Thermal and Rheological Behavior of Polymer Grafted Nanoparticles

Nov 23, 2015 - Department of Chemical Engineering, University of Virginia, 102 Engineers Way, ... Macromolecules , 2015, 48 (23), pp 8651–8659...
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Thermal and Rheological Behavior of Polymer Grafted Nanoparticles Daniel F. Sunday and David L. Green* Department of Chemical Engineering, University of Virginia, 102 Engineers Way, Charlottesville, Virginia 22904, United States ABSTRACT: Composite materials consisting of nanoparticles dispersed in a polymer matrix have potential applications in a wide range of fields. Grafting polymer chains to nanoparticles is a key strategy for controlling the distribution of those nanoparticles throughout the matrix material. The particle distribution is controlled by the relative length of the matrix chains (P) to the graft chains (N) and the density of the graft chains (σ) for particles of a given radius (R). Using differential scanning calorimetry (DSC) and bulk rheological measurements, we examine the interactions between the graft and matrix chains as a function of the three key parameters: σ, P, and N. The results of these measurements are compared to ultrasmall-angle X-ray scattering (USAXS) measurements [Macromolecules 2012, 45, 4007−4011], which were used to determine the phase behavior of the particles. DSC results indicate that the matrix chains must be completely expelled from the graft layer for the autophobic transition to occur. Similar trends are observed from the rheological measurements, where the extracted thickness of the graft layer correlates to changes in the glass transition temperature from DSC. Additional rheological measurements demonstrate the differences in the flow behavior between sparsely and densely grafted particles over a range of shear rates. In particular, as the graft density increases to high levels (0.5−0.7 chains/nm2), the flow behavior of well-dispersed and aggregated particles becomes increasingly similar, as the particle−particle interactions are screened by the thick graft layer.

1. INTRODUCTION Polymer−nanoparticle blends exhibit a rich phase behavior which is directly tied to the thermal, mechanical, and optical properties of the composite system. This has driven an ongoing effort to develop strategies that enable control of the particle distribution throughout the matrix.1−7 One approach for controlling the particle distribution is to alter the particle surface chemistry through the attachment of polymer chains.2,5,8−10 The composition, architecture, and distribution of the graft chains can be carefully designed to tailor interparticle interactions, thereby controlling the dispersion state. In the special case where the chemical composition of the graft and matrix chains are identical, the entropic contributions dominate the thermodynamics.11−14 In a number of studies experimental phase diagrams have been developed for this special case, where the particle miscibility is a function of graft density (σ), the ratio between the molecular weight of the graft (N), and matrix (P) chains (P/N) as well as the particle radius (R).5,9,13−20 At high graft densities the particles undergo a second-order phase transition, becoming immiscible with the matrix with increases in σ and P/N. This is known as the autophobic transition and is caused by the gradual expulsion of the matrix chains from the grafted corona. Here we will expand upon our earlier results where ultrasmall-angle X-ray scattering (USAXS) was used to construct a phase diagram for polystyrene grafted nanoparticles (PS-g-SiO2) in a polystyrene (PS) matrix.9 Thermal and rheological measurements on the same particle set were utilized to better understand how the behavior of the graft layer dictates particle interactions. © XXXX American Chemical Society

Additionally, the effect of the graft density on general trends of the rheological behavior of the composites was examined. Theoretical phase diagrams such as Figure 1 for polymer grafted nanoparticles in a polymer matrix show three distinct

Figure 1. Theoretical phase diagram for polymer grafted nanocomposites in a polymer matrix. σ* indicates the first-order transition from the allophobic region where particles aggregate due to insufficient screening between the particle surface and polymer matrix and the miscible region. σ** indicates the second-order transition into the autophobic region, where particles aggregate due to the expulsion of the matrix chains from the grafted layer. The values for P/N and σ on the scales were determined for 10 nm particles using USAXS.9 Received: May 28, 2015 Revised: October 9, 2015

A

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Macromolecules regions as a function of σ, P/N and R.9 When σ is below the allophobic limit (σ*), the surface coverage of graft chains is low enough that the interactions between the particle and matrix chains are insufficiently screened and the grafted nanoparticles behave similarly to block copolymers, aggregating into distinct morphologies.5,21,22 As σ increases above σ*, the matrix and graft chains interpenetrate, resulting in a wet brush and repulsive interactions between nanoparticles and stable dispersions. Larger values of R, σ, or P/N result in a larger entropic penalty for intermixing due to greater crowding of the graft layer. As the entropic penalty grows, the interpenetration width between the matrix and graft chains decreases until the matrix chains are completely expelled, resulting in attractive interactions and particle aggregation. This occurs above the autophobic phase transition, σ**, the upper line in Figure 1. Studies of PS-g-SiO2 in a PS matrix show that the ratio of P/N which results in a stable dispersion has a maximum around σ = 0.27 chains/nm2 (for particles with R ≈ 9 nm).9 The most direct experimental evidence for the compression of the graft layer with increasing P has been presented through small-angle neutron scattering (SANS). In these experiments the matrix chains were selectively deuterated to contrast match the silica core; as a result the scattering was sensitive to the graft chain conformation. These results show the gradual collapse of the graft layer with increasing P until finally the particles aggregate.23,24 This result directly correlates the change in the conformation of the brush with particle morphology. Rheological measurements have also been shown to be sensitive to the thickness of the graft layer.25,26 When spherical particles are uniformly distributed throughout a matrix, the viscosity increases with particle size. Consequently, the attachment of graft chains results in an increase in the effective particle size and the brush thickness can be extracted from viscosity measurements. In one example, polydimethylsiloxane (PDMS) was grafted onto silica particles (PDMS-g-SiO2) and these were placed into PDMS matrices having a range of molecular weights. An analysis of the zero-shear viscosity was used to obtain the effective particle volume fraction for stable suspensions within the wet brush region (P/N < 1 in this study due to a larger core size, R = 100 nm), resulting in a clear trend of decreasing brush thickness, L, with increasing P/N. The change in the brush thickness was often significant, dropping by roughly 30% when switching to the longer matrix chains (P/N = 0.08−0.5) for stable samples.25 It is unclear if the rheological measurements are sensitive to the entire length of the graft chains. Several studies have proposed that the graft layer should be divided into a concentrated inner core and a diffuse outer corona.27−34 The inner core consists of densely packed chains with highly extended conformations; this is referred to as the concentrated polymer brush region (CPB). After a critical radius (rc), the crowding is sufficiently reduced for the chains to return to a Gaussian conformation; this layer is referred to as the semidilute brush region (SDPB). This hypothesis was recently verified using SANS measurements on PMMA grafted iron oxide nanoparticles. The scattering profile showed evidence of a densely concentrated region around the core of the particle, with the outer portion of the chain relaxing to a Gaussian distribution.30 Rheological measurements are one approach to probing interactions between the particle and the surrounding matrix, but the thermal behavior of the composites also shows sensitivity to the interactions between particles and matrix. The glass transition (Tg) of a polymer chain can be very

sensitive to changes in its local environment. Restrictions on conformational freedom, such as the presence of a surface with an attractive potential, will lead to an increase in Tg for the polymer chains near that surface.35−37 The attractive surface will anchor a fraction of the chains, reducing the overall mobility. Polymer chains grafted to a surface provide a good example of this behavior, where both the anchoring point and the locally high chain density reduce the conformational freedom and contribute to an increase in Tg compared to matrix chains of similar molecular weight.38 Local interactions with a surface can also lead to a depression in Tg. Polymer chains adjacent to a surface with a repulsive potential will experience an increase in free volume, allowing for greater conformational freedom and which drives the reduction in Tg.39 This type of behavior has been previously observed in nanocomposite systems where the extremely high surface area of a nanoscale filler will be capable of having a considerable impact on the composite Tg even at low volume fractions. Bansal and co-workers placed PS-g-SiO2 in PS matrices and studied the glass transition; when P/N ≪ 1 the composite glass transition increased, whereas when P/N ≫ 1 the glass transition decreased.40 Similar results were observed when PS-g-Au nanoparticles were placed in PS matrices; in this study the change in Tg was compared to the change predicted by the Fox equation for simple mixing.41 Stable dispersions of particles had a Tg significantly above what the Fox equation predicted, suggesting that the intermixing between the graft and matrix chains resulted in a higher Tg for the brush/interphase region than was observed in either of the neat components. Based on these results, the Tg of a nanocomposite should be sensitive to the degree of intermixing between the graft and matrix chains. Thus, in the subsequent sections of this article, we explore the connections between effects of the brush/matrix interactions on the thermal, rheological, and microstructural behavior of the PS-g-SiO2/PS nanocomposites.

2. MATERIALS AND METHODS Materials. All materials were purchased from Sigma-Aldrich and used as received unless otherwise noted. Styrene monomer was filtered through a column of basic alumina to remove the inhibitor and then distilled under reduced pressure. The 9 nm silica particles (30 wt % solution in MIBK) were generously donated by the Nissan Corporation. Synthesis. Graft polymers were grown from the particle surface using reversible addition−fragmentation chain transfer (RAFT) polymerization according to previously established procedures.42 PS matrices were also synthesized using RAFT. Composite Preparation. Composites of grafted particles and matrix polymer were prepared according to the following procedures. The desired weight of (PS-g-SiO2) and free polymer were dissolved into toluene with a total solid weight fraction ≈0.05. Once dissolution was complete, the solution was poured into an aluminum dish and the solvent allowed to evaporate at room temperature. The composites were then annealed at 180 °C under vacuum for 10 days. Additional free polymer was annealed concurrently, and the molecular weight was checked before and after annealing in order to ensure the samples had not degraded. The final particle weight fraction in the composite material was checked with thermal gravimetric analysis (TGA). TEM measurements were conducted by microtoming the composite sample with a diamond knife to thickness of ∼150 nm and then floating the sample onto a copper grid. Measurements were conducted on a JEOL 2000FX TEM. Rheology. Rheological measurements were performed on a TA AR 2000 rheometer with a parallel plate geometry, where the plate had a radius of 10 mm. All measurements were performed using an environmental testing chamber at 180 °C under a nitrogen B

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Macromolecules atmosphere. The gap height was set to 800 μm, and measurements were conducted between 600 and 1000 μm in order to ensure that the results were independent of the gap height. Composite samples were molded into solid disks with a radius of 10 mm and a thickness of 1 mm using a homemade hot press. After loading into the rheometer samples were heated to the measurement temperature, and the upper plate was lowered to the gap height, after which the sample was carefully trimmed to avoid edge effects. DSC. Differential scanning calorimetry was performed on a TA QDSC 1000 and analyzed with the Universal Analysis software. Samples were heated to 200 °C at a rate of 10 °C/min and then cooled to 50 °C at 5 °C/min; this heating cycle was repeated five times for each sample. A nitrogen atmosphere was maintained at all times during the experiment. Each reported glass transition temperature was the midpoint of the transition region and is the result of measurements on at least two samples.

wgp wmp 1 = + mp Tg Tg gp Tg f

(1)

ΔTg = Tg − Tg f

(2)

The Fox equation assumes an ideal mixing process that does not result in interactions that can alter the native Tg of either component; therefore, deviations from the predicted Tg can be attributed to conformational changes at the interface of the graft and matrix chains upon mixing. Interpentration of the matrix into the brush reduces the conformational freedom of the chains, resulting in a higher Tg for the composite than that predicted by the Fox equation. However, if the matrix is completely immiscible with the graft layer and forms a free surface, the mobility of the matrix chains adjacent to the particles increases, resulting in a lower glass transition than predicted. The degree of brush−matrix interpenetration will therefore be characterized by measuring the difference between the experimentally determined Tg of the composite and that predicted by the Fox equation (eq 2). For example, SiO2 (σ = 0.1) in 37K PS (P/N = 0.6) is predicted to have a Tg = 101.6 °C (wmp = 0.88 and wgp = 0.12), whereas the measured Tg was 104.5 ± 0.2 °C. This results in a ΔTg of 2.9 ± 0.2 °C, which is a larger increase than would be predicted for a noninteracting mixture, indicating that the matrix chains were interpenetrated into the brush layer. Figure 2 shows the results of this

3. RESULTS AND DISCUSSION Comparison of the Thermal and Microstructural Behavior from DSC and USAXS. The parameters of the materials used in this study (σ, N, and P) were presented in our earlier publication along with the phase diagram derived from USAXS measurements.9 Those results will be cross-referenced with the DSC and rheological measurements presented in this paper in order to better understand the connection between the interactions of the graft and matrix chains with the particle− particle interactions. The Tg for the PS matrices and PS-g-SiO2 are presented in Table 1. The number-average molecular Table 1. Glass Transition Temperatures, Tg, for the PS Matrices and PS-g-SiO2 Used in This Studya PS matrix label (g/mol) 37K 54K 78K 95K 166K a

Tg (°C) 101.3 102.4 104.2 104.9 105.5

± ± ± ± ±

0.2 0.1 0.2 0.2 0.2

PS-g-SiO2 label (σ = chains/nm2) SiO2(σ SiO2(σ SiO2(σ SiO2(σ SiO2(σ

= = = = =

0.1) 0.15) 0.27) 0.51) 0.70)

Tg (°C) 104.0 106.8 108.8 108.6 109.8

± ± ± ± ±

0.1 0.1 0.1 0.1 0.2

The reported Tg’s are the midpoint of the transition region.

weight, Mn, of the matrices ranged from 37 to 166 kg/mol, or 37K−166K, whereas the Mn of the graft polymer was essentially held constant at 60K−65K while increasing the graft density from σ = 0.1−0.7 chains/nm2 for PS-g-SiO2 samples denoted by SiO2(σ = 0.1−0.7). The polydispersity indices (PDI) for the grafts and matrices ranged were ∼1.3, except for the 166K matrix sample where it increased to 1.5.9 The Tg of the matrix PS chains can be seen to increase with molecular weight, rising from 101.3 °C for 37K PS to 105.5 °C for 166K PS. Similarly for the PS-g-SiO2 Tg increases with higher σ. Particles with σ = 0.1 chains/nm2 and Mn = 60K show a Tg of 104.0 °C, just slightly higher than the PS matrix at a similar molecular weight. The most densely grafted particles show a significant increase in Tg, up to 109.8 °C at approximately the same molecular weight, Mn = 60K. The Fox equation, shown in eq 1, has been shown to effectively predict the Tg of an athermal, miscible polymer blend (Tgf represents the predicted glass transition temperature of the mixture, wmp and wgp represent the weight fractions of the matrix and graft PS, and Tgmp and Tggp represent the corresponding glass transition temperatures of the matrix and grafted PS).

Figure 2. ΔTg as a function of σ for P/N ≈ 0.6. ΔTg was calculated using eqs 2 and 3.

measurement for σ = 0.1−0.7 chains/nm2 at a constant P/N ≈ 0.6. As σ increases ΔTg decreases, a result consistent reduced interpenetration of the matrix chains. For this series all of the ΔTg values are positive, which shows that even the highest graft densities on the particles still allow for intermixing with the surrounding brush for a sufficiently low P/N. There are potential contributions to ΔTg from particle−particle interactions, but for well dispersed systems these contributions are likely negligible at the low volume fractions investigated here. Figure 3A plots ΔTg as a function of σ and P/N. There is a maximum in ΔTg for the lowest σ (σ = 0.1 chains/nm2) and P/ N (P/N = 0.6) corresponding to the conditions that should allow for the most intermixing between the chains. The lowest values of ΔTg are observed for high and low values of σ when P/N ≥ 1.6. Systems with more densely grafted chains exhibit small negative values of ΔTg when P/N ≥ 1.6 and σ ≥ 0.5 C

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samples that exhibit reduced but positive ΔTg still demonstrate uniform particle distributions. It should be noted that the magnitude of the negative ΔTg for aggregated samples is small compared to the samples with positive values. This is likely the result of the massive decrease in surface area that occurs when the particles aggregate, which reduces the amount of interacting area between the graft and matrix chains. Interestingly, the aggregated samples at σ = 0.1 chains/nm2 show small positive ΔTg. The driving force for aggregation in this region is the enthalpic penalty associated with particle−matrix contacts due to insufficient screening of the matrix chains from the particle surface.5 It is possible that while the particles are aggregated the graft chains that remain in contact with the matrix layer are still intermixed. The scaling theory proposed by de Gennes classifies samples with sufficiently small σ and high P/N as unstretched and mixed, a state which could lead to the small positive ΔTg observed for the low graft density composites.11 Overall, the correspondence between the thermal behavior of the composites and the particle distributions from USAXS demonstrates the necessity of having interpenetration between the graft and matrix chains in order to promote particle dispersion. The gradual decrease in magnitude of the ΔTg with increasing σ or P/N is strong evidence for the gradual expulsion of the matrix chains as the autophobic transition is approached. Detection of the Allophobic and Autophobic Transitions with Mechanical Rheology. The viscosity of a polymer nanocomposite is a function of both the particle volume fraction and distribution; therefore, in order to compare samples with slightly different particle volume fractions, we examined the effective volume fraction per gram of particles (φg) as a function of P/N at each σ. This analysis enables the use of rheological viscometry as a method to detect the transition from particle stability-to-instability. Within the miscible region uniformly dispersed particles behave as effective hard spheres, which facilitates the determination of the hydrodynamic brush thickness, L, in eq 6 from the zero shear relative viscosity (ηr,0) (for known volume fractions of the particles core (ϕc) and their radius (rcore). ηr,0 is defined as the ratio of the composite viscosity (η) to that of the polymer matrix (ηp) in the limit of zero shear, according to eq 3. The Krieger−Dougherty relation was manipulated to obtain the effective volume fraction (φeff) from eq 4 where [η] is the intrinsic particle viscosity corresponding to the Einstein coefficient of 2.5 for hard spheres, and φmax is the maximum packing fraction for spherical particles, which was set to a random close packing (RCP) configuration of 0.63.43 The value of φeff was then normalized by the particle core weight fraction (wtcore) to give the volume fraction per gram of particle (φg) (eq 5). Of special note, on the basis of eqs 3−6, viscometry yields physically relevant φeff and L for brush-stabilized dispersions; the L and φeff for dispersions within the immiscible regions are nonphysical due to the particle aggregation.

Figure 3. (A) Surface plot of ΔTg as a function of P/N and σ. (B) Contour plot of ΔTg as a function of P/N and σ with an overlay indicating whether the particles where determined to be uniformly dispersed or aggregated via USAXS. ● indicates the particles showed uniform dispersion throughout the matrix; ■ indicates that the particles were aggregated.9

chains/nm2. The less densely grafted chains maintain a small positive ΔTg for P/N ≥ 1.6 and σ = 0.1 chains/nm2. A plateau region is observed at σ = 0.27 chains/nm2 where the rate of decrease of ΔTg with increasing P/N is significantly slower compared to higher or lower graft densities. These results are replotted in Figure 3B as a contour map, with the addition of blue circles to represent samples that were observed to be stable via USAXS measurements and red squares to represent phase-separated aggregates.9 For σ > 0.1 chains/nm2 all of the samples with positive ΔTg coincide with uniform particle distributions. The samples with negative ΔTg values all fall above the autophobic transition and showed signatures of aggregation in the USAXS measurements. The negative ΔTg in the autophobic region offers confirmation that the matrix chains must be completely expelled from the graft layer for the interparticle interactions to become attractive. The

ηr,0 = lim

γ→ 0

η ηp

ϕeff = ϕm(1 − ηr,0−1/[η]ϕm) Φg = D

Φeff wtcore × 100

(3)

(4)

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Macromolecules 1/3 ⎛⎛ ⎞ ϕeff ⎞ ⎜ ⎜ ⎟ L = rcore ⎜ − 1⎟ ⎜⎝ ϕ ⎟⎠ ⎟ core ⎝ ⎠

coinciding with the detection of particle aggregation with USAXS. The samples with σ = 0.15 and 0.28 chains/nm2 are both in the vicinity of the ΔTg plateau region in Figure 2. For particles with σ = 0.28 chains/nm2 in particular, φg for stable particles show much smaller changes in the presence of higher molecular weight matrices, consistent with the predicted ability of the less densely grafted particles to more easily accommodate intermixing with longer matrix chains. This behavior changes toward higher graft densities. For example, at σ = 0.50 chains/ nm2, a sharp decrease in φg occurs as P/N increases from 0.6 to 1.3, plateauing around φg = 0.7 before increasing sharply to particle instability at P/N = 2.7. For particles with the highest graft density (σ = 0.70 chains/nm2) the same general trend is observed, although with an important difference. Unlike the other graft densities the lowest value of P/N results in φg which has an equivalent magnitude to the aggregated composites. While the particle volume fractions were kept low, the grafted layer in this system is very long, and there are likely particle− particle interactions arising from the highly stretched brushes. This results in a high φg in spite of the uniform particle dispersion. Comparison of Wet Brush Graft Polymer Thicknesses to Theoretical Predictions. Figure 5A shows L as a function of P/N and σ for the uniformly dispersed particles (the presence of aggregates prevents the calculation of L from the viscosity measurements for the aggregated systems). Two

(6)

The effective volume fraction per gram of particles φg as a function of σ and P/N is presented in Figure 4. For all samples

Figure 4. Volume fraction per gram of particle, φg, as a function of P/ N for all graft densities. ● indicates the particles showed uniform dispersion throughout the matrix; ■ indicates that the particles were aggregated as determined by USAXS measurements.9

the particle core volume fraction was kept low (φc < 0.03) in order to reduce particle interactions. The general trend in Figure 4 is a decrease in φg corresponding to drop in L with increasing P/N, followed by a dramatic jump in φg upon crossing into the immiscible regions. For σ = 0.10 chains/nm2, φg was below 0.06 for P/N ≤ 0.9, but when placed in a higher molecular weight matrix (P/N = 1.3), φg doubled to 0.12 as a result of particle aggregation. Higher molecular weight matrices collapsed the graft chains, resulting in particle attractions. The jump in φg corresponds to an increase in viscosity, which originates from the presence of aggregates as detected with USAXS, owing to the reduced mixing of the matrix and the grafts chains as inferred from decreases in ΔTg from DSC. A similar trend is observed for systems with σ = 0.15 and 0.28 chains/nm2 where φg decreases with increasing P/N until a sharp increase in φg occurs at P/N = 4.5 and 7.6, respectively,

Figure 5. (A) L as a function of P/N for (∗) 0.10, (■) 0.15, (▲) 0.28, (●) 0.50, and (⧫) 0.70 chains/nm2. (B) Plot of log(LP1/3) against log(σ); the dotted line shows the best fit to the data with a slope of 0.3 ± 0.04. E

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Figure 6. Plots of relative viscosity against the Peclet number (Pe) for both stable (right side) and aggregated (left side) composites for σ = 0.10, 0.15, and 0.70 chains/nm2. Solid circles indicate the up-sweep (increasing shear rate), and open circles indicate the down-sweep (decreasing shear rate). The oscillatory behavior observed with some of the samples is due to periodic temperature oscillations.

interesting features. For particles with σ ≤ 0.28 chains/nm2 there is good qualtiative agreement between the behavior of ΔTg and the brush length, including the apperance of a broad plateua of stability at σ = 0.28 chains/nm2. Slight differences can be observed for the more densely grafted layers when σ ≥ 0.50 chains/nm2. In this region L collapses much faster than for lower σ, with most of the change occurring between a P/N of 0.6 and 0.9, wherease the ΔTg values decrease more gradually. Figure 5B shows the scaling relationship between L, P, and σ. The predicted scaling relationship for L with σ in the moderate coverage regime is shown in eq 7, where a is the statistical segment length.44 The fits to the data in Figure 5B show that L ∼ σ0.3±0.04, which is in good agreement with the predicted scaling of σ 1/3 (for the SDPB). This is in spite of the fact that

general trends are observed: (1) L increases with σ as the crowding of graft chains results in greater interaction and stretching, and (2) L decreases with increased P/N due to the incremental expulsion of the longer matrix chains from the graft layer. The two samples at σ = 0.10 chains/nm2 maintain an approximately constant L at 12 nm. For particles with σ = 0.15 chains/nm2 L is 13.7 nm at P/N = 0.6; this decreases in an approximately linear fashion with P/N until it reaches a value of 9.5 nm at P/N = 1.6. The decline in brush thickness is more gradual for σ = 0.28 chains/nm2, starting at 15.7 nm and eventually reaching 11.5 nm for P/N = 4.3. More densely grafted particles (σ = 0.5 and 0.7 chains/nm2) show a steeper initial decline in L and eventually plateau at P/N > 1.3 and 0.9, respectively. Comparison with the DSC results yields several F

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Figure 7. TEM images of PS-g-SiO2 (σ = 0.7 chains/nm2, N = 60K) distributed in polymer matrices of (A) P/N = 0.6 (stable), (B) P/N = 0.9 (stable), and (C) P/N = 1.6 (aggregated).

much of the polymer brush falls below the critical radius, rc, where it should scale with the CPB regime (with a predicted scaling of σ1/2). Ohno et al. predicted rc using eq 8, where a is the statistical segment length and ν* is the excluded volume parameter divided by (4π)1/2.33 The brush length of the concentrated region (Lc) can then be calcualted from eq 9. For σ = 0.27 chains/nm2 at P/N = 4.3 the hydrodynamic brush measured around the particle had a thickness of 11.8 nm, which is comparable to the predicted Lc of 11.5 nm. Similar results are observed for σ = 0.5 and 0.7 chains/nm2, where the measured brush lengths at the highest stable P/N = 14.1 and 19.7 nm compared to Lc of 14 and 19 nm, respectively. For lower values of P/N the more extended brush will cross the border between the CPB and the SDPB. The origin of the discrepancy between the predicted scaling for L with σ (for brushes which primarily reside in the CPB) and the observed scaling is unclear. It is possible that rheology measurements may not be sensitive to the concentrated brush regime (CBP), or polydispersity in the brush chains may obscure detection of the CBP as the hydrodynamic radius depends on the extent of the chain ends.

L=

aNσ 1/3 P1/3

(7)

rc =

rcoreσ 1/2a v*

(8)

Lc = rc − rcore

(9)

result of mild temperature oscillations in the environmental chamber. The stable colloidal systems in the right column of Figure 6, i.e., Figures 6D−F, reside within the wet brush region of Figure 1. The unstable colloidal systems lie in the left column of Figure 6, i.e., Figures 6A−C. Figures 6B,C are composites that reside in the allophobic region in Figure 1; the flow curve of the top graph, Figure 6A, was obtained from a composite located in the autophobic region of Figure 1. The overlap of the steady-state stress sweep curves for the stable systems in Figures 6D−F is indicative of Brownian hard sphere rheology, wherein Brownian motion imparts a force that induces particle diffusion and the establishment of an equilibrium particle microstructure on time scales greater than the Brownian relaxation time.13,25,26 The steady-state rheology of the brush-stabilized composites exhibit reversible shear and time effects; the measurements in the right column of Figure 6 reached equilibrium values within 90−120 s after switching the applied stress. Accounting for brush thickness, the effective particle volume fraction, φeff, for the suspensions range from φeff = 0.1−0.3, corresponding to particle concentrations spanning the semidilute and concentrated regimes, in which shear thinning, or the drop in ηr with increasing γ̇, is anticipated with particle alignment under flow.13,25,26 The hysteresis in the stress sweeps of the composites in Figures 6B,C, which reside in the allophobic region, are also anticipated. The origin of the hysteresis between the up- and down-sweep curves is due to interparticle attractions from the flocculation of the PS-g-SiO2 particles upon the expulsion of the PS matrix from the brush.13 Accordingly, composites in the allophobic regime exhibit thixotropy, or a continuous decrease of viscosity with time when flow is applied, followed by the recovery of viscosity when the flow is discontinued. Thus, particle attractions have significant effect on particle microstructure and shear history. The flow curves in Figures 6B and 6C are a characteristic respectively of the formation of gels and flocs, which indicate a difference in microstructure upon increases in the matrix molecular in the allophobic region.5,21,22,25 The flow curves for the composite in the autophobic region with σ = 0.70 chains/nm2 and P/N = 1.3 are shown in Figure 6A. While the up- and down-sweep stress curves display a little hysteresis, the flow curves bear greater resemblance to those of composites stabilized by wet brushes in the left column of Figure 6D. As a consequence, the correspondence between the flow curves indicates a similar microstructure for composites in the wet brush and autophobic dry brush regions of the phase diagram of Figure 1. Figure 7 shows TEM images of the silica core distribution for composites with dense grafts of σ = 0.70 chains/nm2. Figures 7A and 7B show composites with P/N =

Steady-State Rheology of Dispersed and Aggregated Composites. To explore further the effect of polymer graft density (σ) on the bulk rheology of the composites in the wet brush and dry brush regions of the experimental phase diagram in Figure 1, the steady state flow behaviors of a representative set of composites were studied. The silica core volume fractions of the composites were dilute at φc = 0.02−0.03, and the neat polymer matrices were Newtonian over the range of shear rates studied. In Figure 6 the reduced viscosity, ηr, is plotted as a function of the Peclet number (Pe), the dimensionless ratio of the rate of fluid advection by flow, γ̇, to the rate of diffusion by Brownian motion of a dilute dispersion, D0, or Pe = γ̇/(D0/R2). Above Pe ≈ 10 the PS matrix and composite samples were observed to irreversibly fracture. Varying σ and P with σ = 0.10, 0.15, and 0.70 chains/nm2 and P = 37K−166K resulted in either uniform dispersion or particle aggregation. The filled circles in Figure 6 represent the steady-state ηr collected with a reduction in shear rate, or down-sweep, and the open circles represent ηr collected with an increase in shear rate, or upsweep. The oscillatations observed in some of the curves are the G

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Macromolecules

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0.6 and 0.9, respectively, for which earlier USAXS measurements confirmed particle stability. Figure 7C shows the P/N = 1.6 where USAXS indicated particle aggragation. The TEM image of the P/N = 1.6 sample in the autophobic region resembles the other two samples in the wet brush region, as the thick layers of graft polymer prevent the close contact of the silica cores. Thus, the correspondence between the particle microstructure and flow rheology for the densely grafted suspensions suggests that high graft densities, e.g., σ = 0.70 chains/nm2, lead to behavior that more closely resembles that of stable colloids.

4. CONCLUSIONS Using DSC and rheological measurements, we examined the interactions between graft and matrix chains for a set of polymer nanocomposites with a well-characterized phase behavior. By comparing the difference between the measured and predicted Tg the degree of interpenetration between the graft and matrix chains can be established. Increasing either σ or P/N resulted in a decrease in ΔTg, corresponding to a reduction in the intermixing. Above the autophobic transition samples, which had been established as aggregated via USAXS measurements, showed a negative ΔTg, indicating that the chains must be completely expelled for aggregation to occur. Rheological measurements were also used to establish the state of the particle distribution and the interactions between the chains. For a given graft density, σ, brush thickness decreased until aggregation occurred, which corresponded with a sharp increase in viscosity. The rheology of the composites in the wet brush and dry brush regions were compared, showing that composites of the highly grafted particles in the autophobic region more easily approached particle stability and equilibrium than the less densely composites in allophobic region, This result has potential implications for processing particle− polymer nanocomposites.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (D.L.G.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge funding from the National Science Foundation (NSF) NSF-CBET-0644890. REFERENCES

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DOI: 10.1021/acs.macromol.5b00987 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.5b00987 Macromolecules XXXX, XXX, XXX−XXX