3 Microscopic three-dimensional mode coupling theory
3.3 Shear and normal stresses
The full 3d calculation is thus quite similar in its γres and decays even a bit slower in vorticity direction. This is reasonable, because the ISHSM averages (heuristically) over all (faster) orientations ofq. The fit with the qy mode describes the final decay slightly better, because its form is a little more compressed exponentially. That the onset of the αdecay is steeper can be explained again with the flattening of theαprocess for steady-state correlators [59]. However, this flattening with increasing tw cannot be verified as clearly from the comparison ofΦq(t) to the experimental steady-stateΦExq (t, tw)as from the comparison to the LJ simulation’s ΦExqz(t, tw) above.
There are two problems apparent in the comparison to the experiment, which compli-cate a clear analysis, e.g. of the mentioned flattening of theαdecay withtw→ ∞. First, the range of experimental Peeffs starts at the upper limit and far exceeds the range of the actual MCT’s Pe0s. Numerical problems, but also the validity of MCT-ITT being restricted to small Pe0 ≪ 1, limit the actual calculations. Second, a yielding master curve is not as clearly approximated by the experiment as by the LJ simulation or by theory. The experimental data rather exhibits a drift of the α decay to smaller strains with decreasing Peeff. There are two possible explanations for this. First, experiments usually exhibit some sort of hopping processes for small Peeff. These could lead to a structural α master curve with fixed decay time τα, which dominates the shear induced yielding master curve, cf. Fig. 3.4. Second, in Ref. [11] it has been claimed that the yielding master curve is not shear-rate independent, as MCT-ITT predicts, but that for the relaxation time it holdsτq ∝γ˙−0.8. MCT-ITT however predictsτq ∝γ˙−1 (Sec.3.2.2) and cannot explain this exponent of −0.8.
3.3 Shear and normal stresses
Macrorheological quantities, viz. shear and normal stresses, calculated with 3d MCT-ITT under shear are presented in this section. They show the full strength of the MCT-ITT formalism, because it yields a tensorial stress tensor σ(t), which is transient, i.e. covers the full range between elastic and viscous strain response, and non-linear, i.e. exhibits a stress overshoot.
3.3.1 Shear stress σxy
For the further discussion, it is useful to first recall Eq. (2.42), it reads σxy(t) = ˙γ
Z t 0
dt′gxy(t′,γ˙), (3.4) with the 3d generalized shear modulus from Eq. (2.43)
gxy(t,γ) =˙
Z d3k 2(2π)3
"
k2xky(−t)ky kk(−t)
S′kSk(′ −t) Sk2
#
Φ2k(−t)(t). (3.5) Equation (3.4) is a constitutive equation for the shear stress component σxy(t) in a 3d system undergoing simple shear after start-up at t0 = 0. The content within brackets
3.3 Shear and normal stresses
in Eq. (3.5) is denoted as vertex, it depends on time only via accumulated strain γt, cf.˙ Eq. (2.25). It is then useful to denote the squared correlators Φ2k(−t)(t) as integration weights.
Fig. 3.9:Transient shear stress σxy as function of accumulated strain γt˙ for several relative packing fractions ε and bare Péclet numbers Pe0 calculated with 3d MCT. Panel a) shows a variation of ε with constant Pe0 = 10−4, while the other panels keep ε constant (see legends for Pe0). Panel b) corresponds to ε =−10−3, c) to ε = 0+, and d) to ε = 10−3. The elastic moduli G∞ and Gc∞, calculated from the plateaus of Fig.3.10, are shown asgrey, dash-dotted lines; find their values in Tab.3.4. Theline styleof the curves is according to Fig.3.11.
Figures 3.9 and 3.10 provide a good qualitative illustration of the transient stress regime, as well as already the quantitative data. Figure3.9shows the shear stressσxy(t) vs strain γt˙ calculated via Eq. (3.4). Figure 3.10 shows the generalized shear modulus gxy(t,γ)˙ from Eq. (3.5). Three regimes of viscoelasticity can be identified from the MCT data. First the stress grows linearly with strain proportional to an elastic shear modulus G∞. Hook’s according law reads σxy = G∞γt. Figure˙ 3.10 shows that in an MCT glass (ε > 0) a plateau in gxy(t,γ˙) is reached after some shear-rate independent β relaxation time τβ. This plateau’s value is G∞. The index ∞ can be understood as to refer to an infinite α relaxation time, which characterizes an elastic solid. Quiescent MCT predicts that the glass structure above φc is persistent and that the plateau does
3.3 Shear and normal stresses
not decay (τα → ∞, ideal glass). The long-time glass structure is in this context the reason for solid elasticity. Shear induced decay melts this plateau after a decay time τq ∝1/γ, see Sec.˙ 3.2.2, and the time integral in Eq. (3.4), viz. the area under gxy(t,γ˙) vs time, does not increase anymore. A steady state value ofσxy(t)is reached, denoted the flow curvevalueσstxy( ˙γ); see Fig. 3.11for the flow curve. For increasing Pe0, the plateau decays earlier and theβ relaxation becomes more important. With thegxy(t,γ˙)plateau disappearing, the elastic regime, whose strain response is proportional to the plateau value, splays out (i.e.σxy(t)gets anγ˙ dependent slope, see Fig.3.9) and becomes rather plastic (there is a loss of memory in the system). The physics behind this is that the β relaxation describes the ‘fluid-like’ motion of the hard spheres inside their structural cages, cf. Sec. 2.3.2.
Between the steady-state regime of the flow curve and the elastic regime occurs a transient regime. From Fig.3.10can be verified thatgxy(t,γ)˙ takes negative values prior to its final decay to zero. This negative area under the gxy(t,γ)˙ curve adds a negative portion toσxy(t)so that the stress decreases onto the steady-state plateau. The emerging bump is calledstress overshoot. The stress is maximal for the peak strain valueγ∗, i.e.γ∗ is the zero ofgxy(t,γ). If, in the fluid phase, the structural relaxation time˙ τα is smaller than the shear induced one τq, i.e. Wi . 1, the stress overshoot vanishes; Fig. 3.9 a) and b) illustrate this. This agrees with a result from linear response theory, viz. that the equilibrium shear modulus gxy(t,γ˙ = 0, ε < 0) is completely monotone [51] and is observed in experiments on colloidal dispersions [42,82]. A more detailed discussion of the interplay between decay times, strains, and the peak strain and height is given below together with Fig.3.12.
The statement that 3d and 2d microscopic MCT yield similar results regarding the macrorheology can be made from anticipating Fig. 5.3 at this point, where 2d MCT’s gxy(t,γ)˙ is shown. To simplify a comparison, the stress axis in Fig.3.10 is rescaled by a factor of vσ∗ = 205. The 3d results differ slightly quantitatively, cf. especially Tab. 3.4 with Tab. 5.1 for γ∗, but agree qualitatively very good in the appearance and features of the transient regime. This is also supported from comparing Fig. 3.9 and Fig. 3.11 with Fig. 5.4 (stress-strain and flow curves). The conclusion can be drawn that the schematic MCT, which is developed in Chap. 5 to mimic 2d MCT, mimics 3d MCT as well without need of further revision. In the schematic MCT, all quantitative differences are incorporated by fit parameters, while the anisotropic effects, which could be different in 2d, are dropped.
Figure3.11shows the flow curve valuesσstxy( ˙γ)of the steady-state regime of the stress-strain curves from Fig. 3.9 and also the long-time shear viscosity ηxy ≡σxyst( ˙γ)/γ. The˙ difference between fluid phase and glass phase in the context of MCT can be seen. In the glass phase and for Pe0 →0, the shear-rate independentβ relaxation hardly contributes to the area undergxy(t,γ˙). Due toτq∝1/γ, the area under˙ gxy(t,γ˙)and thusσxyst( ˙γ)/γ,˙ becomes proportional to1/γ. In consequence, a constant˙ dynamic yield stressσxy0 can be read of directly from the flow curve for Pe0→0. This stress is necessary to keep the glass shear molten and flowing at infinitesimally small shear rates. Because it is non-zero, the viscosity diverges in the glass. This dynamic yield stress must be distinguished from a static yield stress, which would be necessary to overcome a static, elastic threshold of a
3.3 Shear and normal stresses
0 0.1 0.2 0.3 0.4 0.5
10-3 10-2 10-1 100 101 102 103 104 105 106 G∞c
G∞(ε= 10-3)
Pe0= 10-1 10-2 10-3 10-4 10-5 10-6 0
0 5 10 15 20
0.01 0.1 1
ε= -10-3 0 10-3
t
d2/D0 gxy(t,˙γ)/v∗ σ
˙ γt gxy(t,˙γ) kBT/d3
Fig. 3.10: Generalized shear modulusgxy(t,γ), Eq. (2.43), as function of time˙ (main panel)and accumulated strain(inset; every second curve left out for clarity). In the main panel, gxy(t,γ)˙ is rescaled by v∗σ = 205 from Tab. 5.1. The legend provides color coded the strain rates and line-style coded the relative packing fractionε. Elastic shear moduli G∞ can be read of from quiescent curves (Pe0 = 0), withGc∞ for ε= 0+; see Tab.3.4 for the values. Theinset shows subtle differences in the peak positionγ∗, wheregxy(t,γ) = 0, which are caused by˙ γ˙ independent βandαdecays. This becomes most clear for Pe0= 10−6in the fluid phase, where the undershoot disappears.
glass. One could within this context try to identify the overshoot peak stress as static yield stress, but MCT-ITT is here shear-rate and not stress controlled. The static yield stress could be (and is) different if one increases the stress in a controlled manner till the glass starts to flow. The issue of creep does then arise, which can be modeled with a stress controlled MCT [83].
If in the fluid phase (ε <0), the shear-rate independent structural decay after timeτα is much smaller than τq, the integral in Eq. (3.4) becomes shear-rate independent and it isηxy ∝γ; this is called a Newtonian fluid (linear response to shear rate). The viscosity˙ ηxy0 is defined in the limit Wi → 0 and a first Newtonian plateau can be identified in Fig. 3.11 inset. A second Newtonian plateau would arise if only the initial Brownian decay would contribute shear-rate independently to gxy(t,γ)˙ for high Wi, i.e. a glass structure has no time to form up. This is theoretically problematic, because MCT de-scribes the physics of structural arrest and uses for short-times the quiescent Sq and Brownian motion with D0 as input without further considering how shear might affect them prior to structural arrest. Another problem arises, because the actual numerical iteration algorithm, Eq. (A.17), becomes unreliable for Pe0≫0.1. The flow curve yields
3.3 Shear and normal stresses
10-3 10-2 10-1 100 101
10-6 10-5 10-4 10-3 10-2 10-1 100 101 102
ε= -10-2 ε= -10-3 ε= 0 ε= 10-3 ε= 10-2
100 102 104 106
10-5 10-3 10-1 101
Pe0 σst xy kBT/d3
Pe0
ηxy[kBT/(D0d)]
Fig. 3.11: Themain panelshows flow curvesσxy(t→ ∞) =σxyst vs bare Péclet number Pe0 for six shear rates Pe0= 10{−6;...;−1}and for relative packing fractionsεas given in the legend. The Symbolsare connected with straight lines. Forε= 0+, higher Pe0 were calculated; the numerics fails for Pe0>101. The point of Pe0 = 102 was computed without friction kernel in Eq. (2.57) (i.e. just Taylor dispersion). Theinsetshows the long-time viscosityηxy=σstxy/γ.˙
unphysical, non-monotonically decreasing values. Values of Pe0 = 102 can be reached, when regarding just Taylor dispersion in setting the friction kernel mq(t) = 0! in the EQM (2.57). For higher values of Pe0, the wavevector advection in Eq. (3.5) creates numerically an unphysical behaviour, viz. the flow curve becomes non-monotonous and decreases. Regarding the high Pe0 trend in Fig. (3.11), the onset of a second Newtonian plateau within MCT-ITT could be guessed, but cannot be verified finally with the actual numerics.
How a stress overshoot emerges in microscopic MCT has been discussed already for the ISHSM in Ref. [12] and for schematic MCT in Ref. [42], the latter constitutes Secs.5.1.1 and 5.2.1. The correlators Φq(t) in Eq. (3.5) technically depend on strain γt˙ and on timet (strain independently). The former due to wavevector advection, Eq. (2.25), and a decay after accumulated strain γq, see Sec. 3.2.2, the latter due to the structural β and α decays after times τβ and τα. In Eq. (3.5), the squared correlators are a weight for the purely strain dependent vertex in gxy(t,γ˙). If the product of structure-factor deviationsSk′Sk(′ −t)becomes negative for enoughkvalues, whileγt˙ approachesγq, the k-space integral and thusgxy(t,γ)˙ become negative. A new contribution to this discussion is the 3d microscopic dependence of the peak strain γ∗ (the zero of gxy(t,γ)) on shear˙ rate and packing fraction. This dependence is shown in Fig. 3.12. One verifies that in
3.3 Shear and normal stresses
the glass (ε > 0) and for small Pe0, the γ∗ is independent of Pe0. The reason is that τα and τβ play no role in Eq. (3.5), because the weights Φ2k(−t)(t) in Eq. (3.5) decay time-independently at strain γq and the vertex decays at some other specific strain for each wavevector. Section 5.1 introduces γ∗∗ as the strain at which gxy(t,γ˙), i.e. the whole k integral, decays finally to zero. If the weights Φ2k(−t)(t) in the k integration change time-dependently, due to anα decay in the fluid phase, or because the β decay matters (Pe0 & 1), then the values of γ∗ and γ∗∗ change due to their dependence on the whole k integral. The same argumentation with the same quantities holds for the time integral over gxy(t,γ˙). This time integral, Eq. (3.4), yields the flow curve for long times. Therefore it is not surprising, but evident, that γ∗ as function of shear rate and packing fraction, Fig. 3.12, looks qualitatively exactly like the flow curves of Fig. 3.11.
This might be an important result for future experiments, to test the dependence of the overshoot peak strain on the flow curve values.
0.05 0.1 0.2 0.3 0.4
10-6 10-5 10-4 10-3 10-2 10-1 100 101 102
ε= 10-2 ε= 10-3 ε= 0 ε= -10-3 ε= -10-2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-1·10-2 -5·10-3 5·10-3 1·10-2
Pe0= 10-1 Pe0= 10-2 Pe0= 10-3 Pe0= 10-4 Pe0= 10-5 Pe0= 10-6
Pe0 γ∗
ε
σpk xy/σst xy−1
Fig. 3.12: Themain panelshows peak-strain valuesγ∗ as function of bare Péclet number Pe0. The outside legend gives the relative packing fractions ε, note the qualitative correspondence to the flow curves of Fig. 3.12. The γ∗s are read off from the zeros of gxy(t,γ), Eq. (2.43).˙ Symbolsare connected with straight lines whose style is according to Fig.3.11. Theinsetshows the relative peak heightσpk/σst−1 of the MCT stress overshoots for several Pe0 and relative packing fractionsε. The peak-heights increase with shear rate and with packing fraction.
The overshoot’s shape and height is controlled byγ∗−γ∗∗, as long asτα plays no role.
The inset of Fig. 3.12 shows that the relative overshoot height σpk/σst−1, with σpk the peak stress, increases with packing fraction and with Pe0. The former due to the increase of the weights Φ2k(−t)(t) with packing fraction, the latter due to the increasing role of theβ decay with Pe0 (in theβ-decay regime the Φqs are larger, cf. Fig. 3.10). If
3.3 Shear and normal stresses
the accumulated strain is not large enough, prior to the fluid correlator decay, no stress overshoot appears. Note that this behaviour differs from the findings in Ref. [39], where the opposite was measured. A decrease of the relative peak height with packing fraction was attributed to approaching random close packing (RCP). Besides, speculations if ageing [46] played a role in Ref. [39], MCT-ITT does not describe the regime close to RCP, but around the glass transition. It might be that MCT’s prediction fails at much higher or lower densities than atφc.
The comparison of schematic MCT with 2d microscopic MCT in Tab. 5.1shows that γ∗ andγ∗∗ are quite smaller than theγq of Tab. 3.3, which gives the decay strain of the correlator weights Φ2k(−t)(t). Theγ∗s and γ∗∗s are in Sec.5.2also smaller than γc. The latter can be identified as a common γq for all the weightsΦ2k(−t)(t) in the k integral of Eq. 3.5. This means that for large Weissenberg numbers Wi ≫ 1 (i.e. the fluid decay afterτα has a minor role), the vertex in Eq.3.5determines a big amount of the shape of the stress overshoot, because it decays prior to the weights Φ2k(−t)(t). The decay of the correlator Φq(t) cannot be neglected completely, but rather speeds up the approaching of the flow curve. It can be followed that the stress overshoot is clearly a phenomenon related to accumulated-strain and not to time or (the related) flow velocity scales (as long as Pe0 ≪1).
Some rheological quantities from the actual 3d MCT-ITT, which are important for this work, are listed in Tab. 3.4.
ε= (φ−φc)/φc η0xy [kBT /(D0d)] σxy0
kBT /d3
G∞
kBT /d3 γ∗0
−10−2 6.55·102 — Gc∞ —
−10−3 1.99·105 — Gc∞ —
0+ — 2.05 Gc∞= 18.3 0.320
10−3 — 2.43 21.4 0.341
10−2 — 3.26 30.9 0.368
Tab. 3.4:Rheological quantities in the framework of 3d MCT for different relative packing fractionsε, according to the calculations from Fig. 3.9,3.10,3.11, and3.12. The values ofσ0xy, ηxy0 , and γ0∗ are approximated withσxy,ηxy, andγ∗ of Pe0= 10−6, cf. Fig. 3.11and3.12. The shear moduliG∞ are taken from thegxy(t,γ)˙ plateaus of quiescent calculations, cf. Fig.3.10.
3.3.2 Normal stresses σxx, N1, and N2
To illustrate the tensorial nature of the stress tensorσ(t)and of 3d MCT, Fig.3.13shows a calculation of the first and second normal stressesN1≡σxx−σyy andN2≡σyy−σzz. Already Ref. [84] showed numerical and tensorial evaluations of σ(t) via a schematic MCT; see also Eqs. (5.5) and (5.6), where this concept will be generalized to non-linear stresses. However, in schematic MCT it isN2 = 0 under shear by construction. The full 3d numerics is able to calculate the non-zeroN2of a hard-sphere dispersion, which bears a similarity in value to polymeric melts [85].
3.3 Shear and normal stresses
Within MCT under simple shear, σαβ(t) can be calculated analogously to σxy(t), Eq. (2.41), by simply changing the components of some k vectors in Eq. (2.39). Fig-ure 3.13shows the stress on a plane-element perpendicular to the flow direction σxx(t) and the normal stresses as defined above, which are of distinguished rheological interest.
The calculation has been done atε= 0+and Pe0 = 10{−2;−4;−6}, i.e. for a genuine, critical glass behaviour with small Pe0 and large Wi. The first observation is that they all ex-hibit a transient regime, with stress overshoots, which look qualitatively like those of the shear stressσxy(t), cf. Fig.3.9. The overshoots however occurs at strains larger than 0.4, which is larger than allγ∗0, cf. Tab.3.4. At all timest >0, it holdsσxx > σzz > σyy >0, which renders N1>0and N2 <0.
10-3 10-2 10-1 100
0.01 0.1 1
σxx: Pe0= 10-2 10-4 10-6 N1=σxx-σyy: Pe0= 10-6 -N2=σzz-σyy: Pe0= 10-6
˙ γt σxx,N1,−N2 kBT/d3
10-2 10-1
0.01 0.1 1
N1/σxy: Pe0= 10-2 10-4 10-6 -N2/σxy: Pe0= 10-2 γ˙t 0.28·γ˙t
˙ γt
±N
1 2/σxy
Fig. 3.13: Left panel: transient normal stresses σxx (solid lines), first normal stress N1=σxx−σyy (dot-dashed lines), and second normal stress N2 = σyy −σzz (dashed lines, plotted with negative sign) as functions of accumulated strain γt˙ at critical packing fraction (ε = 0+) and for the Pe0s given in the legend (Pe0s arecolor coded for all line-styles). One verifies that all σii > 0 and σzz > σyy. Stress overshoots are apparent, but at at least 25%
higher peak strains than forσxy, cf. Tab. 3.4.
The right panel shows the normal stresses divided by shear stress, N1/σxy and −N2/σxy, to illustrate the validity of Eq. (3.6). Grey dashed and grey quadro-dashed linesare linear fits with slopes 1 and 0.28 (see legend).
The right panel of Fig. 3.13proves that the Lodge-Meissner relationship holds in the elastic regime of 3d MCT. It reads
N1/σxy(t) =γ= ˙γt (3.6)
and states that the slope ofN1 is quadratic in accumulated strain with prefactorG∞. In Ref. [84] this was tested and verified, too, for schematic MCT, but there leavingN2 = 0, because of the isotropic calculation of the generalized shear modulus, cf. Eq (5.7). The full 3d numerics yields, according to Fig.3.13right panel,N2/γ2 ≃ −0.28, which is known to be a sensible value for polymeric melts [85] (there≃ −0.22). The MCT prediction for a hard-sphere dispersion seems to be similar in this sense.