Note pubs.acs.org/Macromolecules
A Phase Diagram for Polymer-Grafted Nanoparticles in Homopolymer Matrices Daniel Sunday,† Jan Ilavsky,‡ and David L. Green†,* †
Department of Chemical Engineering, University of Virginia, 102 Engineers Way, Charlottesville, Virginia 22904, United States Advanced Photon Source, Argonne National Laboratory, 9700 South Cass Avenue, Building 434D, Argonne, Illinois 60439, United States
‡
ABSTRACT: We quantified the stability of polystyrene- (PS-) grafted silica nanoparticles (NPs) in PS matrices with ultrasmall angle X-ray scattering (USAXS) and transmission electron microscopy (TEM) and developed a phase diagram to predict NP dispersion based on the graft polymer density, σ, and the graft and free polymer molecular weights, or N and P, respectively. Using controlled/living polymerizations, polymer nanocomposites were formulated with silica NPs of radius, R = 9 nm where σ = 0.10−0.70 chains/nm2 at an essentially constant N = 61− 68 kg/mol. The matrix molecular weight was varied from P = 37− 465 kg/mol permitting us to vary the swelling ratio, P/N = 0.6−7.7. Using USAXS and TEM, we determined whether the PS-grafted NPs were stable and dispersed uniformly, or were unstable and aggregated within the matrix. From these measurements we developed a phase diagram for NP miscibility with respect to σ, P, and N to determine the allophobic and autophobic transitions that correspond to the wetting the drying of the graft polymer brush which control NP stability in polymer matrices.
I
graft and free polymer molecular weights, N and P, and polymer graft density, σ. To our knowledge, we are the first to scale the regions of NP stability with respect to the autophobic dewetting line, which is a continuous, second-order transition, resulting from the expulsion of the melt from the brush for densely grafted chains (σ(N)1/2 > 1) which should lead to NP aggregation through the attraction between graft layers.1 We also relate NP dispersion to the allophobic transition, a discontinuous, first-order transition at lower graft densities (σ ≤ 0.10 chains/nm2), complementing our previous work that connects NP rheology to wetting and drying of the graft polymer.2 To this end we synthesized PS-grafted silica NPs, or PS-gsilica, as listed in Table 1 using reversible addition transfer fragmentation (RAFT) and atom transfer radical polymerization (ATRP) and blended them with PS matrices to formulate nanocomposites with a wide range of σ = 0.1−0.7 chains/nm2 and P = 37− 465 kg/mol at an essentially constant N = 61− 68 kg/mol. The radius of the silica NPs was R = 9 ± 2 nm as determined with transmission electron microscopy (TEM) and ultrasmall X-ray scattering (USAXS), and the core particle volume fraction was dilute (ϕc = 0.01−0.04) in an attempt to mitigate interactions between stable particles. Moreover, a combination of USAXS, TEM and mechanical rheology were used to determine whether the particles were stable and dispersed uniformly within the matrix or became unstable and aggregated. Representative USAXS and TEM images are presented below
n this article we report our initial results on the stability of polystyrene- (PS-) grafted silica nanoparticles (NPs) in PS matrices with respect to the allophobic and autophobic dewetting transitions in Figure 1, which are a function of the
Figure 1. Illustration of the phase diagram for nanoparticle stability as a function of graft density (σ) and swelling ratio (P/N). Particles at low graft densities encounter the allophobic dewetting transition at σ1. Increasing σ leads to complete wetting of the brush by the melt, stabilizing nanoparticles. Increasing the graft density further leads to the autophobic dewetting transition at σ2 and unstable nanoparticles. Particles are unstable at all graft densities when P/N > (P/N)*. © 2012 American Chemical Society
Received: March 3, 2012 Revised: April 9, 2012 Published: April 23, 2012 4007
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Table 1. Properties of Grafted Particles, Melts and Composites PS label
PS matrix molecular weight (kg/mol)/PDI
37ka 54ka 78ka 95ka 166ka 260kb 465kb
37/1.24 54/1.28 78/1.25 95/1.31 166/1.51 260/1.97 465/1.98
particle label SiO2(σ SiO2(σ SiO2(σ SiO2(σ SiO2(σ
= = = = =
0.1)a 0.15)a 0.27)a 0.51)a 0.70)c
PS graft density-σ (chains/nm2)
graft molecular weight (kg/mol)/PDI
matrix Mn (kg/mol)d
molecular weight ratio - P/Nd
0.10 0.15 0.27 0.50 0.70
61/1.30 61/1.28 60/1.31 68/1.35 60/1.38
37−95 37−460 37−460 37−460 37−260
0.6−1.6 0.6−7.7 0.6−7.7 0.6−7.7 0.6−4.3
a
Synthesized via RAFT. bSynthesized via free radical polymerization. cSynthesized via ATRP. dThe range of melt molecular weights that each particle sample is mixed with, and the corresponding P/N ratios.
while the rheological measurements are omitted for brevity and will be discussed in a later publication. By connecting experiment and theory, we seek to control NP dispersion, which is challenging as strong attractions can cause uncontrollable NP aggregation in polymer matrices. Fine control over NP dispersion and self-assembly becomes increasingly difficult with decreases in particle size, which increases the driving force for phase separation by increasing the contacts between particles and polymers.3 Better control over NP interactions can be obtained by attaching polymers to particles in which the brush conformation governs particle stability as a function of the swelling ratio P/N, as well σ, R, and the Flory parameter, χ.2,4,5 In particular, for χ = 0, or athermal systems of chemically identical graft and matrix polymers, scaling theories and simulations have been used to derive phase diagrams to predict the interactions between the graft and the matrix, a schematic of which is shown in Figure 1. The stability of polymer-grafted NP with brushes chemically identical to the matrix is expected to coincide with the regions in Figure 1 in which the wetting or the penetration of the matrix into the brush leads to uniform NP dispersion in the complete wetting region, and the dewetting or expulsion of the matrix from the brush leads to particle aggregation in the allophobic and autophobic dewetting regions.4,6,7 Allophobic dewetting occurs at low graft densities and results when the matrix prevents the brush from stretching; autophobic dewetting occurs at higher graft densities upon the expulsion of the matrix from the brush. We seek to detect these regions for highly curved surfaces, and depending on interfacial curvature, the dewetting lines intersect at swelling ratios roughly between 1 < P/N* < 4. The lower limit corresponds to planar interfaces and polymergrafted colloids of low radii of curvature (R = 100 nm−1 μm), whereas the upper bound relates to small hairy NPs with highly curved cores (R = 2−16 nm). Thus, increasing particle curvature reduces graft chain crowding, extending the range of swelling ratios over which small NPs can be dispersed in polymer matrices. In contrast, further increases in the swelling ratio above P/N* leads to partial (3 8) of the particles from the matrix, and while not the focus of our study, decreasing the brush/core size below radius of gyration of the melt polymer (R < Rg) will also lead to uniform NP dispersion.8,9 Scaling predictions are used routinely to predict the dewetting transitions primarily for planar surfaces as a function of σ, N,and P. Both de Gennes and Maas et al. forecast that the autophobic transition, σ2 scales as the inverse square root of the matrix molecular weight, P, in eq 1,4,5 σ2 ∼ P−1/2
while Ajdari and Leibler predict a higher power-law dependence in eq 2.1
σ2 ∼ P−2
(2)
The differences in the power laws in eqs 1 and 2 stem from scaling this transition on the basis of different physical underpinnings. For example, de Gennes and Maas et al. derive eq 1 by scaling the expulsion of the matrix from the brush, while Ajdari and Leibler derived eq 2 by calculating the change in sign of the spreading coefficient for a polymer film on a brush layer. Equation 2 is commonly cited as the criteria for phase separation in studies on NP systems even though it was derived for flat substrates. There are currently no corrections to include the effects of particle curvature on the scaling laws. Scaling laws have also been derived for the allophobic transition, σ1, on flat substrates. Equation 3 presents de Gennes calculation for the minimum graft density that would result in the stretching of the graft polymer from the substrate.5 σ1 ∼ P
(3)
Mass et al. derived a different prediction by calculating the transition from a negative to positive spreading in eq 4.4
σ1 ∼
⎛N ⎞−1 ⎜ + N1/3⎟ ⎝P ⎠
(4)
Despite the differences, both results suggest a similar scaling for σ1 with P with a power law of unity. Using light scattering and mechanical rheology, we detected a similar scaling for the allophobic dewetting line in systems of polydimethylsilane (PDMS)-g-silica particles in PDMS over an extensive range of N = 4−36 kg/mol, P = 2−28 kg/mol, σ = 0.09−0.28 chains/nm2 for R = 100 nm and ϕc = 0.10.2 Schadler, Kumar, and Benewicz have experimentally observed and modeled the anisotropic self-assembly of small PS-g-silica NPs with cores of R ≈ 10−13 nm in the allophobic dewetting region at lower graft densities (σ = 0.01−0.10 chains/nm2). Using slightly higher graft densities (σ ≈ 0.2 chains/nm2),10 Chevigny et al used similar-sized PS-g-silica NPs with N = 5− 50 kg/mol in P = 140 kg/mol where particles with the longest grafts (P/N = 2.8) dispersed uniformly, whereas those with the shortest grafts (P/N = 28) phase separated from the bulk, forming spherical aggregates.11 The dispersion of silica NPs with higher graft densities has been investigated in which two sets of PS-g-silica NPs, the first with N = 110 kg/mol and σ = 0.27 chains/nm2 and the second with N = 160 kg/mol and σ = 0.57 chains/nm2 have been shown to disperse at least up to P/N = 2.3 and 1.6, respectively.8,9 However, a full range of higher graft densities have not been explored to define the autophobic dewetting transition.
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Figure 2. (A) Comparison of representative USAXS curves in which the lower curve shows well dispersed particles (SiO2 (σ = 0.15) at P/N = 0.6) having a plateau in the low q, Guinier regime (q < 0.01), and the upper curve shows an aggregated system (SiO2 (σ 0.15) at P/N = 2.7) showing a strong upturn in the low q region, the slope of which is indicative of aggregate structure. (B) PS-g-silica nanoparticles with a graft density of σ = 0.15 in a PS matrix of molecular weight of P = 37 kg/mol, corresponding to a swelling ratio of P/N = 0.6 with a particle core volume fraction of ϕc = 0.04. The TEM image shows well-dispersed particles that correspond to the lower USAXS curve in part A. (C−E) TEM and photographic images of PS-gsilica/PS with σ = 0.15 chains/nm2 at P/N = 2.7, ϕc = 0.04. (C) TEM image showing the border between the phase separated particles and region of lower particle density. (D) TEM of the region adjacent to the phase separated particles showing the presence of small aggregates along with dispersed particles. (E) Photograph of an annealed composite showing visible particle aggregates. These composites correspond to upper curve in the USAXS scans in part A.
P = 37−465 kg/mol as shown in Table 1. The use of these systems permits us to detect the allophobic and autophobic dewetting transitions for highly curved silica cores of R = 9 ± 2 nm. The nanocomposites in Table 1 were formulated by dissolving the
In this communication we show how a wide range of graft densities, graft polymer, and free polymer molecular weights affect the miscibility of PS-g-silica NPs in PS matrices, by varying σ = 0.1−0.7 chains/nm2, N = 61−68 kg/mol, and 4009
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larger than R = 6 μm. Thus, only a small region of the Porod regime of the aggregates is obtained.12,13 The aggregated systems in Figures 2A and 3 have slopes in the low q region of ∼3.35, which suggests the formation of compact aggregates.12,13 Additionally the two systems with P/N ≥ 2.7 in Figure 3 show slight regions of curvature (between q = 0.002 and 0.0006 Å−1 for P/N = 2.7 and between q = 0.0009 and 0.0002 Å−1) indicative of aggregates of finite sizes. The aggregate size in these systems can be determined by fitting multiple levels of structure to the Beuacage unified model.14,15 Parts C−E of Figure 2 show the corresponding TEM images and photograph of the aggregated composite, where Figure 2C shows the border between an aggregate and the bulk composite which contains both smaller aggregates and particles that appear to be well-dispersed. The region adjacent to the aggregate is highlighted in Figure 2D where a combination of well dispersed particles and smaller aggregates are observed. The smaller aggregates range in size from R = 30−100 nm. The photograph of the annealed composite is shown in Figure 2E demonstrating the visible phase separation of visible aggregates. The occurrence of larger and smaller aggregates, in the presence of well distributed particles indicates one of two possibilities: (1) the system is trapped in a metastable state or (2) the system is in equilibrium after annealing for 10 days leading to the formation of equilibrium structures.
particles and polymer in toluene, evaporating off the solvent and annealing the sample at 180 C° under vacuum for 10 days. The particle volume fractions were kept low, between 0.01 and 0.04 to minimize interactions between stable particles. USAXS and TEM were used to evaluate the miscibility of the PS-g-silica NPs in the PS matrices. TEM enabled us to visualize NP dispersion, while USAXS allowed us to evaluate a large sample size in a single measurement. In Figure 2A, we demonstrate the criteria for determining NP stability or instability with USAXS and TEM from the 60k PS-g-silica NPs with σ = 0.15 chains/nm2 in matrices of P = 37 and 77 kg/mol, or P/N = 0.6 and 2.7, respectively. The slit smeared, scattered intensity I(q) of dilute, stable, well dispersed NP in P = 37 kg/mol is displayed in the lower curve in Figure 2 in which we show the full scattering range to facilitate comparisons to the aggregate composite in in the corresponding, offset, upper curve. With uniform NP dispersion in lower curve, I(q) will flatten as the magnitude of the scattering vector, q, approaches low values (e.g., q ≤ 0.2 Å−1) where q = (4π/λ) sin(θ/2) in which X-ray wavelength λ = 1.64 Å while θ is the angle of scattering relative to the incident beam. This behavior is characteristic of scattering from noninteracting NPs from which the primary particle size, R = 9 nm, can be extracted through fitting a spherical form factor to the scattering curve, which agrees well with the size obtained from TEM at R = 9 ± 2 nm. Figure 2B shows that corresponding TEM of the composite, confirming uniform NP dispersion. Further, the surface roughness of the primary particles can be extracted from their Porod slope from 0.03 < q < 0.1 Å−1 in Figure 2A, which has a value of 3.89, indicating that the particles have slight surface roughness as the Porod slope deviates slightly from a value of 4.0, which corresponds to a smooth surface.12 In contrast, the I(q) of aggregated nanocomposites in P = 77 kg/mol, the upper, offset curve in Figure 2A, increases dramatically toward lower q values, indicating the formation of aggregates from the primary particles; similar curves are also shown in Figure 3. The slope I(q) versus q of this region is
Figure 3. USAXS curves for particles with σ = 0.15 chains/nm2 and graft chain molecular weight N = 60 kg/mol in PS matrices ranging from 37 kg/mol to 265 kg/mol corresponding to P/N = 0.6 to 4.3. Particles are well dispersed when P/N < 2.7 and aggregate at and above this value. Stable curves were truncated to aid viewing; their low q region corresponds to the lower curve in Figure 2A.
Figure 4. (A) Phase diagram for R = 10 nm PS-g-silica nanoparticles in PS matrices as a function of graft density (σ) and swelling ratio (P/ N). Blue circles indicate stable dispersions; red squares indicate phase separated aggregates. (B) Phase diagram on a log−log scale with linear least-squares fits of the scaling lines for the autophobic and allophobic phase transitions, or the upper and lower lines, respectively. The power law slope,α, for the autophobic and allophobic transitions are α=-0.71 ± 0.04 and 0.54 ± 0.08, respectively.
equivalent of a Porod slope of a large aggregate. The Guinier regime of the aggregates in which the aggregates can be sized lies at q values outside of the detection limit of the USAXS instrument (q < 0.0001 Å−1), indicating the aggregates are 4010
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From plots like Figure 3 a complete NP stability diagram was developed in which we truncate the stable particle curves for clarity; the stability diagram is shown in Figure 4 where uniform and aggregated dispersions are presented as circles and squares, respectively. Figure 4A shows the unscaled scatter plot, while Figure 4B shows the log−log version used to scaling of the allophobic and autophobic wetting lines for the PS-g-silica NPs in PS matrices. On the basis of Figure 4, parts A and B, the range of P/N where the particles are stable expands from 2.7 at σ = 0.15 chains/nm2 to 4.3 at σ = 0.28 chains/nm2. This appears to be the optimum graft density for NPs of R = 9 nm. As the graft density continues to increase the particles aggregate at increasingly small P/N ratios, showing stability up to P/N of 1.6 for σ = 0.50 chains/nm2 and dropping to 1.3 at the highest graft density, σ = 0.70 chains/nm2. The lower and upper regions of Figure 4B are akin the allophobic and autophobic regions of Figure 1, respectively. Figure 4B shows the phase diagram from Figure 4A on a log−log plot with the fitted scalings for the upper and lower wetting lines, which were determined by linear-least-squares fit to the points on either side vertex of the complete wetting region; thus, the upper line was fit to σ ≥ 0.28 chains/nm2 for autophobic transition whereas the lower line was fit to σ ≤ 0.28 chains/nm2 for the allophobic transition. We find that the lower line scales with σ1 ∼ P0.54±0.08, which deviates significantly from the predictions of de Gennes5 and Maas et al,4 which were derived for flat surfaces on which σ1 ∼ P. Moreover, we find for the autophobic transition that σ2 ∼ P−0.71±0.04, which is close to the power law dependence predicted by de Gennes and Maas et al. of −0.5 but significantly lower than that predicted by Liebler and Ajdari at −2.0. On the basis of our interpretation, our results imply: (1) Particle curvature impacts the power-law dependence of the allophobic dewetting transition at lower graft densities where high curvature permits increased penetration of the matrix into the graft, and (2) particle curvature appears to have less of an effect at higher graft densities where NPs are destabilized by the expulsion of the melt from the melt polymer, i.e., the de Gennes and Maas et al. interpretation. We will follow this note with the submission of a full manuscript that covers in detail our studies on the impact of particle curvature on NP stability in homopolymer matrices.
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AUTHOR INFORMATION
Corresponding Author
*E-mail address:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors would like to acknowledge funding from the following sources: (1) National Science Foundation (NSF) NSF−CBET-0644890; (2) ChemMatCARS Sector 15, which is principally supported by the National Science Foundation/ Department of Energy under Grant No. NSF/CHE-0822838. Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.
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REFERENCES
(1) Ferreira, P. G.; Adjari, A.; Leibler, L. Macromolecules 1998, 31, 3994−4003. 4011
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