Except for the SGR model, all models studied in this thesis involve integrating a differential constitutive equation for the tensorial stress σ — methods for which we provided above in Sections 3.6.1 and 3.6.2. We now outline the numerical methods used to obtain the strain rate response to an imposed stress in the SGR model that was used to produce the results presented in Chapter 4.
In the SGR model, elements ‘hop’ out of their traps22 of energy depth E with a rate dependent on the trap depth and also on the local element strain. A con- ventional Monte Carlo method is not suited this process in the SGR model due to the extremely slow dynamics and long simulation times required: the acceptance of a trial ‘hop’ has a very small probability, resulting in a high number of rejections before a trial ‘hop’ is accepted. Rather, following Refs. [54,57], we use a waiting time Monte Carlo (WTMC) method [15]. This is a ‘rejectionless’ Monte Carlo method as it a priori assumes an element hops, but chooses which element to hop based on the distribution of element hopping rates, and the time at which this hop occurs based on the total hopping rate. That is, the WTCM is an event based algorithm.
Specifically, we model N SGR elements i = 1, ..., N with trap depths Ei and strains li, resulting in the element hopping rate ri = exp−(Ei− 12l2i)/x. The sum of all hopping rates is R =PN
i ri, and the macroscopic strain rate is ˙γ = N1 PN
i rili
[53, 54].
After a hopping event (or initialisation at t = 0) the time at which the next hop
22Note that here, in our terminology, a ‘hop’ out of a trap is equivalent to the ‘yielding’ events
occurs t + δt is chosen from a distribution P (δt) ∝ exp(−δt R) dependent on the sum of all hopping rates. Having calculated the time of the next hopping event, we then need to select which of the N elements will hop. We do so by noting that the probability that any given element will be the one to hop is given by Pihop = ri/R, therefore the element that hops at t + δt is chosen by a search of all the elements weighted by this probability. To perform the ‘hop’, the element strain is set to zero lhopper = 0 and a new trap energy Ehopper is chosen for the element from the distribution ρ(E) = e−E. In the SGR model the total stress is given by the macroscopic strain minus the strain relaxed through hopping events (multiplied by the elastic modulus) [54, 62]. Therefore, in order to maintain a constant total stress in the step stress protocol, the macroscopic strain must be increased after each hopping event: γ(t + δt) = γ(t) + lhopper(t)/N . Since ˙l = ˙γ, all local element strains li must also be updated by the same amount.
We note that all results presented in this thesis have been checked for convergence with respect to the number of elements N ; a typical value required is N = 105 or 106.
4
Step stress protocol
4.1
Introduction
In this chapter we will investigate the rheological response to an imposed shear stress of three classes of material: soft glassy materials and entangled polymeric materials above and below the glass transition. In this experimental protocol a constant shear stress Σ is imposed on the sample for time t > 0 and the shear rate ˙γ(t) varies in time, see Section 2.1.1. We will first derive a criterion for the onset of linear instability to shear rate heterogeneity that is independent of fluid or model type. This depends only on time derivatives of the homogeneous background state: we find growth of shear rate perturbations whenever1 ∂∂t22γ˙/
∂ ˙γ
∂t > 0 (where ˙γ is the shear rate from the homogeneous background state). We will show that this criterion correctly predicts the growth of heterogeneous perturbations during an investigation of the rheological
1Note that from here on we describe this criterion by using the notation ∂2
t ˙γ/∂t˙γ > 0.
response to an imposed stress in four models for various classes of material:
Polymeric fluids: We will show that the rolie-poly (RP) and Giesekus models show qualitatively similar responses of the shear rate ˙γ(t) to a step stress: for imposed stresses with values nearest those on the weakest slope (smallest ∂γ˙Σ) of the constitutive curve the shear rate rises suddenly by several orders of magnitude over a short time period. We find that, consistent with the general criterion, transient shear banding occurs during the upwards curving, upwards sloping region of this rise: i.e., where ∂2
t ˙γ/∂t˙γ > 0 in the homogeneous background state.
Soft glassy materials: As explained in Chapter 2, soft glassy materials (SGMs) usually show an initial ‘creep’ regime in which the shear rate progressively decreases in time in response to an imposed stress. For stresses that exceed the yield stress Σ > Σy, this creep regime is usually followed by a sudden ‘fluidisation’ where the shear rate dramatically rises (in a similar fashion to that of the polymer systems described above) before reaching a steadily flowing state. Various features of the ˙γ(t) curve have been used to define the time at which the system ‘transitions’ between this solid-like (creep) and liquid-like (steady flow) behaviour (often referred to as the ‘fluidisation time’), such as the minimum of the shear rate in time [11, 28] or the inflection point where d ˙γ/dt is maximal [42,64,156]. The general criterion derived in Section 4.2 predicts linear instability to shear rate heterogeneity for times between these two points, which we later refer to as the ‘dip’ and ‘fluidisation’ times, τdipand τf, respectively. Recent experiments [42] and simulations of the SGR model [117] have confirmed that shear banding does arise between these two times. These results have motivated us to determine the relation of the fluidisation and dip times τf, τdip to the imposed shear stress Σ in the SGR model, along with a relation for the shear rate in time during creep. We will focus on two regimes of the effective noise temperature: x < 1 (below the glass point) and 1 < x < 2 (above the glass point). The author is grateful to Dr. Thibaut Divoux for motivating this research, and to Prof. Peter Sollich for collaboration and yield stress data of the SGR model.
Glassy polymers: Finally, we will investigate the rheological response to an im- posed stress in the glassy polymer (GP) model in order to compare results in the shear geometry with recent results in the extensional geometry [58]. These exten-
sional results show that the segmental relaxation time τ (t) (which governs the rate of rearrangement of local segments of the polymer chain) initially decreases as the system begins to fluidise in response to the imposed stress, before reaching a min- imum and rising indefinitely. The authors found that this ‘dip’ in the segmental relaxation time is concurrent with the onset of strain hardening; we will show that these basic features are also found in the shear geometry. We will also explore how shear banding arises during creep, and use the general criterion described above to explain how strain hardening reduces the magnitude of transient shear banding for increasing polymer contribution to the stress. The author is grateful to Prof. Mike Cates and Prof. Ron Larson for collaboration during research on this model.
This chapter is ordered as follows. In Section 4.2 we derive a general criterion for the onset of shear banding during the time-dependent response to a step stress. The rheological response of the RP and Giesekus models are investigated in Sections 4.3 and 4.4, respectively. We explore the ‘creep’ and ‘fluidisation’ behaviour of the SGR model in Section 4.5. Finally, we show the rheological response of the GP model to an imposed shear stress in an analogy to recent extensional load experiments and numerics in Section 4.6. The results of Sections 4.2 and 4.3 are published in Ref. [117], and those of Section 4.6 published in Ref. [59].