Rheology of surface-modified titania nanoparticles ...

Rheology of surface-modified titania nanoparticles dispersed in PDMS melts: The significance of the power law Rose S. Ndong and William B. Russel Citation: J. Rheol. 56, 27 (2012); doi: 10.1122/1.3669646 View online: http://dx.doi.org/10.1122/1.3669646 View Table of Contents: http://www.journalofrheology.org/resource/1/JORHD2/v56/i1 Published by the The Society of Rheology

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Rheology of surface-modified titania nanoparticles dispersed in PDMS melts: The significance of the power law Rose S. Ndong and William B. Russela) Department of Chemical and Biological Engineering, Princeton University, New Jersey 08544 (Received 23 May 2011; final revision received 3 November 2011; published December 15, 2011)

Synopsis We investigated the rheology of titanium dioxide (TiO2) nanoparticles with various surface modifications in neat and binary blends of polydimethylsiloxane (PDMS) homopolymers of different molecular weights (4 k–77 k). The dispersions for bare, octadecyl-(C18), and PDMSgrafted particles reflect different interaction forces. For bare particles, the relative viscosity decreases monotonically with increasing melt Mw or increasing fraction of long chains (f), consistent with thicker adsorbed layers. The octadecyl(C18)-grafted dispersions show no dependence on melt Mw or f, suggesting that the alkyl groups prevent polymer adsorption or bridging. Therefore, the van der Waals attractions are cut off at a separation on the order of twice the thickness of the C18 chains (5 nm), regardless of melt Mw or f. The PDMS-grafted suspensions show an increase in relative viscosity with increasing melt Mw or f, consistent with wetted polymer brushes for P < N and dewetted layers for P > N. The power law we developed previously fits the shear-rate dependent viscosities with a structural relaxation time that scales with the magnitude of the attraction, C 2012 The Society of Rheology. [DOI: 10.1122/ thereby reflecting the microscale dynamics. V

1.3669646] I. INTRODUCTION Colloidal suspensions have wide industrial applications such as creams and gels in cosmetics [Rieger and Rhein (1997)], drug delivery agents in pharmaceuticals [Arora (2002)], and gelatin-based products in foods [Linden and Lorient (1999)]. Colloidal particles are added to polymers to impart or enhance their mechanical, optical, electrical, and thermal properties. By striking a balance between the matrix and filler properties, composites offer properties superior to those obtained from the matrix alone, thus increasing performance and enabling multifunctionality across various applications. Polymer nanocomposites (PNC) present additional advantages over micron-sized particle filled polymers as one can obtain value-added properties without imposing a trade-off between improved properties and processibility, which is usually the case for conventional filled polymers [Vaia and Wagner (2004); Roy et al. (2005); Jain et al. (2008)]. Adding a small amount of nanoparticles might not significantly change the characteristics of the matrix polymer while considerably improving the optical, thermal, and electric properties. For example, incorporating large particles into a resin can render the

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R. S. NDONG and W. B. RUSSEL

composite brittle and opaque, whereas adding well-dispersed nanoparticles makes the composite more resistant to breakage without affecting its transparency. These improved properties stem from the large surface area of the small particles and their strong interactions with the matrix [Tanaka et al. (2004)]. However, the lack of control over dispersion of the particles in these composites is an impediment to industrial applications and commercialization. Controlling colloidal stability is an important aspect in formulation and process design as their final properties and functionality are sensitive to the degree of dispersion. Two recent papers [Krishnamoorti (2007); Kumar and Krishnamoorti (2010)] review theoretical and experimental methods for developing appropriate strategies aimed at achieving well-dispersed nanocomposites [Karim et al. (2002); Bockstaller et al. (2005); Mackay et al. (2006); Vaia and Maguire (2007); Allegra et al. (2008)]. It is well established that particle dispersion is regulated by complex particle–particle and particle-polymer interactions. The magnitude of these forces, whether of Brownian, van der Waals, polymeric or viscous origin, determines the microstructure and rheology of the composites. Optimizing microstructure without compromising rheology requires control of both interparticle forces and processing. The forces present in filled melts are (1) van der Waals attractions, (2) bridging and depletion attractions due to adsorption and the presence of nonbound free chains, respectively, and (3) steric repulsions due to the presence of adsorbed and grafted polymer. Theories [Asakura and Oosawa (1958); Napper (1983); Russel et al. (1989); Israelachvili (1991)] are available for estimating the interaction potentials governing colloidal stability. The net effect of the combined interaction forces, calculated from the derivative of the potentials with respect to interparticle separation (Fi ¼ @Ui=@h where i refers to van der Waals, steric, or depletion) dictates the microstructure and phase behavior of composites. The relationship between the total interaction potential and the microstructure of the dispersion is depicted in Fig. 1. Schweizer et al. [Schweizer and Curro (1994); Chatterjee and Schweizer (1998); Hooper et al. (2004); Hooper and Schweizer (2005); Jayaraman and Schweizer (2008, 2009)] developed a theory for the state of dispersion of bare particles in polymer melts. Their model uses a microscopic polymer reference interaction site model (PRISM) to account for particle–particle, polymer segment-particle, and polymer segment–segment interactions. They found three phase behaviors: (1) macroscopically phase-separated state at low polymer segment-particle interaction due to direct van der Waals attractions, (2) enthalpically stabilized miscible fluid due to bound polymer layers, and (3) microscopically phase-separated state due to local bridging of particles by polymers. Where a filled melt falls among these phases depends on parameters such as the ratio of particle diameter to monomer diameter 2a=l, particle volume fraction /, matrix degree of polymerization P, particle–particle attractions cc and strength (pc) and range (a) of attraction between polymer segments and particle surface. Using PRISM to calculate the particle second virial coefficient B2, Hooper and Schweizer (2006) found that B2 displays a nonmonotonic behavior with increasing strength of polymer segments and particle interaction. The transition from attractive to repulsive behavior (negative to positive B2) depends on parameters such as 2a=l, a and P and cc. Jayaraman and Schweizer (2008) extended the PRISM theory to sparsely grafted particles (1–6 tethers). In this case, competition between tetheredinduced repulsion, polymer-induced depletion, and particle–particle interaction determines the phase behavior. The PRISM theory was developed for polymer mixtures at full equilibrium. Therefore, the nonequilibrium nature of polymer adsorption in our TiO2 filled PDMS systems might render it less adequate for predicting stability criteria. Generalization of the theory to treat densely grafted particles and to account for the nonequilibrium nature of real systems is ongoing.


RHEOLOGY OF TiO2 NANOPARTICLES IN PDMS MELTS

29

FIG. 1. Schematic illustration of the relationship between total interaction potential and dispersion structure. Figure used with permission from Lewis, Journal of American Ceramic Society 83(10), 2341–2359, 2000. Copyright 2000, The American Ceramic Society.

Steric stabilization in a melt via adsorbed polymers is often difficult to achieve and control as adsorbed chains can desorb from the surface and leave the gap. Thus, grafting polymer onto particles is the best technique for colloidal stabilization, though several factors limit the mechanism. There is a growing consensus that the prerequisite for optimal dispersion of polymer-grafted particles is that the melt chains “wet” the grafted layer, which generally requires matrix chains shorter than the grafted chains. The degree of wetting depends on several controllable parameters such as graft density (R, normalized as R0 ¼ Rl2, with l segment length), layer thickness (L), and particle radius (a). Melt chains (P) that are longer than the grafted ones (N) tend to “dewet” from the brush, leading to aggregation. This “dewetting” process can also occur at low graft density when the melt chains are not attracted strongly enough to wet the surface completely (“allophobic” dewetting) or at high graft density when the melt chains are expelled from the brush (autophobic dewetting) [Maas et al. (2002)]. Liu and coworkers [Liu et al. (1994)] were the first to investigate experimentally the autophobic behavior of grafted layers in homopolymer melts and observed complete wetting with P=N < 5 and partial wetting with P=N > 5. On the other hand, self-consistent field calculations (SCFT) by Shull (1991, 1994, 1996) found dewetting to occur when P N. Ferreira et al. (1998) extended Shull’s analysis to include the effects pffiffiffiffi of graft density, finding melt chains pffiffiffito ffi be expelled from 1 for P=N > 1, grafted chains when R0 N > ðP=N Þ1=2 for P=Np whereas the onset of autophobicity occurs when R0 N > N=P2 . However, simulations by Matsen and Gardiner (2001) contradict these predictions. Using a new algorithm in their SCFT calculations, they discovered that an attraction between the brushes always exists, but at low P, that attraction is negligible. The theories and experiments discussed


30


above pertain to large particles or flat surfaces. Xu et al. (2006), who studied the influence of the curvature with a real space implementation of SCFT pffiffiffiffi for N ¼ P, found the interactions to be that of brushes at flat surfaces when a >> l N and that of star polypffiffiffiffi pffiffiffiffi mers when a N, the grafted layer collapses to a thickness between lN1=3 (collapsed free chains in poor solvent) [Halperin and Zhulina (1991)] and lN1=2 (low R0 and P < N) [de Gennes (1980)]. Determining the brush thickness L is important since it provides an estimate for the separation at which the van der Waals attractions are cut-off. Most studies on polymergrafted particles in melts focus on particles with relatively weak van der Waals interactions (i.e. silica), proportional to the Hamaker constant AH. For silica, alumina, and titania dispersed in PDMS, the Hamaker constants are 1.7 1022 J, 3.0 1020, and 2.85 1019, respectively. Therefore, we expect the order of magnitude difference between silica, alumina, and titania to be reflected in the rheological behavior. While most studies on stabilized particles in polymeric media have been conducted in relatively monodisperse homopolymer, industrial applications usually involve polydisperse melts [Kamal et al. (1994)]. Therefore, we also study the effect of polydispersity on rheological properties with binary blends of different molecular weights. Schausberger and co-workers [Schausberger et al. (1987)] measured the linear viscoelasticity of binary blends of long and short PS chains, finding that short chains (Mw < Me) lowered the plateau modulus in proportion to the square of the weight fraction of the long chain component. They attributed this effect to the short chains acting as a diluent and reducing entanglements of the long chains. Nichetti and Manas-Zloczower (1998) developed a simple constitutive model to characterize the effect of polydispersity in decreasing the



31

zero-shear viscosity and exaggerating the shear thinning. Investigations by Kawaguchi and co-workers [Kawaguchi et al. (1984, 1987)] on competitive adsorption in binary blends of homopolymers demonstrate the preferential adsorption of long chains over shorter ones at equilibrium. Therefore, one might expect short chains to not significantly affect the thickness of the adsorbed layer. Consequently, the repulsive force due to the adsorbed layer should be of same magnitude and, as a result, the relative viscosity of the dispersions should be insensitive to the fraction of short chains. However, Devotta and Mashelkar (1995) found an intermediate plateau consisting of long and short chains, before the final equilibrium in the adsorption curve. This secondary plateau arises due to the rate at which the short chains disengage from the surface. Unfortunately, little literature focuses on the influence of polydispersity on the rheology of filled polymer melts. In this paper, we investigate how various surface treatments of TiO2 nanoparticles with strong van der Waals attractions affect the rheology of dispersions in neat and binary melts. Bare, octadecyl-(C18), and PDMS-grafted TiO2 particles were dispersed in neat PDMS melts of molecular weight ranging from 4 to 77 kg=mol and 4k=77k binary blends with various fractions of long chains (f). In addition to van der Waals forces that are present in all suspensions, depletion attraction should be present in binary blends with the C18 particles, whereas attraction due to dewetting should occur when P > N. As for the C18 -treated particles in binary blends, we might expect depletion attraction as the alkyl groups prevent polymer adsorption (nonadsorbing surface) and the short chains act as an ideal solvent [Butt and Kappl (1998)]. The various forces (van der Waals, depletion, and steric) are reflected in the rheological behavior, so we fit our data with the simple two-parameter correlation developed in our previous paper [Ndong and Russel (2011)].

II. MATERIALS AND METHODS A. Materials Rutile titanium dioxide (TiO2) nanoparticles with a specific area of 50 m2=gm (manufacturer’s values) were purchased from SkySprings Nanomaterials, Inc. Transmission electron microscopy (TEM) (Fig. 2) indicates particles with spheroidal morphology and a particle size distribution of 20 6 10 nm. The silane (octadecyltrimethoxysilane) and monohydroxyl terminated PDMS (PDMS-OH) were purchased from Sigma Aldrich, whereas the methylated PDMS (PDMS-CH3) melts were obtained from Gelest. We characterized the molecular weights with gel permeation chromatography and employed Lapp’s [Lapp et al. (1985)] procedure to obtain the true Mw (Table I). All other reagents were obtained from Sigma Aldrich, Fisher, or Gelest and used as-received. The number of segments for the grafted and melt polymers were derived from the molecular weight (Mw), the mass and length per bond (mo and lo, respectively), pand ffiffiffiffiffiffiffiffiffithe characteristic ratio (C1) via l ¼ C1lo and N (or P) ¼ Mw=(C1mo) so that Rg ¼ l N=6. B. Experimental method The as-received TiO2 were dried at 80 C under vacuum for 24 h to remove adsorbed water. Thermogravimetric analysis (TGA) of the dry powder revealed a negligible 0.7% weight loss [Jiang et al. (2003, 2005)]. The thermally treated TiO2 were reacted with octadecyltrimethoxysilane (C18) in toluene under reflux and N2 atmosphere for 24 h. The particles were then subjected to toluene Soxhlet extraction for 24 h and dried under vacuum at 80 C. For PDMS grafting, the dried particles were first dispersed in excess monohydroxyl-terminated PDMS using a Flacktek speedmixer. The homogenized mixture was then placed in a vacuum oven at 150 C for 2 days to condense the hydroxyl


32


FIG. 2. Transmission electron micrograph of commercial titania nanoparticles used in experiment.

end-groups of the polymer with the silanol groups on the particles. The unreacted polymer was removed by a series of centrifugation-sonication until no precipitate formed when the supernatant solution was added to methanol [Prucker and Ruhe (1998)]. The coated particles were dried under vacuum at 80 C for 24 h. The amounts of silane and grafted polymer were determined by TGA using a TA-Q50 in N2 at 10 C=min. The thickness of the alkyl layer is 2.5 nm [Kulkarni et al. (2006)], whereas the brush height as a function of P is estimated from Eq. (1). Rheological measurements of the nanoparticle dispersions at 25 C were performed with a controlled stress Physica MCR 501 rheometer (Anton Paar) with a cone and plate geometry (25 mm diameter and 2 angle). The samples were first homogenized with the speedmixer in trimethyl terminated PDMS of various molecular weights. We performed the rheological tests within 24 h of preparation since sample age can affect the rheological properties [Ziegelbaur and Caruthers (1985); Cosgrove et al. (1997)]. We used the following protocol to reach steady state and assess reproducibility: run for 1000 strain units at each shear rate in a loop from high to low and back to high shear rate. We did not TABLE I. Weight average molecular weight (Mw), polydispersity index (PDI), number of statistical segments (N or P) and viscosity (l). Polymer characteristics: mo ¼ 37 g=mol and l ¼ 0.84 nm [Russel et al. (1989)], C1 ¼ 5.2 [Brandrup and Immergut (1999)]. Mw (kg=mol)

PDI

N

R (chains=nm2)

9.2

1.2

51

0.010 6 0.0004

PDMS-CH3 4k 7k 11 k 24 k 77 k

Mw (kg=mol) 4.7 7.7 11.9 24.1 77.3

PDI 1.20 1.35 1.43 1.75 1.5

P 25 40 62 125 402

l(Pas) 0.022 0.053 0.102 0.37 5.3

8 k (f ¼ 0.1) 17 k (f ¼ 0.3)

8.0a 17.3a

N=A N=A

42 90

0.052 0.22

PDMS-OH 9k

a

Weighted average Mw for the 4 k=77 k blends, calculated from Eq. (5).



33

FIG. 3. Anomalous measurement: The sinusoidal curve usually corresponds to the visual appearance of large clusters on edge interface. Data at 0.04=s ð4Þ are considered invalid whereas variation in the data at 0.4=s (*) falls within experimental error [Ndong and Russel (2011)].

consider data valid if (i) large clusters visibly protruded from the interface at the edge of the plate, (ii) the viscosity oscillated with time or strain as illustrated in Fig. 3, or (iii) steady state was not reached within the 1000 strain units. Rheological characterization was carried out for bare, C18 - and PDMS grafted TiO2 in neat and binary PDMS melts.

III. RESULTS AND DISCUSSION A. Rheological measurements Our objective is to determine the influence of interparticle forces, mediated by surface treatments, on the rheological properties of TiO2 nanoparticle dispersions at 15% solid loading (/ ¼ 0.15). The dispersions of bare, C18- and 9k-grafted TiO2 in PDMS melts of various molecular weights were characterized via measurements of the relative viscosity, _ The shear rate dependence of g=l was similar to that of the g=l, versus shear rate, c. melts filled with large alumina particles in our previous study: (1) a power law dependence on the shear rate at intermediate c_ (shear thinning), (2) a tendency to reach a high shear limit (g1), and (3) the absence of a low shear limit (go). Additionally, the data conform with the two-parameter correlation [Eq. (2)] previously derived [Ndong and Russel (2011)] where the first term accounts for the purely viscous contributions and the second term accounts for the particle interactions. gr g1 _ n : ¼ þ ðs cÞ l l

(2)

The parameter s is a structural relaxation time that depends on the interparticle forces and can be compared with a theoretical relaxation time obtained by equating the viscous forces (6plaU), corrected for lubrication stresses in the gap (a=h), to the maximum attraction. so ¼ 6pla2 =Ftot :

(3)

Here, U is the relative velocity of two particles, h is the surface-to-surface separation, and l is the melt viscosity. At graft densities as low as those in this study (R0 < N1),


34


other investigations [Hasegawa et al. (1996); Ferreira et al. (1998); Yezek et al. (2003); Green and Mewis (2006)] suggest the grafted layers are unable to impart stability as they adopt mushroom or pancake-like configurations to maximize the favorable segmentsurface contacts [Marla and Meredith (2006)], and particles aggregate mainly through van der Waals attraction. At very small separations, the steric repulsion dominates the van der Waals attraction. The maximum attraction at a characteristic distance, hc, reflects a steric repulsion that no longer overcomes the van der Waals attraction but prevents the particles from reaching the deep primary minimum at contact. Thus, the maximum strength of the attraction at hc is set by the steric barrier, and calculated as F ¼ @UvdW=@r with AH the Hamaker constant and r ¼ 2a þ hc the center-to-center separation from the following [Hamaker (1937)]: 2 AH 2a2 2a2 r 4a2 : þ 2 þ ln UvdW ðrÞ ¼ 6 r 2 4a2 r r2

(4)

For TiO2 particles in PDMS, we calculate a nonretarded Hamaker constant AH as 2.85 1019 J using the parameters from Brandrup et al. (1999) for PDMS and Israelachvili (1991) for TiO2. We recognize that the polydispersity in size and the nonspherical nature of the particles introduce uncertainties in our analysis [i.e., Eqs. (3) and (4)]. However, those uncertainties should not affect our overall conclusion since we are looking for semiquantitative trends in our data. 1. Octadecyl-grafted TiO2 nanoparticles in neat and binary PDMS melts

The relative viscosities for the C18-grafted particles dispersed in neat and binary blends show only a modest dependence on melt Mw or long chain volume fraction f (Fig. 4), suggesting that the alkyl chains efficiently prevent adsorption of melt chains. To

FIG. 4. Relative viscosity versus shear stress for octadecyl-treated TiO2 nanoparticles in (a) neat and (b) binary 4k=77 k PDMS melts (15% loading).



35

FIG. 5. Relative viscosity versus shear rate x melt viscosity for octadecyl-treated TiO2 nanoparticles (15% loading) showing collapse of the data onto a single curve. The dotted line is the fit to Eq. (3).

better visualize the lack of dependence on P and f, we remove the effect of melt molecu_ (Fig. 5). The value of so lar Mw on the relative viscosity by plotting the data as g=l vs cl with the C18 layer was determined from Eq. (3) with Ftot ¼ @UvdW=@h and h ¼ 5.0 nm (assuming an impermeable and incompressible layer). From the fit of the data to Eq. (2), we establish a base value s=so 34.4 6 7.6 as a means for determining an apparent separation for the bare and grafted particles. Furthermore, we can estimate a depletion attraction for the binary case. Assuming that the short chains act as an ideal solvent and that the alkyl layer forms an impenetrable barrier to the long chains, we calculate [Vincent (1990); Rao and Ruckenstein (1985)] a depletion attraction at contact of 3 1013N, an order of magnitude smaller than the calculated van der Waals forces. 2. Bare TiO2 nanoparticles in neat and binary PDMS melts

For the bare particles, the relative viscosity g=l decreases with increasing melt molecular weight [Fig.p6(a)], ffiffiffiffiffiffiffi consistent with previous observations that the adsorbed amount is proportional to Mw [Cohen-Addad et al. (1985); Cosgrove et al. (1997); Aranguren et al. (1997)]. pffiffiffi Consequently, the adsorbed layer thickness increases with melt Mw with Lads ¼ l P leading to a larger interparticle separation that weakens the van der Waals attractions. We acknowledge that achieving reproducible results with the bare particles in 4k melt is challenging due to the difficulty in preparing homogeneous samples, with the uncertainty comparable with the distance from the C18 data. For the bare particles dispersed in binary blends [Fig. 6(b)], we observe a decrease in g=l with increasing fraction of long chains. This is surprising as equilibrium adsorption studies [Kawaguchi et al. (1984, 1987)] determine the layer thickness to be that of the longest chains independent of f, implying the same interparticle separation irrespective of f. We attribute the observed rheology to competitive, rather than equilibrium, adsorption of the short and long chains. Since PDMS strongly adsorbs on titania particles, the disengagement rate of the short chains from the surface is very slow, resulting into a secondary adsorption plateau consisting of both components, as discussed by Devotta and Mashelkar (1995). Thus, the average thickness of the adsorbed layer increases with increasing f, thereby reducing the viscosity, as observed in Fig. 6(b). We estimate the average layer thickness from an average molecular weight of the adsorbed chains Mmix as observed by Cohen-Addad (1998)


36


FIG. 6. Relative viscosity versus shear stress for bare titania nanoparticle (15% loading) in (a) neat and (b) binary PDMS melts. The dashed lines are the correlation fits to the data, whereas the dotted line represents the C18 data set.

qffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi Mmix ¼ f Mwl þ ð1 f Þ Mws ;

(5)

where f is the weight concentration of the long chains in the mixture. The adsorbed layer ffi pffiffiffiffiffiffiffiffi thickness for the blend then follows from Mmix as Lads ¼ l Pmix , similar to the neat melt. This square root dependence of the average layer thickness on P is supported by Scheutjens and Fleer (1980), although no mathematical expression was provided. From the PRISM theory, we would expect strong polymer-induced attractions due to bridging since PDMS strongly adsorbs on TiO2. However, at high polymer segment-particle interaction, which is the case for PDMS=TiO2, these attractions lessen and are eventually dominated by the steric repulsion of the grafted layers. Thus, the only source of attraction is the van der Waals forces, consistent with our initial hypothesis. Only the suspension of bare particles dispersed in 77 k PDMS melt displays a low shear Newtonian plateau (go=l of 6 Pa-s) along with the high shear limit. For hard spheres, that corresponds to a volume fraction of 0.38, or a hydrodynamic pffiffiffi layer thickness of 3.6 nm, much less than the adsorbed layer from scaling, L ¼ P. From g1=l, the hydrodynamic layer is 2.7 nm [Ndong and Russel (2011)]. Thus, for bare particles in high melt polymers, adsorption is sufficient to impart stability, though the adsorbed layers are both compressible and permeable.

3. 9k-PDMS grafted TiO2 nanoparticles in neat and binary PDMS melts

The flow curves for the 9 k-grafted PDMS particles dispersed in neat and binary PDMS melts [Figs. 7(a) and 7(b)] indicate an increase in relative viscosity with



37

FIG. 7. Relative viscosity versus shear rate for 9 k-grafted titania nanoparticles (15% loading) in (a) neat and (b) binary PDMS melts. The dashed lines are the correlation fits to the data.

increasing melt Mw, contrary to the C18 and bare particle suspensions. For P < N (4 k and 7 k melts), the layer thickness, accounting for adsorption as L ¼ lP1=2 þ l3R(N P), should increase with P. Therefore, we should expect a decrease in viscosity with increasing P. We attribute the contrary trend to the limited degree to which the melt chains penetrate the grafted layer, which decreases with increasing P. The net effect of increasing brush height and decreasing penetration length as P increases is a reduction in brush stretching. Consequently, the minimum separation decreases leading to stronger van der Waals attractions and higher viscosities. For P > N, the increasing g=l with increasing P (Mw ¼ 11 k – 77 k) is consistent with scaling theories that predict dewetting [Shull (1996); Gay (1997); Ferreira et al. (1998); Matsen and Gardiner (2001); Xu et al. (2006)]. In that region, the melt chains are gradually expelled from the brush and the grafted layer collapses and adopts a flattened conformation to maximize its interaction with the surface. We calculate this collapsed layer (Lc) by equating the volume occupied by the grafted chain which scales with the cube of the root-mean-square end-to-end distance (V ¼ l3N3=2) to the volume of the compressed layer (Lc=R) [Marko and Witten (1992)], giving Lc ¼ 2.2 nm. The resulting steric layer is too thin to counter the van der Waals forces, leading to higher viscosities. 4. Correlation

The high shear viscosities vary from 2.2 to 3.2, with a nonmonotonic dependence on P. de Kruif and co-workers [de Kruif et al. (1985)] correlated the high shear viscosity for bare hard spheres to particle volume fraction as g1=l ¼ [1 (/=/max)]2. With /max ¼ 0.71 and / ¼ 0.15, g1=l for monodisperse suspensions should be 1.61, suggesting that either our suspensions were not homogeneously dispersed or that the steric layer


38


increases the hydrodynamic radius significantly beyond that of the particle radius. Using the maximum packing fraction, /max, as an adjustable parameter, we superimpose our data onto the de Kruif curve to extract an effective volume fraction, /eff ¼ 0.71/=/max, that reflects the additional volume occupied by the grafted layers. The values for /max range from 0.33 to 0.40, consistent with the fact that the additional volume occupied by the grafted and adsorbed chains lowers the maximum packing fraction. We calculate a hydrodynamic brush thickness, Lhydro ¼ a[(/eff=/)1=3 1] and compare with the layer thickness from scaling in Tables II–IV. Note that Lhydro is consistently less than Lscaling, suggesting that the adsorbed or grafted layers allow some fluid motion. By setting s=so ¼ 34.4 as determined from the C18 coated particles, we estimate an apparent separation, happ, between the particles at contact for both bare and grafted particles. For the bare particles (Table III), happ < 2Lscaling, suggesting that the adsorbed layers are partly permeable and compressible. The apparent separation for f ¼ 0.1 (8 k) is larger than that of the 24 k melt, implying that the longer (77 k) melt chains adsorb preferentially. For the 9kgrafted layers (Table IV), the apparent separation suggests three regimes: For P < N (regime 1), happ < 2Lscaling, implying that the polymer brushes form swollen but slightly compressed chains and the swelling decreases as P increases. For P N (regime 3), happ 2 Lscaling, suggesting a completely flattened layer that precludes melt adsorption. At intermediate P (regime 2), there is a gradual transition from regime (1) to regime (3) and the collapsed layer is less flattened with some adsorbing and interpenetrating melt chains. The correlation parameters dependence on melt Mw and surface modification is shown in Fig. 8. The power law index n is mostly constant (0.77–0.94) as a function of Mw regardless of surface treatment, except for bare particles dispersed in high melt Mw (i.e., 77 k and

FIG. 8. (a) The power law index n (empty) and s=l (solid) dependence on melt Mw. (b) the apparent (empty), hydrodynamic (solid) thicknesses estimated from C18 calibration and from scaling (grey). The symbols are: circles for bare, squares for grafted and triangles for C18.



39

f ¼ 0.3) which show a much lower n. Within the range of n ¼ 0 for Newtonian behavior and n ¼ 1 for a yield stress, the values of 0.38 and 0.28 for ( f ¼ 0.3 and 77 k, respectively) are consistent with the nearly Newtonian behavior shown in Fig. 6. The ratio s=l shows a nonmonotonic dependence on Mw with surface modification but increases from alkyl to adsorbed to grafted layers. Though the data scatter on the plots of apparent and hydrodynamic layer thicknesses, we observed a modestly increasing trend in layer thickness for the bare particles, especially since the 77 k layer renders the van der Waals attractions negligible, and a decreasing trend with the grafted layers. Furthermore, the effective layer thickness from the van der Waals is greater than the hydrodynamic thickness, indicative of fluid motion within the layer. Tables II–IV provide the values for n, s, g1=l, /eff, Lhydro, Lscaling and happ as a function of P for C18, bare and 9k-grafted TiO2 dispersions.

TABLE II. Correlation parameter for C18 TiO2 in neat and binary PDMS melts. Mw (kg=mol) 4 7 8(f ¼ 0.1) 17(f ¼ 0.3) 24 77

s

n 0.81 0.84 0.79 0.79 0.79 0.77

0.0005 0.0009 0.0009 0.004 0.0055 0.07

g1=l 2.6 2.6 2.6 2.6 2.6 2.6

/eff

Lhydro (nm)

0.25 0.27 0.27 0.27 0.27 0.27

2.2 2.2 2.2 2.2 2.2 2.2

Lscaling (nm) a

2.5 2.5a 2.5a 2.5a 2.5a 2.5a

happ (nm) N=A N=A N=A N=A N=A N=A

a

Denotes layer thickness for alkyl group.

TABLE III. Correlation parameter for bare TiO2 in neat and binary PDMS melts. NV denotes the fit to the correlation is too poor because of the low n. Mw (kg=mol) 4 8(f ¼ 0.1) 17(f ¼ 0.3) 24 77

n

s

g1=l

/eff

Lhydro (nm)

pffiffiffiffi Lscaling ¼ l N ðnmÞ

happ (nm)

0.80 0.81 0.38 0.62 0.24

0.00035 0.0016 0.003 0.0095 0.008

2.4 2.6 2.8 2.8 2.4

0.25 0.27 0.28 0.28 0.28

1.9 2.2 2.3 2.3 2.3

4.2 5.5 8.0 9.4 16.8

4.9 6.2 NV 5.6 NV

TABLE IV. Correlation parameter for 9 k-grafted TiO2 in neat and binary PDMS melts. Mw (kg=mol) 4 7 8(f ¼ 0.1) 11 17(f ¼ 0.3) 24 77

n

s

g1=l

/eff

Lhydro (nm)

Lscaling (nm)

happ (nm)

0.88 0.82 0.92 0.79 0.90 0.94 0.94

0.0008 0.0035 0.0050 0.0037 0.012 0.028 0.150

2.7 3.3 3.5 3.0 3.0 2.7 2.5

0.29 0.32 0.33 0.30 0.30 0.29 0.26

2.5 2.9 2.9 2.6 2.6 2.5 2.0

4.3 5.4 5.5 2.2 2.2 2.2 2.2

8.0 7.9 8.9 6.5 7.5 8.2 5.5


40


IV. CONCLUSION In this work, we studied the effect of various surface modifications of TiO2 nanoparticles dispersed in neat and binary PDMS melts on rheological properties. The relative viscosities for the octadecyl-grafted particles dispersed in neat and binary blends show only a modest dependence on melt Mw or fraction f of long chains, suggesting that silanization efficiently prevents melt adsorption. For the bare particles in neat melts, the relative viscosity decreases with increasing melt Mw, consistent with the adsorbed amount pffiffiffiffiffiffiffi being proportional to Mw . For the bare particles dispersed in binary blends, we observe a surprising large decrease in relative viscosity with increasing fraction of long chains which we attribute to a preferential adsorption of the higher Mw component and negligible depletion attraction. For the 9k-grafted particles in neat and binary melts, the relative viscosity increased with increasing P or f, consistent with partial wetting for P < N and dewetting for P > N. The van der Waals attractions provide the dominant attraction but steric repulsion determines the cut-off or minimum separation. Our study reveals behavior closer to that of the large alumina particles, so we interpret our results with the simple two-parameter correlation previously developed. By setting s=so ¼ 34.4 from the C18 data, we deduce an apparent separation distance for the bare and 9 k-grafted particles dispersions. This might be the first linkage of the power law behavior with a timescale characteristic of the microscale dynamics.

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