3 Microscopic three-dimensional mode coupling theory
3.2 The transient density correlator
taken for all results of this chapter, unless explicitly noted differently; see Tab.3.2for an overview of this chapter’s generic discretization choice.
π/∆ϑq ∆q d−1
φc
4 0.4 0.476925(5)
6 0.4 0.503853(3)
8 0.4 0.511548(1)
8 0.2 0.511474(6)
16 0.4 0.514909(9)
24 0.4 0.515712(1)
32 0.4 0.515792(8)
Bengtzelius grid
— 0.4 0.51591213(1)
Tab. 3.1:Critical packing fractions φc
for differentq-grid discretizations∆qand
∆ϑq when solving MCT-EQM (2.57) with the BV algorithm, App.A.3.2Eq. (A.17).
φc is rounded up in the seventh digit, i.e. φc describes a glass and φc−10−7 a fluid; find a more detailed discussion in Sec.2.3.2.
φc q ∆q 2π/∆ϕq π/∆ϑq Pe0
0.5157121 [0.2;39.8] 0.4 24 24 10−6
Tab. 3.2:Parameters of theq grid used throughout this chapter, unless noted differently. Even though the bare Péclet number Pe0 is no grid parameter, it is generically used with the given value for illustrative results.
3.2 The transient density correlator
This section presents the transient density correlator Φq(t), Eq. (2.15), which can now be calculated fully anisotropically under shear and in 3d. A closer look at a problem of the MCT-ITT mode-coupling mechanism, viz. Eqs. (2.56) and (2.58), which emerges for small wavevector modes, is focused on in Sec. 3.2.3.
Figure 3.2 shows the parametrization of the wavevector q and the velocity field of simple shear in the coordinate system, which is used for all computations (flow direction is x axis, gradient direction is y axis, vorticity direction is z axis). This figure also illustrates that the strain-rate tensorκof simple shear flow, Eq. (1.1), can be decomposed into a symmetric and an antisymmetric contribution via
κ=κs+κa= 1
2(κ+κT) +1
2(κ−κT). (3.2)
The symmetric term κs gives rise to a deformation field (elongation along the x = y axis) and the antisymmetric term κa to a rotation field. Especially the shape of the deformation field will yield a very convincing argument for the symmetry changes of the structure (factor) regarded in Sec.3.4.
3.2 The transient density correlator
ϑq
ϕq
q
qx
qy qz
ϕq= 45°
ϕq= -45°
ϕq= 45°
ϕq= -45°
elongation rotation simple shear
x x yy
Fig. 3.2:Left panel: spherical coordinates of the wavevectors q (andk, p) with definitions of the azimuthal angleϕq and inclination angleϑq;qdenotes the modulus ofq.
Right panel: simple shear flow in real space, the z-axis points out of the sheet. Elongational and rotational flow correspond to the decomposition of the shear flow into a symmetric and an antisymmetric strain tensor. Elongation takes place along the x =y axis (ϕq = 45◦) and compression along the x=−yaxis (ϕq =−45◦).
3.2.1 Anisotropic Φq(t) in 3d
MCT-ITT claims to be valid for small bare Péclet numbers Pe0 ≪1, because the short time dynamics of the EQM (2.57) implicitly assumes that a quiescent structure factor Sq describe the system for short times (unharmed by strain flow). On the other hand, the dressed Péclet or Weissenberg number Wi= ˙γτα should be larger than one, so that shear induced dynamics can be distinguished from structural α decay. As consequence, a Pe0 = 10−6 and a critical packing fraction ε = 0+, i.e. τα =∞, is the generic choice here to illustrate the features of the theory.
Figure 3.3illustrates the shear induced α decay ofΦq(t) for various orientations of q and for the two first peaks of the non-ergodicity parameters fqc, viz. at q = 7 and q = 12.6, cf. Fig.3.1. The anisotropic α decay is highlighted, because the isotropicβ decay resembles the quiescent case [71]; see Figs. 3.6and 3.7or the discussion in Refs. [12,59].
One verifies different decay-scales for correlator modes of advected wavevectorsq(t)(they have components in flow direction, i.e. qx 6= 0) and neutral correlator modes (qx = 0).
All correlator modes are affected by shear, but the latter ones decay slower; find a closer consideration in Sec.3.2.2. Non-monotonous behaviour with negativeΦq(t)values in the decay region can be seen, but not if q points in the vorticity direction (ϑq = 0). These negative dips are known already from 2d calculations, but are much less pronounced in 3d than e.g. in Fig. 5.11 of Ref. [55], where vast oscillations occur around the peak regions of Sq and fqc. It is easier for particles to circumvent each others in 3d and MCT seems to describe this behaviour.
3.2 The transient density correlator azimuthal anglesϕqand inclination anglesϑq areline-style coded. Theleft panelshows correlator modes of q with components only in gradient and vorticity directions, i.e. for non-advectedq, while the right panel shows modes withq components in flow direction. The line styles of the q= 12.6 curves correspond to those ofq= 7curves, which describe the sameq orientation.
3.2.2 α master curves
Figure3.4shows the α decay of Φq(t)in a fluid (ε=−10−3) and critical state (ε= 0+) for different shear rates. In Fig. 3.5, a glass state with ε = 10−3 is shown. The latter figure proves that a strain rate independent α master curve, or yield scaling function, is approached for small Pe0. This implies an α decay time τq ∝1/γ˙ under strain and thus a shear independent decay strain γq, which depends on q. This is familiar from 2d calculations already [23]. Find a detailed scaling analysis of the α process, i.e. an α master equation, in Refs. [23,41,55]. In the fluid case, an isotropic structural decay time τα can be identified, which dominates for small Wi, i.e. τα . 1/γ. The˙ α master function does then not depend on strain, but on time. Kohlrausch fits to theα master curves of the glass with fqcexp
−( ˙γt/γq)βq
and the fluid with fqcexp
−(t/τα)βα have been performed; find the according Kohlrausch parameters in Tab. 3.3.
The α master curves and Kohlrausch fits allow for the following observations. Corre-lator modes of the shear gradient (qy) and vorticity direction (qz) exhibit a very simi-lar qualitative and quantitative behaviour, i.e. Φq(t) decays almost exponentially with βq ≃ 1. The mode of the flow direction (qx) however decays faster by a factor of 2 to 3 and also strongly compressed exponentially with βqx ≃ 1.8. For increasing Pe0, one can see the emergence of little ‘shoulders’ in the final decay region, especially for the qx direction, which is known from 2d calculations already [22,23]. In general, the observed exponential behaviour of theα master curves is in full accordance with the 2d numerics shown in Refs. [22, 55] and thus puts the actual calculations in a consistent line. The vorticity direction does not exhibit any pronouncedly different behaviour from the gra-dient direction and one might assume at this point already that theqx, qy plane contains
3.2 The transient density correlator color coded (see legend), they correspond to qz, qx, andqy direction. In theleft panel, Pe0 de-creases from left to right in steps of 10, i.e. Pe0= 10{−1;...;−6}. The glassy curve has Pe0= 10−6. For Pe0 = 10−6, the fluid α master curve becomes independent of the q orientation, i.e. fluid decay dominates. The left panel’s legend applies to both panels. Theright panelshows modes of qz and qx direction as functions of accumulated strain γt, while the˙ qy direction is dropped due to its similarity toqz. Theε= 0+ curves are rescaled in strain by 0.01 and inΦq by 0.5 for a better visualization.
Grey linesare Kohlrausch fits to theαmaster curves; find the according parameters in Tab.3.3 and in the legend (t andτα are rescaled byγ˙ = 10−6, which yieldsγ and 0.0564, respectively).
qualitatively the features of the underlying physics for simple shear in 3d; an assumption that comes up again when considering SF distortions in Sec. 3.4.
ε f7 γqx βqx γqz βqz τα(q = 7) βα(q= 7)
−10−3 0 — — — — 5.63895·104 0.78974
0 0.852410 0.630203 1.83444 1.42224 1.06407 ∞ —
10−3 0.87508 0.738626 1.82888 2.01638 1.05480 ∞ —
Tab. 3.3:Kohlrausch parameters for the fits to the αmaster curves in Figs. 3.4and 3.5. Note thatταis forε=−10−3not yetqindependent. According to Ref. [71], it becomesqindependent forε→0, as predicted also by theβ analysis, Chap.4. The unit ofταis[d2/D0].
3.2.3 A small-q problem
Some problems of the actual theory regarding its implementation with small wavevec-tors will be discussed in the following. Investigations of the small-q regime can provide information about the angular dependence of large-distance effects in real space, e.g.
of the pair distribution function g(r) [43]. This is of interest in the context of the
3.2 The transient density correlator (glass phase). The orientations ofqare color coded (see legend), they correspond toqz,qx, and qy directions. Bare Péclet numbers decrease in the left panel from left to right in steps of 10, i.e. Pe0 = 10{−1;...;−6}. The left panel’s legend applies to both panels. The right panel shows modes of theqz andqxdirection as functions of accumulated strainγt, while the˙ qy direction is dropped due to its similarity toqz.
Grey lines are Kohlrausch fits to the yielding master curves; find the according parameters in the legend and Tab.3.3.
shear-transformation-zone theory (STZ) [78], where i.a. the long-range distortion field of so-called Eshelby inclusions [38] is an input quantity and object of studies [79]. Un-fortunately, the actual 3d numerics is not capable of an asymptotic small-q analysis.
Nonetheless, the emergence of hexadecapolar distortions of the structure (l= 4 symme-try in terms of spherical harmonics, Eq. (A.20)) in the plastic regime of deformation can be observed and explained for higher q modes; Sec. 3.4deals with this in detail.
To illustrate the small-q problem, Fig3.6 shows a Φq(t) calculation on a q grid with slightly reduced angular resolution for the sake of faster computation (π/∆ϕq = 8 and 2π/∆ϕq = 16). The correlators of the two smallest wavevector moduli q show strange behaviour, viz. they increase in the qx, qy plane in any direction prior to the α decay for small strains. This deviates from the β analysis of Chap. 4, where a quadrupolar sin(ϕq) cos(ϕq) behaviour is predicted, Eq. (4.7). Quadrupolarity means thatΦq should increase for ϕq = 45◦ and decrease for ϕq = 135◦. Fortunately, all higher q modes do not exhibit this strange behaviour. Forq= 0.2, the correlator Φq(t)becomes even larger than one and is thus unphysical. Figure 3.6 also shows a comparison to a computation, where it is setΦq(t)= 0! forq= 0.2and all times. Thissmall-qreducedgrid has a slightly higher critical packing fraction ofφredc =φc+ 5.26·10−5, with the φc of the π/∆ϕq= 8 grid, see Tab.3.1. Taking the critical packing fraction of both grids (standard and small-q reduced), all correlators, despite those of q = 0.2, are almost identical, as Fig. 3.6 illustrates for the neighboring modes of q = 0.6 and q = 1. This is a relieving result, because the strange behaviour of the small q modes does not even influence the higher q modes quantitatively. The smallest q modes only change the critical packing fraction
3.2 The transient density correlator
slightly. However, the quiescent correlator mode ofq = 0.2 is well behaved, in contrary the Bengtzelius grid, where it decays to a very small plateaufqcat the glass transition, see Fig.3.1. The actual spherical grid is in correspondence with quiescent small-qcalculations of earlier works [80], which yieldedfqc→0 ≃0.4. The smallestq modes can thus be taken into account, as long as no conclusions are drawn from them for the small-q behaviour under shear. panelshows correlator modes according to Fig.3.3. Theq orientations are line-style coded(see legend), whileq moduli arecolor coded (curves that do not appear in the legend correspond in color coding toq moduli ofΦeqq and line style to q angles ofΦ(q,ϕq,ϑq)). The legend applies to both panels. Only equilibrium decays and theqz mode (unadvected) ofΦq(t)are well behaved, while all otherΦq(t)show increasingly strange behaviour with decreasingq.
The right panel shows as double-dashed lines the correlators of a calculation with Φ(0.2,ϕq,ϑq)(t)= 0. When choosing! φc properly, all higher q correlators remain unaffected, the correlators withq ={0.6; 1} are shown exemplary; they perfectly map on the correlators from the left panel (black ongreen,goldenonblue).
The small-q problem can be traced back to the friction kernel in Eq. (2.58). By varying cut-offs (of the grid vectors q,k,p), modulus and angular resolution of q, and interpolation methods of p=q−k, it was possible to exclude that the problem emerges due to the specific coarse discretization or the lower bounds of the q grid used for the computation. Moreover, the unphysical behaviour (Φq(t)>1) reminds of the shoulders that form for increasing Pe0 in Figs. 3.4and3.5and which also occur in 2d calculations [22,23]. It can be identified that the coupling of modes yields a critical feedback for small q. To make this plausible, a closer look at the advected vertex Vqkp(t) of the friction kernel mq(t), Eq. (2.56), is taken, which reads
Vqkp(t) =nq(t)· k(t)ck(t)+p(t)cp(t)
δq,k+p, (3.3)
with advected vectors q(t) = (qx, qy −γtq˙ x, qz) (respectively for k and p). The ck(t) and cp(t) take large negative values for small wavevectors (it must be taken care of the
3.2 The transient density correlator
lower limit of the PY Sq here, because Eq. (A.19) starts to oscillate for small q; this has been accounted for in the computation program by extrapolation of Sq for smallq).
When shear advection becomes relevant, i.e. at the onset of theαdecay, the neighboring (higher) modes of q = 0.2, viz. q = {0.6; 1;. . .}, contribute very large (negative) direct correlation functions ck(t),p(t) to the smaller modes of q = 0.2. This happens because k(t) and p(t) decrease for some angles before they increase again with γt˙ (the squared moduli possess a negative linear term in the strain). The large increase ofVqkp(t)causes an increase of mq(t) and eventually of Φq(t) in the MCT-EQM (2.57) and its iteration discretization Eq. (A.17). Calculations with smaller lowest q moduli of Φq(t) become even more unphysical in these smallestq modes (the increase above one of theseΦq(t)is even bigger; not shown here), which is consistent with the above explanation. This could be an inherent problem in the MCT-ITT EQMs of anα increaseof small q modes under shear. The reason why the qz mode decays normally, cf. Fig. 3.6, while all other modes do not (also not the other neutral modeqy; not shown here), is yet not clear, because via the advected index moduli ofck(t) andcp(t), shear advection enters the kintegration for eachqmode. Maybe wavevector advection must be treated more carefully in the small-q regime, however, a finer discretization of the whole grid (thus also for small q) did not yield an improvement.
These analyses are in an early stage. Further investigations are needed, which means e.g. a small-qevolution of the MCT-ITT equations with an implementation on a small-q grid of high resolution. This evolution could use the actual approach as input for the modes of higher wavevectors. One conclusion must unfortunately be that Φq of q < 1 and for q 6=qez is not trustworthy within the actual computation (the values of q = 1 should be used carefully).
Summing up, small-q modes do not influence the higher ones, which is relieving for this chapter, but they are coupled to the higher modes in an unphysical manner, which causes them to exceed the value of one prior to final decay.
3.2.4 Comparison to simulation and experiment
This section deals with a comparison of the newly achieved 3d transient correlator Φq(t) with both, a simulation and an experiment of a hard-sphere suspensions under shear.
It is necessary to introduce the waiting time twat this point, which denotes some time tw >t0, witht0 the start-up time of shear. A transient correlator after waiting time tw
is defined asΦq(t, tw)≡
ρ∗q(tw)ρq(t,tw)(t+tw)
tw,(t+tw), see Eqs. (2.8) and (2.15), with forward advected wavevectorq(t+tw, tw) =q(t), where the latter holds for constant shear (after start-up) [56]. To calculate Φq(t, tw), a two-times MCT-ITT is needed, which is the subject of Refs. [50,56,59]. With increasing tw,Φq(t, tw) turns more and more into a non-transient, steady-state correlator (eventually fortw → ∞). Then, transient effects, like the stress overshoot and super-diffusive motion vanish.
Figure3.7shows a comparison of the correlatorΦq(t)with the steady-state (tw→ ∞) correlator ΦLJq (t, tw) of a molecular dynamics (MD) simulation of a supercooled binary Lennard-Jones (LJ) mixture below the glass transition at(T−Tc)/Tc ≃ −0.3, which has been published in Ref. [19]. The regarded ΦLJq (t, tw) is an incoherent density correlator
3.2 The transient density correlator
0 0.2 0.4 0.6 0.8 1
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 LJ simulation eq.
γ˙ = 3·10-6,10-5,3·10-5,...,10-1/τLJ MCT eq.
γ˙ = 10-6,...,10-1D0/d2
˙ γt ΦLJ qz,Φqz
Fig. 3.7:Red curves: incoherent steady-state density correlatorsΦLJqz(t, tw)as functions of strain for several shear rates (see legend), with wavevectorq=qez(vorticity direction) andq= 7.1/dAA
(1stSF peak). The quiescent curve (red dashed) is shifted to agree with the one at the highestγ; it˙ shows aging dynamics outside the plotted detail. Coherent transient (tw= 0) 3d-MCT correlators Φqz(t)are plotted asblue dot-dashed curvesfor a comparable range ofγ. MCT curves are taken˙ at critical packing fractionε = 0+ and q = 7 (1st Sq peak); their strain axis is rescaled by a strain parameter ofγres= 0.096. The quiescent correlator (blue dotted) is shifted to agree with the one at the highestγ. Find a similar comparison to the ISHSM in Ref. [53].˙
(cf. Sec.5.1.2; it describes the expectation value of the density modes of a single particle).
Its q is taken at the first SF peak and in vorticity direction (q = qez). A comparison of itsα decay regime to the coherent correlatorΦq(t) is reasonable, because it has been shown in Ref. [23] thatαrelaxation times of coherent and incoherent correlators are very similar at the Sq peak. The simulation’s time scale is in LJ units τLJ, which, because of its Newtonian short-time dynamics, is not comparable to the Brownian short-time dynamics of MCT (see Ref. [19]: τLJ = dAAp
mA/ǫAA, with diameter dAA, mass mA, and interaction strengthǫAAofA-type particles as normalization. It isdBB = 0.88dAA).
However, a comparable range of shear rates is selected. MCT-ITT and the LJ simulation show a good agreement in how their correlator modes approach the yielding master curve as functions of γ. As disagreement, the˙ β decay of MCT-ITT is much too slow (τβ/τq is too large compared to the simulation), which arises due to the different short-time dynamics. In Ref. [59] it has been shown via a two-times schematic model that the α process flattens out withtw → ∞, i.e.Φq(t, tw)decays slower, while τq stays almost the same. This can also be verified very clearly from Fig.3.7.
The qualitative agreement ofΦq(t) with the LJ simulation is high, however, quantita-tively MCT overestimates the α decay strain γq by a factor of about ten. To quantify
3.2 The transient density correlator
this disagreement,γres is introduced as strain rescaling parameter γt˙ →γt/γ˙ res. For the LJ simulation it is γres = 0.096. A similar comparison to the isotropically sheared hard sphere model (ISHSM), which uses a one-dimensional q grid with a heuristic averaging over the anisotropies, can be found in Ref. [53], there with γres = 0.083 (denoted γc
there). The full 3d calculations here yield only a slight increase (i.e. improvement) of γres. dispersion [11] as functions of strain for several effective bare Péclet numbers Peeff (see legend).
Their wavevector hasq= 7.6/d(1st SF peak) and points in vorticity direction (q=qez). The relative packing fraction is εEx ≃ 0.07. For comparison, coherent transient (tw= 0) 3d-MCT correlators Φq(t) are plotted as black lines, see legend for Pe0s. Their parameters are q = 7 (1st Sq peak) and φ= 0.525665(ε≃ 0.0193). The Φq(t)have been linearly interpolated from ε= 2·10−2andε= 1·10−2.
Theleft panelshowsΦqz(t)withq=qez(vorticity direction) and a strain parameterγres= 0.03, while the right panel shows Φqy(t) with q = qey (gradient direction) and a strain parameter γres = 0.037. The quiescent correlator (Pe0 = 0) is shifted to agree with the correlator of Pe0= 10−1. Find a similar comparison to the ISHSM in Ref. [53]
Figure 3.8shows a comparison of Φq(t) with the steady-state incoherent density cor-relator ΦExq (t, tw) of a colloidal poly(methyl methacrylate) (PMMA) dispersion, which was subject to confocal microscopy, at the SF peak. The packing fraction φEx = 0.62 of the experiment corresponds to a glass with about εEx ≃0.07; the data is taken from Ref. [11]. The given effective bare Péclet numbers Peeff = ˙γ d2/Ds of the experiment were estimated with a short-time self-diffusion coefficient Ds ≃ D0/10.1, with D0 the dilute diffusion coefficient, cf. Refs. [53] and [81]. As for the LJ simulation above, the argument for comparingΦq(t)andΦExq (t, tw)at the SF peak is based on Ref. [23]. In the experiment,qis oriented along the vorticity directionqz, but is compared to both neutral direction modes ofΦq(t),qz axis andqy axis. The difference is small, the vorticity direc-tion yields γres = 0.03 and the gradient direction γres = 0.037. Besides this quite large deviation in the strain scale of a factor of about 30, the qualitative agreement is high.
A similar comparison to the ISHSM can be found in Ref. [53], there with γres = 0.033.