Instantaneous dimensionless numbers for transient nonlinear rheology

Abstract

Two instantaneous dimensionless numbers that act as Deborah and Weissenberg numbers are introduced to diagnose flow conditions for transient nonlinear rheology. The utility of the new numbers is demonstrated on the steady alternating large amplitude oscillatory shear response of a colloidal Ludox glass, the soft glassy rheology model, and a viscoelastic wormlike micelle solution. Complex nonlinear trajectories through Pipkin space are observed, from which it is concluded that large amplitude oscillatory shear represents a range of distinct flow types. These results indicate that the observation time may change significantly during a period of oscillation. The complex trajectories observed for all three systems go from close to one axis to close to the other and back in quick succession. Rather than existing in the dominant central area that Pipkin originally marked as “?”, LAOS may simply be the way by which the axes of Pipkin space are dynamically linked.

Appendices

Appendix I: normalization strains

We propose that the normalized recoverable strain be used as an instantaneous Weissenberg number. As discussed in section 1.4, to compare different materials requires some measure of a strain above which nonlinear effects are observed. In this work, we define nonlinear responses as those where G ′ ′, as reported by the rheometry software, changes by 5% in the amplitude sweeps shown here (Fig. 7).

Fig. 7

Amplitude sweeps of the three systems investigated in this work, a Ludox glass, b SGR model, and c wormlike micelle solution. The normalization strains, which are determined as the point where G″ deviates by 5%, are indicated by dashed lines

.

Appendix II: empirical rules

In addition to recent phenomenological studies of LAOS, a number of empirical rheological rules can also be interpreted as supporting the revision of the way we describe transient nonlinear rheological responses. These rules typically relate a frequency-dependent linear viscoelastic measure to an equivalent steady-shear measure at a rate numerically equal to the frequency. In each case, reading these rules within the context of the current discussion leads to the conclusion that the two axes of Pipkin space are linked and that material responses simultaneously exist at two points within Pipkin space. Topologically speaking, these rules suggest that Pipkin space is a noncompact connected manifold equivalent to a cone as shown in Fig. 8. Here, we briefly introduce each rule and discuss its importance to the present work.

The most well-known of Cox and Merz’s three rules (Cox and Merz 1958; Sharma and McKinley 2012) states that the steady shear viscosity can be obtained from SAOS measurements at frequencies equal to the steady shear rates:

$$ \eta \left(\dot{\gamma}\right)\cong {\left|{\eta}^{\ast}\left(\omega \right)\right|}_{\omega =\dot{\gamma}}. $$

(9)

The other two rules proposed by Cox and Merz have the same essential content but relate the dynamic viscosity to the consistency, and the elastic modulus to the viscosity and consistency.

The Rutgers-Delaware rule for materials with yield stress and recoverable strain (Doraiswamy et al. 1991) extends the (first) Cox-Merz rule to say that the complex viscosity vs. maximum (or effective) shear rate (γ₀ω) curve is identical to steady state viscosity vs shear rate:

$$ \left|{\eta}^{\ast}\left({\gamma}_0\omega \right)\right|={\left.\eta \left(\dot{\gamma}\right)\right|}_{\dot{\gamma}={\gamma}_0\omega }. $$

(10)

In 1986, Laun proposed two empirical rules that relate LVE parameters with steady shear equivalents (Laun 1986). The most well-known of the two rules relates the dynamic moduli to the first normal stress difference:

$$ {\left.{N}_1\left(\dot{\gamma}\right)\right|}_{\dot{\gamma}=\omega}\cong 2{G}^{\prime}\left(\omega \right){\left\{1+{\left(\frac{G^{\prime}\left(\omega \right)}{G^{\prime \prime}\left(\omega \right)}\right)}^2\right\}}^{0.7}. $$

(11)

The second of Laun’s rules relates the steady-state recoverable strain during nonlinear steady shearing to the linear-regime dynamic moduli, extending a relation derived from Lodge’s rubber-like liquid(Lodge 1964):

$$ {\left.{\gamma}_r\left(\dot{\gamma}\right)\right|}_{\dot{\gamma}=\omega }=\frac{G^{\prime}\left(\omega \right)}{G^{\prime \prime}\left(\omega \right)}{\left\{1+{\left(\frac{G^{\prime}\left(\omega \right)}{G^{\prime \prime}\left(\omega \right)}\right)}^2\right\}}^{1.5}. $$

(12)

The Cox-Merz rule(s), the Rutgers-Delaware rule, and Laun’s rules state that linear viscoelastic measures from the abscissa of Pipkin space (at all Deborah numbers and zero Weissenberg number) are equivalent to nonlinear viscometric measures on the ordinate (at zero Deborah number and non-zero Weissenberg numbers). The two axes are therefore joined as displayed in Fig. 8, leading to our earlier statement regarding the topology of the space as currently defined.

Fig. 8

Pipkin’s suggestion for the space in which all flows reside is broken into different regions on the basis of the Weissenberg and Deborah numbers (left). Empirical rules such as the well-known Cox-Merz and Rutgers-Delaware rules, as well as Laun’s rules, draw an equivalence between the two axes of Pipkin space under certain circumstances, making the space a connected manifold equivalent to a cone

While these rules are acknowledged as not always applying (Pedro E.D. Augusto et al. 2013b; Augusto et al. 2013a; Berg 2004; Gleissle and Hochstein 2003; Ianniruberto and Marrucci 1996; Lin et al. 2013; Manero et al. 2002; Marrucci 1996; Venkatraman and Okano 1990; Wen et al. 2004; Winter 2009; Wood-Adams 2001), and in some cases being purely phenomenological, their existence suggests that materials responses can exhibit some sort of superposition of states by simultaneously existing at two distinct locations in Pipkin space as traditionally defined. While Poole and Dealy have noted that many researchers mix the definitions of Deborah and Weissenberg numbers, and Reiner has cautioned us to always use the right Deborah number, these equivalences clearly show that the conceptual mixing of Deborah and Weissenberg may not be so easy to separate. We take inspiration and use this discussion as motivation for revisiting the definitions of the most useful dimensionless groups for transient nonlinear rheology.