The Role of Nanoparticle Rigidity on the Diffusion of Linear

Nov 12, 2015 - ... affect polymer dynamics taking into account the ratio of Rg,matrix and RNP,(19, 22, 43) polymer MW,(16, ..... Bahr , J. L.; Tour , ...
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The Role of Nanoparticle Rigidity on the Diffusion of Linear Polystyrene in a Polymer Nanocomposite Brad Miller,† Adam E. Imel,† Wade Holley,‡ Durairaj Baskaran,† J. W. Mays,†,‡ and Mark D. Dadmun*,†,‡ †

Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996, United States Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States

‡

S Supporting Information *

ABSTRACT: The impact of the inclusion of a nanoparticle in a polymer matrix on the dynamics of the polymer chains is an area of recent interest. In this article, we describe the role of nanoparticle rigidity or softness on the impact of the presence of that nanoparticle on the diffusive behavior of linear polymer chains. The neutron reflectivity results clearly show that the inclusion of ∼10 nm soft nanoparticles in a polymer matrix (Rg ∼ 20 nm) increases the diffusion coefficient of the linear polymer chain. Surprisingly, thermal analysis shows that these nanocomposites exhibit an increase in their glass transition temperature, which is incommensurate with an increase in free volume. Therefore, it appears that this effect is more complex than a simple plasticizing effect. Results from small-angle neutron scattering of the nanoparticles in solution show a structure that consists of a gel like core with a corona of free chain ends and loops. Therefore, the increase in linear polymer diffusion may be related to an increase in constraint release mechanisms in the reptation of the polymer chain, in a similar manner to that which has been reported for the diffusion of linear polymer chains in the presence of star polymers.

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INTRODUCTION In the pursuit of new and improved engineering materials, researchers are more frequently turning to nanotechnology and nanocomposites for solutions. Polymer nanocomposites are of particular interest because of their potential to translate many of the desirable properties of nanoscale fillers into processable bulk materials. However, frequently this is not a straightforward task. In many instances, fillers will aggregate, or the nanocomposite otherwise fails to achieve the predicted enhancement in mechanical, electrical, optical, or thermal properties desired for the bulk material. Improving dispersion in polymer-based nanocomposites has been extensively investigated, where intermolecular interaction between polymer matrix and nanoparticle can be achieved by either covalent attachment of polymers onto the nanoparticle1,2 or noncovalent interactions between polymer and nanoparticle.3−5 Covalent attachment of polymers to nanoparticles can be detrimental to the electronic or mechanical properties of some nanoparticles.6 Therefore, noncovalent interactions offer a promising route to achieve spatial uniformity of nanoparticles in polymer nanocomposites.7−9,4,10−15 Recent reports of anomalous melt dynamics in nanoparticlefilled polymer matrices16−25 have also sparked significant interest in how the presence of the nanoscale fillers alters the dynamics of the surrounding polymer chains. These results exemplify that a greater understanding of how inclusion of a nanoparticle (NP) filler impacts the chain dynamics of a © 2015 American Chemical Society

polymer matrix is required to aid in the rational design of polymer/NP systems, including the identification of processing conditions that translate NP properties into bulk material properties more efficiently. Previous studies have noted anomalous dynamics of the polymer chain in the presence of nanoparticles. For instance, Winey et al. used elastic recoil detection to monitor the diffusion of deuterated polystyrene (dPS) tracer films into polystyrene (PS)/carbon nanotube (CNT) composites.17−19 The researchers reported that as the CNT concentration increased, the dPS tracer diffusion coefficient decreases until a minimum at ∼0.4 vol %. Beyond the minimum (up to 4 vol %) the PS diffusion coefficient increased toward the diffusion coefficient of the pure polymer.17,18 A follow-up study using CNTs of various diameters showed that the presence and depth of the minimum were strongly dependent on the ratio of the radius of gyration (Rg,dPS) of the dPS to the radius of the CNT (RCNT).17,19 More specifically, it was found that if Rg,dPS > RCNT, then a minimum in the PS diffusion with concentration was observed, but if Rg,dPS < RCNT, no minimum was observed. These observations led these researchers to propose a new model to describe the diffusive behavior polymer chains near CNTs.16−18 The model accounts for the dependence of the Received: September 8, 2015 Revised: October 18, 2015 Published: November 12, 2015 8369

DOI: 10.1021/acs.macromol.5b01976 Macromolecules 2015, 48, 8369−8375

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Macromolecules Table 1. Molecular Properties of Soft PS Nanoparticlesa sample

xb (mol %)

Mwc (106 g/mol)

Tg (°C)

⟨Rc⟩n (nm)

⟨Rp⟩n (nm)

τsurfd (nm)

ξe (nm)

NP1 NP2

0.81 1.91

0.78 0.81

108 111

3.80 5.49

9.34 9.75

2.77 ± 0.1 2.13 ± 0.1

3.96 ± 0.8 3.99 ± 1.5

a ⟨Rp⟩n = mean particle radius, ⟨Rc⟩n = mean core radius, 2τsurf = corona width, and τsurf = (⟨Rp⟩n − ⟨Rc⟩n)/2. bComposition of cross-linker (DVB) in emulsion feed. cApparent molecular weight determined by SANS. dHalf-width of fuzzy surface layer. eNetwork correlation length or mesh size between cross-links.

obtained from Polymer Source and used as received. Polystyrene soft nanoparticles (nanoparticle 1 (NP1) and nanoparticle 2 (NP2)) were synthesized by microemulsion polymerization of styrene and pdivinylbenzene (DVB).44 As determined by small-angle neutron scattering (SANS) in solution and atomic force microscopy (AFM) on mica, the PS soft nanoparticles can best be described as fuzzy gel particles with a homogeneously cross-linked core and a corona of free chain ends and loops. Details of the particle synthesis and characterization can be found elsewhere.44 Table 1 lists selected properties of the two soft nanoparticles studied here, NP1 and NP2, and lists the characteristics of the fuzzy-gel particle structure characterized by Holley et al.44 The listed divinylbenzene (DVB) content (x) is that of the microemulsion feed and is a nominal measure of the cross-link density of the nanoparticle and correlates to the “softness” or “rigidity” of the final nanoparticle, where the higher cross-link density results in more rigid nanoparticle. The glass transition temperature (Tg) was determined by differential scanning calorimetry (DSC), and the weight-average molecular weight (Mw), mean particle radius (⟨Rp⟩n), mean core radius (⟨Rc⟩n), fuzzy surface half-width (τsurf), and network correlation length (ξ) were determined from solution SANS. The toluene (HPLC grade), concentrated sulfuric acid, and 30% hydrogen peroxide were obtained from Fisher Scientific and used as received. The C60 fullerenes were purchased from Bucky USA and used as received. Silicon wafers used for the reflectivity experiments were undoped and varied from 3 to 6 mm thick. Preparation of Bilayers. Silicon wafers were cleaned by soaking in a piranha solution (3:1 sulfuric acid to 30% hydrogen peroxide) for 2 h, then rinsed with ultrapure water (resistivity greater than 18 MΩ), and dried with a stream of nitrogen gas. To ensure a uniform SiOx layer, wafers were then subjected to 15 min of ozonolysis in a Jelight Model 144AX UV-ozone cleaner. Bilayer samples were prepared by spin-casting a PS matrix layer directly onto the cleaned silicon wafers and float coating a dPS matrix layer on top. Films were spun-cast from toluene solutions containing 2.6 wt % polymer with respect to solvent and (except the control sample) 1 wt % nanoparticles with respect to polymer. Samples of each solution were spun-cast at 2500 rpm for 30 s onto the wafers. Next, the deuterated polystyrene (dPS) solutions (also containing nanoparticles when applicable) were spun-cast on to polished, monolithic, sodium chloride salt plates. This layer is then floated onto a trough of ultrapure water. The silicon wafer coated with the corresponding PS film is lowered beneath the surface of the water and slowly raised up at an angle below the floating dPS layer, capturing it on top of the PS layer. The freshly prepared bilayers were dried under vacuum overnight at room temperature. Neutron Reflectivity. Neutron reflectivity (NR) experiments were performed on the Liquids Reflectometer at Oak Ridge National Laboratory’s Spallation Neutron Source (SNS). Measurements were taken at seven angles for an effective q-range of 0.009−0.2 Å−1. For specular reflection, q is defined as the scattering vector normal to the sample surface and is given by q = 4π sin(αi)/λ, where αi is the incident angle and λ is the radiation wavelength. Interlayer diffusion was monitored by acquiring NR profiles of samples that are as-cast and after annealing at 150 °C for 5, 10, 15, 23, 30, and 60 min. Annealing was completed under vacuum in a preheated Lindberg Blue M oven with thermal mass added for temperature stability during sample exchange. Upon removal from the annealing oven, samples were immediately quenched on a chilled metal block.

diffusive minimum on Rg,dPS:RCNT by splitting dPS diffusion into separate modes: diffusion perpendicular to the CNT and parallel to the CNT.19 These results suggest that the size of a nanofiller relative to the polymer chains is a crucial factor in the observation of anomalous diffusion in low-loading nanocomposites. Anisotropic diffusion of polymers near linear CNTs, however, may differ from any anomalous melt dynamics in composites containing spherical NPs. For instance, Cosgrove et al. observed that hard, spherical, polysilicate nanoparticles in PDMS cause an apparent reduction in polymer entanglement and speed up chain dynamics below a critical filler concentration (∼30 vol %) for high molecular weight polymer.20 Beyond this critical filler content, an abrupt shift to reinforcement is observed. This behavior was absent in low molecular weight polymers,21 where reinforcement is observed at all filler concentrations. Combined with a rigorous follow-up study,22 these researchers showed that an observed maximum in diffusion was only observed when the matrix was sufficiently far above its entanglement molecular weight. The observed diffusive maximum is distinctly different from the minimum found by Winey et al.17−19 in CNT composites and highlights how NP geometry must be considered when interpreting composite melt dynamics. Cosgove et al. also show the dependency of anomalous chain diffusion in composites on the level of entanglement in the base polymer as well as on the ratio of Rg,matrix and RNP. Numerous techniques have been used to study polymer segmental dynamics and diffusion in the melt. Some of the most common techniques include rheology,26−29 nuclear magnetic resonance (NMR) relaxation,20−22,30,31 ablative ionbeam profiling techniques,18,19,23,31−34 and light, X-ray, or neutron scattering.35−42 From the body of research currently available, reasonable qualitative predictions can be made for how hard noninteracting NPs will affect polymer dynamics taking into account the ratio of Rg,matrix and RNP,19,22,43 polymer MW,16,20,21 and NP aspect ratio.18,20 However, it is still unclear what effect particle softness has on the matrix dynamics in composites containing soft NPs. To this end, it is the goal of this study to examine the dynamics of polystyrene in composites that contain a new class of soft, polystyrene-based nanoparticles24,25,44 with controlled rigidity to examine the effect of the NP softness (or rigidity) on matrix diffusion. These nanoparticles are unique in their inherent miscibility with their linear analogues and the ability to tailor the stiffness of the particles synthetically. This study uses neutron reflectivity to probe matrix diffusion in thin film composites containing soft NPs of equivalent size and varying stiffness. For comparison, the diffusion of neat PS and in samples containing C60 fullerenes was also studied.

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EXPERIMENTAL SECTION

Materials. Deuterated polystyrene (dPS, Mn = 525K, PDI = 1.09) and protonated polystyrene (PS, Mn = 535K, PDI = 1.2) were 8370

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Macromolecules The data for multiple incident angles were reduced and combined into a single reflected intensity (R) vs q (Å−1) curve for each sample run, using software provided at the SNS facility. A scattering length density (SLD) profile of each sample was extracted by fitting the specular reflectivity data to a substrate supported bilayer model using the Microsoft Excel based fitting module Layers, which was provided by the Liquids Reflectometer instrument team. The original thickness of each film is determined by ellipsometry and input as an initial parameter in the fitting procedure. Each fitted SLD profile was also tested for conservation of mass for all annealing times of a given sample. This is accomplished by verifying that the area under the SLD depth profile of a bilayer is constant at all annealing times.

βh (t ) = βh0 +

(βd0 − βh0)hd hd + hh

⎛ −hd ⎞ exp⎜ ⎟ ⎝ 4Dt ⎠

(1)

where βd(t) and βh(t) are the SLD at time, t, of the dPS and PS layers, respectively and hd and hh are the thicknesses of the dPS and PS layers, respectively. cE is a fitting constant used in generating the SLD profile in Layers, and σa and σs are the roughness at the air−dPS interface and PS−substrate interface. The derivation of this equation from a common presentation of the equation for the volume fraction profile that follows the solution to Fick’s second law18,19 is given in the Supporting Information. The scattering length densities of the as-cast layers are represented as βd0 and βh0. D, as written, is the mutual diffusion coefficient, which is equivalent to the tracer diffusion coefficient of the polystyrene molecules under the conditions of this study (MWdPS ≈ MWPS∴DdPS ≈ DPS = D). The terms βd(t) and βh(t) in eq 1 are designed to ensure proper mass balance convergence as t → ∞ based on the thickness of each layer. Additionally, they account for the variation of the SLD of each layer with time as the model transitions from pure layers above and below a narrow dPS−PS interface to slowly converging dPS−PS mixtures on either side of a broad interface. It should be noted that this relationship assumes mixing is athermal. These expressions can be converted to the volume fraction profiles of dPS using the relationship

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RESULTS Neutron Reflectivity. All reflectivity data were fit as described above. A summary of the SLD and layer thickness obtained for each as-cast bilayer can be found in Table S1 of the Supporting Information. The SLD depth profiles obtained from the fitting process were converted to depth profiles of the volume fraction of dPS (ϕd) by normalizing each profile using the SLDs obtained from the as-cast bilayers. The evolution of the depth profile with annealing for the NP1 bilayer can be seen in Figure 1, while the same data for the pure PS/dPS, NP2, and C60 bilayers are presented in Figures S1−S3, respectively, of the Supporting Information.

ϕd(x , t ) =

β(x , t ) − βh0 βd0 − βh0

(2)

The diffusion coefficient after time, t, of annealing can then be found from either the SLD or ϕd depth profiles using eq 1 or eq 2, respectively, and fitting for D. Figure 2 shows two

Figure 1. Time evolution of the dPS composition depth profiles for NP1 sample that is annealed at 150 °C.

Since the dPS and PS used in these experiments are of equivalent molecular weight, they will interdiffuse symmetrically. Given this fact, the time dependence of the SLD, β, at depth, x, and time, t, can be expressed in accordance with Fick’s second law as β (x , t ) = +

β h (t ) 2

βd (t ) 2

Figure 2. SLD depth profiles of control bilayer after 5 and 60 min. Solid lines are the fits of β(x,t) at t = 300 s (D = (1.077 ± 0.034) × 10−15 cm2 s−1) and t = 3600 s (D = (0.764 ± 0.01968) × 10−15 cm2 s−1).

⎛ xcE ⎞ β (t ) − βh (t ) ⎛ x − hd ⎞ erf⎜ erf⎜ ⎟ ⎟+ d ⎝ 4Dt ⎠ σ 2 ⎝ a ⎠

representative SLD depth profiles and their corresponding fits. A summary of the diffusion coefficient for each bilayer as a function of annealing time is presented in Figure 3. It should be noted that the uncertainty in the reported diffusion coefficients is derived from the uncertainty in the interfacial roughness that stems from the fitting the experimental reflectivity curves to the model bilayer.

⎛ (h + hh − x)cE ⎞ erf⎜ d ⎟ σs ⎝ ⎠

βd (t ) = βd0 −

(βd0 − βh0)hh hd + hh

⎛ − hh ⎞ exp⎜ ⎟ ⎝ 4Dt ⎠ 8371

DOI: 10.1021/acs.macromol.5b01976 Macromolecules 2015, 48, 8369−8375

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Macromolecules

Δ(w2)/4(Δt) in the t1/2 regime of the pure dPS/PS sample gives a diffusion coefficient (D(t1/2)) of 8.90 × 10−16 cm2 s−1, which is consistent with the values reported in Figure 3 above. Moreover, this diffusion coefficient can be used to estimate the time that the chains transition from Rouse to reptative dynamics using eq 3: τr =

Nb2 3π 2D

(3)

In this equation, N is the degree of polymerization and b is the segment length (b = 0.67 nm). This calculation yields a reptation time (τr) of 876 s, which corresponds nicely with the transition from t1/4 to t1/2 time dependence observed in Figure 4. Similarly, analysis of the time dependence of the interfacial width between the PS and dPS layers in the 1 wt % NP2 composite yields an average diffusion coefficient of 1.44 × 10−15 cm2 s−1and a reptation time 541 s. This diffusion coefficient value is in good agreement with Figure 3, and the reptation time roughly aligns with the t1/4 to t1/2 crossover in Figure 5.

Figure 3. Plot of instantaneous tracer diffusion coefficient, D, of polystyrene as a function of annealing time for all samples.

Self-Consistent Analysis of Interfacial Width. The time dependence of the interfacial width between the PS and dPS layers with thermal annealing can also be analyzed to extract information about the molecular level dynamics that occur to broaden the interface. Fickian diffusion predicts that the interfacial width (w) will increase with time as w = (4Dt)1/2. To test this prediction, the interfacial width of the PS−dPS interface is plotted as a function of annealing time at 150 °C in Figures 4−6 for the neat polymer bilayer, the NP2 bilayer, and

Figure 5. Plot of log(interfacial width) vs log(time) for NP2 bilayer. Solid lines illustrate t1/4 and t1/2 power-law dependencies.

Finally, Figure 6 shows the time dependence of the interfacial width for the NP1 bilayer. Inspection of this plot shows that the width increases as the square root of time from the beginning of the data set, implying that the onset of reptative diffusion occurs at an earlier time. The diffusion coefficient that is estimated from the slope of this plot in the t1/2 regime provides D = 4.68 × 10−15 cm2 s−1, which is consistent with the values reported in Figure 3. The absence of an observed Rouse regime is confirmed by estimating the reptation time from the diffusion coefficient and eq 3, giving a value of τr = 163 s, which is before the first annealed time. Interestingly, Figure 6 also indicates a change of the time dependence of the width to a ∼t1/3 dependence as the dPS−PS interfacial roughness, σ, approaches the thickness of the thinnest layer and the Parratt formalism fails. Beyond this point, obtaining an accurate value for the interfacial roughness value becomes increasingly difficult. Fortunately, at long times eqs 1 and 2 become less dependent on interfacial width in determining D and more dependent on the mass balance. Since

Figure 4. Plot of log(interfacial width) vs log(time) for pure dPS−PS bilayer. Solid lines illustrate t1/4 and t1/2 power-law dependencies.

the NP1 bilayer, respectively. For the neat polymer and the sample with NP2, the interfacial width first increases as t1/4 at early annealing times, which is consistent with Rouse-like dynamics of the polymer chain. The time dependence then transitions to a t1/2 dependence at later annealing times. For the neat polymer bilayer, this transition occurs at ∼900 s of annealing time. Such a transition is in agreement with theory and has been reported in simulation45−48 and time-resolved neutron reflectivity experiments.38,42 Calculating the value of 8372

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Another possible explanation for the increase in the diffusion coefficient would be that the addition of the NP enables constraint release mechanisms in the polymer diffusion that follows the reptation model. Constraint release occurs when the rapid motion of shorter chains that produce entanglements move on a time scale that is faster than that of the reptating polymer chain, allowing the entanglement to disappear before the polymer reptates through it and thus modifying the structure of the reptation tube.49,50 Moreover, constraint release mechanisms have been invoked to explain the dynamics of star/ linear polymer blends, which are separated into three time scales.50,51 At short time scales, the motion of the long linear chains is dominated by Rouse modes and contour length fluctuations. In the intermediate time range, the short-lived constraints that exist from entanglements that incorporate the arms of the star polymers may move. The linear polymer can then explore the larger tube. Furthermore, it appears that the arm retraction of the stars enables constraint release at loadings below ∼2%, which is in agreement with our results. During the longest time scales, the arms of the stars relax in the larger tube, coinciding with the constraint release of other stars. Similar mechanisms are likely to occur in our system as well, where the heterogeneous surface of the soft nanoparticle consists of lightly cross-linked loops and chain ends, which produces a corona with many fast chain segments at the periphery of the nanoparticles. These faster chain segments should therefore behave analogous to the star polymer with a high number of arms. Further experiments are underway to test this hypothesis and provide further insight into this anomalous behavior. Finally, the impact of the cross-link density of the nanoparticle on the diffusion coefficient of the linear polymer is also intriguing. The results in Figure 3 show that the less cross-linked NP1 increases the diffusion coefficient of the surrounding matrix more than the more cross-linked NP2 nanoparticle does. Thus, it appears that the softness of the nanoparticle is a crucial factor in determining the impact of the presence of the nanoparticle on linear polymer diffusion, where a “softer” nanoparticle increases the extent of the rise in the polymer diffusion rate. Therefore, one way to envision the process by which the addition of a soft NP alters the diffusion of the linear chain is to extrapolate this behavior to a “zero cross-link” nanoparticle. The “zero cross-link” nanoparticle will be a linear polystyrene chain that is the same size (i.e., Rg) as the soft nanoparticles used in this experiment. The radius of gyration (Rg) of a linear polystyrene chains in the melt can be estimated from its molecular weight using eq 4:52

Figure 6. Plot of log(interfacial width) vs log(time) for 1 wt % NP1 bilayer. Solid lines illustrate ∼t1/2 and ∼t1/3 power-law dependencies.

the mass balance is more dependent on the effective SLD and thickness of each layer, the Fickian analysis described by eqs 1 and 2 may help extend the time during which polymer diffusion in symmetric bilayers can be effectively probed by neutron reflectivity. Based on these analyses, it is clear that the timedependent scaling terms incorporated into eqs 1 and 2 provide excellent fits to the depth profiles and provide tremendous agreement to both theory and literature within the Rouse and reptation regimes.

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DISCUSSION Impact of Soft Nanoparticles on PS Diffusion. As can clearly be seen in Figure 3, inclusion of 1 wt % NP1 increases the diffusion coefficient of polystyrene by a factor of 4, while the presence of NP2 increases D by a factor of ∼2. This is a stimulating result in that it is not commonly observed in polymer nanocomposites; more often, the diffusion of a polymer chain is hindered by the inclusion of surrounding nanoparticles.35−40 Curiously, Mackay et al. have observed comparable results with similar but smaller PS nanoparticles.9 They found that inclusion of polystyrene soft nanoparticles into a linear polystyrene matrix decreases the viscosity, which they interpreted as the result of an increase in the free volume of the sample, which was confirmed by measurement of the glass transition temperature (Tg). As shown in Table 2, preliminary DSC results show a modest increase in the glass transition of the nanocomposites with addition of 1% of NP1 and NP2 nanoparticles to the polymer matrix, which is counter to an increase in free volume. These results, therefore suggest that the increased polymer mobility must have its origin in other processes.

Rg =

sample

Tg 105.8 °C (±0.1) 107.3 °C (±0.5) 107.4 °C (±0.4)

(4)

where N is the degree of polymerization, and a is the monomer length (0.67 nm for PS). Using eq 4, the Rg for the 535 kDa PS is ∼20 nm, while the average particle radii of NP1 and NP2 are 9.34 and 9.75 nm, respectively. These radii of the NPs correspond roughly to the Rg of a 130 kDa linear PS chain. Thus, a sample that mimics a nanocomposite with a “zero cross-link” nanoparticle will consist of a blend of 535 kDa PS (99%) and 130 kDa PS (1%), where this maintains the relative sizes and loadings of the linear polymer and NP. Given this perspective, understanding the diffusion behavior of this ideal “zero cross-link” nanocomposite (i.e., blend) will provide insight into the process by which the soft nanoparticle alters the diffusive behavior of the linear polymer matrix, and the “zero

Table 2

535 K polystyrene 1% NP1 nanocomposite 1% NP2 nanocomposite

a N 6

8373

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The inclusion of hard C60 nanoparticles also results in a, roughly, doubling in the diffusion coefficient of the polystyrene chain over the annealing time range available. Though the data set is limited, this trend appears significantly different from both the diffusion of neat PS and soft NP bilayers. This deviation in diffusive behavior is likely due to the small size of C60 (RNP < 0.5 nm), allowing it to diffuse readily though the matrix during casting and annealing. Though more data are needed to form longer range trends and draw firm conclusions, an overall increase in diffusive dynamics at low C60 loading is in agreement with recent literature observations that at low loading a high Rg/RNP size ratio will result in faster chain dynamics in high molecular weight PNCs.18−20,22,24,25

cross-link” nanocomposite represents an upper limit of the diffusion coefficient, D, of the polystyrene in the presence of a soft NP. Using the standard reptation model for diffusion coefficient scaling with molecular weight (M),18 D = D0M−2, the selfdiffusion coefficient of a 130 K polystyrene chain at 150 °C can be estimated to be 1.5 × 1014 cm2/s from the measured diffusion coefficient of the pure polystyrene sample. Combining these self-diffusion coefficient values, and the fast-mode theory of diffusion of miscible polymer blends,32,38 provides a method to estimate the mutual diffusion coefficient of a blend that consists of 1% 130 K PS and 99% 535 K PS, which gives a value of 1.48 × 1014 cm2 s−1. Using this value for the diffusion of PS in the presence of a 0% cross-linked soft nanoparticle that is 10 nm in size and the experimental data, the self-diffusion coefficient of the PS chains in the presence of the soft nanoparticle is plotted as a function of mol % DVB (i.e., nominal cross-link density) in Figure 7.

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CONCLUSIONS The results of this study provide insight into the role of nanoparticle rigidity on the alteration of the dynamics of a linear polymer chain in the presence of a soft nanoparticle. The neutron reflectivity results reported here clearly show that the presence of a ∼10 nm soft polystyrene nanoparticle will increase the rate of diffusion of a larger linear polymer chain (∼20 nm). The extent by which the presence of the nanoparticle increases polymer diffusion is dependent on the softness of the nanoparticle, as quantified by its nominal crosslink density. The soft nanoparticle with approximately 1% cross-link density increases the polymer diffusion by a factor of 4, while a soft nanoparticle with ∼2% cross-link density only increases the diffusion rate by a factor of 2. Surprisingly, the increased polymer chain diffusion does not correlate to a commensurate increase in free volume, and thus this does not appear to be a simple plasticizing effect. We postulate that the structure of the soft nanoparticle, which comprises of a gel like core surrounded by a corona that consists of lightly cross-linked loops and chain ends, enables constraint release mechanisms in the polymer diffusion that speeds up the molecular motion. Further experiments are underway to provide further insight into this interesting behavior.

Figure 7. Plot of diffusion coefficient as a function of mol % crosslinker in the soft nanoparticle. The dotted line illustrates a possible exponential trend to the data.

From this plot, the diffusion coefficient of the polystyrene chain appears to decrease rapidly with an increase in the cross-link density, which corresponds to an increase in the stiffness and density of the NPs. It appears that the increased cross-linking of the soft nanoparticle quickly modulates the increase in diffusion of the matrix polystyrene by the soft nanoparticle. The line is a fit of the data to an exponential decay, which is not meant to denote that this functionality uniquely models the data, but rather that it is a reasonable fit and accentuates the rapid decrease of the diffusion coefficient with increased cross-link density. Indeed, a true exponential dependency of the diffusion coefficient on soft nanoparticle cross-link density precludes reinforcement, and therefore it is likely that a more complex relationship will emerge as composites with NPs of higher cross-link density are studied. The existence of critical cross-link density for the onset of matrix reinforcement will be examined in future experiments. Because of the nature of the particles, increasing cross-link density will not only stiffen the NPs but may also cause the NPs to transition from a penetrable, deformable, fuzzy particle to an impenetrable rigid particle, which may significantly alter the interaction of the nanoparticle with and the dynamics of the surrounding polymer matrix. Therefore, it will be important to study composites of soft NPs with a wide range of cross-link density to more fully detail the impact of soft nanoparticle cross-link density on polymer matrix diffusion.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b01976. Figures S1−S5, Table S1, and eqs 1−8 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (M.D.D.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research is supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering.

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REFERENCES

(1) Ying, Y.; Saini, R. K.; Liang, F.; Sadana, A. K.; Billups, W. E. Org. Lett. 2003, 5, 1471−1473. (2) Bahr, J. L.; Tour, J. M. J. Mater. Chem. 2002, 12, 1952−1958. 8374

DOI: 10.1021/acs.macromol.5b01976 Macromolecules 2015, 48, 8369−8375

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DOI: 10.1021/acs.macromol.5b01976 Macromolecules 2015, 48, 8369−8375