Effects of Tethered Polymers on Dynamics of Nanoparticles in Unentangled Polymer Melts

Abstract

Polymer-tethered nanoparticles (NPs) are commonly added to a polymer matrix to improve material properties. Critical to the fabrication and processing of such composites is the mobility of the tethered NPs. Here we study the motion of tethered-NPs in unentangled polymer melts using molecular dynamics simulations, which offer a precise control of the grafted chain length N_g and the number z of grafted chains per particle. As N_g increases, there is a crossover from particle-dominated to tethered-chain-dominated terminal diffusion of NPs with the same z. The mean squared displacement of loosely tethered NPs in the case of tethered-chain dominated terminal diffusion exhibits two sub-diffusive regimes at intermediate time scales for small z. The first one at shorter time scales arises from the dynamical coupling of the particle and matrix chains, while the one at longer time scales is due to the participation of the particle in the dynamics of the tethered chains. The friction of loosely grafted chains in unentangled melts scales linearly with the total number of monomers in the chains, as the frictions of individual monomers are additive in the absence of hydrodynamic coupling. As more chains are grafted to a particle, hydrodynamic interactions between grafted chains emerge. As a result, there is a non-draining layer of hydrodynamically coupled chain segments surrounding the bare particle. Outside the non-draining layer is a free-draining layer of grafted chain segments with no hydrodynamic coupling. The boundary of the two layers is the stick surface where the shear stress due to the relative melt flow is balanced by the friction between grafted and melt chains in the interpenetration layer. The stick surface is located further away from the bare surface of the particle with higher grafting density.

Graphical Abstract

1. Introduction

Grafting to nanoparticles (NPs) polymer chains with well-controlled molecular weights, grafting density, and composition provides a versatile pathway to design polymer nano-composites with desirable material properties.^1–3 However, the addition of grafted chains alters the mobility of NPs in viscoelastic polymer matrices^4,5 and thus affect the fabrication and processing of polymer nano-composites. An understanding of the effects of tethered chains on the dynamics of NPs is thus critical to the design of polymer nanocomposites.

Recently, Rutherford backscattering spectrometry has been used to examine the diffusion of poly(methyl methacrylate) (PMMA)-grafted NPs in unentangled to weakly entangled PMMA melts.⁵ The experiments showed that the mobility of NPs with densely grafted chains is strongly suppressed with respect to that of bare NPs and the larger effective particle size agrees with the size of the NP core plus the extended grafted chains. Scaling theory by Ge and Rubinstein⁶ has systematically described the mobility of polymer-tethered NPs in melts of unentangled polymers. For NPs with loosely grafted chains, the theory predicts that the mobility of NPs is dominated by either the bare particle or the grafted chains, whichever possesses a higher friction coefficient or lower mobility. For densely-grafted NPs, the theory approximates their hydrodynamic radii as the sum of the particle size and the size of grafted chains as a result of the hydrodynamic coupling of the particle and grafted chains.

Numerical simulations complement experiments of tethered NPs in polymer matrices. In molecular simulations, one can control the size and shape of bare NPs, the length and density of grafted chains, and the length of matrix chains, allowing an exact evaluation of each part with access to the microscopic details that are not directly observable in experiments. Simulations cover a wide range of time and length scales including the intermediate time scales relevant for the interplay of NPs, grafted chains, and matrix chains, while experiments are usually limited to the terminal diffusion of tethered NPs over distances larger than the particle size. Numerical simulations, experiments, and theory provide a powerful combination to elucidate the microscopic picture of the dynamics of tethered NPs in a polymer matrix.

In this article, we present the results from molecular dynamics (MD) simulations of polymer-tethered NPs in melts of unentangled polymers. For loosely grafted NPs, we observe a crossover from particle-dominated to tethered-chain-dominated diffusion with increasing length N_g of the tethered chains. In the tethered-chain-dominated case with z = 1 and z = 2 chains per particle, there are two sub-diffusive regimes in the mean squared displacement 〈Δr²(t)〉 of NPs at intermediate time t prior to long-time diffusion. The initial sub-diffusion arises from the dynamical coupling of the particle and melt chains, while the second sub-diffusive regime at later times results from the participation of the particle in the dynamics of the grafted chains. The friction of loosely grafted chains at small z without hydrodynamic coupling is proportional to the total number of monomers in the grafted chains. With increasing z, hydrodynamic interactions emerge, resulting in an inner non-draining layer of hydrodynamically coupled chain segments surrounding the particle and an outer free-draining layer of chain segments with no hydrodynamic coupling. The boundary between the two layers is the stick surface where the shear stress due to a relative melt flow is balanced by the friction between grafted and melt chains in the interpenetration layer. The distance of the stick surface from the particle center r_stick increases as the number z of grafted chains increases.

2. Models and Methods

The simulations employed the canonical bead-spring model of polymers.^7–9 Monomers in the matrix chains and grafted chains are both of size σ and mass m. All the monomers interact via the Lennard-Jones (LJ) potential with interaction strength ϵ, cut-off distance r_c = 2.5σ, and characteristic time $\tau =\sigma \sqrt{m/ϵ}$. Neighboring monomers in a polymer chain are connected by the finitely extensible nonlinear elastic (FENE) potential. Each polymer chain in the matrix melt consists of N = 16 monomers. The root mean square (RMS) end-to-end size of a matrix chain 〈R²〉^1/2 = 4.7σ. The number density of monomers in a melt of pure matrix chains is ρ = 0.89σ⁻³ at temperature $T=ϵ/k_B$ and pressure P = 0. The diffusion coefficient of matrix chains of length N = 16 in the melt D_melt = 0.0044σ²/τ. The zero-shear-rate viscosity of the melt η_melt = 15.7τϵ/σ³ as determined by a non-equilibrium MD simulation. The number of monomers in a chain grafted to a NP ranges from N_g = 2 to 64, and the number of grafted chains per particle ranges from z = 1 to 64.

NPs of diameter d = 5σ are modeled as smooth spheres with mass density ρ_m = m/σ³.^10,11 As in previous simulations of bare NPs in polymer melts,^10–14 the interactions between NPs are purely repulsive with the Hamaker constant $A=4{\pi }^2ϵ\approx 39.5ϵ$. NPs interact with monomers of polymer chains through a weakly attractive potential with the Hamaker constant A = 100ϵ, which corresponds to a weak binding energy of about 1.4ϵ per monomer, which is sufficient to disperse the bare NPs in the polymer matrix.^12,13 Each grafted chain has one end monomer fixed at the surface of a NP. For z = 2, the grafted points are located at two opposite poles of the sphere. For z = 4 or 8, the grafted points are the vertices of a regular tetrahedron or cube inscribed in the sphere. For z ≥ 16, the grafted points are equi-distributed on the NP surface, meaning that the separation between adjacent points is the same in two orthogonal direction.¹⁵ The distribution of grafted points is identical for all NPs with the same z. End monomers at the grafted points are rigidly attached to the NP cores.

The number of matrix chains varies from 72, 576 to 76, 356 in different samples, while the number of NPs is fixed at 27. All samples were equilibrated using a Nosé-Hoover thermostat/barostat^16–18 at temperature T = ϵ/k_B and pressure P = 0. Periodic boundary conditions were applied in all three directions. The simulation box is cubic with dimensions L³. At equilibrium, L varied from 110σ to 112σ for different samples. The volume fraction of the NP cores ϕ_core = 27 × (πd³/6)/L³ ≈ 0.0013 in all samples. The volume fraction of grafted NPs is estimated as ϕ = 27×(πd³/6+zN_g/ρ)/L³, where ρ is the number density of monomers in the melt of pure matrix chains. ϕ = 0.003–0.02 for different samples. Assuming a tethered NP as an effective sphere with radius $r_{eff}=d/2+{〈h_{e-e}^2〉}^{1/2}$, where ${〈h_{e-e}^2〉}^{1/2}$ is the RMS size of the end-to-end vector of a grafted chain projected to the normal vector of the spherical surface at the grafting point, the volume fraction of the effective spheres ϕ_eff = 0.004–0.04 for different combinations of z and N_g, corresponding to separated tethered NPs. Representative loosely and densely grafted NPs in polymer matrices are visualized in Figure 1. For the simulations of the diffusion of tethered NPs, the equilibrated samples were maintained at a fixed volume and a constant temperature T = ϵ/k_B using a Langevin thermostat^19,20 with damping time 10τ. Depending on z and N_g, the simulations were run from 10⁵τ to 10⁶τ with a time step 0.01τ using the LAMMPS simulation package.²¹ The root mean square (RMS) displacement of a NP during the run time is at least 2r_eff.

Figure 1:

Snapshots of 27 NPs each loosely grafted with z = 2 polymer chains of length N_g = 32 (left) and densely grafted with z = 64 and N_g = 16 (right) in polymer melts of chains of length N = 16. Nanoparticle cores (diameter d = 5σ) and grafted chains are illustrated by blue spheres and yellow chains, respectively. For clarity, the melt chains are not shown. Periodic boundaries of simulation boxes are indicated by the black lines. One tethered NP from each simulation is enlarged in insets for a clear view.

3. Results

We quantify the motion of NPs using the mean squared displacement (MSD) 〈Δr²(t)〉 of NPs as a function of time t. Figure 2(a) shows the MSDs of NPs loosely grafted with z = 2 chains for different lengths N_g and the corresponding MSD of bare NPs in the same polymer matrix. With increasing N_g, the MSD at the same time t is reduced, and the crossover to the terminal diffusive regime is delayed. To demonstrate clearly the effects of grafted chains, 〈Δr²(t)〉/6t is plotted in Figure 2(b). This plot is motivated by the fact that the long-time limit of 〈Δr²(t)〉/6t is the diffusion coefficient D. In all samples, 〈Δr²(t)〉/6t approaches a plateau that corresponds to the terminal diffusive regime. The crossover to the diffusive regime is fit to the function $〈\Delta r^2(t)〉/6t=D{\left[{\left(t/{\tau }_c\right)}^{-(1-\alpha )\gamma }+1\right]}^{1/\gamma }$, which describes the crossover from 〈Δr²(t)〉 ~ t^α for the sub-diffusive motion prior to the diffusion to 〈Δr²(t)〉 = 6Dt for the terminal diffusion. τ_c is the time at which the crossover occurs, and γ characterizes the shape of the crossover function. A simultaneous fit of 〈Δr²(t)〉/6t in the different scenarios to the crossover function with the same α and γ yields α = 0.79±0.01 and γ = 8.2 ± 0.6. The best-fit values of τ_c and D from the simultaneous fit are shown in Table 1. Based on the simultaneous fit to the crossover function, 〈Δr²(t)〉/6t is extrapolated to larger t to yield D, as shown in Figure 2(b). For bare NPs, the diffusion coefficient D_bare = (2.12 ± 0.007) × 10⁻³σ²/τ. Using the Stokes-Einstein relation D = k_BT/(fπηd) with the full-slip condition f = 2 for the boundary between a particle of diameter d and surrounding medium of viscosity η, the effective viscosity experienced by a bare NP in the simulations is η_bare = k_BT/(2πD_bared) = 15.2τϵ/σ³, which is comparable to the measured melt viscosity η_melt = 15.7τ/σ³. The full-slip condition is appropriate because the particle radius d/2 = 2.5σ is smaller than the estimated slip length²² L_s ≈ b(η_melt/η₀) ≈ 16σ, where b ≈ σ is the monomer size, and η₀ ≈ τϵ/σ² is the viscosity of monomers. Since the bare particle size d = 5σ is slightly larger than the RMS end-to-end size of matrix chains 〈R²〉^1/2 = 4.7σ, the result η_bare ≈ η_melt confirms the scaling ansatz²³ that the effective viscosity for the diffusion of a NP in an unentangled polymer melt is comparable to the melt viscosity if the particle size d exceeds the average melt chain size.

Figure 2:

(a) MSD 〈Δr²(t)〉 of bare NPs with diameter d = 5σ and NPs loosely grafted with z = 2 chains of indicated chain length N_g in a melt of polymer chains with N = 16 monomers per chain. (b) Solid lines show 〈Δr²(t)〉/6t for the NPs in (a). Dashed lines show extrapolation of the simulation data to larger t using the crossover function $〈\Delta r^2(t)〉/6t=D{\left[{\left(t/{\tau }_c\right)}^{-(1-\alpha )\gamma }+1\right]}^{1/\gamma }$. The dark yellow solid and dashed lines are 〈Δr²(t)〉/6t and [〈Δr²(t)〉/6t]/2 for the monomers in a free chain with N = 64. The blue dotted lines indicate two sub-diffusive regimes for N_g = 64.

Table 1:

Best-fit values of D and τ_c from simultaneously fitting MSD/6t for bare NPs and NPs tethered with z = 2 chains of length N_g in the polymer melt with N = 16 to the crossover function $〈\Delta r^2(t)〉/6t=D{\left[{\left(t/{\tau }_c\right)}^{-(1-\alpha )\gamma }+1\right]}^{1/\gamma }$.

	D [10−3σ2/τ]	τc [102τ]
Bare NP	2.12 ± 0.007	4.4 ± 0.2
Ng = 2	2.00 ± 0.007	4.5 ± 0.2
Ng = 4	1.90 ± 0.007	5.4 ± 0.3
Ng = 8	1.76 ± 0.005	7.1 ± 0.4
Ng = 16	1.41 ± 0.004	22 ± 1
Ng = 32	1.07 ± 0.007	120 ± 10
Ng = 64	0.703 ± 0.0004	740 ± 50

As shown in Table 1, the reduction of D with respect to D_bare is smaller than D_bare/2 for tethered NPs with z = 2 and N_g ≤ 16. As a result, the friction coefficient ζ = k_BT/D for the terminal diffusion is controlled by the bare particle friction ζ_bare = k_BT/D_bare. For tethered NPs with z = 2 and N_g = 32, D ≈ D_bare/2, indicating that almost half of ζ is from the friction coefficient ζ_g of the grafted chains. For tethered NPs with z = 2 and N_g = 64, D < D_bare/2, and the terminal diffusion is dominated by the grafted chains. The crossover from particle-dominated to tethered-chain-dominated diffusion with increasing N_g in Figure 2(b) and Table 1 agrees with the prediction of the scaling theory by Ge and Rubinstein.⁶

The scaling theory⁶ approximates the friction coefficient of a loosely grafted NP as the sum of the friction of the bare particle and the monomeric friction coefficient times the number of monomers in grafted chains. We evaluate the validity of this approximation in the simulations. The friction coefficient in the simulations is calculated as ζ = k_BT/D, and the result is shown in Figure 3. The increase of ζ of tethered NPs with respect to the bare particle friction ζ_bare scales linearly with the number of monomers in grafted chains. ζ_g = ζ − ζ_bare is fit to ζ₀[2(N_g − 1)] with the friction coefficient per grafted monomer ζ₀ as the fitting parameter. Note that (N_g − 1) corresponds to the exclusion of the end monomer that moves rigidly with the NP. The best fit with ${\zeta }_0=(8.3\pm 0.4)ϵ\tau /{\sigma }^2$ is shown by the red dashed line in Figure 3. The friction coefficient per grafted monomer is smaller than the monomeric friction coefficient ${\zeta }_{0,\mathit{\text{melt}}}=\left(k_{B}T/D_{\mathit{\text{melt}}}\right)/N=14.3ϵ\tau /{\sigma }^2$ in a pure polymer melt. One possible reason for the reduction of ζ₀ with respect to ζ_0,melt is the hydrodynamic coupling between the particle and nearby sections of grafted chains. This coupling could make ζ smaller than ζ_bare +ζ_0,melt[2(N_g −1)], which corresponds to the uncoupled dynamics of the bare particle and grafted chains.

Figure 3:

Friction coefficient ζ for NPs loosely grafted with z = 2 chains and N_g monomers per chain (black circles) in a melt of chains with N = 16 monomers per chain. The black dash-dotted line indicates ζ_bare. Additional friction from the grafted chains ζ_g = ζ − ζ_bare (red diamonds) is fit to ζ₀[2(N_g −1)], where the fitting parameter ζ₀ is the friction coefficient per grafted monomer. The best fit is shown by the red dashed line.

The MSD of a loosely-grafted NP with tethered-chain-dominated diffusion exhibits two sub-diffusive regimes at intermediate time scales prior to terminal diffusion. The two sub-diffusive regimes for N_g = 64 are indicated by the blue dotted lines in Figure 2(b). The first regime is similar to the sub-diffusion of bare NPs (see the solid magenta line in Figure 2(b)), which results from the dynamical coupling of the particle and matrix chains. In the first sub-diffusive regime, the local minimum of the log-log slope α = dlog〈Δr²(t)〉/dlogt is 0.77±0.01, as indicated by the first dotted blue line. This value is close to α_min = 0.81±0.01 for a bare NP. The theoretical prediction is α = 1/2 for both bare NPs prior to terminal diffusion and tethered NPs in the first sub-diffusive regime.^6,23 In the simulations, α_min > 1/2 is related to the crossover from the sub-diffusive regime with α = 1/2 to the diffusive regime with α = 1. Note that there is a small difference between the MSDs of bare and tethered NPs at small t, as the permanently grafted monomers increase the inertia of tethered NPs by $2m/\left({\rho }_m\pi d^3/6\right)=3\%$ for z = 2. The second sub-diffusive regime arises from the participation of the particle in the dynamics of grafted chains, which is absent in the motion of bare NPs. The second dotted blue line indicates α_min = 0.80±0.02 for the second sub-diffusive regime. The scaling theory⁶ approximates the MSD of tethered NPs in the second sub-diffusive regime and the subsequent terminal diffusion as the MSD of monomers in free chains of length N = N_g in the same melt divided by z. Motivated by the scaling theory, 〈Δr²(t)〉/6t and [〈Δr²(t)〉/6t]/2 for monomers of a free chain of length N = 64 in the melt of chains with N = 16 is shown by the dark yellow solid and dashed lines in Figure 2(b). For monomers in the free chains, the log-log slope α continuously increases from 0.6 at t ≈ 600τ to 1 in the terminal diffusive regime. For tethered NPs in the second sub-diffusive regime, 0.6 < α_min < 1 reflects the slowdown of NP motion caused by the dynamics of grafted chains. The MSD of tethered NPs in the second sub-diffusive regime and the terminal diffusive regime is not identical to 1/z = 1/2 of the MSD of monomers in the free chains. This difference may be related to the small but not negligible friction of the bare particle, the hydrodynamic coupling between the particle and nearby sections of grafted chains, and also the dynamic modes across the particle from one grafted chain to the other, all of which are not captured by the scaling theory.⁶ It would be interesting to investigate these aspects by theoretical methods beyond the scaling description and by experiments as well.

As shown in Figure 3, friction coefficients of individual monomers for z = 2 tethered chains are additive. In general, this additivity holds for loosely grafted chains in unentangled melts. Figure 4 shows the additional friction ζ_g = ζ − ζ_bare due to loosely grafted chains with different combinations of z and N_g. For a given N_g, ζ_g increases linearly with z in the loosely-grafted regime, as shown in the inset of Figure 4. ζ_g in different systems collapse onto a universal linear function ζ_g = ζ₀[z(N_g −1)], where ${\zeta }_0=(7.9\pm 0.3)ϵ\tau /{\sigma }^2$ is the friction coefficient per grafted monomer.²⁴ The additivity of the frictions of individual monomers results from the screening of the hydrodynamic interactions between loosely grafted chains by the polymers of the matrix melt.

Figure 4:

ζ_g = ζ − ζ_bare scales linearly with z(N_g − 1) for loosely-grafted NPs with different combinations of z and N_g. The best fit of ζ_g to ζ₀[z(N_g −1)] is shown by the black solid line. The inset shows the corresponding ζ_g as a function of z for different N_g. The dashed lines with slope 1 are guides to eyes.

The existence of two sub-diffusive regimes for particles with tethered-chain-dominated terminal diffusion requires that the second sub-diffusive regime is well separated from the first one. For tethered NPs with the bare particle size d comparable to the size of unentangled matrix chains, as in the present simulations, the first regime ends at the Rouse relaxation time τ_R of the melt. According to the scaling theory,⁶ the second regime starts at the crossover time τ*, at which the friction ζ₀z (τ*/τ₀)^1/2 of z grafted chain segments with (τ*/τ₀)^1/2 coherently moving monomers each is comparable to the friction ζ_bare ≈ dη_melt ≈ (d/b)Nζ₀ of a bare NP. This crossover time is τ* ≈ τ_R(d/b)²/z². Two separate sub-diffusive regimes exists if τ_R < τ*, or equivalently z < d/b. Another necessary condition for the existence of two sub-diffusive regimes is z > (d/b)(N/N_g), resulting from the requirement that the additional friction zN_gζ₀ due to the grafted chains is larger than ζ_bare ≈ (d/b)Nζ₀. As a result, particles with tethered-chain-dominated terminal diffusion exhibit two sub-diffusive regimes if (d/b)(N/N_g) < z < d/b is satisfied. In the simulations, two sub-diffusive regimes are observed for particles grafted with z = 1 chain of length N_g = 64, and particles grafted with z = 2 chains of N_g = 32 and N_g = 64 (see the symbols enclosed by the black circles within the triangle outlined by the orange dashed lines in Figure 9).

Figure 9:

Summary of the simulated samples of NPs with diameter d = 5σ grafted with z chains consisting of N_g monomers each in a polymer melt with N = 16 monomers per chain. Green circles, cyan squares, and red stars indicate three distinctive regimes. Terminal diffusion is dominated by the particle in Regime I, while by the tethered chains in Regimes II and III. Hydrodynamic interactions between grafted chains are screened by the melt polymers in Regimes I and II, while hydrodynamic coupling allows the emergence of a non-draining layer surrounding the particle in Regime III. The black circles enclose three samples where two sub-diffusive regimes in 〈Δr²(t)〉 are observed. The blue line shows z = ζ_bare/[(N_g − 1)ζ₀] with ${\zeta }_{\mathit{\text{bare}}}=k_{B}T/D_{\mathit{\text{bare}}}=472ϵ\tau /{\sigma }^2$ and ${\zeta }_0=7.9ϵ\tau /{\sigma }^2$, which is the boundary that separates the particle dominated Regime I and tethered-chain dominated Regime II. The magenta dashed line $z=1024N_g^{-3/2}$ indicates the onset of hydrodynamic interactions between Regimes II and III. The orange dashed lines indicate the boundary of the triangular region (d/b)(N/N_g) < z < d/b with d = 5σ and b = 1.3σ, where two sub-diffusive regimes can exist.

To motivate further study of loosely-grafted NPs in polymer melts, both theoretically and experimentally, we summarize the key observations in the molecular simulations of loosely-grafted particles. These observations include (1) the crossover from particle dominated diffusion to tethered-chain dominated diffusion with increasing tethered chain length, (2) the existence of two qualitatively different sub-diffusive regimes for the MSD of loosely-grafted particles with (d/b)(N/N_g) < z < d/b, (3) the similarity of the MSD in the first sub-diffusive regime and the MSD of the bare particle in the sub-diffusive regime, (4) the similarity of the MSD in the second sub-diffusive regime and the MSD of the monomers in a free chain divided by the number of grafted chains z, and (5) the linear scaling of the additional friction of grafted chains with the number of monomers in the grafted chains and ζ₀ per grafted monomer being smaller than ζ_0,melt. These simulation results are consistent with the predictions of the scaling theory,⁶ supporting the scaling description of the effects of loosely grafted chains on the motion of nanoparticles.

As more chains are grafted to a particle, the dynamics of tethered NPs are expected to change with respect to the loosely grafted regime. Figure 5(a) shows the MSDs of NPs grafted with increasingly more chains of length N_g = 16 and the corresponding MSD of bare NPs. At small t, the reduction of the MSD of tethered NPs with respect to that of bare NPs results from the increase of particle inertia due to z permanently grafted monomers. This reduction is up to a factor of $\left({\rho }_m\pi d^3/6+zm\right)/\left({\rho }_m\pi d^3/6\right)\approx 2$ for z = 64. In the terminal diffusive regime at large t, the MSD of tethered NPs is reduced because the friction ζ is increased by the grafted chains. Figure 5(b) shows the increase of ζ_g = ζ − ζ_bare (symbols) with z for tethered NPs with N_g = 8, 16, and 32. The increase deviates from the linear dependence on z and becomes sub-linear with increasing z.

Figure 5:

(a) MSD 〈Δr²(t)〉 of bare NPs of diameter d = 5σ and NPs grafted with z chains of length N_g = 16 in a melt of polymer chains of length N = 16. (b) ζ_g = ζ − ζ_bare as a function of z for indicated N_g.

The deviation from the linear z-dependence of ζ_g with increasing z is expected to correlate with the structural change of the interpenetration layer between grafted and melt chains. As shown in Appendix A, the average extension of grafted chains normal to the particle surface increases by less than 20% as z increases towards the maximum possible number of grafted chains, which is approximately 80 for d = 5σ. While there is no strong extension of grafted chains with increasing z, the structure of the interpenetration layer changes. To quantify the change, we calculate the volume fractions of monomers belonging to grafted chains and melt chains, ϕ_g and ϕ_m, as functions of the distance r from the particle center. The results are shown in Figure 6(a), (c), and (e) for N_g = 8, 16, and 32, respectively. The corresponding products of the two volume fractions ϕ_gϕ_m quantify the mixing of grafted chain and melt chains in the interpenetration layer, and are shown in Figure 6(b), (d), and (f). Two length scales are extracted from ϕ_gϕ_m. One length scale r_peak is the distance of the peak of ϕ_gϕ_m from the particle center. It corresponds to the maximum mixing of grafted and melt chains. The peak occurs at ϕ_g = ϕ_m = 0.5 with ϕ_gϕ_m = 0.25 in most cases, while in the close vicinity to the particle surface in two cases (N_g = 16 and z = 8 in Figure 6(d) and N_g = 32 and z = 8 in Figure 6(f)). The other length scale is r_half−width at which φ_gφ_m = 0.125, which is half of the maximum possible value 0.25, ϕ_g ≈ 0.15, and ϕ_m = 1 − ϕ_g ≈ 0.85. r_half−width represents the outer boundary of the layer with significant overlap of grafted and melt chains. r_peak and r_half−width are marked on the plots of ϕ_gϕ_m in Figure 6(b), (d), and (f), and shown by the red and green symbols in Figure 7. Both r_peak and r_half−width increase with increasing z for the same N_g, indicating that the interpenetration layer extends into the melt matrix.

Figure 6:

Volume fractions of monomers in grafted chains ϕ_g and in melt chains ϕ_m as functions of the distance r from the particle center for (a) N_g = 8, (c) N_g = 16, and (e) N_g = 32. The product ϕ_gϕ_m that quantifies the interpenetration zone and the positions of various length scales characterizing the structure and dynamics of grafted chains for (b) N_g = 8, (d) N_g = 16, and (f) N_g = 32.

Figure 7:

Various length scales characterizing the structure and dynamics of grafted chains as functions of the number of grafted chains per particle z for (a) N_g = 8, (b) N_g = 16, and (c) N_g = 32.

One additional length scale r_dry is determined from $(4\pi /3)r_{dry}^3-(4\pi /3){(d/2)}^3=\left(z{N}_g\right)/{\rho }_{\mathit{\text{melt}}\cdot r_{\mathit{\text{dry}}}}$ corresponds to the sum of the bare particle radius d/2 and the thickness (r_dry − d) of a completely dry layer of grafted chains at melt density ρ_melt in the absence of any mixing with the melt surrounding the particle. The positions of r_dry are marked in Figure 6(b), (d), and (f), and the values are shown by the black symbols in Figure 7. For the same N_g, r_dry increases with z, corresponding to the increase of the dry layer thickness. The dry layer serves as a reference state for the study of the mixing of grafted and melt chains. Driven by the mixing entropy, the grafted chains from the dry layer would mix with melt chains. The mixing extends the grafted chains beyond the thickness of the dry layer, and the outer boundary of the half-height section of the mixing layer is at r_half−width > r_dry. As shown in Figure 7, r_peak < r_dry < r_half−width is observed for all N_g. The relation r_peak < r_dry indicates that the amount of melt monomers inside the inner part of the interpenetration layer with r < r_peak is smaller than that of grafted monomers in the outer part of the interpenetration layer with r > r_peak, reflecting an asymmetric mixing of grafted and melt chains around r_peak. If the mixing were symmetric around r_peak, r_peak = r_dry, and the amount of melt monomers in the region with r < r_peak = r_dry would be the same as that of grafted monomers in the region with r > r_peak = r_dry.

The extension of the interpenetration layer into the melt suggests that the effective size of a tethered NP grows with increasing z. In both experiment⁵ and theory,⁶ a densely grafted NP has been approximated as a spherical hydrodynamic object enlarged with respect to the bare particle. For the tethered NPs in Figure 6, we extract an effective hydrodynamic radius r_H from the friction coefficient ζ. According to the Stokes law with stick boundary condition, r_H = ζ/(6πη_melt). We compare r_H with the structural length scales r_peak, r_half−width, and r_dry in Figure 7. The increase of r_H with z is shown by the magenta symbols in Figure 7. r_H(z) is closer to r_dry(z) for small N_g = 8 in Figure 7(a), while becomes closer to r_half−width(z) for larger N_g = 16 in Figure 7(b) and N_g = 32 in Figure 7(c). In general, we observe that the hydrodynamic radius increases towards the outer boundary of the interpenetration layer as N_g increases.

The increase of r_H with z is related to the hydrodynamic interactions between grafted chains, which make the friction coefficient ζ smaller than free-draining ζ_bare + ζ₀z(N_g − 1) with no hydrodynamic interactions, as shown by the concave shape of ζ_g(z) in Figure 5(b). To further understand the hydrodynamic coupling of grafted chains, we estimate the position r_stick of the stick surface below which strong hydrodynamic coupling takes place. As illustrated in Figure 8, due to the hydrodynamic coupling, the monomers of the melt chains located between the bare particle surface at r = d/2 and r_stick on average move together with the grafted chains and the particle with no relative velocity. By contrast, with no hydrodynamic coupling, the monomers of the melt chains outside the stick surface on average move at a relative velocity v(r). Here, we use a planar geometry to simplify the description and assume a linear profile for the relative velocity, $v(r)=\dot{\gamma }\left(r-r_{\mathit{\text{stick}}}\right)$, where $\dot{\gamma }$ is the constant velocity gradient due to the assumed relative motion of the melt with respect to NP at a small constant velocity.

Figure 8:

Schematic illustration of the average velocity of monomers of melt chains with respect to the stick surface.

The estimate of r_stick is based on the balance of the shear stress on the stick surface due to the relative melt flow and the friction between the grafted and melt chains in relative motion. The shear stress on the stick surface is ${\sigma }_{\mathit{\text{melt}}}={\eta }_{\mathit{\text{melt}}}\dot{\gamma }$. The friction force per unit area of the stick surface ${\sigma }_f={\int }_{r_{\mathit{\text{stick}}}}^{\infty }{\varphi }_g(r){\varphi }_m(r){\zeta }_{0}v(r)dr$, in which the friction coefficient per monomer is ζ₀. From σ_melt = σ_f,

$${\int }_{r_{\mathit{\text{stick}}}}^{\infty }{\varphi }_g(r){\varphi }_m(r)\left(r-r_{\mathit{\text{stick}}}\right)dr={\eta }_{\mathit{\text{melt}}}/{\zeta }_{0,\mathit{\text{melt}}}$$ (1)

where ζ₀ is set as the monomeric friction coefficient ζ_0,melt of the melt chains. Eq.(1) is numerically solved with the monomer volume fractions ϕ_g(r) and ϕ_m(r) presented in Figure 6. Numerical solution for r_stick does not exist in certain cases of low grafting density and short grafted chains, meaning that the stick surface is below the particle surface and there is slip at the particle surface. We find that there is no solution for N_g = 8 with z = 16 and 32 and N_g = 16 with z = 8, while a solution exists for other systems with larger z and N_g in Figure 6. In the cases where a numerical solution exists, r_stick is marked in Figure 6(d) and (f), and shown by the blue symbols in Figure 7. r_stick(z) lies below r_peak(z) for N_g = 16 in Figure 7(b), crosses r_peak(z) but remains below r_dry(z) and r_half−width(z) for N_g = 32 in Figure 7(c).

For both N_g = 16 and N_g = 32, the stick surface radius r_stick is smaller than the corresponding r_H by roughly 2σ, as shown by the cyan symbols in Figure 7 (b) and (c). This suggests that instead of being approximated as a sphere with hydrodynamic radius r_H, a tethered NP in these systems is more accurately modeled as a sphere with a core-shell structure, where the core of radius r_stick > d/2 consists of the bare particle and a non-draining layer of grafted chain segments that are coupled by hydrodynamic interactions, while the shell outside the core is a free-draining layer of grafted chain segments with no hydrodynamic coupling. The additional friction of this free-draining layer accounts for the additional 2σ of the hydrodynamic radius r_H in comparison with the stick surface radius r_stick. This two-layer description is a step-function approximation of the hydrodynamic coupling continuously decreasing with increasing r. In the systems where r_stick ≥ d/2 does not exist, a non-draining layer surrounding the particle cannot form, and the hydrodynamic radius r_H is above the particle radius by less than 2σ.

Following the analysis of the simulation results, we present a scaling estimate of the condition for the emergence of a non-draining layer of hydrodynamically coupled chain segments surrounding the bare particle. Consider chain segments with n monomers per chain in the free-draining layer. The friction per unit area is μ_n ≈ ζ₀n(z/d²) where z/d² is the number of grafted chains per unit area of the particle surface. Slip length at the outer boundary of the free-draining layer is estimated as λ_s ≈ η_melt/μ_n, where the melt viscosity η_melt ≈ ζ₀N/b. This slip length should equal the average end-to-end size of the chain segments in the free-draining layer outside the non-draining layer, λ_s ≈ bn^1/2, which is consistent with no slip on the stick surface between the two layers. As a result, n ≈ (Nd²/zb²)^2/3. The existence of a non-draining layer requires only a fraction of a tethered chain in the free-draining layer, i.e., n < N_g, leading to $z>\left(N{d}^2/b^2\right)N_g^{-3/2}$. The dashed magenta line $z=1024N_g^{-3/2}\approx 4.3\left(N{d}^2/b^2\right)N_g^{-3/2}$ in Figure 9 indicates where a non-draining layer emerges in the (N_g,z) parameter space of the simulations with N = 16, d = 5σ and b = 1.3σ. Numerical solution for r_stick in Eq.(1) exists for the systems indicated by the red stars in Figure 9.

4. Conclusions

We have used molecular dynamics simulations to investigate the effects of tethered polymer chains on the dynamics of NPs in melts of unentangled polymers. As summarized in Figure 9, depending on the number of monomers N_g per tethered chain and the number z of tethered chains per particle, there are three qualitatively different regimes.

(1). In Regime I, the friction ζ_g of loosely grafted chains is smaller than the friction ζ_bare of the bare particle, and therefore the terminal diffusion of tethered NPs is dominated by the particle. There is no hydrodynamic coupling between the grafted chains, and there is a full slip at the particle surface as the slip length L_s is larger than the bare particle radius d/2.

(2). In Regime II, ζ_g > ζ_bare, the tethered chains dominate the terminal diffusion. The boundary between Regime I and Regime II corresponds to ζ_g = ζ_bare (blue line in Figure 9). Note that the cyan symbols in Figure 9 indicate both the systems with ζ_g > ζ_bare and ζ_g = ζ_bare. In Regime II, there is no hydrodynamic coupling of grafted chains that can give rise to a non-draining layer surrounding the bare particle. No numerical solution for r_stick that satisfies Eq.(1) with r_stick ≥ d/2 exists. The boundary between Regime II and Regime III (magenta dashed line in Figure 9) corresponds to r_stick ≈ d/2. Across Regime II, the boundary condition on the bare particle surface changes from full slip to stick as z increases at the same N_g, and the slip length drops from L_s > d/2 to L_s = 0.

(3). In Regime III, the diffusion of tethered NPs is dominated by the grafted chains with hydrodynamic interactions. There is a non-draining layer of hydrodynamically coupled chain segments surrounding the bare particle. Outside the non-draining layer, there is a free-draining layer of other chain segments with no hydrodynamic coupling. Across Regime III, the radius r_stick of stick surface between the two layers increases from the particle radius d/2 to the interpenetration layer between grafted and melt chains as z increases. The two-layer description is a simplification of the actual hydrodynamic coupling that decreases continuously with increasing distance from the particle center.

In both Regime I and Regime II, the hydrodynamic interactions between grafted chains are effectively screened by the melt polymers, and the friction of loosely grafted chains scales linearly with the number of monomers in the grafted chains. Note that none of the systems used in Figure 4 that shows ζ_g ~ z(N_g −1) corresponds to a red star of Regime III in Figure 9. For z = 1 and z = 2, the predicted two qualitatively different sub-diffusive regimes of particle MSD in the tethered-chain-dominated case are observed for sufficiently large N_g, as indicated by solid symbols enclosed by black circles in Figure 9. The condition on the scaling level for the existence of two sub-diffusive regimes is (d/b)(N/N_g) < z < d/b (region of the diagram in Figure 9 outlined by the orange dashed lines).

The results of the simulations foster the understanding of the mobility of tethered NPs in unetangled polymer matrices. They are consistent with the recent scaling theory⁶ and existing experiments.⁵ The microscopic description based on the scaling theory and the molecular simulations needs further confirmation by experiments. We note that the distribution of grafted points on a particle surface in experiments is not as regular as in the simulations. However, the randomness does not change the behavior of tethered particles significantly, as long as it does not alter the non-interacting dynamics of chains in Regimes I and II or the hydrodynamic coupling of chains in Regime III.

Appendix

A. Size of Grafted Chains

We examine the variation of the average extension of grafted chains normal to the particle surface ${〈h_{e-e}^2〉}^{1/2}$ as the number of grafted chains per particle z increases. As shown in Figure A.1(a), ${〈h_{e-e}^2〉}^{1/2}$ of a grafted chain at z = 2 is only slightly above ${\left[r_{e-e}^2/3\right]}^{1/2}$, where $〈r_{e-e}^2〉$ is the mean square size of a free chain with N_g monomers in the same melt. For N_g = 8 and 32, ${〈h_{e-e}^2〉}^{1/2}$ remains almost constant for z up to 32. For N_g = 16, ${〈h_{e-e}^2〉}^{1/2}$ does not change much up to z = 32, and increases by less than 20% at z = 64. Note that z = 64 is close to the maximum possible number of grafted chains z_max = 80 on a particle with surface area πd² ≈ 80σ². We conclude that there is no strong extension of grafted chains as z increases towards z_max in the simulations. This conclusion is for the particular particle size d = 5σ and matrix chain length N = 16. A general theoretical description of the conformation of grafted chains in polymer melts is presented in a recent theory paper.⁶

Figure A.1:

The root mean square (RMS) radial component of the end-to-end vector of a grafted chain ${〈h_{e-e}^2〉}^{1/2}$ (symbols and error bars) as a function of z for indicated N_g. The solid lines show ${\left[r_{e-e}^2/3\right]}^{1/2}$, where $〈r_{e-e}^2〉$ is the mean square end-to-end size of free chains with N_g monomers per chain in the same melt. Error bars are smaller than the thickness of the lines.