Fitting parameters for Figure 4.5.4
Table 4.4 shows values of the fitting parameters (found using a nonlinear fitting function in Grace [1]) for the dashed lines in Figure 4.5.4.
x C α Ef Edip Ff Fdip
0.1 0.019 1.11 — — — —
0.3 0.096 1.10 541 100 7.67 7.67
0.5 0.124 1.08 33.0 10.4 5.25 5.43
Table 4.4: Fitting parameters for Figure 4.5.4. C, α are parameters for the fluidi- sation time fit using: τf/tw = C(Σ − Σy)−α for (a, c, e) of the figure. Edip,f, Fdip,f are parameters for the dip (dip) time and fluidisation (f) time fit using: τdip,f/tw =
Edip,fexp(−Fdip,fΣ) for (d, f) of the figure.
Discussion on the origin of the critical stress Σ
cHere, we discuss the origin and value of the critical stress Σc that divides the small and large stress regimes that result in different relations (Eqns 4.5.5 and 4.5.6, respectively) for the fluidisation time τf on the imposed stress Σ.
In fact, the separation of these regimes appears to originate in whether the fluidisation time occurs at short (∆t tw) or long (∆t tw) times. This can be clearly seen by comparing the smallest/largest fluidisation time τf for which the Eqns 4.5.5 and 4.5.6 (dashed lines) fit the data in Figure 4.5.4. [Refer to a schematic of the short and long time creep regimes in Figure 4.8.2 (below). In this schematic that assumes Σ < Σc, fluidisation occurs in the long time regime: τf tw. For larger stresses Σ > Σc fluidisation occurs in the short time regime: τf tw.] The critical stress Σc then describes the stress at which fluidisation occurs at τf ∼ tw. By considering Eqn 4.5.5 in the limit τf/tw → 1 from above, we find that:
Σc= Σy + C1/α. (4.8.1)
values from Figure 4.5.2 can be found in Table 4.4 above. (Recall that Σy is also dependent on x, see Chapter 3.)
Convergence of α in Figure 4.5.4 (e) w.r.t. t
w4 5 6 7 log10t w -1.3 -1.2 -1.1 -1 α
Figure 4.8.1: Convergence of the exponent α from Eqn 4.5.5 with respect to the waiting time tw in the SGR model, for x = 0.5.
Discussion of τ
dipin the small stress regime Σ < Σ
cHere, we discuss the difficulty in obtaining a meaningful τdip during the response to small stresses Σ < Σc.
Firstly, recall from Section 4.5.2 that there exists a short (∆t tw) and long (∆t tw) time creep regime. For imposed stresses below the critical stress Σ < Σc the fluidisation time occurs during the long time creep regime τf tw. We show this in a schematic of the response in the SGR model in Figure 4.8.2. Here, both creep regimes are seen before fluidisation occurs.
We find that the cross over between the short and long time creep regimes is accompanied by a ‘pseudo-minimum’ that can be seen at ∆t ∼ tw, and appears to result from the number of elements that have not yet yielded (since the onset of deformation) decreasing to zero. This pseudo-minimum is shown in the schematic of Figure 4.8.2 at ∆t ∼ tw; we find that the ‘pseudo-minimum’ apparently persists even as Σ → Σy. Ideally, we could still extract the time of the dip τdip that actually corresponds to the transition to steady flow. However, unless a sufficient separation between τdip and tw exists, we find that the minimum corresponding to the onset of
log ∆t log γ . tw τdipτf µ φ test test
Figure 4.8.2: Schematic of generic response (valid for all x < 2). Two power law creep regimes exist (with exponents µ, φ), a transition between the two occurs at time tw. For stresses Σ > Σy the system will fluidise at a time ∆t ∼ τdip — this time depends on the imposed stress so that, for some stresses, only the short time creep regime might be seen τdip< tw.
fluidisation τdip becomes lost in the ‘pseudo-minimum’ at ∆t ∼ tw, so that extract- ing a meaningful τdip is very difficult. We therefore do not attempt to associate τdip with τf of Eqn 4.5.5.
5
Shear startup protocol
5.1
Introduction
In this chapter we will investigate the rheological response of soft glassy materials and entangled polymeric materials above and below the glass transition during shear startup. In this protocol a constant shear rate ˙γ is imposed on the material and the dynamics of the total shear stress Σ(t), are measured as a function of time t, or equivalently, strain γ = ˙γt. Recall Figure 2.1.3 of Chapter 2.
As explained in Chapter 2, there is much need of a criterion for the onset of time- dependent linear instability to shear heterogeneity in this time-dependent protocol with the same fluid-universal status as that of the well known steady state criterion of a negative slope in the constitutive curve1 ∂
˙
γΣ|γ→∞ < 0 [160]. We will derive
1Recall that the ‘constitutive curve’ is defined as the relation (with homogeneity in the flow
gradient direction enforced) of the steady state total stress Σ( ˙γ)|t→∞ to the imposed shear rate.
The ‘flow curve’ is the same curve with heterogeneity in the flow gradient direction allowed.
such a criterion in Section 5.2, for which credit is given to Dr. Suzanne Fielding. By considering large strain and large shear rate limits we will identify ‘elastic’ and ‘viscous’ terms that contribute to linear instability to shear banding. We will show that the ‘elastic’ term is consistent with numerous experimental findings in entangled polymeric and soft glassy materials showing time-dependent shear banding during the negative slope of shear stress as a function of strain during shear startup [18, 19, 43, 45, 77, 136, 138, 162].
We will show how this general criterion applies widely in models for entangled polymeric fluids using the rolie-poly and Giesekus models. (Examples of these ma- terials include concentrated solutions or melts of high molecular weight polymers, and concentrated solutions of wormlike micelles or DNA.) We will investigate age- dependent transient shear bands in the scalar fluidity model, for which the consti- tutive curve is monotonic. We also show these to arise in the glassy polymer model, albeit with a reduced magnitude due to the effect of strain hardening. The author is grateful to Prof. Mike Cates and Prof. Ron Larson for collaboration during research on this model.
The results of Sections 5.2 and 5.3 are published in Ref. [117], Section 5.6 in Ref. [116], and Section 5.7 in Ref. [59].