Article Cite This: Macromolecules XXXX, XXX, XXX−XXX
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Effect of Copolymer Sequence on Local Viscoelastic Properties near a Nanoparticle Alex J. Trazkovich,†,‡ Mitchell F. Wendt,† and Lisa M. Hall*,† †
William G. Lowrie Department of Chemical and Biomolecular Engineering, The Ohio State University, 140 W 19th Ave., Columbus, Ohio 43210, United States ‡ Cooper Tire & Rubber Company, 701 Lima Ave., Findlay, Ohio 45840, United States
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S Supporting Information *
ABSTRACT: We simulate a simple nanocomposite consisting of a single spherical nanoparticle surrounded by coarse-grained polymer chains that are composed of two monomer types differing only in their interactions with the nanoparticle. We measure the atomic stress fluctuations and use them to estimate the local stress autocorrelation as a function of distance from the nanoparticle. This local stress autocorrelation is substituted into the well-known relationship between the bulk stress autocorrelation and the bulk (frequency-dependent) dynamic modulus, and the result is treated as an estimate of the local dynamic modulus. This allows us to examine the effect of adjusting copolymer sequence on estimations of local storage and loss modulus as a function of distance from the nanoparticle. Notably, we find certain blocky copolymer sequences can lead to a higher tan(δ) (hysteresis) in the interphase than either homopolymer system, suggesting that tuning the copolymer sequence could allow for significant control over nanocomposite dynamics.
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temperature lower.19 The polymer−filler interaction strength is dependent on the polymer type and nanoparticle surface chemistry, so interphase properties can be adjusted by functionalizing the nanoparticle surface or changing the polymer chemistry. Because of the high surface-to-volume ratio of the nanoparticles, a significant fraction of polymer chains may be incorporated in the interphase even at low nanoparticle loading. As the nanoparticle loading is increased, the overall composite properties usually begin to more closely resemble those of the interphase (up to a certain threshold, above which composite properties become increasingly dependent on filler−filler network effects).31,32 The relationship between interphase properties, the nanoparticle volume fraction, and the overall composite properties has been characterized using analytical theories.33−36 Therefore, in order for material designers to understand and predict overall composite behavior, it is beneficial to have an understanding of the properties of the interphase. One material property of significant interest to material designers is the elastic modulus, which measures a material’s stiffness and resistance to deformation under static load. Generally, introducing nanofiller with favorable interactions with the polymer matrix increases modulus, and adjusting surface chemistry to increase the polymer−filler interaction
INTRODUCTION Nanoparticles are incorporated into polymers to form nanocomposite materials with enhanced properties for many industrial, commercial, and scientific applications. Significant research effort has focused on understanding the connection between polymer and nanoparticle features (e.g., size and chemical interactions) and composite properties, including dynamic mechanical properties such as modulus and energy dissipation, to allow for rational design of improved materials. Multiple molecular simulation studies have made progress in this area,1−12 and particular attention has been paid to the polymer−nanoparticle interphase region. It is clear that local dynamics and polymer chain conformations in the interphase are significantly different than in the bulk polymer, which is one reason why adding nanoparticles has such a large impact on the overall composite properties.13−23 The properties of the polymer−nanoparticle interphase are heavily influenced by the polymer−filler interaction strength. In most commercial applications, polymer−filler interactions are favorable, which helps prevent nanoparticle aggregation that would otherwise occur due to entropic depletion. When interactions are favorable, polymers adsorb on the nanoparticle, which results in slower dynamics and an increased glass transition temperature in the interphase.24−29 Adsorbed chains may also dynamically couple with nearby nonadsorbed chains, slowing their dynamics and extending the effective range of the interphase.30 Meanwhile, when components are selected such that interactions are unfavorable, interphase dynamics may be faster than the bulk and the glass transition © XXXX American Chemical Society
Received: October 4, 2018 Revised: November 20, 2018
A
DOI: 10.1021/acs.macromol.8b02136 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules strength tends to increase modulus further.33,37 Several analytical theories exist to predict composite modulus, often using three-phase models that account for the moduli of the polymer matrix, the filler, and the interphase region as well as the relative volume fractions of each.38−41 Elastic modulus may be measured locally using atomic force microscopy, and experimental studies have used this technique to demonstrate that modulus increases in the interphase between polymer and hard surfaces, provided that the polymer has good affinity for the surface.42−44 The static modulus may also be measured locally in simulation, where a local sum of interatomic forces can be used to calculate the “Born term”, the theoretical instantaneous elastic modulus that would be experienced in response to a uniform, affine, infinitesimal strain.45,46 Alternatively, the second derivative of free energy with respect to strain can by used to derive a more complex formulation of the local modulus which is the sum of a Born term, a kinetic energy term, and a stress fluctuation term that accounts for nonaffine local particle movement. A substantial body of work has developed and used this technique to study molecular scale mechanical heterogeneities in polymer glasses and crystalline materials.47−50 Although these simulation studies have provided a clear method to measure local static modulus, and meanwhile experimental studies have shown that static modulus increases near nanoparticles, none of this particular body of work considered local dynamic modulus. The dynamic modulus, which varies as a function of the frequency of applied strain, can be described as the sum of a storage modulus and a loss modulus, which respectively are the components of the response in phase and out of phase with the applied oscillatory strain. The ratio of the loss modulus to the storage modulus gives the ratio of the amount of energy lost to the amount of energy returned in a single cyclic excitation, and it is therefore a measure of the material’s viscoelastic hysteresis. Dynamic moduli and, in particular, hysteresis are important in any application where a polymer nanocomposite is subject to cyclic or changing loads. In filled rubbers, low hysteresis is desirable when the goal is to reduce energy loss and heat buildup, while high hysteresis is desirable in applications where the goal is to dissipate energy under cyclic load. In some applications, the design challenge may be even more complex, requiring low hysteresis under certain operating conditions while maintaining high hysteresis under a different set of operating conditions. For example, in car tires, maintaining low rolling resistance (and therefore good fuel economy) requires low hysteresis at the relatively low loading frequencies associated with tire rotation,51−53 while achieving good traction requires high hysteresis at the relatively high loading frequencies associated with tread surface deformation during sliding across road surface asperities.54,55 These types of design challenges have led material designers to employ a variety of methods to control hysteresis and other viscoelastic properties. Introducing nanofiller is known to impact energy dissipation, and material hysteresis can be either increased or decreased depending on the filler size, loading, and polymer−filler chemistry. When multiple polymer components interact near the nanoparticle surface, the resulting dynamics include interesting and sometimes surprising effects. For example, a significant body of work from the Akcora group has studied nanoparticles coated in highly adsorbing chains of a homopolymer with relatively high glass transition temperature (Tg) that are dispersed in a matrix of a
different homopolymer with a lower Tg.56−58 When the temperature is above the Tg of the bulk chains but below the Tg of the adsorbed chains, the adsorbed chains are in a glassy state and so are dynamically decoupled from the surrounding matrix. Meanwhile, at temperatures above Tg of both polymers, the adsorbed chains are mobile and so dynamically couple with the surrounding chains, dramatically increasing reinforcement. The result is a system that increases in modulus and transitions from a more liquidlike to an elastic state as temperature is increased, demonstrating one striking result that can be achieved when two polymer components with different affinities for a nanoparticle interact near a surface. Another strategy that has been used to adjust mechanical properties has been the development of tailored blocky copolymers.59−62 In copolymer−nanoparticle composites, as in nanoparticle-reinforced blends of homopolymers, some monomers in the system may interact more favorably with the nanoparticle than others,63 although unlike in homopolymer blends, monomers with different affinities for the nanoparticle may be present on the same chain. In these conditions, the specific copolymer sequence would be expected to affect the adsorption of the polymer chain to the filler surface. Because a crucial mechanism underlying viscoelastic properties is the adsorption−desorption process,20,64 adjusting sequence in copolymer−nanoparticle composites potentially provides a way to significantly alter material dynamics. Thus, we seek to develop a better understanding of viscoelastic properties in the polymer−filler interphase, especially in terms of their dependence on copolymer sequence. In the current work, we consider nanoparticles incorporated into copolymer systems with two monomer types in various sequences. Several previous simulation studies have considered nanocomposites with adjustable copolymer sequences, with the majority of these examining the effect of grafted copolymer chain sequence on nanoparticle interactions and selfassembly.65−67 It is well-known that in blocky copolymer systems strong enough unfavorable interactions between unlike monomers may lead to microphase separation, which may in turn affect the polymer−nanoparticle interphase, the composite modulus,68,69 and nanoparticle dispersion.70−74 A small number of simulation studies have specifically examined properties of the copolymer−nanoparticle interphase and have determined that structure and chain conformations around a nanoparticle depend on both the relative interaction strength of the two monomer types75 and the specific block sequence.76 While most studies include unfavorable interactions between unlike monomers, some prior theoretical work has shown that even when all monomer−monomer interactions are equal and monomers differ only in their interactions with nanoparticles (as is the case in the current work), copolymer structure near nanoparticles still depends significantly on copolymer sequence.77−80 We also previously considered such an idealized system, as this allows us to focus on the effect of adsorption strength alone without the confounding effects of bulk microphase separation. These systems can also be considered to be relevant to heterogeneous systems with low binary interaction parameters such as copolymers of poly(methyl methacrylate) (PMMA) and poly(ethylene oxide) (PEO) or poly(vinyl acetate) (PVA) and PEO, where the polymer components are very miscible with one another but may exhibit significant differences in their interactions with a nanoparticle. However, unlike in those B
DOI: 10.1021/acs.macromol.8b02136 Macromolecules XXXX, XXX, XXX−XXX
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monomers adsorb strongly to the nanoparticle while A monomers do not. We use m, σ, and ε as our reduced units of mass, length, and energy, which are defined based on the length scale and strength of the interaction between nonbonded monomers. This further defines the reduced unit of time as τ = σ(m/ε)1/2. All monomers have mass 1.0m, and the nanoparticle has mass 1000m. The mapping of ε, σ, and m to real experimental units depends on the system being modeled, the temperature considered, and the degree of coarse-graining, but the process is relatively straightforward. Mapping τ to real time units in coarse-grained systems is much more complex, since coarsegraining removes degrees of freedom, shortening time scales relative to those of atomistic simulations or experimental systems.85,86 If the beads in our system were considered to represent approximately the mass and length scales of Kuhn segments of polybutadiene, and if the temperature modeled was considered to be close to room temperature, then ε, σ, and m would take on values of approximately 2.5 kJ/mol, 0.99 nm, and 113 g/mol, respectively. 87 If one uses the most straightforward mapping (simply applying the equation τ = σ(m/ε)1/2), τ would then be ∼2 × 10−10 s. Because this simplistic mapping does not consider the effect of lost degrees of freedom, the result is effectively a lower bound on τ. These numbers are provided only as a reference; since our goal in this work is to understand basic physical mechanisms, we have intentionally not selected a specific experimental system to model, so we will proceed using reduced units only. Monomers are bonded with the finitely extensible nonlinear elastic (FENE) potential:
experimental systems, our two model components do not have different glass transition temperatures. We consider the same model as in our previous simulation study81a simple nanocomposite consisting of a single nanoparticle incorporated in a copolymer melt. We use two monomer types that differ only in their strength of adsorption to a nanoparticle, and the blockiness of the copolymer sequence is adjustable (Figure 1). Previously, we found that
Figure 1. Representative snapshots of selected polymers from two simulated systems. Pink beads adsorb more strongly to the nanoparticle (purple) than cyan beads.
copolymer sequence affected ordering of monomers in the interphase and polymer radius of gyration as well as end-to-end autocorrelation, bond vector autocorrelation, and self-intermediate scattering function relaxation times, all of which tended to increase with proximity to the nanoparticle and with copolymer block length. Because a clear method to characterize local dynamic modulus and hysteresis is currently lacking, we now develop a new method to analyze atomic stress fluctuations and estimate the dynamic mechanical properties in the interphase, and we use this method to show that copolymer sequence has a significant effect on interphase material properties. By binning groups of atoms based on their distance from the nanoparticle, we calculate a local stress autocorrelation function. We then apply a method analogous to a method commonly used to relate the bulk stress autocorrelation to the bulk modulus. We treat the result as an estimate of the local dynamic modulus, which allows us to develop and analyze the resulting estimations of local storage modulus, loss modulus, and hysteresis as a function of copolymer sequence and distance from the nanoparticle. Although a true measurement of the local dynamic modulus is frustrated by the fact that particles move during the course of the simulation window associated with a given frequency of excitation, our results still provide perspective about the effect of copolymer sequence on the local properties in the polymer−nanoparticle interphase.
UFENE
ÄÅ É l 2Ñ ÅÅ o o ij r yz ÑÑÑÑ Å 1 o 2 Å o o o− kR 0 lnÅÅÅÅ1 − jjjj zzzz ÑÑÑÑ r ≤ R 0 =o m 2 ÅÅÇ k R 0 { ÑÑÖ o o o o o r > R0 o n0
(1)
where r is the distance between the monomers, R0 is the bond cutoff distance, set to 1.5σ, and k is a constant that sets the energy of the bond, for which we use the standard value of k = 30ε/σ2 to prevent chain crossing or scission.83 Monomer−monomer pairwise interactions follow a standard cutoff and shifted Lennard-Jones (LJ) potential:
ULJ, ij
É ÅÄÅ σ 12 l 6Ñ o ÅÅij ij yz o ij σij yz ÑÑÑÑ o Å o j z j z 4 ε − o ijÅÅÅj z j z ÑÑÑ + δ r ≤ rc r =o m k { k r { ÑÑÖ Å Å Ç o o o o o o r > rc n0
(2)
Here, εij is the interaction strength, σij is the interaction length scale, and the subscripts i and j refer to the monomer types. The cutoff distance, rc, is set to 21/6σij for bonded monomers and 2.5σij for nonbonded monomers. δ is a vertical shift factor selected so that U(rc) = 0. Monomer sizes and monomer− monomer interaction strengths do not depend on monomer type, so σAA = σAB = σBB = 1.0σ and εAA = εAB = εBB = 1.0ε. Unlike monomer−monomer interactions, monomer−nanoparticle interactions depend on monomer type. Here, we use a radially shifted LJ potential, which has been used to govern polymer−nanoparticle interactions in several prior simulation studies of coarse-grained polymer nanocomposites.8,13,17,88 This potential is defined as
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METHODS We use a standard attractive Kremer−Grest bead−spring model82,83 in which polymers are freely jointed chains of coarse-grained monomer beads. Our systems consist of 400 linear chains of length N = 100 (which are expected to have fewer than two entanglements per chain84). The simulation box has periodic boundary conditions, and in the center of the simulation box is a nanoparticle with an effective diameter 10 times the size of an individual monomer. We consider two monomer types, A and B, which differ only in that B C
DOI: 10.1021/acs.macromol.8b02136 Macromolecules XXXX, XXX, XXX−XXX
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UNM
l r−Δ≤0 ∞ o o o o ÄÅ É o o 12 6Ñ Å Ñ o ÅÅi σ yz o i σ yz ÑÑÑ zz − jjj zz ÑÑ + δ 0 < r − Δ ≤ rc =o m 4εNMÅÅÅÅjjj o o r − Δ Å k { k r − Δ { ÑÑÖÑ o Å Ç o o o o o o0 r − Δ > rc n
default equations of motion.93 Initial chain conformations were generated as random walks with 0.96σ between bonded monomers, and monomer locations were rejected and regenerated if a location would fall inside the nanoparticle. Initial overlap between monomers was eliminated using a short soft push-off phase preceding equilibration. A time step of δt = 0.01 was used throughout this study. Equilibration was performed in an isobaric−isothermal (NPT) ensemble using a Nosé−Hoover thermostat (damping parameters 1.0) and barostat (damping parameter 10.0) to hold the reduced temperature and pressure at 1.0 and 0, respectively. Equlibration ran for 200000τ. This was 10 times as long as the bulk end-to-end vector autocorrelation function relaxation time (∼1.9 × 104τ), and during this time, the monomers’ root-mean-square displacement reached 16.8σ more than 3 times the average radius of gyration (∼5.1σ). This suggests that the polymers had sufficient time to explore the simulation box and find their preferred conformations. After equilibration in NPT, the system was switched to a microcannonical (NVE) ensemble by removing the thermostat and barostat and fixing the volume at the average of the previous 1 million timesteps. After this switch, the length of each side of the cubic simulation box was ∼36σ. An additional equilibration of 50000τ was performed after changing ensembles and before saving data for analysis to ensure that the system had time to adjust to the new fixed volume. The size of the simulation box was sufficient to ensure that in all systems studied, with the nanoparticle defined as the center of the simulation box, the radius of gyration was within 2% of its bulk value for polymer whose centers were 6σ from the box edge, and the A and B monomer−nanoparticle pair distribution functions were within 2% of 1.0 by 2σ from the box edge. Data were saved for analysis at every time step in five trajectories which each ran for 100τ (10000 timesteps) and whose initial configurations were separated by 50000τ, well past twice the end-to-end vector autocorrelation function relaxation time. At each time step, atomic stresses on each monomer are calculated according to
(3)
where σNM is the effective diameter of the nanoparticle, set to 10σ, and the shift factor Δ = (σNM − σ)/2 = 4.5σ. The interaction strength, εNM, is a function of monomer type, M, with εNA = 1 and εNB = 5. Thus, A monomers have the same affinity for the nanoparticle as for other monomers, while B monomers adsorb strongly to the nanoparticle. Note that the nanoparticle−A monomer interaction is effectively slightly repulsive: since the nanoparticle is larger than the monomer, an A monomer approaching the nanoparticle surface from the bulk loses multiple interactions with other monomers but gains only one interaction with the nanoparticle. We study A homopolymer and B homopolymer systems as well as a number of AB copolymer systems. Each copolymer chain consists of an equal number of A and B monomers, which are arranged in different configurations depending on the system. Initially, we study a 50−50 random compolymer system as well as a series of regular multiblock copolymer sequences with form [AxBx]y, where y = 100/(2x) and x is the length of each block. Depending on the system, x is set to 1, 2, 5, 10, 25, or 50. Hereafter, we refer to these systems by their block length as “BL = x”. These systems are inspired by experimental multiblock copolymer sequences with adjustable block length, which have been successfully synthesized using a variety of methods in recent years. Accessible multiblock copolymer chemistries include poly(styrene-b-butadiene),89,90 poly(lactide-b-butadiene),91 and poly(styrene-b-methyl methacrylate).92 In the random copolymer system, chain sequences are chosen by individually randomizing the order of a set of 50 A and 50 B monomers. Therefore, in the random system, sequences may vary between chains, which is not the case in the other systems; however, each individual chain still contains exactly 50% of each monomer type. In the particular set of random sequences generated by this procedure and used in this work, the average length of a block selected by choosing a random B monomer was 2.99, the average length of the longest B block on each chain was 5.86, and the longest B block in the entire system was 13 monomers long. Figure 2 shows a schematic of a few representative copolymer sequences from our study. Molecular dynamics (MD) simulations were conducted using the open-source package LAMMPS and applying the
σiab = −miviavib −
1 2
∑ (ri Fij a
j≠i
b
+ rj Fji ) a
a
(4)
where a and b take on the values x, y, z to calculate the six components of the symmetric stress tensor, σiab is the stress on atom i in direction ab, mi is the bead mass, vi is the bead velocity, ri is the position of bead i, and Fij is the force that particle j exerts on particle i. The summation is performed over all other particles in the system, including the nanoparticle. When a = b, the stress component is tensile, and when a ≠ b, the stress component is shear. Our goal is to estimate local dynamic modulus using atomic stresses, but no method to do this has been well-established in prior work. However, recall that bulk dynamic modulus can be measured from MD simulations relatively easily using extensively studied nonequilibrium85,94,95 or equilibrium96−98 methods. In the nonequilibrium method, cyclic strain of a given frequency is applied to a simulation box and the resulting stress is measured (or vice versa), and from the magnitude and phase shift of the response, the dynamic storage and loss modulus at that frequency are determined. Alternatively (and more relevant to this work) dynamic modulus can also be measured from an equilibrium MD simulation using
Figure 2. Schematic of some of the copolymer sequences used in this work. For visual clarity, each segment shows one-half of an N = 100 polymer chain. D
DOI: 10.1021/acs.macromol.8b02136 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules fluctuation dissipation relationships. Specifically, the complex modulus is calculated from the stress autocorrelation function (SACF): G′(ω) + iG″(ω) = iω
V kBT
∫0
+∞
width of a single shell during the course of a single autocorrelation time window. (Over sufficiently long time scales, our approximation of the local SACF would converge to the bulk SACF regardless of the location of the shell since the monomers would eventually explore the entire simulation box regardless of initial location.) Bearing these caveats in mind, we hereafter refer to this approximation of the local SACF as the LSACF. Because the system is isotropic, the three tensile components of the stress tensor are effectively equivalent. Therefore, to improve the statistics of the measurement for each shell, we average the LSACF of the three on-diagonal (normal) components of the stress tensor. By the same logic, we also average the LSACF of the three off diagonal (shear) components. After this averaging, the LSACF is still quite noisy; other researchers have noted the bulk SACF is inherently noisy when calculated from MD simulation.96 To further reduce the noise level, we apply a running average filter similar to the one applied by Sen et al.,96 such that the final LSACF data at each time t is the average of all raw data points from 0.8Δt to 1.2Δt. This filter uses a wider time window than Sen and colleagues since our system is divided into shells and therefore has fewer atoms contributing to each ensemble. The effect of this filter on the magnitude of the final results is small, and a comparison between results obtained with and without the filter can be found in the Supporting Information. The final challenge with our analysis arises from that fact that we truncate the autocorrelation data at 8τ. As a result of taking a Fourier transform of this finite time window with a discontinuity at its end, the calculated storage and loss moduli exhibit substantial oscillations in the frequency domain. This effect could be reduced by lengthening the time window considered; however, we avoided this for two reasons: first, as discussed above, using a longer time window necessarily makes the LSACF a less localized description of stress fluctuations, and second, since the stresses on and location of every atom must be recorded and analyzed at every time step, the data files required to perform this analysis become prohibitively large to store and analyze for more than an order of magnitude or two beyond 8τ. Instead, after filtering and prior to applying the Fourier transform, we artificially extend the raw LSACF data by performing a power law fit on the existing data above 1τ and then extrapolating the data out to 1000τ. This has the effect of nearly eliminating the oscillations in the dynamic modulus while leaving the overall magnitude relatively unchanged in the frequency range of interest (which is above 2π/8 rad/τ and thus is primarily influenced by the LSACF data from times shorter than 8τ). The Supporting Information contains a comparative example of dynamic modulus data calculated with and without extending the raw LSACF data.
e−iωt ⟨σab(0)σab(t )⟩ dt (5)
where σab(t) is the bulk stress at time t, ⟨σab(0)σab(t)⟩ is the SACF, V is the system volume, and kBT is the thermal energy. Here, a Fourier transform has been used to decompose the modulus into the storage (G′) and loss (G″) moduli, which are functions of the frequency of excitation, ω. Compared to nonequilibrium methods, this method is more computationally efficient since the modulus can be calculated for a continuous range of frequencies from a single simulation. This technique has been used to compute dynamic moduli for a variety of polymer systems.96−102 The works referenced above measured the bulk SACF and calculated the bulk modulus. In contrast, our goal is to estimate local dynamic modulus as a function of distance from the nanoparticle. To do this precisely, it would be necessary to have a measurement of the SACF that could be defined locally and would vary as a function of position. However, it is not obvious how to precisely define and calculate “local SACF” because the SACF is a function of time, and atoms are not stationary in time. Here, we have developed a method to approximate the SACF on a local scale by dividing the system into concentric spherical shells centered on the nanoparticle and calculating ⟨σ(0)σ(t)⟩s, an estimate of the average stress autocorrelation function of monomers in shell s. The result is then used with eq 5 to estimate the dynamic storage and loss modulus for each individual shell. Specifically, for each shell and at each time step t′, the stress on each bead, i, whose center is in the shell at t′ is calculated according to eq 4, and then a stress autocorrelation σi(t′)σi(t′ + Δt) is calculated for each possible time window length Δt from 0.01τ to 8τ (1 to 800 timesteps). At each starting t′, this quantity is averaged over all monomers in each shell, yielding [σ(t′)σ(t′ + Δt)]s, the average stress dissipation for atoms initially residing in shell s at time t′. [σ(t′)σ(t′ + Δt)]s is further averaged across all possible times, t′, yielding ⟨σ(0)σ(Δt)⟩s for each shell. Therefore, our estimate of ⟨σ(t′)σ(t′ + Δt)⟩s is the average of 10000 − Δt/δt subwindows within each trajectory. The results are further averaged across the five trajectories, ensuring that the system is sampled at several configurations that are separated by time scales longer than the end-to-end vector relaxation time. Critically, atoms assigned to each shell are not necessarily the same at each autocorrelation window start time step, and moreover, atoms may move between shells during the course of the autocorrelation time window, possibly leaving the shell to which they were assigned. Therefore, this formulation of the SACF measures the average stress dissipation of atoms that initially reside in a given shell, meaning that it is an approximation of the true “local SACF”. As discussed above, some level approximation is unavoidable; the SACF is necessarily measured using information from particles that move over time and therefore does not have a precise local meaning. To reduce the degree of approximation, the longest autocorrelation time window we consider is Δt = 8τ (800 timesteps). This time was selected such that the square root of the monomer mean-squared displacement is 2 × 10−10 s for a system at room temperature with length and mass scales similar to polybutadiene, the tan δ peak would occur at a frequency of less than 7 × 108 Hz. Experimentally, loss peaks have been measured around ∼106 Hz in polybutadiene-containing copolymers somewhat below room temperature using dielectric spectroscopy by Vo et al.104 The authors of that work attributed their loss measurements to relaxations in the polymer−nanoparticle interphase, so it is plausible that we are measuring a similar phenomenon. For comparison, Figure 4 reports the same mechanical properties as a function of distance from the nanoparticle in the A homopolymer system. Although the trends in |G*| are similar, the homopolymer A system exhibits significantly less mechanical reinforcement than the homopolymer B system. Despite the fact that the monomer−nanoparticle interaction is slightly unfavorable relative to bulk monomer−monomer
Figure 5. Complex modulus (a) and hysteresis (b) of the BL = 1 system as a function of distance from the nanoparticle center, d. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. For comparison, the dotted lines reproduce the results from the shell closest to the nanoparticle in the B homopolymer system.
two homopolymer systems. Modulus increases more sharply with proximity to the nanoparticle than in the homopolymer A system but less sharply than in the homopolymer B system. The interphase hysteresis exhibits a peak around the same frequency as in the homopolymer B system, but the peak is lower and the interphase of increased modulus is narrower than in that system. The fact that the dynamic mechanical properties of the BL = 1 system fall somewhere between the
Figure 4. Complex modulus (a) and hysteresis (b) of the A homopolymer system as a function of distance from the nanoparticle center, d. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. For comparison, the dotted lines reproduce the results from the shell closest to the nanoparticle in the B homopolymer system. G
DOI: 10.1021/acs.macromol.8b02136 Macromolecules XXXX, XXX, XXX−XXX
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systems: each shell’s peak hysteresis and each shell’s degree of dynamic mechanical reinforcement relative to the bulk. We calculate peak hysteresis simply by recording the maximum tan δ in the studied range, and we calculate dynamic mechanical reinforcement by choosing a frequency and taking the ratio between the magnitude of the shell’s complex modulus and the magnitude of the bulk complex modulus. The results for mechanical reinforcement are relatively independent of the choice of frequency, and we select 9 rad/τ because it is close to the frequency at which the peak hysteresis occurs. In Figure 7,
homopolymer A and homopolymer B systems in fairly intuitive since the chains are composed of alternating A and B monomers. Figure 6 reports the dynamic properties of the interphase of a system with much longer blocks, BL = 25. Both the modulus
Figure 6. Complex modulus (a) and hysteresis (b) of the BL = 25 system as a function of distance from the nanoparticle center, d. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. For comparison, the dotted lines reproduce the results from the shell closest to the nanoparticle in the B homopolymer system.
Figure 7. Dynamic mechanical reinforcement under shear stress at 9 rad/τ (a) and peak hysteresis under shear stress (b) for the various copolymer and homopolymer systems, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries).
and peak hysteresis of the interphase are significantly higher than in the BL = 1 system, with the trends being much more similar to those of the homopolymer B system. In fact, the peak hysteresis of the two closest shells is actually higher than that of the homopolymer B system, and in the closest shell, this difference is statistically significant. This result is especially striking in light of the fact that the homopolymer B system contains 50% more adsorbing monomers than the BL = 25 system. The results for the other copolymer systems are qualitatively similar to those of the BL = 1 and BL = 25 systems and are available in the Supporting Information. To understand the quantitative effect of blockiness in sequence, we focus on the two features that appear to most significantly differentiate the
we plot peak hysteresis and reinforcement under shear loading for each shell as a function of distance from the nanoparticle (analogous results for normal loading are reported in the Supporting Information). Generally, the homopolymer A system exhibits the lowest interphase reinforcement, the homopolymer B system exhibits the highest, and the copolymer systems fall in between. In the closest shell, reinforcement is relatively independent of copolymer sequence, with all copolymer systems having reinforcement similar to the homopolymer B system. In the second and third shells, reinforcement usually increases slightly H
DOI: 10.1021/acs.macromol.8b02136 Macromolecules XXXX, XXX, XXX−XXX
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The authors of the latter work attributed the observed effect to increased mobility of the shorter adsorbed chains, which they speculated allowed the chains to experience stronger dynamic coupling with the surrounding matrix. Note that Senses and co-workers studied PMMA-adsorbed nanoparticles dispersed in PEO, while Yang et al. studied PVA-adsorbed nanoparticles in PEO, so the different effects observed are likely related to the chemical differences between the two systems. Although the increase in hysteresis observed when shortening adsorbed chain lengths in the works discussed above may be analogous to our results that show hysteresis increasing with reduced adsorbed block length, we do note that since the prior work studied homopolymer blends whereas we study copolymers, some of the relevant dynamic mechanisms may be different. In our case, segments of adsorbed monomers are necessarily coupled to nonadsorbing segments via bonds, which does not happen in blend systems. However, adsorbed segments in our systems may also couple dynamically with other segments via nonbonded interactions, a mechanism that is more relevant to the prior work on blends. Overall, more investigation is needed to confirm our hypothesis that hysteresis very close to the nanoparticle depends on the sharpness of the B−A phase transition. To begin to test this theory, we have considered three additional systems aimed at maximizing the sharpness of the B−A phase transition to see how hysteresis is impacted. The systems consist of short B blocks of defined length attached to either end of a single long A block such that the total chain length is equal to 100. Thus, these additional sequences are triblock copolymers with form BxAyBx, where y = 100 − 2x. x is the length of the B blocks ends, set to 2, 5, or 10 depending on the system. We refer to these new systems as BE = x, and for brevity, we will refer to the regular multiblock copolymer systems that we considered earlier collectively as “BL systems” and the new triblock copolymer systems as “BE systems”. A schematic of the BE systems is shown in Figure 8.
with block length, with the random copolymer system falling close to behavior of the BL = 5 and BL = 10 systems. There may be some correlation between the reinforcement and the percentage of B monomers residing in a given shell, which we have previously reported for the same system.81 However, other than the systems with the shortest block lengths, there is relatively little difference in the reinforcement between the different copolymer systems. This is in contrast to the peak hysteresis, which shows notable differences between copolymer systems. In the closest shell, all copolymer systems except the BL = 1 system exhibit more hysteresis than even the homopolymer B system, and hysteresis tends to decrease with increasing block length. This rank order largely reverses in the second shell, with hysteresis tending to increase with block length and all of the copolymer systems having interphase properties between that of the homopolymer systems, with the exception of the BL = 25 system. In contrast to the reinforcement results, these trends are not easily explained by local monomer composition. In an attempt to explain this phenomenon, we note that in all the copolymer systems the region nearest the nanoparticle always has a higher relative concentration of B monomers (above 90% except in the BL = 1 system), and this B-dominant region is, in turn, surrounded by a region that has a higher relative concentration of A monomers.81 The longer the block length, the wider the initial B-rich region and the more gradual the B−A transition. When the B−A transition is sharp, it impacts the mobility of B monomers very near the nanoparticle surface, since it is energetically less favorable to swap an adsorbed B monomer with a nearby A monomer that it is to swap it with a nearby B monomer. Meanwhile, since a hysteresis peak is evident in the hompolymer B but not in the homopolymer A system, it is reasonable to conclude that hysteresis also increases to some degree with the relative concentration of B monomers in a copolymer system. We speculate that very near the nanoparticle (where monomers are directly adsorbed to the surface) the effect of the sharpness of the B−A transition, which tends to increase with decreasing block length, dominates the hysteresis. Slightly farther from the nanoparticle, the relative concentration of B monomers, which tends to increase with block length, may dominate hysteresis. This may explain why the trend in hysteresis reverses with respect to block length from the first to second shell. Some mechanisms contributing to this trend may be related to the results from the Akcora group,56−58 which studied blends of adsorbing and nonadsorbing polymer. Notably, a portion of that work examined the effect of adjusting adsorbed chain length, and their results indicate that in sufficiently loaded composites bulk hysteresis was higher in systems with shorter adsorbed chains,56,57 suggesting that shortening adsorbing chain length increased hysteresis in the polymer− nanoparticle interphase. Although the authors do not directly discuss this effect in terms of hysteresis, Senses and colleagues57 do comment that shorter adsorbed chains experience fewer interactions with nonadsorbed chains, thus decreasing interphase storage modulus. As we noted above, the tan(δ) peaks that we observed correspond mathematically to a relative reduction in storage modulus, rather than a change in the loss modulus, so it is possible that mechanism discussed in Senses et al. is relevant to our results. On the other hand, in Yang et al.56 decreasing adsorbed chain length actually increased composite storage modulus, although tan(δ) still increased slightly because the loss modulus increased as well.
Figure 8. Schematic of BE sequences introduced in the second half of this work. For visual clarity, each segment shows one-half of an N = 100 polymer chain (chain sequences are mirrored about the right side of the figure).
We characterize the structure of the interphase in terms of PB(d), the percentage of monomers at distance d from the center of the nanoparticle that are type B. PB can be calculated according to gBNb/(gANa + gBNb), where gAN and gBN are the A monomer−nanoparticle and B monomer−nanoparticle pair correlation functions, respectively, and a and b are the bulk fraction of A and B monomers. Figure 9 reports PB as a function of distance from the nanoparticle surface for the BE systems, and for comparison, it also includes results for reference BL systems (results for the remaining copolymer systems are qualitatively similar and can be found in ref 81). I
DOI: 10.1021/acs.macromol.8b02136 Macromolecules XXXX, XXX, XXX−XXX
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Figure 9. PB, the fraction of monomers at distance d that are type B, for the BE systems and selected BL systems for comparison, as labeled.
Near the nanoparticle surface, each system exhibits a phase with a very high portion of B monomers, and the width of this phase increases with the length of the adsorbed B blocks. Slightly farther from the nanoparticle, the BL systems exhibit a secondary A-rich phase outside this initial B phase before settling out at 50% B in the bulk. Close to the nanoparticle, the BE systems tend to have similar structure to the BL systems at the same B block length. Farther from the nanoparticle (at a distance that increases with BE), the structures of BE and BL systems diverge, and in the BE systems, PB trends sharply toward the bulk concentration, which is predominantly A (PB = 2BE/100). As intended, the BE systems exhibit significantly sharper B−A phase transitions than the BL systems. We calculate the magnitude of the complex modulus and tan(δ) of the BL systems according to the procedure defined above. Figure 10 reports these viscoelastic properties for the BE = 2 system. The magnitude of the complex modulus in the closest shell of the BE = 2 system is comparable to that of the majority of systems examined previously. Modulus decays rapidly to the bulk value with increasing distance from the nanoparticle. Maximum hysteresis in the closest shell is significantly higher than in any of the systems studied previously. The shell is nearly purely viscous, with tan(δ) over 200. Meanwhile, as with the complex modulus, the hysteresis in shells farther from the nanoparticle is only minimally different than the bulk. Figure 11 reports the magnitude of the complex modulus and tan(δ) in the BE = 5, system. Compared to that of the BE = 2 system, the complex modulus of the BE = 5 system is higher, and the region of increased modulus appears to extend farther from the nanoparticle. The tan(δ) peak of the closest shell in the BE = 5 system is less wide than that of the BE = 2 system, but it is still dramatically higher than in the B homopolymer or any of the BL systems. The peak has a maximum value of ∼60, less than that of the BE = 2 system. However, hysteresis in the second shell is higher than in the BE = 2 system. Viscoelastic properties are reported for the BE = 10 system in Figure 12. Its complex modulus is slightly higher than that of the BE = 5 system. Compared to the BE = 5 system, the hysteresis results continue the trend that was observed from the BE = 2 system to the BE = 5 system; hysteresis is lower in the shell nearest the nanoparticle and higher in the second shell for longer BE.
Figure 10. Complex modulus (a) and hysteresis (b) of the BE = 2 copolymer system as a function of distance from the nanoparticle center, d. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. In the closest shell, the hysteresis peak is over 200 and is cropped for visual clarity. For comparison, the dotted lines reproduce the results from the shell closest to the nanoparticle in the B homopolymer system.
As was done for the first set of systems in Figure 7, Figure 13 reports relative reinforcement and peak tan(δ) for the BE systems as well as the reference BL systems and the B homopolymer system for comparison. Reinforcement decreases with decreasing BE and is lower in the BE = 2 and BE = 5 systems than in the BL = 2 and BL = 5 systems, respectively. However, reinforcement in the BL = 10 and BE = 10 systems is comparable. Reinforcement is also lower in the new copolymer systems than in the B homopolymer system. Overall, among these systems, reinforcement appears to decrease with decreasing local percentage of B monomers. Peak hysteresis in the shell closest to the nanoparticle decreases with increasing BE even more dramatically than it does for increasing BL. As was noted earlier, peak hysteresis in the first shell is significantly higher in the BE systems than in the BL systems. In the second shell, the rank order of hysteresis reverses among the BE systems, such that systems with higher values of BE also have higher hysteresis. This is similar to the rank order reversal that was noted in the BL systems. J
DOI: 10.1021/acs.macromol.8b02136 Macromolecules XXXX, XXX, XXX−XXX
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Figure 11. Complex modulus (a) and hysteresis (b) of the BE = 5 copolymer system as a function of distance from the nanoparticle center, d. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. In the closest shell, the hysteresis peak is ∼60 and is cropped for visual clarity. For comparison, the dotted lines reproduce the results from the shell closest to the nanoparticle in the B homopolymer system.
Figure 12. Complex modulus (a) and hysteresis (b) of the BE = 10 copolymer system as a function of distance from the nanoparticle center, d. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. For comparison, the dotted lines reproduce the results from the shell closest to the nanoparticle in the B homopolymer system.
local hysteresis is not necessarily appropriate. Overall, although these additional sequences have provided some additional evidence that hysteresis very near the nanoparticle is related to the sharpness of the B−A phase transition, more study is still needed to fully understand the relationship of structure and sequence to hysteresis. As a final note, we briefly consider the apparent width of the dynamic interphase, which is a factor of interest in polymer− nanoparticle composites and which is known to depend on the molecular property used to define it.25,27 Here, to estimate the width of the interphase according to reinforcement, we examine Figure 7a to see, for each system, the closest shell from the nanoparticle in which reinforcement differed from the bulk value by
