Morphology of Blends with Cross-Linked PMMA Microgels and Linear

Article pubs.acs.org/Macromolecules

Morphology of Blends with Cross-Linked PMMA Microgels and Linear PMMA Chains M. Schneider,† R. Michels,† V. Pipich,‡ G. Goerigk,§ V. Sauer,⊥ H.-P. Heim,⊥ and K. Huber†,* †

Physical Chemistry, University of Paderborn, 33098 Paderborn, Germany Outstation FRM-II, Jülich Centre for Neutron Science, 85747 Garching, Germany § Helmholtz Centre Berlin, 14109 Berlin, Germany ⊥ Institute for Materials Engineering−Polymer Technology, University of Kassel, 34125 Kassel, Germany ‡

S Supporting Information *

ABSTRACT: The present work investigates PMMA colloids in a polymer matrix of linear chains as a simple and suitable system for complex nanocomposites. The investigation was based on SANS experiments, which were enabled by the use of deuterated colloids immersed in a matrix of linear hydrogenated chains. Cross-linked deuterated PMMA-colloids were synthesized with two different sizes (70 and 140 nm) by means of the surfactant-free emulsion polymerization method and the swelling behavior adjusted by varying the amount of added cross-linker (1.5 and 15.0 mol-%) at each size respectively. Colloid−polymer blends were prepared from colloid−polymer solutions. SANS experiments on these blends consistently revealed that colloids with a low cross-linking density could be homogeneously distributed throughout the matrix of linear chains. Fits with model form factors indicated the structure of fuzzy spheres for these molecularly dispersed microgels, which are slightly swollen with respect to their size and shape in H2O. Contrary to this, colloids with a high cross-linking density form aggregates in the blends. Despite this aggregation, we succeeded to unravel the shape and the size of single colloids by preparation of colloid− polymer blends where the colloid component is a binary mixture of deuterated and hydrogenated colloids both with a high crosslinking density. SANS on the latter blends suggested a core−shell particle in a lattice cell of an aggregate.

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INTRODUCTION Nanocomposites or colloidal filler dispersions are a new class of materials where polymer chains act as a matrix for colloidal particles. Currently, an increasing interest can be observed in such hybrid materials because they are expected to offer significant mechanical improvements1−3 and further developments in the field of rubber technology,4 electrical properties,5 flammability,6 and optical properties of plastic materials.7,8 The performance of such materials beyond doubt is governed by the spatial distribution of the colloids within the hosting polymer matrix and aggregation of such colloids will certainly counteract the advantages provided by the added micro and nanofillers. Therefore, progress in preparing homogeneous hybrid-materials with optimally distributed fillers is an issue of utmost importance in nanocomposite research. Preparation of filler dispersions considers two main aspects. One aspect focuses on a modification of the filler surface in order to increase its compatibility with the hosting polymer matrix. This includes formation of colloidal brushes with polymeric hairs as compatibilizers9 or the coverage of the colloids with reactive groups ready to bind covalently to the polymer matrix. The other aspect is to control the mode of mixing of the two components, where several feasible alternatives have been established meanwhile. These alter© 2013 American Chemical Society

natives comprise the dispersion of the colloid in a liquid established by the monomer which later forms the polymer matrix thereby arresting the proper state of dispersion10 or preparation of two solutions, one with the nanofiller and one with the polymer and successively removing the solvent or precipitating the mixed solution in a nonsolvent for both components. It is only the morphological characterization of the composite which makes accessible a significant structure− property relationship between the morphology of the filler distribution on one side and the performance of the filler dispersion in the anticipated application on the other side. Hence, appropriate characterization of the resulting morphology is just as essential for establishing polymer composite materials as is the preparation techniques.11 Physiochemical methodologies frequently applied to investigate the composite morphology include solid-state NMR and small angle neutron scattering (SANS). Antonietti et al.12 have prepared blends from cross-linked polystyrene microgels with linear polystyrene (PS) chains. Received: September 12, 2013 Revised: October 17, 2013 Published: November 7, 2013 9091

dx.doi.org/10.1021/ma401889k | Macromolecules 2013, 46, 9091−9103

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deuterated linear PS. According to Mackay et al.19 the melt viscosity of the blends are governed by two features, entanglement of the linear matrix chains and the averaged distance between neighboring nanogels compared to the size of the matrix chains. As long as the matrix chains are shorter than the size required for chain entanglement, addition of nanoparticles increases the melt viscosity in close agreement with earlier findings.16 In the presence of chain entanglement in the polymer matrix, a drop in the melt viscosity could be revealed once the nanoparticles are closer together on average than the size of a polymer matrix coil, i.e. once the matrix chains get confined by the dispersed nanoparticles. An instructive example of model colloid−polymer composites which is at the edge of having relevance for technical applications is the investigation of the reinforcement of elastomers in order to improve the performance of tires.20 By means of small angle scattering and NMR experiments on simple binary mixtures of an elastomer with a silica based filler, the effect of a loose filler network interpenetrating the elastomer matrix was investigated, whereby the results offered the following reasoning. The colloidal filler particles are surrounded by a layer of polymer adopting a glassy state covered by a second layer with intermediate chain mobility. This shell of glassy and immobilized polymer segments act as a glue among the particles and rupture of those joints may be responsible for the so-called Payne effect.21 The latter corresponds to the striking decrease of the elastic modulus occurring under oscillatory shear beyond a certain small value of the strain amplitude. In the present work, we will prepare deuterated PMMA microgels, which shall be dispersed in a matrix of linear PMMA chains. The use of PMMA for the hosting polymer matrix and the added filler provides a system as simple as possible thereby greatly helping to reveal significant information on selected aspects of an otherwise highly complex system. Deuteration of the colloidal microgels shall enable us to investigate by SANS the structure of the microgels within the hosting matrix and along with this the degree of homogeneity of its spatial distribution in the PMMA matrix. Detailed characterization of the size and shape of the very colloids in solution state prior to its incorporation into the PMMA matrix will provide the necessary reference data in order to better judge differences inferred by the PMMA matrix. To this end two different solution states will be considered, (i) a dispersion of the deuterated colloids in water, where a highly compact state is expected and (ii) a dispersion in THF, which acts as a good solvent for PMMA and hence will swell the microgels. Preparation of colloids with two different radii, one close to 70 nm and another one close to 140 nm each at two different percentages of cross-linker respectively will enable us to investigate the impact of the filler size and of the degree of cross-linking on the dispersion state in the PMMA matrix. Three different protocols will be applied to generate suitable filler dispersions. All three protocols start with the codissolution of the colloid and the PMMA matrix at the boiling temperature of the respective solvent. After codissolution, the solvent will be removed according to the following three alternative routes: (i) slow evaporation of the solvent (SE method) leaving us with a composite film ready to be analyzed by SANS; (ii) freezedrying of the mixed solution (FD method) and succeeding film formation by injection molding; (iii) precipitation via adding a nonsolvent to the mixed solution (RP method) and succeeding film-formation by drying and injection molding of the

Since both components were chemically identical, the blends could easily be prepared by mixing the two component solutions and freeze-drying the mixed solution (FD method). A detailed NMR-study revealed that considerable penetration of the microgels could only be observed if the linear matrix chains are smaller than the network strands of the microgels. If the cross-linking density got high enough, penetration by the chains could be prevented, yet the microgels still exhibited a molecularly dispersed state. Schärtl et al. succeeded to prepare homogeneously dispersed composites of highly cross-linked polyorganosilaxane microgels in a PS matrix by grafting linear PS chains onto the microgels13,14 and by dispersing block-copolymer micelles with a PS core and a polybutadiene (PB) corona in a matrix of linear PB chains.15 Preparation of the blends were performed by solution evaporation (SE method) from a mixed solution of both components in THF in the case of the polyorganosiloxane microgels13,14 and by freeze-drying (FD method) a mixed solution in benzene in the case of the micelles.15 Homogeneity could be achieved once the chains grafted to the microgels or once the PB-blocks in the corona of the micelles were equal or larger in molecular weight than the molecular weight of the respective hosting matrix chains. This result is closely related to the observation made by Antonietti et al.,12 which indicates a penetration of matrix chains into the microgels, once the matrix chains get smaller than the mesh size in the microgels. Schärtl et al.16 successively investigated the rheological behavior of selected blends marked by the feature that the coil size of the matrix material kept below the entanglement regime. Addition of the spherical brushes to the linear matrix increased the zero shear viscosity of the blends, yet with some surprising trends. Counterintuitively, the degree of swelling of microgels with a high grafting density increased with increasing volume fraction of the microgels. This feature cannot be reconciled with spheres with a soft repulsion but refers to more complex interaction potentials between the brushes, which are controlled by the grafting density and length of the hairy PS chains. Montes et al.17 have prepared filler dispersions of modified silica particles in a polymethylmethacrylate (PMMA) matrix and analyzed the resulting dispersions by means of SANS. Dispersion was achieved by first covering the silica nanoparticles with a suitable agent compatible with the hosting PMMA elastomer or alternatively with functional residues ready to bind covalently to the hosting PMMA matrix and then embedding these model colloids in liquid acrylate monomers designed to form the cross-linked PMMA elastomer,18 which acts as the matrix. Succeeding characterization of the dispersion state revealed well distributed fillers if the covering residues include double bonds ready to polymerize with the acrylate monomers of the matrix. A depletion effect induced an aggregation of the colloids once this coverage is replaced by a nonpolymerizable group. Mackay et al.19 extended our knowledge on model composites from cross-linked PS in linear PS chains by preparing extremely small PS nanoparticles, which have radii smaller than 10 nm. This enabled them to analyze in detail the rheology of the melts at variable ratios of the size value of the nanoparticles to the size value of the linear matrix chains. An optimal mixing of the components was achieved by combining chemically identical components in a THF solution and rapid precipitation (RP method) of the solution in a nonsolvent for PS. Successful dispersion of the nanogels on a molecular level has been verified by SANS of the largest nanogel available in 9092


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15.0 mol-% of MMA) respectively. The compositions of the batches applied to prepare all deuterated colloids used in the present work are summarized in Table 2. The two hydrogenated colloids were prepared according to the same protocol as used for the synthesis of the deuterated colloid D70-LC and D70-HC. Only the reaction volume was scaled-up to V = 0.4 l, while keeping the concentration of the hydrogenated MMA, the cross-linker and the initiator respectively constant. Sample nomenclature was selected as follows: The first latter indicates deuterated (D) or hydrogenated (H) colloid, the number gives the size of the colloid in water (70 or 140 nm) and the two letters following the minus sign signifies colloid with a low (LC) or high cross-linking (HC) density. Blend Preparation. Three methods were applied to prepare blends of deuterated PMMA colloids and linear PMMA-6N: solvent evaporation (SE), rapid precipitation (RP), and freeze-drying (FD). In all cases, the deuterated colloid was dissolved at the boiling temperature of the respective solvent. The linear PMMA-6N was added in portions after complete dissolution of the colloids while the solution was still kept at its boiling point. The final solution was stirred for at least 2 days for complete homogenization. In the case of the SE method the colloid and the linear PMMA-6N were dissolved in chloroform. The resulting polymer-colloid solution was filled in a specimen with 15 cm × 4 cm × 0.1 cm. After 1 day of drying at room temperature, the film was detached from the specimen. For the RP method the colloid and the PMMA were dissolved in THF and precipitated in iso-hexane. The material was dried overnight at 70 °C. In the case of the FD method a colloid−polymer solution was prepared in benzene and freeze-dried. A white solid was obtained in the case of the RP and the FD method. For both methods foils were generated by mold-compression at 180 °C with a heatable molding press. Foils of purely hydrogenated PMMA were prepared according to the SE method and the FD method in order to provide appropriate background samples for the SANS experiments with colloid−polymer composites. Turbidity Measurements. Transmission experiments were performed with a Thermo Scientific Genesys 20 spectrometer at a fixed wavelength of λ = 500 nm. The transmission experiments were performed to analyze the phase behavior of mixed polymer-colloid solutions. All experiments were carried out in glass cuvettes with a path length of 1 cm. A pure PMMA-6N solution in chloroform was used as reference resulting in the transmitted intensity, I(PMMA-6N). This reference intensity is compared with the intensity I(PMMA-6N + colloid) of the beam transmitting the mixed polymer-colloid solutions. The mass concentrations of the PMMA-6N and of the colloid applied for each colloid−polymer solution were 19.9 wt % and 0.1 wt % respectively. The resulting transmission T is expressed as T = I(PMMA-6N + colloid)/I(PMMA-6N). Characterization of PMMA-6N by Size Exclusion Chromatography (SEC). The absolute molar mass distribution of the linear PMMA-6N, used as matrix material, was determined by a size exclusion chromatography device for high temperature measurements (Waters GPC Systems). Signals were collected by a coupled RI- and viscosity-detector, which enables universal calibration. For calibration, we used 8 narrow distributed polystyrene standard samples from PSS Polymer Standard Service (Mainz, Germany). BHT was the internal standard and TCB served as the solvent. For the separation, a sequence of three columns with 106, 105, and 103 Å pore size were applied. Then, 20−40 mg of PMMA-6N was dissolved in 10 mL of

precipitate. The investigation is designed to answer the following questions. Will the structure of single PMMA colloids dispersed in the PMMA matrix depend on the cross-linking density of the colloids? Does the degree of cross-linking exert an influence on the degree of homogeneity of the filler distribution throughout the dispersion? Does the application of different mixing protocols lead to significantly different filler dispersions?

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EXPERIMENTAL SECTION

Materials. Benzene, butylhydroxytoluene (BHT), chloroform, ethylene glycol dimethacrylate (EGMA), methyl methacrylate (MMA), methyl methacrylate-d8 (MMA-d8), tetrahydrofuran (THF), potassium peroxodisulfate (KPS), and 1.2.4-trichlorobenzene (TCB) were purchased from Sigma-Aldrich (Taufkirchen, Germany). All monomers were destabilized by a flush column with basic aluminum oxide Woelm B- Super I from Woelm Pharma (Bad Honnef, Germany) before use. Water was distilled twice before use. Matrix Polymer. PMMA-6N, which was used as the linear matrix, was granted from Evonik Industries AG (Marl, Germany). The PMMA-6N granulate was dried overnight at 75 °C prior to preparation of blends. A detailed analysis of the sample PMMA-6N has been performed with size exclusion chromatography (SEC) and static light scattering (SLS). The results are summarized in Table 1.

Table 1. Characterization of PMMA-6N by SEC and SLS sample PMMA6N

Mn(SEC)/ (g/mol)

Mw(SEC)/ (g/mol)

PDI (SEC)

Mw(SLS)/ (g/mol)

Rg/nm

64200

91500

1.43

82600

10.0

The results of both methods agree well within the uncertainty of the characterization methods. The weight-averaged molar mass is Mw = 85000 g/mol and the particle size of the polymer coil in THF is Rg = 10 nm. These results indicate that the polymer is entangled in the bulk phase, since the critical molar mass for entanglement is approximately 10000 g/mol, which is smaller by an order of magnitude than the Mw of PMMA-6N. Surfactant-Free Emulsion Polymerization (SFEP). All deuterated colloids were prepared by surfactant-free emulsion polymerization in the presence of the cross-linker EGMA according to a procedure published by Zentel et al.22 For a typical polymerization a 250 mL two-neck round-bottom flask was equipped with a reflux condenser and two septums. The flask was filled with 185 mL bidistilled water, heated up to 85 °C and purged with nitrogen for 30 min. After the nitrogen flow was stopped 1.98 mL (20 mmol) MMAd8 and 0.05 mL (0.3 mmol) EGMA were added in one shot. The polymerization was initiated by adding a solution of 13.5 mg KPS in 2 mL degassed water. After 2 h the reaction was stopped by opening the reaction vessel. The solution was further stirred for 30 min at 85 °C to remove nonreacted monomers by evaporation. Aggregates were removed by filtration though a folded filter (Machery-Nagel, Düren, Germany) with 12.5 cm diameter. The final solution was dried in an oven at 75 °C leaving a white powder. The particle size and crosslinking density was adjusted by varying the monomer concentration (c = 0.1 and 0.5 mol/L) and the amount of added cross-linker (1.5 and

Table 2. Compositions Applied for Preparation of the Deuterated and Hydrogenated PMMA Colloids with SFEP batch

V(H2O)/L

c(MMA)/(mol/L)

V(MMA)/mL

cross-linker/mol % MMA

V(EGMA)/mL

m(K2S2O8)/mg

D70-LC D70-HC D140-LC D140-HC H70-LC H70-HC

0.185

0.1

1.98

0.5

0.4

0.1

0.05 0.53 0.05 0.53 0.11 1.15

13.5

0.037

1.5 15.0 1.5 15.0 1.5 15.0

4.27

9093

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Table 3. Results from SLS and DLS Experiments with Deuterated Colloids in Water and THF H2O

THF

batch

Rg/nm

Rh/nm

ρ

⟨varz⟩

Rg/nm

Rh/nm

ρ

S

D70-LC D70-HC D140-LC D140-HC

73.56 77.8 136.1 153.3

96.36 94.9 162.4 181.1

0.76 0.82 0.83 0.85

0.006 0.008 0.0007 −0.007

82.9 92.9 185.8 174.4

189.67 142.3 336.6 232.2

0.43 0.65 0.55 0.75

7.63 3.37 8.90 2.11

TCB and heated up to 165 °C for complete dissolution of the polymer. The injection volume was 220 μL. Each sample was measured at least 2 times at 155 °C. The universal calibration enabled determination of the absolute molar mass distribution of PMMA-6N and a calculation of its mean values represented as the number averaged molar mass Mn and the mass averaged molar Mw. The polydispersity (PDI) was calculated by Mw/Mn. The results from SEC are summarized in Table 1. Static Light Scattering Characterization of PMMA-6N. The characterization of PMMA-6N by static light scattering (SLS) was performed with a combined static and dynamic light scattering (DLS) instrument model 5000e goniometer system from ALV-Laser Vertriebsgesellschaft (Langen. Germany). The light source was a Nd:YAG laser with an output of 100 mW operating at a wavelength of λ = 532 nm. Cylindrical quartz glass cells from Hellma (Germany) with 20 mm outer diameter were used to perform SLS/DLSmeasurements in the range of 30° ≤ θ ≤ 150°. The cell housing of the goniometer system was thermally adjusted by a Haake thermostat. The solvent THF and PMMA-6N solutions in THF were filtered through PTFE syringe filters (Machery-Nagel, Düren, Germany) with 0.2 μm pore size. Four solutions of PMMA-6N in THF were prepared in the regime of 1 g/L ≤ c ≤ 4 g/L. The SLS-measurements were performed at 25 °C. The absolute weight averaged molar mass (Mw) and radius of gyration (Rg) of the linear PMMA-6N was established from the SLS data following Zimm’s approximation24

Characterization of the Colloids by SLS and DLS. The deuterated colloids from all four batches were analyzed with the combined SLS and DLS instrument model 5000e goniometer system from ALV-Laser Vertriebsgesellschaft (Langen, Germany) under the same instrumental conditions that were applied for the SLS characterization of the PMMA-6N. Each colloid was analyzed in water and THF. All solvents were filtered through PE (water) or PTFE (THF) syringe filters (Machery-Nagel, Düren, Germany) with 0.2 μm pore size. Colloid solutions in water were filtered through a 0.45 μm (D70-LC and D70-HC) or a 1.2 μm (D140-LC and D140HC) syringe PE-filter (Machery-Nagel, Düren, Germany). All THF solutions were filtered through a PTFE of 1.2 μm pore size (MacheryNagel, Düren, Germany). Starting from a stock solution with 1 g/L, the solutions of colloids in water or THF were diluted to a final concentration of approximately c = 0.01 g/L. The two hydrogenated colloids H70-LC and H70-HC were characterized according to the same procedure as the four deuterated colloids. The colloidal particles were characterized using the Guinier approximation,26 which is valid in the range of qRg < 2:

ln ΔR(q) = ln ΔR(q = 0) −

(1)

2 4π 2 ⎛⎜ dn ⎞⎟ n0 4 λ 0 NA ⎝ dc ⎠

4πn sin(θ /2) λ0

varz =

For small values of q, the following approximation

(4) 2

was used for 1/P(q), where Rg is the z-averaged mean-square radius of gyration, simplifying eq1 to:

Rg 2 2 Kc 1 = + q + A2 c ΔR(q) Mw 3M w

(7)

μ2 Γ

2

=

⟨D2⟩z − ⟨D⟩z 2 ⟨D⟩z 2

(9)

The variance of the diffusion coefficient is commonly used as a measure for the polydispersity of the polymer sample. Correlation functions were evaluated by cumulant analysis with a 90% and a 69% decay of the amplitude [g2(τ) − 1]1/2. The latter decay is suitable for the determination of varz.28 Since no angular dependence of apparent diffusion coefficients was observed in the range of 30° ≤ θ ≤ 80°, we calculated the mean value of the apparent diffusion coefficients in order to extract the z-averaged particle diffusion coefficient D0. The same procedure was applied for the determination of varz, where again no angular dependence was observed. Some varz values were negative, which is physically impossible, but can be attributed to the experimental uncertainty of a value close to zero. A detailed graphical outline of the trends of varz with q is given in the Supporting Information. The mean values of varz, which were averaged over all q-values, are given in Table 3. The mean values of ⟨varz⟩ nicely illustrate that the distributions of the diffusion coefficient are extremely narrow corresponding to monodisperse particles. In addition, the standard deviation of varz was 10 times

(3)

Rg 2 2 1 ≅1+ q P(q) 3

(6)

In eq8 Γ is the z-average of the decay rate and μ2 is the second moment of the diffusion coefficient. The latter leads to the variance of the z-average of the diffusion coefficients expressed as

(2)

with λ0 the wavelength of the laser, NA Avogadro’s number, n0 the refractive index of the solvent and the refractive index increment of the polymer in the respective solvent dn/dc. Here, a dn/dc of 0.0871 mL/g was applied for PMMA-6N in THF.25 The scattering vector is dependent on the wavelength of the laser and the observation angle θ:

q=

q2

were converted into the corresponding field-time-correlation-functions g1(τ). The field-time-correlation function g1(τ) was evaluated with the second-order-cumulant method.27 Where the field-time-correlationfunction g1(τ) was approximated by μ ln g1(τ ) = ln β1/2 − Γτ + 2 τ 2 (8) 2

where K is the contrast factor, c the concentration of polymer in g/L, ΔR(q) the excess Rayleigh ratio of the particle solution with respect to the solvent, P(q) the form factor and A2 the second osmotic virial coefficient. The contrast factor is defined as K=

3

DLS experiments were performed simultaneously with the SLS measurements. Under the assumption that the Siegert-relation is valid, the resulting normalized intensity-correlation-functions g2(τ)

g2(τ ) = 1 + β |g1(τ )|2

Kc 1 = + A 2c ΔR(q) MP(q)

Rg 2

(5)

The resulting scattering curves were extrapolated to q = 0 and c = 0 in order to extract Mw and Rg. The results of the SLS analysis are given in Table 1. 9094


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higher than the corresponding mean value ⟨varz⟩. Hence, we concluded that the deuterated colloids reveal finite values of varz from DLS, small enough to enable the conclusion that the polydispersity parameter ⟨varz⟩ does not differ significantly from zero. Furthermore D0 can be used to calculate the hydrodynamically effective radius Rh with the Stokes−Einstein-equation Rh =

kT 1 6πη D0

(10)

with k the Boltzmann constant, T the temperature, and η the (dynamic) viscosity of the respective solvent. The calculated radius of gyration and the hydrodynamic radius were used to determine the structure sensitive factor ρ.

ρ=

Figure 1. Guinier plot and respective data analysis of the neutron scattering experiment with sample D140-HC in water. Data points selected for the fit are marked with black circles. The radius of gyration resulting from the indicated slope is Rg = 145 nm.

Rg Rh

(11)

For homogeneous spheres the structure sensitive factor is ρ = 0.77529 and for swollen microgels values of 0.3 ≤ ρ ≤ 0.50 were observed.30−32 Small Angle Neutron Scattering Experiments (SANS). All SANS experiments were carried out with the KWS-3 focusing-mirror high-resolution small angle scattering instrument at the Jülich Centre for Neutron Science (JCNS) located at the FRM-II research reactor in Garching, Germany. Samples were characterized at sample-to-detector distances of L = 9.5 m and L = 1.2 m with a wavelength of λ = 12.8 Å. The wavelength spread of the selector was Δλ/λ ≈ 20%. With both setups a total q-range 2.5 × 10−3 nm−1 ≤ q ≤ 2.0 × 10−1 nm−1 was accessible. The reduction of raw data was performed by the routine qtiKWS33 including corrections for detector sensitivity, background noise and the empty cell. The scattering data were expressed in absolute units (cm−1) with the calibration being based on the direct neutron beam.34,35 The absolute differential scattering cross section dΣ/dΩ(q) was calculated according to

degree of smearing to the model scattering curves. This was performed with the SasView library37 using an appropriate resolution function R(q,q′) for pinhole SANS. This resolution function modulates the scattering intensity, which can be represented for particles with spherical symmetry as38

⎛ dΣ ⎞ 2 ⎟(q) = nΔρ Δ⎜ ⎝ dΩ ⎠

D(R , R̅ , σ )V 2(R )Psp(q′, R )S(q′) (14)

where n is the number density of particles, Δρ is the difference in scattering length density between colloid and sample background (solvent or polymer matrix), V is the volume of the spherical particle with radius R, D(R,R̅ ,σ) is a suitable distribution of radii R with an average R̅ and a standard deviation σ, Psp(q,R) is the form factor of particles with a spherical symmetry, S(q) is the structure factor and R(q,q′) is the instrumental resolution function. It is this resolution function that has to be established now. One procedure to establish the influence of smearing can be performed with scattering data with well-known model character. In the present work these data are available from our SANS measurements performed with samples D70-LC and D140-HC in water, where all deuterated colloids behave as hard spheres. This model character could be nicely demonstrated by our combined SLS and DLS analysis and by SEM: SLS and DLS verified the hard sphere behavior in water by revealing values for the structure sensitive factor ρ defined by eq11, which are close to ρ = 0.78 for all four deuterated samples (Table 3) and at the same time demonstrated that the samples have a negligible polydispersity in size, since varz defined in eq9 is close to zero. SEM-images confirmed the spherical shape and the narrow size distribution. Further information are given in the Supporting Information. Hence, the form factor of a monodisperse sphere defined as39

(12)

Indices s, sb, ec, dc, and eb refer to, sample, sample background, empty cell, dark current of the detector, and electronic background respectively. The flux I0, which is incoming to the sample, is determined by the direct beam measurement of the empty cell and includes a correction for the electronic background. The sample (s) was a colloid solution or a blend from a colloid with the linear PMMA matrix. The respective sample background (sb) was the solvent (water or THF) or the pure linear PMMA matrix. Since, the last preparation step of RP and FD method were equal, we used the FD foil as background for the RP and the FD colloid−polymer composites. The parameters T, a, h, I(q), and L denote the transmission, the pixel size, the sample thickness, the scattered intensity and sample-to-detectordistance. Finally, the differential scattering cross section dΣ/dΩ(q) was radially averaged. The scattering curves of the blends and the pure polymer matrix were corrected for varying foil thickness within 0.5 ≤ h ≤ 1.1 mm. The absolute scattering cross section Δ(dΣ/dΩ) of colloids was calculated by subtracting the cross section of the sample background from the cross section of the sample:

⎛ dΣ ⎞ ⎛ dΣ ⎞ ⎛ dΣ ⎞ ⎟ = ⎜ ⎟ − ⎜ ⎟ Δ⎜ ⎝ d Ω ⎠ ⎝ d Ω ⎠s ⎝ d Ω ⎠sb

∞

R(q , q′) dR dq′

dΣ L2 (q)s , sb = 2 ([I(q)s , sb − I(q)dc ] dΩ a Ts , sbhs , sbI0 − Ts , sb[I(q)ec − I(q)dc ])

∞

∫0 ∫0

⎡ 3[sin(qR ) − 3qR cos(qR ) ⎤2 ⎥ Psp(q) = ⎢ (qR )3 ⎣ ⎦

(15)

is used for Psp(q) in eq14, with the distribution D(R,R̅ ,σ) of particle size being neglected. Any residual polydispersity, if existing, is inevitably plugged into the resolution function R(q,q′). Further on, we set S(q) ≈ 1, because low enough concentrations of colloids were used for the preparation of solutions. Under the assumption that the distribution of the scattering vector q′ around the value q of the respective data point obeys a Gaussian distribution37,40 with the standard deviation σ, the resolution function R(q,q′) can be expressed as follows

(13)

If not stated otherwise, the resulting scattering curves from both detector distances were merged with the qtiKWS routine.33 Neutron scattering curves were further evaluated using eq6 for the calculation of Rg whereby ΔR was simply replaced by Δ(dΣ/dΩ). Figure 1 shows a representative Guinier plot. The Smearing of SANS Data. All neutron scattering curves were smeared due to the instrumental resolution of the small angle scattering diffractometer. To correct for such a smearing either the raw scattering data can be desmeared or model curves can be smeared using a q-dependent resolution function R(q,q′).36,37 We attributed a

R(q , q′) =

⎡ − (q − q′)2 ⎤ 1 ⎢ ⎥ exp 2σ 2 ⎣ ⎦ (2πσ 2)1/2

(16)

We have used for a smearing of our model scattering curves the simplified smearing algorithm of a pinhole geometry. The latter applies for the wavelength spread Δλ/λ, the cross section of the beam Δxbeam 9095


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and the size of one detector pixel Δxpixel. The beam divergence as fourth parameter of a pinhole based smearing was not taken into account due to the focusing neutron optics of the KWS-3. This neutron optics establishes a nondivergent i.e. convergent focusing beam. Accordingly, the q-dependent uncertainty of KWS-3 was defined by eq17

Δq ≈ q

2 2 ⎛ Δλ ⎞2 ⎛ 2π ⎞2 (Δxbeam) + (Δxpixel) ⎟ × ⎜ ⎟ + ⎜ ⎝ λ ⎠ ⎝ λ ⎠ (Lq)2

(17)

The ratio Δq/q from eq17 will successively enter eq16 via σ = (Δq)2. In a further step, we had to consider, which resolution for Δq/q is suitable to describe the smearing. Application of eq17 to our experimental setup based on a sample-to-detector distance of L = 9.5 m and 1.2 m, a pixel size of Δxpixel = 0.4 mm and a beam width of Δxbeam = 2 mm is shown in Figure 2. At low q-values, the instrumental 2

Figure 3. Scattering data (○) recorded at a sample-to-detectordistance of 9.5 and 1.2 m for samples D70-LC (A) and D70-HC (B) at a colloid concentration of 0.1 g/L. In addition, a fit based on a monodisperse sphere using an uncertainty of Δq/q = 0.13 (blue line) of each q-value for smearing is shown.

DLS in water and THF. The swelling ratio S was determined from the hydrodynamic radii determined in water (collapsed state) and THF (swollen state) by means of DLS-experiments.41 Figure 2. Normalized instrumental uncertainty Δq/q as a function of the scattering vector q for the detector-to-sample-distance of 9.5 m (black line) and 1.2 m (red line). The gray area represents the qregime of the 1.2 m sample-to-detector distance, which was omitted for the fitting procedure due to the high smearing of the respective qvalues. The resulting Δq/q-values are normalized by a wavelength spread of Δλ/λ = 0.13.

S=

R h , THF 3 Vswollen ≈ Vcollapsed R h , H2O 3

(18)

The data from light scattering, which are listed in Table 3, show that the synthesized deuterated colloidal particles revealed the anticipated size values and swelling behavior. The swelling is compatible with the behavior observed with an extended series of hydrogenated colloids also prepared in our laboratory by following the same synthetic protocols. Details on the comparative swelling study of deuterated and hydrogenated colloids are given in the Supporting Information. The structure sensitive factor ρ observed for all four deuterated colloids in water is close to the value of ρ = 0.78 expected for hard spheres.29 It is noteworthy that all four values measured in THF were below this limit, in agreement with earlier findings. 30,31 This may provide a hint for an inhomogeneous swelling, which apparently has a lower effect on Rg than on Rh. As expected the degree of swelling is larger for colloids with the low degree of cross-linking. Additional SANS experiments of the colloidal particles in both solvents complete our knowledge on the reference system and will enable us to appropriately investigate the swelling of the respective deuterated colloids in the linear PMMA-matrix. SANS with the deuterated colloids were performed with concentrations c = 1.0 g/L in both solvents. Since, the small colloids D70-LC and D70-HC showed a slight tendency to aggregate in water, we repeated the SANS-investigation of solutions containing the smaller colloids in water also at concentrations of 0.1 g/L. Figure 3 and 4 represent the SANS data of the deuterated colloids in water at a concentration of c = 0.1 and c = 1.0 g/L. All scattering curves measured in water revealed a satisfactory fit with the model of the monodisperse hard sphere including a smearing based on Δq/q = 0.13. The plotted data of the large colloids D140-LC and D140-HC indicate that the first minimum of the sphere form factor is located at the overlap regime of the curves recorded at 9.5 and 1.2 m detector-to-

uncertainty in q, which was normalized by Δλ/λ depends on the scattering vector q and is decaying to a value, which is proportional to Δλ/λ once q is high enough. Noteworthy, this limiting value is approached at a higher q-value in the case of the shorter distance with L = 1.2 m. At a q-value of q = 0.03 nm−1, to give but one example, the instrumental resolution at a detector distance of 1.2 m is 2.3 times larger than the instrumental resolution at the detector distance of 9.5 m. Despite the strong upturn at low q, we decided to use a fixed value of Δq/q for all model fits to be applied to our SANS data and to omit the regime of 2.8 × 10−2 nm−1 ≤ q ≤ 3.8 × 10−2 nm−1 from the short distance, if necessary, since the upturn toward low q-values is most pronounced in this regime for L = 1.2 m. Justification for this procedure is based on the following argument. For the distance at L = 9.5 m Δq/q has decayed to values smaller than 2 already beyond q ∼ 4.7 × 10−2 nm−1 but the same low level is achieved for L = 1.2 m only at q > 3.8 × 10−1 nm−1. For the evaluation of a fixed Δq/q, which is suitable to capture the optimal overall smearing by the instrument KWS-3, the scattering curve of the deuterated colloid D70-LC in water was examined in detail. Figure 3A shows the scattering curve and the best model-fit with the form factor of a monodisperse sphere based on eqs14, 15 and 16 with D(R,R̅ ,σ) = 1 and S(q) = 1. The experimental curve was fitted at variable instrumental smearing Δq/q covering a regime of 0.10 ≤ Δq/ q ≤ 0.16. The same q-range and number density of particles had been applied for all fits. The resulting χ2-values exhibit a minimum at Δq/q = 0.13. Hence, a value of Δq/q = 0.13 will be applied for all fits with model form factors in the present work.

■

RESULTS AND DISCUSSION Colloid Characterization. Deuterated colloids were prepared with MMA-d8 as monomer and EGMA as crosslinker and first characterized by means of combined SLS and 9096


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Article 2 ⎡ ⎛ −(σ q)2 ⎞⎤ 3[sin(qR ) − qR cos(qR )] surf ⎢ ⎥ ⎜ ⎟ Pfuzzy(q) = exp⎜ ⎟⎥ ⎢⎣ 2 (qR )3 ⎝ ⎠⎦

(19)

in addition to model fits with the hard sphere model. Equation 19 describes a spherical particle with a blurred surface-layer, which is compatible with a highly cross-linked core and a shell with decreasing cross-linking density. The model, denoted as fuzzy sphere, is determined by two parameters, the width of the fuzzy surface σsurf and a hard core with the radius Rbox with R = Rbox + 2·σsurf. The scattering length density ρ decreases with increasing distance from the center of the particle reaching a values of ρ = 0 at R + 2σsurf. The scattering curves with the respective fits in Figure 5 nicely show that the fuzzy sphere model improves the

Figure 4. Scattering data (○) recorded at a sample-to-detectordistance of 9.5 and 1.2 m for samples D140-LC (A) and D140-HC (B) at a colloid concentration of 1.0 g/L. In addition, a fit based on a monodisperse sphere using an uncertainty of Δq/q = 0.13 (blue line) of each q-value for smearing is shown. The data points marked in red were omitted for the fitting procedure.

sample-distance. Since smearing is very high at the corresponding low q-regime of 1.6 × 10−2 nm−1 ≤ q ≤ 3.8 × 10−2 nm−1 for L = 1.2 m, the first minimum was partly (D140-HC) or entirely (D140-LC) missed by data recorded at L = 9.5 m and could not be resolved by data recorded at L = 1.2 m for the large colloids. This problem did not affect the scattering curves of the small colloids D70-LC and D70-HC, because the first minimum of the form factor is shifted toward higher q-values and could thus be resolved by data recorded at L = 9.5 m once it appeared. Therefore we omitted the data recorded at 2.8 × 10−1 nm−1 ≤ q ≤ 3.8 × 10−1 nm−1 with the distance of L = 1.2 m for model fits to the large colloids D140-LC and D140-HC. The sphere radius R was directly obtained as a model parameter of the respective fit based on eqs14 and 15. It can be converted into the radius of gyration of the respective model Rg(model) by Rg(model) = R/(5/3)1/2 (Table 4).

Figure 5. SANS data (○) recorded in THF at a colloid concentration of 1.0 g/L for batches D70-LC (A), D70-HC (B), D140-LC (C) and D140-HC (D) with a best fit based on the model of a monodisperse sphere (blue line) and a fuzzy sphere (green line). All fits included a smearing based on Δq/q = 0.13 for all q-values.

description of the data considerably compared to the hard sphere model particularly for q > 4 × 10−2 nm−1 for D70-LC and D70-HC and q > 2 × 10−2 nm−1 for D140-LC and D140HC. Independent of the degree of cross-linking, all particles are swollen in THF. Only the degree of swelling is varying with the amount of added cross-linker, which is expressed in terms of the fuzziness parameter σ surf. Noteworthy, σ surf varies consistently with the swelling ratio S determined from DLS (see Table 3). In addition, the radius of gyration in THF measured by SANS increased compared to the respective value in water. However, the Rg values of D70-HC increased stronger than the Rg values of D70-LC contrary to our expectations. The resulting fit parameters for the colloids in THF are summarized in Table 5. The goodness of the fit is expressed by the parameter χ,2 defined as37

Table 4. Radius of Gyration Determined from SLS and SANS in Comparison to the Radius of Gyration Rg(model) and the Outer Sphere Radius R Calculated from the Monodisperse Sphere Fit to the SANS Experiments in Water batch

Rg(SLS)/nm

Rg(SANS)/nm

Rg(model)/nm

R/nm

D70-LC D70-HC D140-LC D140-HC

73.6 77.8 136.1 153.3

79.1 71.2 134.0 144.6

77.3 70.2 125.4 128.6

99.7 90.6 161.9 165.9

SANS experiments of all four colloids in the good solvent THF completed the swelling analysis of the colloids. The colloid concentration was 1.0 g/L for all SANS experiments in THF. Since the structure sensitive ratio determined by SLS/ DLS experiments was smaller than the limit of ρ = 0.775, we hypothesize that the PMMA colloids are inhomogeneously swollen in THF.23 Probably a higher degree of swelling occurs in an outer shell than in an inner core. Therefore the scattering intensity was modeled by the fuzzy sphere form-factor Pfuzzy(q) by Stieger et al.38

⎛N χ 2 = ⎜∑ ⎜ ⎝ i=1

dΣ (fit) dΩ

− σ

dΣ (exp) ⎞ dΩ ⎟

⎟ ⎠

/(N − p) (20)

where dΣ/dΩ(fit) represents fitted values, dΣ/dΩ(exp) are experimental data, N is the number of data points, σ is the associated uncertainty, and p is the number of free fit parameters. In order to compare the radii of gyration from light scattering Rg(SLS) and neutron scattering Rg(SANS) with the respective 9097


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Table 5. Radius of Gyration Determined from SLS and SANS of the Experiments in THF in Comparison to the Corresponding Fit Parameters of the Fit with the Sphere Model and the Fit with the Fuzzy Sphere Modela Rg(SLS)/nm

Rg(SANS)/nm

model

Rg(Fit)/nm

R/nm

σsurf/nm

χ2

D70-LC

82.9

83.8

D70-HC

92.9

95.8

D140-LC

185.8

176.9

D140-HC

174.4

165.8

sphere fuzzy sphere sphere fuzzy sphere sphere fuzzy sphere sphere fuzzy sphere

78.8 80.4 102.3 100.9 180.7 176.7 163.9 165.2

104.2 99.3 124.1 121.1 210.2 195.2 196.0 189.6

− 16.0 − 14.3 − 42.1 − 24.0

8.55 8.20 8.47 7.54 133.55 45.8 63.29 43.4

batch

a

In the case of the sphere model R denotes the outer sphere radius, and in the case of the fuzzy sphere model the parameter R is defined by R = Rbox + 2·σsurf .

values of the model fits, we applied a Guinier-evaluation according to eq6 also to the fitted model curves and thus calculated the radius of gyration Rg(fit). SANS Investigation of Polymer−Colloid Blends. With the detailed set of reference data presented above, proper analysis of SANS data from the blend is now accessible. To this end, blends were prepared according to three different methods: evaporation of chloroform (SE), precipitation of a THF solution in isohexane (RP) and freeze-drying of the benzene solution (FD).13,19,12 The first method directly resulted in a thin foil. In the case of the second and third method thin foils were obtained after mold injection with a heatable press. All SANS experiments on colloids in polymercolloid-blends were performed with PMMA-6N as linear matrix material. The most striking result, to begin with, refers to the degree of homogeneity achieved with the blends. Blends with colloidal particles with a low cross-linking density are well dispersed, since a plateau at low q-values is observed for SANS-curves in Figure 6A and C. On the other hand, the lack of such a plateau toward low q-values indicates a partial aggregation of the highly cross-linked deuterated colloidal particles in the blends (Figure 6B and D). The tendency to form aggregates depends also on

the preparation method. As can be seen in Figure 6B and D, the upturn at low q-values is strongest for the SE method. The ratio of colloid size to chain size is 7:1 for the D70-LC and D70-HC colloids and 14:1 for the D140-LC and D140-HC colloids and the size of the matrix chains lies well above the entanglement threshold. Hence, according to Mackay et al.19 a phase separation of colloidal particles should be observable for all blends. Yet, the phase separation was only present in the case of the blends that consist of highly cross-linked colloids. This behavior requires further explanations. In order to shed further light on this issue, we more closely scrutinized the form factors of the two colloids D70-LC and D140-LC (Table 6) with the low cross-linking density, which could be well dispersed in the polymer matrix. To this end the scattering curves were interpreted with the model of a homogeneous sphere39 and with the model of a fuzzy sphere.38 Table 6. Results from the Fits with Model Form Factors in Comparison to the Radius of Gyration Established with the Guinier Approximation to the SANS Data at Low q for Blends Containing the Colloids D70-LC and D140-LC, both Having a Low Cross-Linking Densitya sample preparation SE

RP

FD

SE

RP

FD

Figure 6. SANS curves of blends with 0.5 wt % of deuterated colloid in a PMMA-6N matrix for colloids: D70-LC (A), D70-HC (B), D140-LC (C), and D140-HC (D). The symbols denote the preparation method of the blends: SE method (○), RP method (red ○), and FD method (blue ○).

Rg(SANS)/nm

model

R/nm

σsurf/nm

Data for Blends with D70-LC Colloids 70.8 sphere 88.5 fuzzy 88.5 sphere 69.4 sphere 76.2 fuzzy 85.1 sphere 62.2 sphere 78.4 fuzzy 76.9 sphere Data for Blends with D140-LC Colloids 130.3 sphere 145.9 fuzzy 144.7 sphere 123.3 sphere 146.0 fuzzy 143.8 sphere 121.4 sphere 147.4 fuzzy 146.3 sphere

χ2

− 0.5

4.36 4.36

− 7.6

3.20 3.00

− 7.6

5.25 4.59

− 9.5

320.9 314.4

− 13.5

318.1 307.8

− 10.5

158.9 151.7

a

In the case of the sphere model R denotes the outer sphere radius, and in the case of the fuzzy sphere model the parameter R is defined by R = Rbox + 2σsurf. The quality of fit is given in terms of χ2 defined by eq 20. 9098


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Figure 7 depicts the scattering intensity for blends with the colloids D70-LC and D140-LC, in comparison with the sphere

Figure 8. Ratio of fitted scattering cross-section and experimental scattering cross-section of blends with the deuterated colloids D70-LC (left column) and D140-LC (right column) for all three preparation methods: the SE method (1), the RP method (2), and the FD method (3). The color of the lines refers to the form factor model, which was used for the calculation of the scattering ratios: the sphere form factor (blue line) and the fuzzy sphere form factor (red line). The gray area was omitted for fit procedures, since the experimental data in this range were affected by an uncertainty in q, which was significantly larger than Δq/q = 0.13.

Figure 7. Scattering curves (○) of blends with 0.5 wt % of the colloid D70-LC (A) and of the colloid D140-LC (B). The numbers refer to the sample preparation: SE method (1), RP method (2) and FD method (3). The models are marked as follows: (blue line) the sphere model and (red line) the fuzzy sphere model. The data points of Figure 8B, which are marked in red, were omitted for the fitting procedure. The second and third scattering curve in parts A and B are shifted by a factor of 10 and 100 respectively.

water. The same procedure has been applied for colloids with low cross-linking density, where H70-LC with Rg = 74 nm and Rh = 96 nm (in water) complements the pair of colloids. Blends of these colloids were prepared at an overall mass fraction of 5.0 wt % colloid with different ratios of the hydrogenated to the deuterated colloids. This mixing of hydrogenated and deuterated colloids in a polymer matrix, where the hydrogenated colloid is invisible for SANS-experiments, because it has nearly the same contrast as the hydrogenated polymer matrix, is expected to provide a “dilution series” of deuterated colloids suitable to give access to the particle shape of the deuterated colloids within colloidal aggregates via extrapolation of the data to infinite dilution of the deuterated component. The initial SANS-experiments on blends with binary mixtures of colloids, where the blends consisted of 95.0 wt % PMMA-6N, 4.5 wt % H70-LC or H70-HC and 0.5 wt % D70LC or D70-HC, are shown in the Figure S7 and S8 of the Supporting Information. All blends were prepared according to the FD method. The scattering curves in Figure S7 showed the same trends as the scattering curves in Figure 6A and B, indicating a homogeneous distribution in blends with the low cross-linked colloids and an aggregation of colloids in blends with the highly cross-linked colloids. The direct comparison of the scattering data for the blends with the pure highly crosslinked deuterated colloid D70-HC and the binary mixture of the highly cross-linked colloids D70-HC and H70-HC nicely outline that the slope toward low q-values was significantly decreased for the blend with the binary mixture (Figure S7). Based on these preliminary SANS-experiments, we prepared four additional blends by the FD method and characterized the “resulting dilution series” by means of SANS. All blends were composed of 95.0 wt % PMMA-6N and 5.0 wt % of highly cross-linked colloid, but the amount of the deuterated colloid D70-HC was further decreased from 0.4 wt % to 0.1 wt %. The scattering curves, which are presented in Figure 9 show that the initial slope at low q-value decreased with decreasing content of

and fuzzy sphere model fits. For all fits performed with data from D140-LC the q-regime of 2.8 × 10−2 nm−1 < q < 3.8 × 10−2 nm−1 recorded at a detector distance of L = 1.2 m has again been discarded due to the high smearing of the experimental data in this regime. Except for the sample D70LC prepared with the SE method, all data at q > 3 × 10−2 nm−1 in Figure 7 revealed a marginal improvement once the model of a fuzzy sphere was applied. For both colloids the degree of fuzziness seemed to be the most pronounced in blends prepared by the FD method. Comparative judgment of the fit quality is supported by a representation of the corresponding trends for χ2, which is shown in Figure 8. Here, we plotted the ratios of the fit curves and the experimental data of blends with D70-LC and D140-LC versus the scattering vector q. The scattering curves from the blends with highly crosslinked samples D70-HC and D140-HC were dominated by aggregates at q < 2 × 10−2 nm−1 and it turned out to be too difficult to identify a suitable model for these aggregates. In order to still provide information on the morphology of those highly cross-linked single colloids in blends the following experiment had been designed. We prepared blends with the FD method that again consisted of PMMA-6N as matrix material. As filler, we applied a binary colloid mixture, where the two colloid components differ only in the scattering contrast, i.e., values for size and cross-linking density were kept as close to each other as possible. The size was set to Rg ≈ 70 nm. Since this corresponds to the size of the small deuterated D70-HC colloids, we selected the deuterated colloid D70-HC as the deuterated component and prepared in addition a hydrogenated colloid denoted as H70-HC. Sample H70-HC turned out to have size values of Rg = 83.4 and Rh = 104.2 in 9099


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SANS in the dilution series of our blends are not sphere-like colloids isolated in the linear matrix, but marked colloids filling lattice cells in an aggregate. Space filling may be less effective in the outer zones adjacent to neighboring colloids. The simplest model able to account for this feature of a sphere whose density decays to the faces of neighboring spheres may be a core−shell structure. The normalized core−shell (CS) form factor PCS(q) reads as follows26 Figure 9. SANS data of blends with 95.0 wt % PMMA-6N and 5.0 wt % of highly cross-linked colloids. The highly cross-linked colloid is a mixture of the deuterated colloid D70-HC and the hydrogenated colloid H70-HC, where both colloids have radii of Rg ≈ 70 nm. The symbols indicate the amount of deuterated colloid in the blend: 0.4 wt % (red ○), 0.3 wt % (blue ○), 0.2 wt % (green ○), and 0.1 wt % (○).

PCS(q) =

[sin(qRc) − qRc cos(qRc)] n⎡ ⎢3Vc(ρc − ρs ) Vs ⎣ (qRc)3 [sin(qR s) − qR s cos(qR s)] ⎤ ⎥ + 3Vs(ρs − ρsolv ) (qR s)3 ⎦

2

(23)

the deuterated colloid. All scattering curves show parallel trends at high q-values, where a second and a third shoulder appears. For further data treatment, we divided the scattering crosssection Δ(dΣ/dΩ) by the deuterated colloid mass fraction α and extrapolated the scattering data to the value at infinitely low concentration of deuterated colloid. The extrapolation was performed according to the following equation ⎛ dΣ ⎞ ⎛ dΣ ⎞ ⎟ / α = Δ⎜ ⎟ lim Δ⎜ ⎝ dΩ ⎠ ⎝ d Ω ⎠α = 0

α→0

where Vs is the volume of the outer shell, Vc the volume of the core, Rs is the thickness of the shell, Rc the radius of the core, ρc is the scattering length density of the core, ρs is scattering length density of the shell and ρsolv the scattering length density of the solvent. The extrapolated scattering data were described by PCS(q) using the following fit parameters: a core and shell radius of Rc = Rs = 52.2 nm and the scattering length density of the core with ρc = 2.91 × 10−4 nm−2, which was higher than the scattering length density of the shell with ρs = 2.753 × 10−4 nm−2. As is shown in Figure 10, this spherical core−shell model indeed describes the entire scattering curve and is thus superior to the simple hard sphere model. Like in all other fits an uncertainty of Δq/q = 0.13 has been used in order to account for the smearing of the data. Inspired by this success, we selected in addition a modified core−shell form factor PSO(q) with a homogeneous spherical core26 and an octahedral shell,42,43 in order to alternatively reproduce the shape of the deuterated colloids with high crosslinking density sitting in lattice cells of colloidal aggregates formed in the polymer matrix. The octahedral shell has a lower scattering length density than the spherical core, in close analogy to the simple spherical core−shell model, which already succeeded to describe the experiment shown in Figure 10. The surfaces of the polyhedron represent the touching interfaces among the neighboring colloids. We are fully aware that in a dense packing of spheres any particle has rather 12 instead of 8 such interfaces. However, the scattering amplitude of an octahedron is much easier to handle than the scattering amplitude of a dodecahedron,42,43 but may give already an adequate description. For the numerical calculation, we adopted the scattering length densities of the spherical core−shell fit with the scattering density of the core with ρc = 2.91 × 10−4 nm−2 and the scattering length density of the octahedral shell with ρo = 2.74 × 10−4 nm−2. The resulting fit values for the radii of the spherical core Rc and the octahedral radius Ro amounted to Rc = 52.2 nm and Ro = 144.4 nm. A detailed representation of the model form factor PSO(q) and its application is given in the Supporting Information. As expected, the octahedron with a spherical core and a smearing of Δq/q = 0.13 is also able to satisfactory describe the experimental scattering curve of a colloid being placed within the aggregate (Figure 10). Clearly, the discrepancy between spheres or fuzzy spheres and the experiment can be attributed to a modification of the spherical colloids while being incorporated into lattice cells of the colloidal aggregates.

(21)

with α=

mD70 ‐ HC

mD70 ‐ HC + mH70 ‐ HC + mPMMA ‐ 6N

(22)

where mx is the mass of the respective component x. Figure 10 depicts the result of this extrapolation in comparison to fits with

Figure 10. Experimental data from Figure 9 extrapolated to c = 0 (○) in comparison with different model form factors: monodisperse sphere form factor39 (blue line), fuzzy sphere form factor38 (red line), spherical core−shell form factor26 (green line) and octahedral core− shell form factor42,43 (pink line). All fits included an instrumental uncertainty of Δq/q = 0.13.

various model form factors such as monodisperse sphere39 and fuzzy sphere38 with the anticipated smearing of Δq/q = 0.13. The sphere model and the fuzzy sphere model describe well the initial decay of the form factor, but yield values, which lay significantly lower than the experimental data for q > 4 × 10−2 nm−1. Most strikingly, the decay of the experimental form factor leads to a shoulder, which is four times higher than that of a sphere or fuzzy sphere and the maximum is much more pronounced than that of a sphere and fuzzy sphere with Δq/q = 0.13. This unexpected result may be explained only after a careful look on the cartoon of Scheme 1. The colloids to be seen by 9100


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Scheme 1. Design of the SANS Experiments on Blends with a Binary Mixture of Deuterated and Hydrogenated Colloids with High Cross-Linking Density, Where the Hydrogenated Colloid Has Nearly the Same Contrast as the Surrounding PMMA-6N Polymer Matrixa

The SANS data of blends with a binary mixture of D70-HC and H70-HC were interpreted for the “visible” D70-HC colloid in terms of a spherical or an octahedral core-shell structure. a

potential among the colloids with a range extending to the size of the polymer coils. The coils gain accessible volume while the colloids are aggregating. This effect seems to hold also beyond the overlap concentration of the linear chains, although this increase of the chain concentration may decrease the range of the attractive potential from the size of a coil down to the correlation length typical in semidilute solutions.46 Apparently this tendency to exhibit a depletion induced aggregation of the colloids in solution is suppressed once the cross-linking density is low enough, since this enables polymer chains to penetrate the colloids thereby stabilizing situations where the colloids are kept apart. Such a stabilization was not considered in the theoretical work on depletion just mentioned,44,46 since hard spheres are used as colloids in the respective models. Parallel to these optical transmission experiments, we prepared identical solutions with both types of the large, deuterated colloids (D140-LC and D140-HC). After different periods of storing time a solution probe was removed to prepare a blend according to the SE method and the resulting blend was characterized by SANS-experiments at a sample-todetector-distance of L = 9.5 m. All SANS curves of the precursor solution recorded at variable shelf life times, which cover a regime of 20 days, are given in the Supporting Information. The SANS curves from the blends with colloids with low (D140-LC) and high (D140-HC) cross-linking density show the same behavior as the scattering curves, which were shown in Figure 6C and D, respectively; i.e., D140LC predominantly exists as isolated particles and D140-HC forms aggregates. These results confirm that the scattering experiments are reproducible. Just as important, the results did not vary with aging time of the precursor solutions. Whereas this can be expected for the case of stable dispersions of colloids with low cross-linking density D140-LC, it is remarkable for sample D140-HC, which shows aggregation in solution and aggregates in blends, since in the case of the highly cross-linked colloid D140-HC the same strong aggregation is observed for all aging times, in close agreement with the trend shown in Figure 6D. Apparently, the long-term phase separation eventually observed with optical transmission measurements for the colloids with a high cross-linking density during their shelf life is accelerated in a reproducible way while solvent is removed in order to cast the foils with the SE method. Obviously the result of this acceleration is independent of the age of the precursor solution and the resulting blends show the

In order to establish the origin of aggregation observed with the colloids with a high cross-linking density in polymer-colloid blends, we performed a series of optical transmission experiments with the mixed precursor solutions. The transmission data, which were recorded at a wavelength of λ = 500 nm showed that the polymer-colloid solutions containing low cross-linked colloids are stable over 75 days, since no significant decrease of transmission was observed. Contrary to this observation, a time dependent decrease of transmission was monitored for polymer-colloid solutions with highly crosslinked colloids. In addition, the polymer-colloid solution with small colloids D70-HC having a high cross-linking density was more stable than the polymer-colloid solution with the corresponding large colloid D140-HC. Transmission data are given in Figure 11. The results can be nicely reconciled with a depletion effect44 as a typical feature of polymer−colloid solutions in the limit45 where the polymer chains are significantly smaller in size than the colloidal particles. The polymer coils induce an attraction

Figure 11. Time dependent transmission experiments at λ = 500 nm on polymer-colloid mixtures in chloroform performed with (A) small colloids D70-LC (○) and D70-HC (red ○) and performed with (B) large colloids D140-LC (○) and D140-HC (red ○). The transmission T is based on the ratio of the transmitted intensity of the polymercolloid solution and the transmitted intensity of the respective pure polymer solution. 9101


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density gets low enough, since linear chains may more and more penetrate the colloids and thereby preventing those colloids from aggregation. In order to confirm this hypothesis, we performed supplementary time dependent optical transmission experiments on precursor polymer-colloid solutions and SANS on the respective blends. The results of the transmission experiments showed that solutions including low cross-linked colloids are stable over 75 days while solutions with highly cross-linked colloids precipitated after 24 or 72 days depending on the particle size of the colloid. SANS-experiments on blends, which were prepared from precursor colloid−polymer solutions differing in their shelf life showed similar aggregation in the blend, independent of the age of the precursor solution. Therefore the aggregates formed in the blends with highly cross-linked colloids can be unambiguously attributed to a depletion effect induced by the matrix chains in solution. This depletion effect is accelerated by the formation of the blends via removal of the solvent, which suppresses a possible impact from shelf life of the precursor solutions. In order to learn more about the structure of the highly cross-linked colloids within the aggregates, we characterized blends of deuterated D70-HC and hydrogenated H70-HC colloids with high cross-linking density at variable ratio of the two colloid species. The resulting scattering curves were extrapolated toward dilution of the deuterated colloid D70-HC and modeled with different form factors. We found that the intensity of the undulation at q ≥ 5 × 10−2 nm−1 of the extrapolated curve could not be modeled by spheres39 or fuzzy spheres.38 We therefore applied in addition a simple spherical core−shell model26 and an octahedron with a spherical core.42,43 Both models were able to equally well reproduce the data. This indicates that colloids with high cross-linking density being placed within colloidal aggregates will not adopt an ideal spherical shape, but a core−shell structure imposed by the polyhedron like boundaries of the “lattice sites” offered by the aggregate.

same extent of aggregation as observed with the blend of D140HC in Figure 6D.

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CONCLUSIONS We prepared deuterated and hydrogenated PMMA colloids by surfactant-free emulsion polymerization.22 As was shown by combined SLS, DLS, and SANS experiments, the particle size and swelling behavior of these colloids could be adjusted successfully by varying the concentration of MMA and the amount of added cross-linker respectively. SANS of all deuterated colloids in water revealed monodisperse hard spheres.39 Once the colloids were transferred into THF, the model of a fuzzy sphere38 turned out to describe the scattering curves more adequately than the model of a homogeneous sphere,39 which confirms a considerable degree of swelling of colloids in THF. The SANS experiments of colloids, which were dissolved in water and THF were used successively as a reference system for the investigation of the distribution and swelling behavior of deuterated colloids in blends with linear PMMA chains. Three different methods for blend preparation were applied. The SANS experiments on the blends revealed that the blends with low cross-linking density were homogeneously distributed in the polymer matrix, independent of the preparation method. Using the appropriate form factor of a fuzzy sphere,38 we could conclude that the low cross-linked colloids are partially swollen by linear chains and that the degree of swelling was smaller in comparison to the swelling of the respective colloids in THF. In contradiction, the highly cross-linked colloids always formed aggregates in the resulting blends. Hence, the colloids with a low cross-linking density behave similar to colloids equipped with a shell of linear chains.13,14 A fuzzy shell established by linear chains or a lower degree of cross-linking favors dispersion in linear matrix chains. The observations for blends including the colloids with low and high cross-linking density are graphically depicted in Scheme 2. All blends were prepared from precursor solutions with mixed polymers and colloids, which are sensitive to depletion.44 Depletion inevitably generates aggregates of colloids also in the resulting blends. Obviously, the influence of such a depletion can be suppressed once the cross-linking

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ASSOCIATED CONTENT

S Supporting Information *

Estimation of polydispersity of deuterated colloids, swelling behavior of colloids, scanning electron microscopy on deuterated colloids, SANS investigations of blends of colloids, calculation of the form factor of an octahedron with a spherical core, and SANS experiments on blends from aged precursor solutions. This information is available free of charge via the Internet at http://pubs.acs.org/

Scheme 2. Graphical Illustration of the Colloid Distribution in Blends of Linear PMMA-6Na

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AUTHOR INFORMATION

Corresponding Author

*E-mail: (K.H.) [email protected]. Telephone: (+49) 5251 602125. Fax: (+49) 5251 604208. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work benefited from DANSE software developed under NSF Award DMR-0520547. This work was funded by a grant from the German Research Foundation (DFG) within the project “Modifizierung von linearem PMMA mit vernetzten PMMA-Kolloiden zur gezielten Veränderung der Werkstoffeigenschaften”.

a

Colloids with a low cross-linking density are molecularly dispersed. Colloids with a high cross-linking density form aggregates. 9102


Macromolecules

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(41) Bartsch, E.; Kirsch, E.; Lindner, P.; Scherer, T.; Stölken, S. Ber. Bunsen-Ges. Phys. Chem. 1998, 102 (11), 1597−1602. (42) Li, X.; Shew, C. Y.; He, L.; Meilleur, F.; Myles, D. A. A.; Liu, E.; Zhang, Y.; Smith, G. S.; Herwig, K. W.; Pynn, R.; Chen, W.-R. J. Appl. Crystallogr. 2011, 44, 545−557. (43) Stokes, A. R.; Wilson, A. J. C. Math. Proc. Cambridge Philos. Soc. 1942, 38, 313−322. (44) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22 (7), 1255−1256. (45) Turnier, R.; Rieger, J.; de Kruif, C. G. Adv. Colloid Interface Sci. 2003, 103, 1−31. (46) Fuchs, M.; Schweizer, K. S. J. Phys.: Condens. Matter 2002, 14, 239−269.

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Morphology of Blends with Cross-Linked PMMA Microgels and Linear

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