Palierne model

However, these mechanisms are obviously not dominant when the fillers reside in the matrix, as shown through estimations of the interfacial tension using the Palierne model [44]. First of all, it should be kept in mind that when the filler partitions in one phase, the reduction in dispersed particle size may be attributed to a compositional effect. In the presence of the nanofiller, the ratio of compositions is altered. For example, 5 wt% of a filler localized in the matrix will correspond to a higher ratio of matrix over the dispersed phase, which in turn may affect the morphology. This effect should be more pronounced for relatively high filler concentrations that are not commonly encountered. 

The dynamic behavior of polymer blends under low strain has been theoretically treated from the perspective of microrheology. Table 2.3 lists a summary of this approach. These models well describe the experimental data within the range of stresses and concentrations where neither drop-breakup nor coalescence takes place. The two latter models yield similar predictions as that of Palierne. The last entry in the Table 2.3 is an empirical modification of Palieme s model by replacement of the volume fraction of dispersed phase by its efiective quantity (Eq. (2.24)), which extends the applicability of the relation up to 0 < 0.449. However, at these high concentrations the drop-drop interactions absent in the Palierne model must complicate the flow and coalescence is expected. The practical solution to the latter problem is compatibilization, but the presence of the third component in blends has not been treated theoretically. 

Starting with Palierne model the authors derived an expression between elonga-tional viscosity ratio and strains ... 

Here GJo jpogj ion represents complex shear modulus of the blend without interfacial effects while GJ p is the interface contribution, which comprises the extra elasticity originating in drop deformability. The former contribution was computed from the Kemer model [equivalent to Eq. (2.23) with v 2 = 0], while the second was calculated from the Palierne model. The computed dependencies for PP with ethylene-vinyl acetate (EVAc) copolymer and Si02, hydrophilic or hydro-phobic, well represented the frequency dependencies of G, G", and rf. Also, the addition of 3wt% silica reduced EVAc droplet size from 2.2 to about 0.5 m. The hydrophilic silica migrated to EVAc, while hydrophobic silica migrated to the PP phase. 

The influence of compatibilizer concentration on the two relaxation times also was analyzed by Van Hemelrijck et al. (2004). They have fitted the Palierne model with an interfacial shear modulus by introducing a concentration gradient of the block copolymer along the interface. Figure 1.3 presents the dynamic moduli of compatibilized blends—polydimethylsiloxane (PDMS)/polyisoprene (PI)—at various ratios of compatibilizers versus angular frequency (Van Hemelrijck et al. 2005). 

Very good emulsion effective medium) models that average over the continuous and dispersed phases to describe the continuum rheology have been developed to describe the linear viscoelasticity of polymer blends. The Palierne model, for example, provides a mixing rule for the complex modulus G = G + i G " in terms of the moduli for the dispersed and matrix phases as... 

A number of other models and theories have been proposed for evaluating viscosity data. Two models that are referred to as emulsion models predict the complex modulus or viscosity of an immiscible blend with spherical inclusions of one phase in a continuous phase (Oldroyd [263] and Paherne [264] models). The emulsion models can predict a positive deviation as noted in Fig. 6.21. Application of the Palierne model showed good agreement for viscosity data for EVAc/PE blends [265,266]. Another emulsion model proposed by Choi and Schowalter [267] is based on a cell model composed of a viscous matrix with viscous dispersed spheres (droplets). The viscosity of these models in the limit of zero shear viscosity can be expressed by the following equations. 

Experimental data for 10/90 and 90/10 PS/PMMA blends clearty demonstrate that the increase in blend s elasticity and the longer relaxation times observed in the terminal zone are due to the deformability of the suspended droplets. On the other hand, results obtained for PMMA/Rubber and PS/Rubber blends illustrate limitations of the Palierne model. For these blends, the model does not even qualitatively predict the secondary plateau arising at low frequencies for high rubber contents. The model does not account for particle-particle interactions. For volume fraction of rubber larger than 15 %, the particles form a network-type structure. For rubber particles concentration of 15 % and larger the elasticity of the network structure is satisfactorily described by the percolation theory. For PS/Rubber blends, a network is observed at a particles concentration of 10%. This is not predicted by the percolation theory. 

At the same time, it should be noted that the Palierne model, having an unsymmetrical form with respect to Gm and Gd (cf. eqn [75b]), is formulated only for the droplet/ matrix-type blends and cannot apply to blends having a co-continuous morphology at rest. Several models applicable to such co-continuous blends have been proposed. ° ° Among these models, the model proposed by Yu et can be most easily utilized for the... 

Low viscosity mixtures of PDMS and PI, with X = 0.155, 0.825, and 4.02 were studied at room temperature [Kitade et al., 1997]. The dynamic data were analyzed using Eq 7.69. Good agreement was found. However, for a = 4.02 system the drops were insensitive to the flow field — they neither broke nor coalesced. Similar observations were reported for PDMS/PIB system [Vinckier et al, 1996]. The latter authors also observed that agreement with Palierne s model worsens for blends pre-sheared at higher shear rate, i.e., blends with finer drop dispersion. 

Reprocessing was found to be effective also in PP/EPDM blends (Lee et al. 2012). In this case, the mbber particles in the PP matrix are progressively smaller with positive influence on the mechanical properties. Moreover, the interfacial tension, estimated by using Palierne and Choi-Schowalter models, was lower in multiple processed materials. 

A linear viscoelastic constitutive model of dilute emulsion viscoelastic properties was proposed by Oldroyd [111, 112]. The model considered low deformation of monodispersed drops of one Newtonian liquid in another, with an interphase. Choi and Schowalter [113] extended their cell model to dilute emulsions with Newtonian matrix and viscoelastic drops under infinitesimally small oscillatory deformation. Oldroyd s model was modified by Palierne [126, 127] for dilute viscoelastic hquids emulsions with polydispersed spherical drops (thus, subject to small deformations) with constant interfacial tension coefficient, Vu, at concentrations below that where the drop-drop interactions start complicating the flow field, that is, < 0.1 ... 

While the Choi and Schowalter [113] theory is fundamental in understanding the rheological behavior of Newtonian emulsions under steady-state flow, the Palierne equation [126], Eq. (2.23), and its numerous modifleations is the preferred model for the dynamic behavior of viscoelastic liquids under small oscillatory deformation. Thus, the linear viscoelastic behavior of such blends as PS with PMMA, PDMS with PEG, and PS with PEMA (poly(ethyl methacrylate))at <0.15 followed Palierne s equation [129]. From the single model parameter, R = R/vu, the extracted interfacial tension coefficient was in good agreement with the value measured directly. However, the theory (developed for dilute emulsions) fails at concentrations above the percolation limit, 0 > (p rc 0.19 0.09. 

Graebling, D., Muller, R., and Palierne, J.F. (1993) Linear viscoelastic behavior of some incompatible polymer blends in the melt interpretation of data with a model of emulsion of viscoelastic liquids. Macromolecules. 26 (2), 320-329. 

P.J. (1997) Linear viscoelastic behavior of molten polymer blends a comparative study of the Palierne and the Lee and Park models. Rhed. Acta, 36 (4),... 

Bousmina, M., Palierne, J.F., and Utracki, L.A. (1998) Modeling of immiscible polymer blends flow in laminar shear field. Polymer Engng Sci., (in press). 

The viscoelastic properties of partially miscible mixtures can be analyzed by applying incompressible emulsion models (Palierne 1990). Various studies (Graebling et al. 1993 Lacroix et al. 1998 Vinckier and Launn 1999) have confirmed the success of this model, providing information on the complex, typically bimodal, terminal relaxation of two immiscible or partially miscible polymers subjected to phase separation and capable of attaining dynamic equilibrium. Assuming that the droplet size is uniform, the complex modulus G (co) can be written as... 

Consequently, two major factors affect the rheological properties of the mixture during phase separation (i) the change of composition in the epoxy-rich matrix and (ii) the variation of viscoelastic behavior of the phase-separated blend. The authors employed a two-phase suspension model as proposed by Graebling and Palierne [52] to explain the effect of viscoelastic behavior on the phase-separated mixture ... 

For incompatible blends, the slope of log G versus log G" plots in the terminal region was less than 2. The slopes of the sonicated and compatibilized samples were higher than that of untreated samples in the linear viscoelastic region (see Figure 8.28), which meant that the compatibility could be enhanced by ultrasonic irradiation as well as by a compatibilizer. Palierne [97] has developed a model that can predict the linear viscoelastic behavior of a polymer emulsion, by considering the droplet size in a matrix and the interfacial tension between the components. From this model the unknown interfacial tension between the matrix and the dispersed phase could be estimated, and the predicted values decreased by ultrasonic irradiation due to ultrasonic compatibilization. 

PP PA6 PP-g-MA added for compatibUization rheology data compared with Palierne emulsion model 296... 

Palierne JE. Linear rheology of viscoelastic emulsions with interfacial tension. Rheol Acta 1990 214 204-14. Vermant J, Cioccolo G, Golapan Nair K, Moldenaers P. Coalescence suppression in model immiscible polymer blends by nano-sized colloidal particles. Rheol Acta 2004 43 529-38. 

For the sake of simplicity we retain here the simpler form of the emulsion model proposed by Palierne [2]. For a dispersion of viscoelastic incompressible inclusions in a viscoelastic impressible... 

Comparison between experimental data and Palierne s model predictions for the 10/90 PMMA/PS blend at 200 C. 

The results reported in this work on some selected polymer blends demonstrate that the melt rheology of multiphase systems is very complex, even in the linear viscoelastic domain. The main features exhibited by molten immiscible polymer blends are an increase of elasticity at low frequencies and longer relaxation times compared with that of the matrix. The linear viscoelastic properties of blends are satisfactorily described by the Palierne emulsion model and the enhancement of elasticity is ascribed to the deformability of the minor phase s droplets. 

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