind that reader that the bead–spring model is not parametrized for a particular polymer chemistry and that results of this conversion may be misleading.

Throughout this study, the filler loading percentage ϕ_NPs was kept fixed at 25 vol %, a commonly employed filler loading level in many industrial applications. , The values of r _f/R _g in this study range from 0.7 to 9.8, covering all three regimes (r _f/R _g < 1, r _f/R _g ≈ 1, and r _f/R _g > 1). Each PNC system was subjected to an initial high temperature (T = 10.0) simulation for approximately 1.0 × 10⁴ τ (equivalent to 1.0 × 10⁷ time-steps) to ensure rapid chain relaxation and effective mixing of the polymeric chains and the NPs. This was followed by cooling to T = 1.0 at a cooling rate Γ̇ = ΔT/Δt ≈ 1.0 × 10^–3 during which the simulation box contracted to the final polymer density ρ ≈ 0.87, as illustrated in Figure d. An equilibration-run of length 1.0 × 10⁵ τ was then performed, with the state of equilibrium monitored through various structural indicators, such as average chain end-to-end distance, radius of gyration, and radial distribution function. During this phase the NPs diffused slowly and displayed a tendency to aggregate, leading to the formation of small clusters within simulation time-range of t = 0.1 to 10 × 10² τ, with the cluster-size distribution stabilizing at longer times. The working definition of an NP cluster is specified more clearly in Section .

To determine whether this equilibration trajectory is sufficiently long for the polymers in this study, we consider the distance between polymer entanglements (entanglement length, l _e), the time for a polymer segment to diffuse a length l _e (entanglement time, τ_e), and the time required for the entire polymer strand to escape its confining tube (reptation time, τ_rep). For linear and fully flexible bead–spring polymers like those in this study, l _e ∼ 84 , and τ_e ∼ 10⁴ τ. These values set a chain-length upper bound of l ∼ 200 such that the characteristic reptation times ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{p}}\sim {\tau }_{\mathrm{e}}{(l/l_{\mathrm{e}})}^3$ remain comfortably within the total equilibration time used in these simulations.

Following equilibration, an NVT production run of 1.0 × 10⁵ τ was conducted for each system. Additionally, for comparison, we simulated neat polymer melts of density ρ ≈ 0.87 and temperature T = 1. The initial configurations and equilibration procedures for these neat systems followed the standard methods outlined in Auhl et al. The final equilibrium analyses were performed by averaging over the last 5.0 × 10⁴ τ of the production run.

To probe the rheological response, each PNC system was subjected to uniaxial tension, with elongation increasing at a constant engineering strain rate $\mathrm{ϵ̇}$ = 10^–3. Meanwhile, constant pressure (p = 0.01) was maintained in the orthogonal directions to ensure maintenance of system density via NPT dynamics. Throughout the simulation, we monitored the stress in the direction of elongation to analyze the strain–stress and flow-viscosity characteristics of PNCs.

Although performing KG simulations in dimensionless units is a widely accepted procedure, it is worthwhile to attach physical estimates of length-scales, time-scales, and strain/flow rates represented by the simulations presented in this work. To this end, we select values that correspond to the base polymeric system of our interest, i.e., PDMS, for which we estimate σ = 4.6 Å, k _B T = 4.1 × 10^–21 J, and m = 1.23 × 10^–25 kg, which allows us to approximate $\mathrm{ϵ̇}$ ∼ 10⁸ s^–1 and η ∼ 10^–4 Pa s. This strain rate is significantly larger than corresponding experimental values (∼10^–3 s^–1) but typical for tensile testing simulated by MD. Similarly, the viscosity is lower than experimental values, ∼0.1 to 1.0 Pa s for linear PDMS, a discrepancy that results from limitations in both simulation time and strain rate.

3. Results and Discussion

3.1. Structure of PNCs under Weak NP Clustering

Even the weak clustering behavior of NPs can significantly affect the structural, dynamic, and rheological properties of PNCs. Hence, we first focused on determining the average number of NPs per cluster ⟨N _c⟩, as illustrated in Figure . A cluster was defined to be a collection of NPs where each NP has at least one neighboring NP within a cutoff distance. The cutoff distance r _cut, defined in terms of pairwise center-to-center distance of two NPs, was adjusted based on the NP radius, with cutoff values chosen to be 1σ larger than the diameter of the filler particles, representing interparticle surfaces in close contact via nonbond interactions like van der Waals or H-bonding. Thus, the values for r _cut were chosen to be 11σ, 21σ, and 41σ for r _f = 5σ, 10σ, and 20σ, respectively. This choice also ensures that the cutoff fully spans the first peak of the NP–NP radial distribution function, and we verified that the observed cluster-size trends are not sensitive to small, reasonable variations in the selected cutoff distance. Our initial analysis confirmed the formation of only small clusters of NPs, as expected under weak, nonspecific NP–NP interactions. These findings align with those of Hagita et al., who studied both clustered and dispersed states of NPs using different parameters for NP–NP interactions. In their work, they observed weak clustering at lower interaction strengths and strong clustering at higher values.

2.

(a) Average cluster size ⟨N _c⟩ for NPs as a function of NP radius r _f. (b,c) show the corresponding cluster size distribution P(N _c) profiles for polymer chain-lengths l = 20 and 200. Vertical error bars represent standard deviation in each quantity determined by analyzing the last 50% of the production-run trajectory.

Figure a indicates a weakly decreasing trend in the average cluster size (in terms of number of NPs) ⟨N _c⟩ with increasing NP radius r _f, regardless of the polymer chain-length l. However, there is a large fluctuation, up to 16% of the corresponding mean values (measured in terms of standard deviation computed from the last 50% of the production-run trajectory), which makes such a trend statistically not significant. Nonetheless, the downward trend in the mean ⟨N _c⟩ indicates that smaller NPs tend to aggregate more than larger ones, likely due to higher mobility of smaller NPs allowing them to diffuse through the polymer matrix and find other NPs, irrespective of the polymer chain length. However, in terms of absolute size, larger NPs form spatially larger aggregates. For a given filler size, r _f, ⟨N _c⟩ was observed to increase with a longer polymer chain length. Shorter polymer chains, due to their high mobility, disrupt the formation of larger clusters, while longer chains facilitate aggregation. Subsequently, we examined the cluster-size distributions P(N _c) for each system and found that longer chains lead to broader cluster-size distributions compared to shorter chains (see Figure b,c). Similar broad distributions were observed by Mélé et al.

We now turn our attention to whether the NP size and polymer chain length affect the packing of NPs within clusters. To explore this question, we calculated the radial distribution function g(r) for NP–NP pairs as defined by g(r) = dn/(4πr ²dr·ρ), where dn is the average number of beads at distance between r and r + dr from the center bead, 4πr ²dr is the volume of the spherical shell of thickness dr, and ρ is the average number-density of beads in the system. The results of this analysis are presented in Figure . For both 5σ and 10σ NPs we observed peaks at approximately 1σ larger than 2r _f, corresponding to the distance to the nearest neighboring NP. For comparison, theoretical calculations were performed assuming NPs arranged in a face-centered cubic (fcc) lattice at ϕ_NPs = 25%, which yielded interparticle distances of ∼14.4σ, 28.7σ, and 57.4σ for the three filler sizes. These results indicate that the first few peaks, particularly the first two, in Figure a,b correspond to cluster formation. In contrast, the r _f = 20σ NPs exhibited two distinct peaks at approximately 1σ and 2σ more than 2r _f, irrespective of chain lengths l. This two-peak signature is probably due to the trapping of polymer chains within clusters of large NPs. Given the slow dynamics of the large NPs, it is also possible that the two-peak structure is a finite-size, finite-simulation effect; longer runs with a larger simulation cell may result in a single broad peak. Additionally, peak broadness and intensity of the first peak increase with r _f. We also analyzed the g(r) for NP–polymer interactions to understand polymer bead distribution around the NPs (Figure S1, Supporting Information). For both 5σ and 20σ NPs, an intense first peak corresponds to polymer beads at the NP surface, while the second and third peaks represent polymer bead distributions further from the NP surface. Interestingly, polymer chain length has a minimal effect on polymer bead packing around the NP, consistent across all three NP sizes studied.

3.

NP–NP radial distribution function g(r _NP–NP) for different chain lengths and NP sizes: (a) r _f = 5σ; (b) r _f = 10σ (inset r _f = 20).

To gain further insights into the effects of r _f and l on NP aggregation within the simulation cell, we calculated the NP–NP structure factors S(q), presented in Figure . The S(q) of the NPs is defined as $S(q)=N^{-1}⟨{|\sum _{j=1}^N\exp (iq·r_j)|}^2⟩$ where r _j is the position vector of the center bead of the jth NP and N the total number of NPs in the system. The momentum transfer q is expressed in units of 2π/σ.

4.

Structure factor S(q) vs momentum transfer (q) for filler NPs for different filler sizes: (a) r _f = 5σ; (b) r _f = 10σ (Inset: r _f = 20σ).

In Figure a,b, the signature peak at the low q region is observed at 0.57σ^–1, 0.29σ^–1, and 0.14σ^–1 for r _f = 5σ, 10σ, and 20σ, respectively. A similar trend was also reported by Starr et al. in their filler structure factor measurements within PNCs. Recalling that the nearest-neighbor distances for an fcc lattice for the three filler sizes are at ∼14.4σ, 28.7σ, and 57.4σ, which correspond to q ∼ 0.44σ^–1, 0.22σ^–1, and 0.11σ^–1, the observed S(q) peaks at relatively higher q values can be attributed to the formation of NP clusters.

To investigate whether the presence of filler particles affect the overall chain conformations in the polymer, Figure a plots the logarithm of the root-mean-square end-to-end distance log­( ${⟨R_{\mathrm{e}\mathrm{e}}^2⟩}^{1/2})$ and the logarithm of the root-mean-square radius of gyration log­( ${⟨R_{\mathrm{g}}^2⟩}^{1/2})$ vs log­(l) (inset).

5.

(a) Chain end-to-end distance log­( ${⟨R_{\mathrm{e}\mathrm{e}}^2⟩}^{1/2})$ vs chain length log­(l). The inset is the corresponding radius of gyration log­( ${⟨R_{\mathrm{g}}^2⟩}^{1/2})$ for chains. Plots (b–d) respectively display the distribution function P(R _ee) of chain end-to-end profiles for l = 20, 100, and 200.

Theoretically, polymer chains are known to exhibit ideal Gaussian behavior in the absence of a solvent (i.e., in the melt state), with the end-to-end distance scaling as ${⟨R_{\mathrm{e}\mathrm{e}}^2⟩}^{1/2}\sim \sqrt{N}$ . Figure a shows that even in the PNC polymer chains maintain this ideal Gaussian behavior for all NP sizes, which is consistent with results from previous studies of similar bead–spring polymer melts. , The distribution of chain end-to-end distances, as plotted in Figure b–d, also displays little change from that in the corresponding neat polymer melt, confirming that polymer conformations in PNCs are not much affected by the NPs and follow trends similar to those in neat polymer melts.

Additionally, to quantify the state of chain entanglement in the PNCs we used the Z1+ code from Kröger et al. to analyze the mean contour length of the primitive paths ⟨L _pp⟩ and the mean number of kinks per chain ⟨Z⟩ for different NP sizes. The plots in Figure reveal a clear and interesting trend: under the nonspecific interactions considered here, NPs exhibit a plasticization effect that reduces the degree of entanglement, and larger NPs produce a more pronounced response. In Figure , the average number of kinks per chain ⟨Z⟩ appears to be higher than what is expected from the theoretical entanglement length of l _e ∼ 84, with the discrepancy being nearly a factor of 2 for the neat polymer. This discrepancy arises from differences between topological calculations (e.g., Z1+) and rheological measurements of entanglement, as discussed and noted in previous works. , Additionally, given that the chains are only slightly longer than those of l _e, the average number of kinks reported by Z1+ should be interpreted with appropriate caution. Even so, the consistent trends in ⟨L _pp⟩ and ⟨Z⟩ with increasing filler size r _f are compelling and merit emphasis.

6.

Plots are corresponding to (a) mean contour length of the primitive paths ⟨L _pp⟩ and (b) mean number of kinks per chain ⟨Z⟩ for r _f = 5σ, 10σ, and 20σ. The black dashed line is for neat systems.

3.2. Mobility of Chains and NP Fillers

Both simulation and neutron scattering studies have shown that the presence of NPs significantly impacts the mobility of polymer chains. This section focuses on understanding translational diffusion using MSD. Figure presents the computed MSD of the center of mass (COM) for polymer chains and NPs separately, for various r _f and l. Regarding polymer dynamics, we see the following notable trends from Figure a: (1) chain mobility in PNCs is somewhat slower relative to the mobility in the neat polymer melt, especially for small fillers. For large fillers, the mobility is nearly the same as in the melt. This suggests that at a constant volume fraction and with nonspecific interaction potentials small fillers disrupt mobility within the melt due to their larger number and surface area, as compared to larger NPs; (2) chain COM mobility is (expectedly) slower for longer chains. However, it is difficult to quantitatively cast these results in terms of standard polymer diffusion models because l = 100 and 200 are neither completely unentangled nor deeply entangled. Additionally, for the longest chains, the total simulation time was just on the order of the reptation time, which led to the diffusive regime (i.e., MSD ∝ t) barely being reached toward the end of the run. Overall, we see MSD decrease superlinearly as a function of chain-length l, which is consistent with impedance effects due to entanglements (for the longer chains).

7.

(a) MSD of the center of mass of the polymer chains for various filler sizes (r _f) and chain lengths (l). Neat polymer results are indicated by dashed curves; (b) MSD of the filler NPs for various r _f and l.

As for filler dynamics, we unsurprisingly see the slowing of NP mobility with increasing r _f. The NP mobility is also lower in PNCs with longer chains due to the presence of entanglements, which impedes motion of both polymers and fillers. However, it is difficult to cast these results in terms of particle diffusivity (e.g., Stokes–Einstein or related models) because in some cases we see the MSD vs t slope different from 1. It must also be kept in mind that not all NPs are isolated bodies diffusing within the polymer melt, but many are part of a larger NP cluster, which can slow down motion.

3.3. PNCs under Tensile Deformations

In this section, we explore the stress–strain and flow viscosity response of the PNCs as a function of r _f and l. Figure a illustrates the simulation box deformation at different levels of engineering strain (ϵ, expressed in %). Figure b plots the quantitative stress–strain responses (where engineering stress P is expressed in terms of extensional viscosity η = P/ $\mathrm{ϵ̇}$ ), as a function of strain. The inset, which plots the engineering stress (P) at very low strain values, shows that all systems initially exhibit a linear response up to ∼0.1% strain that is characteristic of elastic behavior. A peak stress level is reached at a few percent extension, which is followed by yielding at higher strain levels, marking the transition from elastic behavior to plastic flow of the non-cross-linked system. The simulations also show two expected effects, i.e., (1) modulus enhancement (at low strain levels) relative to the neat polymer and (2) higher stress levels (and, therefore, higher viscosity) for systems with longer chains, which can be attributed to entanglements.

8.

(a) 2d slice displaying spatial distribution of filler particles in PNCs under tensile deformation for various strain levels, with snapshots obtained using Ovito. Polymers are represented in red, while NPs are shown in blue. (b) The extensional flow viscosity (η = P/ϵ̇) as a function of engineering strain (ϵ) for filled and unfilled polymers of various chain-lengths for a fixed filler size (10σ). The inset plot shows the engineering stress P at very low strain levels. Note that in (a) all filler particles are of the same size (r _f = 10σ), although a 2d projection on the slice plane can make them appear to be of different sizes.

An important motivation of this work was to investigate the effect of filler size on the extensional viscosity of the PNCs. Figure a demonstrates that for both shorter (l = 20) and longer polymer chains (l = 200), the viscosity increases with increasing r _f. This behavior can be attributed to (1) the fact that at a fixed filler fraction (25% by volume in this work) the level of NP aggregation is effectively higher for larger filler sizes, which leads to spatially larger clusters and lowering of configurational entropy of the polymeric chains and (2) the lower mobility of larger NPs leading to enhanced friction to chain mobility and thus higher flow viscosity.

9.

Computed properties of PNCs under tensile strain (ϵ): (a) viscosity η = P/ϵ̇, (b) average absolute chain end-to-end distance ⟨R _ee⟩, and (c) average number of NPs per cluster ⟨N _c⟩. (d) Elastic modulus E = P/ϵ (at ϵ < 0.1%). Dashed lines indicate corresponding results for a neat (i.e., unfilled) polymer.

Figure b,c, respectively, display the average absolute end-to-end distance ⟨R _ee⟩ and average cluster size ⟨N _c⟩ as a function of strain. (b) shows that chain extension is insensitive to filler size and that longer chains undergo a larger increase in ⟨R _ee⟩ with uniaxial extension. These trends align well with observations from nonequilibrium MD simulations conducted by Li et al. The decrease in ⟨N _c⟩ with ϵ in (c) demonstrates that the clusters fragment with increasing strain, and this fragmentation is more pronounced for smaller NPs. Zhu et al. hypothesized a similar breakup of filler–filler associations under strain. Figure d shows the elastic modulus E extracted from the slope of initial linear elastic region of the stress–strain curves (ϵ < 0.1%, cf. Figure b). Compared to the neat polymer, the PNCs exhibit significantly higher elastic modulus values, with a marked increase in modulus as a function of NP size, and a more modest increase with polymer chain length.

4. Conclusions

With the aim of exploring structural, mechanical, and rheological properties of non-cross-linked, filled PNC ink systems, we carried out coarse-grained bead–spring simulations at a fixed filler loading (25% by volume) under weak, nonspecific interactions, which are representative of a wide class of technologically important systems. The present study considered polymer chains of lengths (l) both below and (slightly) above the entanglement length (l _e ∼ 84) and filler sizes (radius r _f) below and of the order of the radius of gyration of the polymer chains. The simulations revealed several interesting results. First, the equilibrium chain-configuration within the PNC remains Gaussian-like as in an unfilled melt, with the end-to-end distance ${⟨R_{\mathrm{e}\mathrm{e}}^2⟩}^{1/2}$ scaling with the square-root of chain-length. Second, in the absence of energetic driving forces, configurational entropy leads to the formation of small clusters, with such a clustering tendency being stronger for smaller filler particles and longer chains. Third, larger fillers lead to a notable reduction in entanglement density (for chain lengths above l _e) and accelerate chain mobility, thereby acting as plasticizers. Fourth, fillers enhance the tensile response modulus, with the effect being stronger for larger particles. Given that larger fillers (under a fixed loading fraction) can be considered an extreme form of aggregation of smaller filler particles, this is consistent with known results for filler-induced mechanical reinforcement as a function of secondary and tertiary structure of filler aggregates. We also subjected the PNC systems to a time-varying tensile strain ϵ̇ and studied the breakage of NP clusters, yielding, stretching of polymer chains, and the resulting evolution (thinning) of extensional flow viscosity.

The aim of the present study was to demonstrate the utility of coarse-grained simulations to provide semiquantitative insights into processing-structure–property correlations for filled-polymer ink systems. It is not difficult to see that the design phase space is vast, with significantly diverse choices for polymer functional groups and controlled modifications thereof, chain architecture, the chemical nature of the filler surface, etc. Each of these choices has the potential to alter the polymer–polymer, polymer–filler, and filler–filler interactions, which in our coarse-grained simulations can be represented with an appropriate choice of nonbond interaction parameters and bead–spring topology. Finally, advances in computational methods and resources will allow more detailed representations of polymer–filler interactions to be explored at larger, more realistic system sizes and simulated durations.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.5c11248.

• g(r) for the NP–polymer to understand polymer bead distribution around the NPs (PDF)

The authors declare no competing financial interest.