Jagannathan T.
Kalathi
ab,
Sanat K.
Kumar
*a,
Michael
Rubinstein
c and
Gary S.
Grest
d
aDepartment of Chemical Engineering, Columbia University, New York, NY 10027, USA. E-mail: sk2794@columbia.edu
bDepartment of Chemical Engineering, National Institute of Technology Karnataka, Surathkal, KA 575025, India
cDepartment of Chemistry, University of North Carolina, Chapel Hill, NC 27599, USA
dSandia National Laboratories, Albuquerque, NM 87185, USA
First published on 20th April 2015
Large-scale molecular dynamics simulations are used to study the internal relaxations of chains in nanoparticle (NP)/polymer composites. We examine the Rouse modes of the chains, a quantity that is closest in spirit to the self-intermediate scattering function, typically determined in an (incoherent) inelastic neutron scattering experiment. Our simulations show that for weakly interacting mixtures of NPs and polymers, the effective monomeric relaxation rates are faster than in a neat melt when the NPs are smaller than the entanglement mesh size. In this case, the NPs serve to reduce both the monomeric friction and the entanglements in the polymer melt, as in the case of a polymer–solvent system. However, for NPs larger than half the entanglement mesh size, the effective monomer relaxation is essentially unaffected for low NP concentrations. Even in this case, we observe a strong reduction in chain entanglements for larger NP loadings. Thus, the role of NPs is to always reduce the number of entanglements, with this effect only becoming pronounced for small NPs or for high concentrations of large NPs. Our studies of the relaxation of single chains resonate with recent neutron spin echo (NSE) experiments, which deduce a similar entanglement dilution effect.
In our previous studies, using molecular dynamics (MD) simulations, we showed that the shear viscosity of a polymer melt can be significantly reduced when it is filled with small energetically neutral NPs (smaller than roughly half the entanglement mesh size).19 We deduced that small NPs act akin to solvent molecules and reduce the viscosity of a polymer melt in this “plasticization” limit. This effect is reversed for larger NPs, in which case they increase the viscosity of the polymer, as may be expected using classical theories such as those formulated by Einstein and by Batchelor.20 The reduction of viscosity seen for small NPs can also be overcome by increasing the attractive strength of NP–polymer interactions.
The diffusivities of the NP in a polymer melt are also found to be strongly dependent on their size.21 For NPs smaller than the polymer's entanglement mesh size, the relaxation times and NP diffusivity are described by the Stokes–Einstein relationship, where the viscosity is set by the segment of polymer chain with end-to-end distance comparable to the NP diameter. However, for NPs with diameters larger than the entanglement mesh size it appears that the competition of full chain relaxation vs. NP hopping through entanglement gates controls NP diffusion22 – however, there is no ready means to apply the Stokes–Einstein formula here.
Schneider et al.10 experimentally studied the relaxation of entangled poly(ethylene-alt-propylene) (PEP) chains (tube diameter ∼ 5 nm) filled with silica NPs (average diameter ∼ 17 nm). The silica volume fraction was varied in the range 0 ≤ ϕNP ≤ 0.6, where the ϕNP is calculated from the measured weight fraction of silica in the nanocomposite, and by assuming silica and polymer densities of 2.2 and ∼1 gm cm−1,3 respectively. Chain dynamics in these nanocomposites, with non-attractive interactions, are explored using neutron spin echo spectroscopy (NSE) and the resulting collective dynamic scattering function data analyzed using the idea of a tube-like confinement for chain relaxation below the reptation time. This procedure yields the following primary conclusions: (i) the monomeric relaxation rates (see below for definition) are unaffected by the addition of NPs, even at high particle loadings; (ii) chain conformations remain Gaussian for all loadings considered; and (iii) the tube diameter determined from analysis of NSE data decreases monotonically with added NPs. It is argued that there are two contributions to overall chain dynamics, and how they are affected by the addition of the NPs. The number of topological chain–chain entanglements decreases with increased NP loading, i.e., the chains disentangle from each other since a part of the system volume is occupied by the NPs. This is (more than) compensated by the geometric constraints that NPs present to chain dynamics. Since the second factor dominates at large loadings, the NSE reports an increase in chain relaxation time, while at the same time a reduction in the number of intra-chain entanglements.
Several of these experimental deductions have been considered by Li et al.,17 who conducted MD simulations on melts of well-entangled chains of length N = 500 with a single sized (10σ, where σ is the diameter of the chain monomers) NP (comparable to the size of the tube diameter). They used a primitive path analysis (PPA) assuming that the NPs were “phantoms” – that is the NPs are penetrable in the PPA and hence do not interfere with the chain “straightening” inherent in this calculation. The simulation-derived collective scattering functions were used to deduce the net effective tube diameter, which defines this collective motion (and presumably convolutes the effects of the NPs in chain dynamics and also chain–chain entanglements) in apparently good agreement with the experiments of Schneider et al. However, there is some uncertainty about the role of the NPs in the PPA: while Li et al., use a phantom description where the chains can penetrate the NPs (which will naturally yield an entanglement dilution effect as seen in the experiments), there is a second possibility where the NPs are held fixed and impenetrable to the polymer chains, which would yield an increase in the number of entanglements. The latter scenario should clearly be operative if the chains are strongly adsorbed (strongly favorable NP–polymer interactions). However since attractive NP–polymer interactions are necessary to ensure the miscibility of the mixture,23,24 absent the experimental results, it is a priori unclear which of these descriptions is accurate. Clearly, a method that does not involve the PPA would help to unequivocally clarify this point.
A related issue is the role of polymer–NP attractions. Specifically, Smith et al.18 carried out MD simulations and found slower chain dynamics in attractive PNCs, compared to repulsive systems. This has been attributed to heterogeneity in relaxation of chains arising from polymers adsorbed on the NPs.
The main objective of this work is to examine the role of NPs on chain relaxation in nanocomposites for both unentangled and entangled melts. NPs ranging in size from the chain monomer to ∼1.5 times the tube diameter are studied, particularly because it has been conjectured that NP diffusion (and hence presumably dynamics) change dramatically in character when their size goes from well below the entanglement mesh size to well above it.25 We use the Rouse modes of the chains (which are equivalent to the normal modes of short chain melts) to show that the addition of weakly attractive NPs always reduces interchain entanglements, with these effects having different origins for small vs. large NPs. For NPs larger than the entanglement mesh size the dominant effect comes from the fact that some part of the volume is taken up by the NPs (“entanglement dilution”). For small NPs, there is both a reduction of monomer friction and a decrease of entanglement density. As an experimental direction, we propose that probing the self-intermediate scattering function of the polymer might give direct evidence into this entanglement dilution effect. This would directly complement previous NSE measurements and could help to separate out the NP induced entanglement dilution effects from the NP confinement effect on chain motion. The role of the interaction strength between a NP and the polymer is also studied. We present a scaling model explaining some of the observed phenomena.
The Rouse modes, p = 0, 1, 2,…, N − 1, of a chain of length N are defined as:28. The p = 0 mode describes the motion of the chain center-of-mass, while the modes with p ≥ 1 describe internal relaxations with a mode number p corresponding to a sub-chain of (N − 1)/p segments. The autocorrelation of the Rouse modes, is predicted to decay exponentially and independently for each mode p for an ideal chain with a relaxation time τp where and . Simulations of homopolymer melts have found that the Rouse mode autocorrelations are better described by a stretched exponential form:14,16,17. The effective relaxation times of mode p can be obtained by integrating this relaxation function:14,16,17,29 where Γ(x) is the gamma function. The effective monomeric relaxation rate is , and for the Rouse model this quantity should be independent of mode number and only depend on the monomer friction, temperature and the statistical segment length b.
It is well known that the Rouse modes are not the correct normal modes of long polymer melts in the entangled regime.30 They clearly violate the fundamental principle of mode decoupling in the Rouse model, but provide a useful description for comparing chain relaxation in polymer nanocomposites to homopolymer melts. More pertinently, experimentalists often model chain dynamics in the language of the Rouse model. Understanding experimental results therefore leads us to analyze the simulations in the same manner. In our previous work29 on neat melts of short unentangled chains we found that the stretching exponent that defines the Rouse mode autocorrelation functions, βp, increases from ∼0.8 for large p to ∼1 for the p = 1 mode. The situation for entangled chains is quite different. For long well-entangled chains, the large p modes have a stretching exponent ∼0.8. However, βp decreases with decreasing mode number, reaching a minimum of ∼0.5 for modes that are in the vicinity of the entanglement length Ne. Li et al.16 also found that the minimum in βp occurs for N/p ∼ Ne. Previous work by Padding and Briels16 and by Shaffer14 suggest that this minimum in βp is due to kinetic constraints on the chains.31,32 To summarize, there are two essential results that we shall employ here to understand the role of NPs on chain dynamics. First, for well-entangled chains the minimum value of βp ∼ 0.5 which occurs for N/p ∼ Ne. Second, as one decreases the chain length towards Ne, the minimum value of βp no longer occurs at βp ∼ 0.5 (it becomes progressively higher for shorter chains), and the location of this minimum, i.e., N/p, is no longer at Ne, but rather at some smaller value.
The NPs are modeled as bare smooth spheres of diameter, σNP, composed of uniformly distributed monomers of the size of a polymer segment with a mass density, ρNP* = ρNPσ3 = 1. Under these assumptions, the Lennard-Jones interactions between a polymer segment and the NP-segments are integrated over all the NP spheres to obtain the effective interaction:37–40
The Hamaker constant for NP–NP interactions is Ann = 4π2εnnρNP2σ6. Since we use the same LJ potential for the interactions between two polymer-beads and between two beads comprising a NP, εnn = ε, we have Ann = 39.48ε. We employed a cut-off distance rc = σNP so that inter-NP interactions are purely repulsive. In a similar vein, NP-chain monomer interactions are governed by Anp = 24πεnpρNPσ3. While εnp = ε yields Anp = 75.3ε, we find NP agglomeration for this interaction energy and a cut-off distance .23 Instead, we use a larger Anp = 100ε for σNP > 3σ and 120ε for σNP = 3σ (and smaller NPs) to avoid NP agglomeration,23 but the NPs are still neutral to the polymer as evident from NP–polymer radial distribution functions, shown below. We have also considered higher interaction strengths Anp in a few cases to study its effect on chain relaxation.
Most of the results presented here are for NPs of diameter σNP = 1–15σ at fixed NP loading , where N, MNP and MC are the chain length, number of NPs and number of polymer chains, respectively (Table 1). Though we have studied N = 10, 20, 40, 60, 80, 100, 200 and 400 we mainly discuss N = 40, 100 and 400. To test the effect of increasing NP loading, we also simulated a system with ϕNP = 0.6 for N = 400 and σNP = 10. All of the simulations are carried out using the large scale atomic molecular massively parallel simulator (LAMMPS).41 The initial configurations of neat and NP-filled polymer systems are prepared at random at a constant number density while allowing for overlaps among beads. The overlaps are removed by initially using a soft potential between polymer monomers, and then by gradually increasing the strength of the potential. After all overlaps are removed, the LJ interactions between polymer monomers is turned on and the volume of the simulation cell is allowed to adjust at a constant reduced pressure P = 0. Systems of chain length N = 100–400 are equilibrated following the double-bridging procedure.34 The shorter N melts are equilibrated by running isobarically, and then at constant volume until the chains have moved their own size multiple times. After equilibration, the systems are run at constant volume at temperature with a Langevin thermostat with damping constant Γ = 0.1τ−1. For longer chain lengths we find that the average pressure P = (0 ± 0.05)ε/σ3, whereas for shorter chains P = (0 ± 0.1)ε/σ3. The number of chains and NPs, and volume of simulation cell of the system studied in the present work along with static properties of chains are listed in Table 1.
NP diameter σNP | Chain length N | Number of chains MC | Number of NP MNP | Volume fraction of NP φNP | Length of simulation box L (σ) | 〈Rg2〉1/2 (σ) |
---|---|---|---|---|---|---|
15 | 10 | 15188 | 5 | 0.10 | 57.11 | 1.58 |
15 | 20 | 7594 | 5 | 0.10 | 56.88 | 2.36 |
15 | 40 | 3800 | 5 | 0.10 | 56.80 | 3.45 |
15 | 60 | 2500 | 5 | 0.10 | 56.52 | 4.27 |
15 | 80 | 1880 | 5 | 0.10 | 56.55 | 4.97 |
15 | 100 | 1800 | 6 | 0.10 | 60.03 | 5.58 |
15 | 200 | 1215 | 8 | 0.10 | 66.30 | 7.95 |
15 | 400 | 600 | 8 | 0.10 | 66.02 | 11.26 |
10 | 10 | 9000 | 10 | 0.10 | 48.07 | 1.58 |
10 | 20 | 4500 | 10 | 0.10 | 47.87 | 2.36 |
10 | 40 | 2000 | 10 | 0.11 | 46.04 | 3.44 |
10 | 60 | 1500 | 10 | 0.10 | 47.74 | 4.27 |
10 | 80 | 2250 | 20 | 0.10 | 60.13 | 4.96 |
10 | 100 | 900 | 10 | 0.10 | 47.72 | 5.57 |
10 | 150 | 900 | 15 | 0.10 | 54.62 | 6.87 |
10 | 200 | 500 | 11 | 0.10 | 49.41 | 7.95 |
10 | 400 | 500 | 23 | 0.10 | 62.29 | 11.39 |
10 | 400 | 500 | 300 | 0.60 | 75.81 | 11.33 |
8 | 10 | 9210 | 20 | 0.10 | 48.51 | 1.58 |
8 | 100 | 500 | 11 | 0.10 | 39.29 | 5.72 |
8 | 400 | 500 | 45 | 0.10 | 62.39 | 11.37 |
5 | 10 | 3000 | 27 | 0.10 | 33.58 | 1.57 |
5 | 20 | 2000 | 40 | 0.11 | 36.92 | 2.36 |
5 | 40 | 1000 | 40 | 0.11 | 36.84 | 3.44 |
5 | 60 | 500 | 27 | 0.10 | 33.36 | 4.27 |
5 | 100 | 500 | 45 | 0.10 | 39.54 | 5.58 |
5 | 150 | 500 | 67 | 0.10 | 45.21 | 6.88 |
5 | 200 | 500 | 90 | 0.10 | 49.77 | 7.97 |
5 | 400 | 517 | 200 | 0.11 | 63.49 | 11.57 |
3 | 10 | 1000 | 42 | 0.10 | 23.54 | 1.57 |
3 | 20 | 500 | 42 | 0.10 | 23.45 | 2.36 |
3 | 40 | 1000 | 185 | 0.11 | 37.31 | 3.45 |
3 | 100 | 500 | 206 | 0.10 | 39.94 | 5.59 |
3 | 200 | 500 | 411 | 0.10 | 50.29 | 7.99 |
3 | 400 | 500 | 825 | 0.10 | 63.37 | 11.26 |
1 | 10 | 500 | 555 | 0.10 | 18.75 | 1.58 |
1 | 20 | 500 | 1111 | 0.10 | 23.56 | 2.38 |
1 | 40 | 500 | 2300 | 0.10 | 29.65 | 3.48 |
1 | 100 | 500 | 5555 | 0.10 | 40.14 | 5.81 |
1 | 200 | 500 | 11111 | 0.10 | 50.56 | 8.31 |
1 | 400 | 500 | 26000 | 0.12 | 64.07 | 11.94 |
Neat | 10 | 500 | — | — | 17.95 | 1.58 |
Neat | 20 | 1000 | — | — | 28.37 | 2.36 |
Neat | 40 | 500 | — | — | 28.28 | 3.45 |
Neat | 60 | 500 | — | — | 32.38 | 4.40 |
Neat | 80 | 500 | — | — | 35.62 | 5.10 |
Neat | 100 | 500 | — | — | 38.37 | 5.74 |
Neat | 150 | 500 | — | — | 43.91 | 6.91 |
Neat | 200 | 500 | — | — | 48.33 | 8.20 |
Neat | 400 | 500 | — | — | 60.80 | 11.43 |
The amplitudes of the autocorrelation function of the Rouse modes for different chain lengths fall on a master curve for chain lengths N = 20–400 (Fig. 1(e)). We find that this master curve follows where C∞ = 2.05 and k2 = 4.69. Here we remind the reader that the Rouse model predicts that independent of p, and hence the amplitudes depend purely on static chain conformations. Consistent with this fact, the become asymptotically independent of p, and the characteristic ratio C∞ ∼ 2.05 is obtained asymptotically for . When chain bending (stiffness), and excluded volume interactions affect this function. However, the important conclusion here is that chain conformations are essentially unaffected by the NPs at all length scales.17,42
There are also several pieces of information that can be deduced on the role of NPs on chain dynamics. First, we focus on the stretching exponent, βp. Fig. 1(f) shows that a minimum value of βp ∼ 0.5 is attained in all cases, consistent with the fact that the N = 400 chains are long enough to proper delineate the role of the NPs on entanglements at this particle concentration. The location of this minimum changes from ∼50 ± 4 (neat melt) to ∼65 (for NPs with σNP = 1 and 3) and then ∼55 for all larger NPs (5 ≤ σNP ≤ 15) (see Table 2). The results from the larger NPs are consistent with the notion that we have added 10% by volume of the NPs. These NP must result in a reduction in the number of entanglements, and our results suggest that this effect is about a 20% increase in Ne, as quantified in this fashion. More generally, these findings are consistent with our previous conjectures that small NPs act as plasticizers and reduce the entanglement density (and thus increase the entanglement length). For σNP = 1 and 3 we find that Ne is about 40% larger than the value obtained for large NPs.
NP size | N/p | Uncertainty (+/−) |
---|---|---|
Neat melt | 50 | 4 |
1σ | 66 | 3 |
3σ | 64 | 3 |
5σ | 56 | 4 |
8σ | 54 | 3 |
10σ | 56 | 3 |
15σ | 54 | 3 |
To independently verify these conclusions, we compare the relaxation times of the different modes for chains in the PNCs for three different degrees of polymerization N filled with NPs of different sizes for ϕNP = 0.1 to neat melts without NPs [Fig. 2(a)–(c)]. For the shortest chain length (N = 40), which corresponds to an unentangled melt, the relaxation time ratio of the pth modes are effectively independent of mode number p except for the smallest NP size which show a decrease of this ratio. The smallest NPs are expected to reduce the monomeric friction, i.e., they act as a solvent for the polymer. These effects decrease with increasing particle size, and for the largest NPs the monomeric relaxation time ratio is effectively equal to unity. These findings for the shortest chains are in good agreement with the experiments of Schneider et al.11
Fig. 2 Normalized effective relaxation times of p-th mode for chains in nanocomposites for different NP sizes at ϕNP = 0.1: (a) N = 40; (b) N = 100; (c) N = 400. (d) Effect of NP loading for N = 400, σNP = 10σ (Closed triangles correspond to σNP = 10σ in N = 500 at similar NP loading from ref. 17). (e) Corresponding plot for the stretching exponent βp. |
The trends for the long chains are richer. For small N/p, we see a plateau for that depends on NP size. We believe that this plateau is related to the effect of NPs on monomer friction (Fig. 2(c)). There also appears to be a monotonic decrease of the relaxation time ratio for larger N/p, and finally a plateau for .19 We assume that the relaxation time for a chain follows the crossover ansatz that smoothly bridges between the Rouse model and reptation dynamics: , where τ0 is the monomer relaxation time. In this ansatz, the large p modes directly yield information on the monomer friction and how it is modified by the addition of the NPs. In contrast, in the limit of p = 1, the plateau is directly proportional to the ratio of τ0/Ne in the PNC compared to that in the pure melt. Our results for the longest chains clearly show that the monomeric relaxation times are decreased by (∼30%) by the addition of σNP = σ NPs. However, this effect disappears for larger NPs, i.e., 3σ and larger. This is in good agreement with our previous results.19 More interesting are the findings that are apparent in the low p modes: for small NPs, which act as a diluent, there is an additional speedup, which we attribute to a reduction in entanglements. Our relaxation time ansatz suggests that , from which it follows that Ne,melt/Ne,NP ∼ 0.9 in this regime for N = 400, which is in very good agreement with the results obtained from the stretching exponents discussed above.19 Our results also show that the entanglement length recovers its melt value with increasing NP size, but that even for σNP = 15σ, Ne,NP is ∼10% larger than the pure melt, Fig. 2(c) and (e).
These results are also in reasonable agreement with our previous work, where a primitive path analysis (PPA)19 with penetrable NPs, suggested that the presence of NPs smaller than the tube diameter increased the entanglement molecular weight even at low concentrations. These results are also consistent with the effect of NP size on zero-shear viscosity where smaller NPs act as plasticizers and reduce the viscosity. These effects disappear once the particle size becomes comparable to the tube diameter, where the additional confinement effects of the NPs on chain motion results in an increased viscosity relative to the pure melt (see Discussion for more on this topic).
The monomeric relaxation rates, Weff, of different chain lengths melts with the smallest and the largest NP size studied are compared with that of neat melts in Fig. 3. It is evident that small NPs plasticize the melts and hence the relaxation rates are always higher than that of the neat melt. However, the largest NP do not affect the relaxation of chains and Weff of the neat and filled melts overlap for all modes to within the uncertainties in the simulations.15,43
Fig. 3 Monomeric relaxation rates for different chain lengths at ϕNP = 0.1 for (a) σNP = 1σ and (b) σNP = 15σ. Solid lines correspond to neat melts. |
A second point that emerges is that the NSE experiments, though relevant to understanding the composite effect of NPs on polymer dynamics, are not sensitive to exclusively probe inter-chain entanglement effects in the presence of NPs. Our results clearly show (as do the previous work of Li et al.) that the Rouse modes characterizing the internal dynamics of single chains are extremely sensitive to chain–chain entanglements without any interference from the confining effects of the NPs. Since the motion of chains are best characterized in the framework of the van Hove (or the self-intermediate scattering) function, we propose that experiments that focus on other methods for delineating this quantity might shed new light on the relaxations of these systems.
It is also interesting to focus on the novelty of the current work relative to the published literature. Li et al.17 for example considered a very similar situation, but they only considered a single NP size, but then varied the NP loading. Our work, in contrast, allows for the polymer length, NP size and NP loading to vary, thus providing a much more complete picture of the dynamics of these filled systems. Smith et al.18 studied the role of attractions on chains dynamics. Our previous work in this area looked at the effect of NP on the viscosity of polymer melts,19 and also the Rouse modes of neat melts29 – the current work looks at the Rouse modes of chains in the presence of NPs.
Finally, we focus on understanding the consequence of our results on the viscosity of nanocomposites filled with NPs. In this work we split this effect into NP contributions to the relaxation time and to modulus. The product of these two effects yields the role of NP on viscosity. The viscosity of the nanocomposites can be written as an integral of the stress relaxation function which can be split into short-time modes with wavelength smaller than the particle size: and long-time modes . The effective viscosity of a neat melt of polymers of size equal to the particle size is hardened by particles smaller than tube diameter σNP < a with volume fraction ϕNP as described by Einstein equation . This hardening occurs due to an increase of modulus of PNC without a change of relaxation rates of short time modes. For long-time modes NPs act as diluents and their contribution can be estimated as where with bulk viscosity of neat melt η0. Thus, the total viscosity of the nanocomposites is η = η0 + ϕnp[(6π + 1)ηeff − η0] that can be either larger or smaller than viscosity η0 of the neat melt depending on the relative value of (6π + 1)ηeff and η0 (or relative value of particle and polymer size). This conclusion is in good agreement with our previous simulation results,19,29 which showed that the critical variable for determining the effect of NP on viscosity was the NP size relative to a characteristic chain size. For short, unentangled chains this corresponds to the radius of gyration of the chains, while for longer chains it is the entanglement mesh size.
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