Strain recovery of model immiscible blends without compatibilizer

Abstract

Strain recovery after the cessation of shear was studied in model immiscible blends composed of polyisobutylene drops (10–30% by weight) in a polydimethylsiloxane matrix. Blends of viscosity ratio (viscosity of the drops relative to the matrix viscosity) ranging from 0.3 to 1.7 were studied. Most of the strain recovery was attributable to interfacial tension, and could be well-described by just two parameters: the ultimate recovery and a single retardation time. Both these parameters were found to increase with the capillary number of the drops prior to cessation of shear. For blends that had reached steady shear conditions, the ultimate recovery decreased with increasing viscosity ratio, whereas the retardation time increased with increasing viscosity ratio. The retardation time was well-predicted, but the ultimate recovery was over-predicted by a linear viscoelastic model developed previously by Vinckier et al. (Rheol Acta 38:65–72, 1999).

Appendix

Figure 6 shows a typical dynamic mechanical frequency sweep measurement for a blend with 10 wt% drops. Also shown is the G′ expected from the components (calculated using the Palierne model with the interfacial tension set to zero). The most obvious feature is the pronounced shoulder in the measured G′ of the blend that is entirely absent from the components. This shoulder has been attributed to the interfacial tension and its characteristics can be related to the size of the drops in the blends (Graebling et al. 1993a, 1994; Vinckier et al. 1996). Such frequency sweep data have been analyzed extensively in past publications (Graebling et al. 1993a, 1994; Vinckier et al. 1996; Kitade et al. 1997; Velankar et al. 2001). Here we will follow the analysis outlined by Velankar et al. (2004), which was specifically devised for the situations in which the shoulder in G′ is prominent and well-separated from any higher frequency relaxations. The G′ expected from the components was first subtracted from the measured G′ of the blend. The remainder, which may be regarded as the interfacial contribution to the G′, was fitted to a sum a few (up to 3) Maxwell modes:

$${G}\ifmmode{'}\else$'$\fi(\omega) = {\sum\limits_{k = 1}^{n} {\frac{{\omega^{2}\;\exp (a_{\rm k} + 2t_{\rm k})}}{{1 + \omega^{2}\;\exp (2t_{\rm k}).}}}}$$

(17)

Fig. 6

Analysis of dynamic mechanical data. Open circles are the measured G′ of a blend with 10 wt% drops, p=1.1, sheared for 53912 s at 120 Pa. The solid line corresponds to Eq. 17 with n=2 i.e. a sum of two Maxwell modes

Fits were performed using the free gnuplot software as described previously (Velankar et al. 2004). A sample fit has been shown in Fig. 6, and additional examples have been shown previously (Velankar et al. 2004). In all cases, the Maxwell mode corresponding to the shoulder in G′ was separated from any other modes by at least one decade in frequency. Thus, for all practical purposes, the shoulder can be captured by only one Maxwell mode. The corresponding relaxation time, multiplied by the shear rate prior to cessation of shear, yields the dimensionless relaxation time λ ^*_F1 . Equation 8 is then used to obtain the Ca.