3 Microscopic three-dimensional mode coupling theory
3.4 Structure-factor distortion
3.4 Structure-factor distortion
The mode-coupling equations can be used to study three-dimensional, transient struc-tural distortions δSq(t) = Sq(t,γ)˙ −Sq, Eq. (2.46), of a MCT glass under shear. All presented MCT results of this section have been calculated forε= 0+ (glass phase) and Pe0= 10−6. One important conclusion from the following is that the qualitative change ofδSq, which occurs correlated to the transient stress regime, happens most pronounced in the shear flow, shear gradient (qx, qy) plane. It is a change in the symmetries from purely quadrupolar to partially hexadecapolar, which can be showed to arise naturally in the flow field of simple shear.
Before proceeding, it is recommended to have a look again on Fig. 3.2, where the parametrization of theqvector is shown, together with a sketch of the simple-shear flow-velocity field. The flow field can be decomposed into a deformational and a rotational part, which corresponds to symmetric and antisymmetric decomposition of the strain-rate tensorκ, Eq. (3.2). Especially the symmetry of the deformational field is important for the mechanism of structure distortion in the qx, qy plane, viz. elongation along the x = y diagonal and compression along the x = −y diagonal. In Fourier space, length scales are inverted, i.e. compression and elongation switch. Recall, the qx axis points into the flow direction, theqy axis into the shear gradient direction, and the qz axis into the vorticity direction.
0 0.5 1 1.5 2 2.5 3 3.5 4
0 2 4 6 8 10 12
q= 7
5 5.8
12.6
3.8
8.2
9.4 9.8
q d−1 Sq
PY forε= 0+, cf. Tab.3.1
Fig. 3.14: Isotropic PYSq discretized as used for all 3d microscopic MCT calculations of this chapter. The critical packing fraction used for this plot isφc = 0.515712(1), cf. Tab. 3.1 and Fig.2.2. Arrows mark importantqvalues of this chapter and label theircolor coding. The firstSq
peak is atq= 7, the second one atq= 13, butq= 12.6(the secondfcpeak, Fig.3.1) is used in-stead for convergence with the rest of this chapter. The firstSqminimum corresponds toq= 9.8.
3.4 Structure-factor distortion
Figure 3.14 shows the isotropic PY structure factor (SF) Sq at the critical packing fraction of the MCT glass transition. Some wavevectors are marked with colored arrows;
this color coding is used in the following for distinguished wavevectors.
The distortion δSq is inversion invariant under ϑq → π−ϑq as well as under ϕq → π+ϕq, as it inherits the symmetry of Φq(t) in Eq. (2.46). This means also that δSq
can possess only such spherical symmetries, in terms of real spherical harmonics Yl,m, Eq. (A.20), that are even inlandm. The l= 0 term is isotropic and denotedmonopolar moment, l = 2 terms are denoted quadrupolar moments and l = 4 terms are denoted hexadecapolar moments. Higher order symmetries are not considered in the following.
3.4.1 The flow, gradient plane
Figure3.15shows SF distortionsδSq in theqx, qy plane for different strain values, which represent the elastic, transient, and steady state regimes. The quadrupolar structure (Y2,−2, i.e. ∝sinϕqcosϕq, Eq. (A.20)) of the elastic regime can be verified. It changes subsequently to a structure with four negative poles on a circle of fixed modulus q, when passing through the transient into the steady-state regime; find a more detailed discussion on this together with Fig. 3.18. A change of the overall color mood can be observed between the elastic regime and the steady state. While red and blue color are balanced in the elastic regime, blue dominates in the steady state. Because red labels positive δSq and blue negative δSq. The change of the color mood implies that on average an isotropic decrease of Sq is created by shear; Fig. 3.16 also gives a more quantitative impression in showing δSq+Sq as function of modulus q around the first peak region. Note that another purely isotropic contribution to the SF distortion, cf.
Eq. (2.44), has been neglected in Eq. (2.46) for reasons (of being unphysical in the large q limit, i.e. Sq→∞( ˙γt → ∞) 9 1), which were discussed in Ref. [53]. A 2d calculation and evaluation of this isotropic term can be found in Refs. [55, 64]. Another, but most important observation is that the SF gets compressed on theqx=qy axis and stretched on theqx =−qy axis, see also Fig. 3.16, which is consistent with the 2d calculations of Refs. [22, 64]. This corresponds precisely to the real-space deformational flow field of simple shear, Fig. (3.2), but in a Fourier space, where length scales are inverted. At first sight, it seems that this deformation increases monotonically when shearing through the transient regime. Hence, the panel C of Fig. 3.15, which illustrates the peak strain γ∗, does not seem to be special in any way. However, as is discussed below and in Sec. 3.5, with accumulating this distortion, hexadecapolar symmetries are accumulated and the quadrupolar symmetry is decreased. The latter yields contributions to the stress, the former do not. This way, a stress overshoot emerges within the homogeneous MCT for Wi&1, which is one of the important findings of this work.
3.4 Structure-factor distortion (ϑq= 90◦, shear-flow, -gradient plane) for different strains as labeled in thetop left panel. They are taken at valuesγ= ˙γt={0.001; 0.155; 0.318; 0.481; 3.44}of the generalized modulusgxy(t,γ)˙ atε= 0+ and Pe0= 10−6, Fig.3.10. The chosen strains correspond to elastic – A, non-elastic andγ < γ∗ – B,γ=γ∗– C,g( ˙γ, t)minimum –D, and steady-state regime – E.
3.4 Structure-factor distortion
Figure3.16shows the distorted SFSq(t) =δSq(t) +Sq in the steady state as function of modulus q in the qx, qy plane for the qx axis and the qx = ±qy diagonals. It can be verified that the region of the first peak is deformed towards smaller q for qx =qy and to larger q for qx = −qy, as (inversely) according to simple-shear flow. For qx = −qy
the jump of the first SF peak to the next grid point of q can even be resolved on the discretization ∆q = 0.4 of the modulus. Also for the qx axis, a shift to smaller q can be guessed. In Ref. [15], the latter has been measured in the qx, qz plane. With the actual discretization of∆q= 0.4, it is hardly possible to resolve exactly a shift in the first peak on the qx axis. Fitting some Gaussian to the peak and deducing the shift of its center (as done in Ref. [15]) is not convenient here. Depending on theq range, which is chosen to be fitted by the Gaussian, rather arbitrary results can be produced, because the peak region exhibits a big change in hight and relative width. Section3.4.3deals more detailed with Ref. [15] and presents a comparison to MCT’s qx, qz plane.
For all orientations of q, the peak region exhibits a pronounced decrease on average (over q). Thus the hard-sphere dispersion can clearly be regarded as fluid when being shear molten.
0 0.5 1 1.5 2 2.5 3 3.5
2 4 6 8 10 12 14
E
q d−1 S(q,ϕq,ϑq)
PYSq forε= 0+ S(q,0◦,90◦) S(q,45◦,90◦) S(q,−45◦,90◦)
Fig. 3.16: Distorted structure factor S(q,ϕq,ϑq)(t) = δSq+Sq in the qx, qy plane according to Fig.3.15E(steady state) forqy= 0 (ϕq = 0◦),qx=qy (ϕq = 45◦), and qx=−qy (ϕq =−45◦).
The quiescent PYSq from Fig. 3.14is plotted as reference; see the legend for thecolor coding.
A shift of the first peak to smallerq can be guessed forϕq = 0◦ and ϕq = 45◦ modes, but is evident forϕq =−45◦. The peak hight clearly decreases for allq.
The consequences of the elliptical distortion of δSq with increasing strain, which is observed in Fig. 3.15, can be studied better when using a logarithmic scale for δSq. Thus, Fig.3.17 shows the elastic and steady state regimes of Fig. 3.15again, but using
3.4 Structure-factor distortion
a logarithmic color scale for δSq. It is easier to see that the deformation of δSq along the deformation field of simple shear happens on all length scales. The δSq is at first quadrupolar and of symmetry qxqy ∝sinϕqcosϕq in the elastic state, then it develops a symmetry of qx2qy2 ∝ sin2ϕqcos2ϕq in the steady state, which is even better seen in Fig. 3.18. This can also be justified from a small strain evolution of δSq in Eq. (3.10).
The reason is that Sq and its derivative Sq′ as factor in δSq, Eq. (2.46), are oscillatory functions in q. If an oscillating quadrupole gets deformed elliptically, hexadecapolar symmetries emerge. This can be verified from considering circles of constant q radii in Fig.3.17. They change their colors subsequently from red color to blue color in the elastic state (quadrupole). In the steady state they exhibit a sequence of just red color or just blue color for properly chosenq radii. Becauseq2xq2y ∝sin2ϕqcos2ϕq = 18(1−cos(4ϕq)) is a decomposition into a constant plus a cos(4ϕq), a hexadecapolar symmetry emerges (viz. the cos term), which corresponds to Y4,4, Eq. (A.20). It does not contribute to the shear stress, only the quadrupolar qxqy symmetry does, as is explained explicitly in Sec.3.5Eq. (3.9). This symmetry change starts at the white circles in Fig.3.17 A, which are the zeros of δSq, which stem from the zeros of Sq′. The latter are the turning points of Sq included the first Sq peak. The region around this first peak yields the largest contributions toσxy in Eq. (2.48), because Sq′ takes the largest values there.
10 20
Fig. 3.17: Logarithmic illustration of panels A and E of Fig. 3.15, viz. the structure-factor distortion δSq in the qx, qy plane in the elastic regime (left panel – A) and steady state (right panel–E). In theleft panel, a collage of the fullqrange together with a detail ofq620is shown.
A cut-off value of10−9for the absolute value|δSq|is used. The cut-off in theright panelis10−8. Figure 3.18 shows the SF distortion δSq in the qx, qy plane as function of azimuthal angle ϕq for several fixed q moduli. One verifies clearly the sinϕqcosϕq symmetry for all q in the elastic regime and the sin2ϕqcos2ϕq symmetry for some q in the steady state regime, especially for the important SF peak value of q = 7 (on the chosen q grid). Some q values keep their qxqy symmetry, they are in the middle of the colored bands in Fig. 3.17. That means that δSq has a relative maximum there as function of q, while at points where δSq oscillates through zero (color switching, white circles), the hexadecapolar symmetry gets strong. It can also be seen that it is δSq 6= 0on the
3.4 Structure-factor distortion
qx axis (see also Fig. 3.16). This could be connected to the rotational flow of simple shear, Fig.3.2. However, contradicting this last assumption, it always holdsδSq = 0 for theqy axis (qx= 0), as is given by the theory and can be verified from Eq. (2.46).
1 -10-2 -10-4 0 10-4 10-2 1
0 0.5π π 1.5π 2π
elastic q = 3.8 7.0 8.2
12.69.8 steady q = 7.0 A
ϕq
δS(q,ϕq,90◦)
-1 -10-1 -10-2 -10-3 0 10-3 10-2 10-1 1
0 0.5π π 1.5π 2π
E
ϕq
δS(q,ϕq,90◦)
Fig. 3.18: The symbols (connected with straight lines) give δSq from Fig. 3.17 A and E for various constant radiiq. The legend gives asline styles the strain regime (elastic –A– dashed, or steady state –E –solid) and encodes withcolors theq moduli; it applies to both panels. In panelA, thered symbolsare fitted with∝cos(ϕq) sin(ϕq), which corresponds toY(2,−2). Note the deviation atϕq ={0;π}, i.e. q=±qex. Panel Eshows aq2xq2y symmetry forq={7; 9.8; 12.6}, while intermediateδSq modes remainqxqy like. The cut-offs for the logarithmic illustrations are 10−6 in theleftand10−4in the right panel.
3.4.2 Considering three dimensions; the vorticity direction
The actual 3d calculations allow for the first time to investigate SF distortions δSq in planes with the vorticity direction (qzaxis) as spanning vector, denotedϕplanes, because ϕq is constant. Figure 3.19 shows contour plots for various strains in the different flow regimes of Fig.3.15, but for theqx, qzplane (ϕq= 0◦), theqx =qy plane (ϕq= 45◦), and the qx =−qy plane (ϕq =−45◦). One verifies again the symmetry of aY2,−2 spherical harmonic in the elastic regime, which has an inclination term ∝ sin2(ϑq), Eq. (A.21), while its azimuthal term describes the qx, qy plane. This form can be deduced also from Eq. (4.7) of the new, anisotropicβ analysis of Chap.4, which holds in the elastic regime.
An isotropic decrease, when approaching the steady state, can be verified like for theqx, qy plane, because the color mood changes to blue, which means negativeδSq dominate on average. The values of δSq in the ϕq= 45◦ andϕq =−45◦ planes are much larger than in the qx, qz plane. According to the Y2,−2 symmetry, there should be no distortion in the qx, qz plane, like there is no distortion in the qy, qz plane, cf. Eq. (2.46). Hence, the azimuthal contribution on the qx axis is plastic and consequently becomes relevant only for the transient and steady-state regimes. However, also in the qx, qz plane, there arises an inclination symmetry ofsin2(ϑq) (then∝Y2,2). The overallsin2(ϑq)symmetry does not seem to change a lot till the steady state is reached, which is different to the
3.4 Structure-factor distortion
ellipsoidal change of symmetry in the qx, qy plane. This implies that the flow field of
ϕq= 0°
Fig. 3.19: Contour plots of the asymmetric SF distortionδSqfor strains corresponding to panels A,C,Eof Fig.3.15and with the same parameters. Left panels: qx, qzplane (ϕq = 0◦, shear-flow, vorticity plane), centric panels: qx =qy plane (ϕq = 45◦, denoted ’NO’ in Ref. [55]), and right panels: qx=−qy plane (ϕq =−45◦, denoted ’SO’ in Ref. [55]).
3.4 Structure-factor distortion
simple shear acts mainly in theqx, qy plane, which does not contradict physical intuition.
Find a closer discussion about the symmetry change in theϕplanes around the first SF peak together with Fig.3.21 below.
The largest distortions δSq in the steady state of Fig.3.19E happen around the first SF peak and also the peak is reduced strongly. From the color distribution aroundq = 7 it can be guessed what is shown in Fig. 3.16 already, viz. a shift of the peak to the inside on the ϕq = 45◦ and qx axes and to the outside on the ϕq =−45◦ axis; see also Fig.3.16. To identify higher symmetries of spherical harmonics thanl= 2, a closer look on the change of otherqmoduli than just the peak region is necessary, i.e. the logarithmic scaling turns out to be useful again in Fig.3.21.
Figure 3.20 shows δSq for fixed q moduli as function of inclination angle ϑq in the ϕq = ±45◦ planes in the elastic regime. The sin2(ϑq) symmetry is verified on a linear scale and one can also get familiar with the use of the logarithmic scaling and the shape of alog10sin2(ϑq).
-1·10-3 0·100 1·10-3 2·10-3 3·10-3 4·10-3 5·10-3
0 π/4 π/2
3.8 7.0 elastic q = 8.2 9.8 12.6
A ϕq= 45°
ϑq
∝sin2(ϑq)
δS(q,45◦,ϑq)
-10-2 -10-4 0 10-4 10-2
0 π/4 π/2
A ϕq= -45°
ϑq
δS(q,−45◦,ϑq)
Fig. 3.20: Symbols: SF distortionδSq from Fig.3.19for several constant moduli q(legend) in the elastic regime –A. Theleft panelshows a linear plot of theqx=qy plane (Fig.3.19centric), while theright panelshows a logarithmic plot of the qx=−qy plane (Fig. 3.19right hand side) with a cut-off of ±10−6. All curves are fitted with a function δS(q,ϕq=±45◦,ϑq=90◦)·sin2(ϑq) (dashed lines); the agreement between fit and symbols is perfect. The legend applies to both panels.
Figure3.21showsδSq for fixedqas function of angleϑq in theϕq=±45◦ plane in the steady state. The attempt is to verify thatl= 4 symmetries also emerge in theϕplanes when approaching the steady state. They can be identified, but not reduced to at least two real spherical harmonics. A fitting with more Yl,m would of course be successful, because spherical harmonics are a complete set, but this would not yield more insights at this point. Three real spherical harmonics with fitting parameters A, B, C shall be
3.4 Structure-factor distortion
shown explicitly for further use below, they read Y4,4=Asin4(ϑq) cos(4ϕq), Y4,−2=B 7 cos2(ϑq)−1
sin2(ϑq)2 cos(ϕq) sin(ϕq), Y2,−2=Csin2(ϑq)2 cos(ϕq) sin(ϕq).
(3.7) According fit parameters A, B, C to Fig. 3.21 are given in the legend of this figure.
Figure 3.21 yields that for some q moduli the distortions δSq remain Y2,−2 like in the steady state (like in the qx, qy plane). Other moduli, especially the quiescent SF peak modulusq = 7, can only be described qualitatively when using at least the hexadecapolar functions Y4,4 and Y4,−2. The former, to have the same sign of δSq on the qx = ±qy
diagonals and the latter, to create a bump for inclination angles ϑq <90◦ in one plane qx =±qy, but not in the other one qx =∓qy. The finding of this fitting is that there is somel= 4symmetry emerging in the ϕplanes, however, it is yet not elaborated, if their consequences on the shear stress are as pronounced as pointed out above for the qx, qy plane. Because all distortions δSq scale as some power of asin(ϑq)function from theqz axis to theqx, qy plane, the contributions of the distortions from the vicinity of theqx, qy
plane are dominant. In conclusion, it seems that the contribution ofδSq as function of ϑq is either proportional to the δSq value in theqx, qy plane, or small.
-10-1 -10-3 0 10-3 10-1
0 π/4 π/2
3.8 7.0 steady q= 8.2 9.8 12.6
E ϕq= 45°
ϑq δS(q,45◦,ϑq)
-10-1 -10-3 0 10-3 10-1
0 π/4 π/2
A= 1.6 B= 0.01 3.9·10-2 1.8·10-3 -7.0·10-2 2.8·10-3 C= -0.10
2.1·10-3
E ϕq= -45°
ϑq
δS(q,−45◦,ϑq)
Fig. 3.21: Symbols: SF distortionδSq from Fig.3.19for several constant moduliq (legend) in the steady state –E. Both panels show logarithmic plots with cut-offs of±10−5. Theleft panel shows theqx=qyplane and theright paneltheqx=−qyplane. Thedashed curvesare a fit with Y(2,−2), Eq. (3.7), while thedot-dashed curvesare fits with a superposition ofY(4,4)andY(4,−2); find in the right panel’s legendA,B, andC. Both legends apply to both panels.
The qx, qz plane is also worth a closer look, which is combined with a comparison to experimental data in the following Sec.3.4.3.
3.4.3 Comparison with experiment
Recent experiments have made it possible to measure at the same time and in the same sample rheological quantities, viz. stress-strain curves, and SF distortions in the flow,
3.4 Structure-factor distortion
vorticity plane (qx, qz) via X-ray scattering [15]. Dmitry Denisov and his co-authors from Ref. [15] kindly provided unpublished measurement data from the context of Ref. [15]
to compare to the actual 3d MCT SF distortion data. The experiments used silica particles with a hard-core diameter of 50nm and a Debye screening length of 2.7nm for electrostatic interactions, which is used to give the particles diameter as d ≃ 55.4nm.
The packing fraction wasφ≃0.58, i.e. at the hard-sphere glass transition pointφg. The used rheometer however was stress controlled and therefore only the steady state, where stress and shear rate do not change anymore, is yet comparable to the data of this work.
The steady state yields the flow curve, see Sec. 3.3.1. In Ref. [15], a highlight was set on non-homogeneous flow, viz. shear banding, which does not allow for a comparison to the flow curves of this work. However, one of the measured shear rates ofγ˙ = 3·10−4/s is assumed to produce homogeneous shear flow, see Fig. 3 of Ref. [15], and can be used here. The corresponding Pe0 is much smaller than one, which allows for a comparison with the MCT data at Pe0 = 10−6 and ε= 0+ of this section, which has a steady-state stress of 2.05kBT /d3, Tab. 3.4. A steady-state stress of σSixy ≃ 2.07kBT /d3 has been reported in Ref. [15] (σxySi ≃ 49.3Pa, normalized with room temperature of ≃ 20◦C), which perfectly suits the value of MCT here. Because the structural distortionδSq is the main contribution to thek-space integral of Eq. (2.48), which yields the stress, one might assume that MCT’sδSqis of the same size in theory and experiment, which unfortunately turns out to be not true. The theory overestimatesδSq by about a factor of 3. Note that 3d MCT overestimates the peak strain γ∗ compared to colloidal experiments also by a factor of 3–5. This can be verified from comparison of Tab.3.4with Tab.5.2. Thus MCT has more time to accumulate stress and SF distortion and the assumption is that MCT’s SF distortion is therefore overestimated. That the flow curve values still coincide might have various reasons, the stress controlled driving or a less pronounced stress-overshoot decay on the flow curve in the experiment. It can hardly be examined at this point, as long as the whole transient stress-strain curve is not available (in shear-rate controlled flow). After all, the order of magnitude fits already very well.
Figure 3.22 shows a qualitative comparison of the qx, qz plane of δSq with the SF distortion of the silica particles δSSiq. The former is normalized to the q grid’s SF peak S7 = 3.48, the latter to the average peak of a sufficiently relaxed state state Sre6.09 (relax-ation time = 10min). Its unit is arbitrary and its wavevector is given in silica-particle diameters. The peak wavevector value of 6.09 is interpolated. The experimental δSSiq plane seems to be rotated a little, which is yet not determined to be a physical effect or maybe due to the measurement (here the latter is assumed and accounted for in Fig.3.23 with an offset). Green rings show the position of the quiescent (relaxed state) SF peaks.
The qualitative agreement is very high. Both, experiment and 3d MCT seem to shift the SF peak to the inside, cf. Fig. 3.16, but MCT decreases the peak value of Sq(t) stronger; cf. discussion on Fig. 3.23. The outer region of the experimental plot is noisy, the MCT data takes very small values there. However, the negative portion ofδSq seems to dominate in MCT, while for the experiment the color mood of red and blue is more balanced. This and the finding of a positive value of δSSiq on the qz axis in Fig. 3.23, are indicators that the isotropic contribution to the SF distortion, cf. Eq. (2.44) and Ref. [53], must be reconsidered in future comparisons.
3.4 Structure-factor distortion
The right panel shows δSSiq in the qx, qz plane, also in the steady state. Both distortions are normalized to their quiescent/relaxed SF peak values. The green circles show the position of the quiescent/relaxed SF peaks. A contraction of the SF along theqx axis can be identified.
The right panel shows δSSiq in the qx, qz plane, also in the steady state. Both distortions are normalized to their quiescent/relaxed SF peak values. The green circles show the position of the quiescent/relaxed SF peaks. A contraction of the SF along theqx axis can be identified.