5 Schematic mode coupling theory
5.5 Mean square displacement and rheology in schematic MCT and MD simulations
The universality of the schematic model is paid-for with a certain arbitrariness due to nine fitting parameters (ε, v∗σ, Γ, η∞, γc, γ∗, γ∗∗, α, D0, while a and b as vertex parameters are not even counted). This might bring to mind the phrase of John von Neumann “. . . with four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” [100]. However, all the parameters represent distinct features of microscopic MCT and using it helps also to identify differences between microscopic MCT and experiments/simulations, e.g. that MCT overestimatesγ∗,γ∗∗, andγc and that the vertex decay dominates the correlator decay in gxy(t,γ), Eq. (3.5). To validate the˙ F12( ˙γ)parameters in context of fitting MSDs, it is however desirable to have flow curves, stress-strain data, and MSDs of the same experimental setup and for identical shear rates. The molecular dynamics (MD) simulations of Jürgen Zausch [12,88] provide this.
More MSD fits of the author with the F12( ˙γ)model on BD simulations and experiments on PMMA colloids can be studied in Ref. [13]. They are not as convenient in the above sense, because MSD and rheological data were not available for the same systems, thus they are not repeated here.
Simulation details
The MD simulation describes the ballistic motion of a binary mixture of (almost) hard spheres. The interaction of the particles is modeled by a truncated Yukawa potential. A thermostat with a small damping parameter serves to equilibrate the system [101]. As normalization serve diameterdAA= 1, interaction energy scale ǫAA = 1 =kB, and mass mA = 1of the A-type particles, in the following denoted as d, ǫMD, and m. It follows a time-scale τMD=dp
m/ǫMD. The more interested reader is referred to Refs. [12,88].
5.5.1 Ballistic schematic model
The MD simulation exhibits an almost purely ballistic short-time dynamics, with a very weak friction term as part of a thermostat [12], but the actual microscopic and schematic MCT describe an overdamped Brownian short-time dynamics. Fortunately, the F12( ˙γ)model has the flexibility that Brownian and ballistic short-time dynamics can be switched easily with hardly no effect on the intermediate and long time behaviour, as this section demonstrates by comparison. The microscopic theory does not possess this flexibility, because the implementation of shear flow would need an completely dif-ferent ansatz for Newtonian dynamics; recall, MCT starts from an overdamped Smolu-chowski Eq. (2.1).
To account for the ballistic short-time motion, each first term in Eqs. (5.1) and (5.17) are replaced by their first order time derivatives, which is in full accordance with New-tonian quiescent MCT [51, 102]. A ballistic MSD equation can easily be derived from Ref. [94] in the manner as Eq. (5.17). The ballistic schematic EQM and MSD equation
5.5 Mean square displacement and rheology in schematic MCT and MD simulations
(for shear gradient and vorticity directionsry andrz) read 1
Ω20∂t2Φ(t) + Φ(t) + Z t
0
dt′m(t−t′)∂t′Φ(t′) = 0, with (5.23) m(t) = v1Φ(t) +v2Φ2(t)
1 + ( ˙γt/γc)2 and
∂t r2α(t)
+D0
Z t
0
m(0)(t−t′) r2α(t′)
= 2D0t. (5.24)
The newly introduced F12( ˙γ)parameter Ω0 > 0 replaces Γ as short time-scale. In mi-croscopic MCT, Ω20 is proportional to the inverse particle mass. It would be possible to conserve the first order derivative of Φ(t) with time-scale Γ in Eq. (5.23), but for the prize of raising the number of model parameters even more. The first order term would then represent a kind of effective diffusion coefficient due to particle collisions on a time-scale larger than the ballistic motion, but smaller than the structural arrest.
The MD simulation contains such a friction term to equilibrate the system via a thermo-stat [12,88]. However, the influence is small and neglected in the actual considerations to not to overparameterize the model. The notionballisticrefers in the following to the use of Eqs. (5.23) and (5.24), whileBrownianrefers to Eqs. (5.1) and (5.17) in the calculation of theF12( ˙γ)fits. Deviating from the Brownian EQM (5.1), the ballistic EQM (5.23) is ini-tialized with Φ(t) = cos(Ω0t), cf. Sec. 5.1.1. The flow curve is shifted horizontally with Ω20, as can be proved by rescaling t→ Ω0tand γ˙ →γ/(Ω˙ 0γc) (cf. Ref. [96], F12( ˙γ)master scaling).
5.5.2 Comparison of schematic MCT and MD simulation Fitting procedure
The fitting procedure of the F12( ˙γ)model to the MD data worked as follows. Figure 5.15 shows flow curves and Fig.5.16stress vs strain curves of the MD simulation, fitted with the F12( ˙γ)model. As data of linear-response shear moduli is lacking, the linear increase of stress prior to the overshoot in Fig. 5.16 is used to fit v∗σ. The topmost flow curve from Fig.5.15is taken into consideration andΓorΩ20,ε, and a preliminaryγc are fitted.
This flow curve accords to T = 0.14ǫMD. Because stress vs strain curves for the other two flow curves (T = 0.15ǫMD and T = 0.21ǫMD) is lacking, fitting these curves is an overparameterized problem. Therefore fits were made with keeping all parameters of T = 0.14ǫMD fixed, despiteε. Thevσ∗ was kept constant in units ofǫMD, because keeping it constant in units of kBT /d3 did not yield satisfactory results. This seems to be a sensible outcome as viscosities and elastic moduli typically also decrease with increasing temperature. In consequence, the decrease is inverse to temperature here. As there is no high frequency loss modulusG′′ or high shear rate flow curve available, which would determine η∞, this parameter is set to zero. Also, the solvent viscosity in a Newtonian hard-sphere gas does not exist. The γ∗ is taken as stress-peak position and finally γc is fitted for all curves, andγ∗∗for each curve, like for the PNiPAM experiments in Sec. 5.3.
5.5 Mean square displacement and rheology in schematic MCT and MD simulations
Illustration of the comparison
Figure 5.15 shows flow curves of the MD simulation together with F12( ˙γ)fits (ballistic and Brownian). Stress vs strain data was available only for the lowest temperature of T = 0.14ǫMD, which is assumed to be at (slightly above) the glass transition [88]. This cannot clearly be verified from the actual data, moreover, the MSD data in Fig. 5.17 suggests a structural decay with time, possibly due to hopping. The other two temper-atures (T = 0.15ǫMD and T = 0.21ǫMD) clearly possess a structural α decay and are in the fluid phase.
0.01 0.1 1
10-6 10-5 10-4 10-3 10-2 10-1
= 0.14 0.15 0.21
˙ γ
τ−1 σxy kBT/d3
kBT /ǫMD
Ballistic F12( ˙γ) Brownian
Fig. 5.15: Flow curves of the MD simulation (symbols) for temperatures as shown in the legend.
The stress-strain curves of Fig.5.16correspond toT = 0.14ǫMD(glassy state). BrownianF12( ˙γ)fits are shown as solid lines, ballistic F12( ˙γ)fits as dashed lines. The F12( ˙γ)parameters are given in Tab.5.7.
Figure 5.16 shows stress-strain curves corresponding to the T = 0.14ǫMD curve of Fig.5.15. A fit with both, ballistic and Brownian MCT works very well. The outcome of Sec.5.3.3 is strengthened by the insets of Fig. 5.16, viz. thatγ∗ can be coupled linearly to γ∗∗. The γ∗s drift with increasing shear rates to larger values. This validates again the analyses of Fig. 3.12 and 5.5 that γ∗ is qualitatively behaving like the flow curve, viz. it increases with shear rate and here decreasing temperature (which is equivalent to increasing packing fraction). Other decay strain scales, e.g. γ∗∗ are linearly coupled to this behaviour.
Finally, msq→0(t) =! αk3πd
BTg(t,γ), with constant˙ α, is used to calculate the incoherent memory kernel and fit MSDs. With the simulation data of stress vs strain curves and MSDs, this ansatz can now be tested. Figure5.17 shows F12( ˙γ)fits to the MSDs in
gradi-5.5 Mean square displacement and rheology in schematic MCT and MD simulations
Fig. 5.16: Stress-strain curves of the MD simulation (symbols) corresponding to shear rates as given in the legend. The corresponding temperature isT = 0.14ǫMD, cf. Fig.5.15. Solid lines show fits with theF12( ˙γ)model, Brownian in the left panel, ballistic in theright panel. Theinsets show the correspondingγ∗ and γ∗∗ valuescolor coded. They lie perfectly on a linear curve. In Tab.5.7theF12( ˙γ)parameters and theγ∗-γ∗∗ relation can be found.
ent directionry, which have been calculated in the MD simulation. The proportionality parameterαis used to fit the long-time diffusivities of the simulation data. All long-time diffusivities under shear fit quantitatively well, which can be taken as a validation of the GSE for long times and in the regarded regime. The equilibrium MSD is matched
10-3
Fig. 5.17: Mean square displacements in gradient direction ofB-type particles (dBB= 1.2d) of the MD simulation atT = 0.14ǫMD(symbols), with corresponding shear rates given in the legend.
Black crossesdenote the equilibrium curves (γ˙ = 0). F12( ˙γ)fits are shown assolid lines. Theleft panelshows BrownianF12( ˙γ)fits, theright panelballisticF12( ˙γ)fits. Thegrey, dash-dotted linesshow a linear, diffusive long-time behavior
r2y
= 2Dt. Find theF12( ˙γ)parameters in Tab. 5.7.
5.5 Mean square displacement and rheology in schematic MCT and MD simulations
poorly by the theory. Obviously there is some equilibrating process active, which might be structural fluid decay or hopping. The latter is suggested, as a fit with ε <0 yielded very bad results for the rheological Figs. 5.15 and 5.16 (not shown here). Around the peak strain γ∗, superdiffusive motion can be verified in theory and simulation in quali-tative agreement, however, it is much more pronounced in the simulation. This is also a finding of Ref. [13]. Additionally, the Brownian MCT exhibits a more pronounced superdiffusive motion than the ballistic calculation. The intermediate plateau, which could be interpreted as ‘cage size’, is overestimated quantitatively by MCT. It cannot be matched exactly, whenαis used to fit the long-time diffusivities; the qualitative behavior is catched however. Without ballistic term in Eq. (5.24), the short-time MSD is linear in time in Brownian MCT and does not fit the quadratic dependence of ballistic motion.
The fit turns out to be well with ballistic MCT, the quantitative value is of course just fitted with another parameter D0 and should not be overinterpreted.
BallisticF12( ˙γ) vσ∗ = 150kBT /d3 Ω0= 18.0/τMD ε= 5·10−4 γc = 0.576 D0= 0.406d2/τMD α= 0.178 τMD=p
md2/ǫMD
˙ γ
τMD−1
6·10−5 3·10−4 6·10−4 3·10−3
γ∗ 0.078 0.078 0.088 0.112
γ∗∗ rel. γ∗∗= 1.58γ∗−1.30·10−2
BrownianF12( ˙γ) vσ∗ = 150kBT /d3 Γ = 40/τMD ε= 1.5·10−3 γc= 0.34 D0= 0.1d2/τMD α= 0.178
˙ γ
τMD−1
6·10−5 3·10−4 6·10−4 3·10−3
γ∗ 0.078 0.078 0.088 0.112
γ∗∗ rel. γ∗∗= 1.83γ∗−2.48·10−2
Tab. 5.7:Parameters of the schematic model in the fits to the MD simulations, shown in Figs. 5.15–5.17. For all fits η∞ = 0. Note, the scaling of the parameters is different from Ref. [13], where a comparison among many different experimental and simulated systems via Wi s was highlighted.
5.5.3 Comparison of MD simulation and colloidal experiment
The reader shall be referred to other literature for a detailed discussion of the actual MD simulation with colloidal experiments and microscopic MCT. In Ref. [12], an extensive study of this MD simulation together with PMMA colloidal experiments and ISHSM MCT can be found. Reference [13] shows this MD simulation with parts of the above schematic fits together with fits to PMMA experiments and BD simulations. A highlight was set on the comparison via Weissenberg (dressed Péclet) numbers (performed by Marco Laurati). Event driven BD simulations in comparison with the PNiPAM discussion of Sec. 5.3within the actual schematic framework can be found in Ref. [42].
Even tough the actual MD simulation under shear is of ballistic short-time dynamics, it is not surprising that it coincides well with shear experiments of colloidal dispersions,
5.6 Summary
because it is designed to do so (of course with all the advantages, which a simulation possesses, e.g. being in control of all parameters). The actual F12( ˙γ)model proofs this co-incidence. A comparison of Tab.5.2with Tab.5.7shows the similarities. The parameter of vσ∗ is larger by a factor of 2 compared to the PNiPAM experiments, but within the same order of magnitude. The stress-peak strains γ∗ are within the same range and as well smaller than microscopic MCT by a factor of 3–5, cf. Tabs. 3.4and 5.1. Especially with the ballistic short time motion, the values of Γ or Ω0 are not sensibly comparable.
However, when comparing the Weissenberg numbers of PNiPAM, Tab. 5.6, with those of the MD simulation from Ref. [13] (5.86Wi6288), one verifies that the shear rates and long-time scales lie within the same range, which eases a comparison compared to the rather small Weissenberg numbers that were estimated for Vitreloy 1 in Sec.5.4.3.
In conclusion, the PNiPAM experiments and MD simulation of Secs. 5.3 and 5.5 do not contradict each others with inconsistent F12( ˙γ)parameters. Thus they validate the use of the actual schematic model with the newly elaborated, time-dependent vertexvσ(t,γ˙) for colloidal dispersions under shear (and according simulations).
5.6 Summary
A schematic model of mode coupling theory, viz. the F12( ˙γ)model, was generalized to de-scribe consistently flow curves, linear response shear moduli and nonlinear stress-strain curves, especially the stress overshoot. The generalization holds in a universal, homoge-neous straining geometry, with a focus on simple-shear and compressional flow. This was done by implementing a time-dependent vertex functionvσ(I(t))in the generalized shear modulus g(t,[B]), which automatically incorporates via the Finger tensor B the strain geometry. Schematic MCT remains isotropic, but dependent on the rotational invariants ofB via a scalar functionI(t), on which the isotropic vertex also solely depends.
The new vertex is in accordance with microscopic MCT as long as bare Péclet numbers are asymptotically small and the correlator decay is governed by strain. For higher Péclet numbers, model parameters must be varied to mimic a influence ofαandβ decay on the new vertex vσ(I(t)). Effectively just one new rheological parameter, the peak strain γ∗, at which the stress is maximal, is added to the model, because a second parameter, γ∗∗, turned out to be linearly coupled to γ∗. The γ∗∗ is the decay strain of the new vertex and describes the accumulated strain after which the stress relaxes on the flow-curve value. A comparison to experiments and MD simulations showed that microscopic MCT overestimates γ∗ by a factor of 3–5.
Together with experiments on metallic glasses, the F12( ˙γ)model was applied to fit com-pressional flow. It turned out that the parameters of the model are universal to different flow geometries when a rescaled strain rate is defined that captures the prefactors from the invariants ofB. This explains why the peak strain is shifted to smaller values under compression, compared to shear flow. A comparison of the colloidal experiment with the metallic glasses via schematic MCT revealed that a universal structural behavior under strain in these very different systems can be identified and that it is captured by MCT’s structural decay. Decay-strain and energy scales are of the same order when renormalized
5.6 Summary
properly by flow geometry and thermal energy per particle size.
Via a coupling constant, single-particle mean square displacements could be described in using the (transient) viscosity of the F12( ˙γ)model in a generalized Stokes-Einstein re-lation. Describing the long-time limit of a MD simulation of hard spheres worked well this way. A superdiffusive motion in start-up shear is described by MCT just around the strain of the stress overshoot, which the simulation yielded as well. However, the simulation exhibited a more pronounced super-diffusivity and a long-time diffusivity in the quiescent glass state, which is absent in (idealized) MCT. It was also proven that schematic MCT under shear can easily be applied to fit ballistic short time motion, which is a big obstacle for the actual microscopic MCT-ITT.
For the future, it will be interesting to directly couple all decay strains, including γ∗, directly to the flow curve value, which the comparison to the microscopic theory implies, because γ∗ increases monotonically with packing fraction and shear rate outside the asymptotic regime (but close to the glass transition). A simple linear coupling might or might not reveal fascinating physics about the connection of steady state and transient quantities.
6 Zusammenfassung
Jedes Kapitel endete mit einer englischen Zusammenfassung und Ideen die Theorie weiter zu verfeinern. Im folgenden werden diese Ergebnisse noch einmal auf deutsch zusam-mengefasst.
Zuerst wurden in Kapitel 2die Grundlagen der Modenkopplungstheorie [51] mit Inte-gration der transienten Dynamik (MCT-ITT) [53] wiederholt und in einer aktuellen, ten-soriellen Weise dargestellt [56]. Sie beschreibt die Rheologie und Struktur von Hartkugel-dispersionen unter Deformation mittels Dichte-Paarkorrelatoren, deren Bewegungsgle-ichungen (EQM) mikroskopisch hergeleitet werden (durch Projektion von Dichtefluktua-tionen mittels Zwanzig-Mori-artigem Formalismus auf die Smoluchowskigleichung). Die Darstellung erfolgte im Hinblick auf einen zeitlich konstanten, homogenen, aber sonst allgemeinen Deformationsfluß, der zum Zeitpunkt t0 = 0 eingeschaltet und über eine Deformationsrate γ˙ kontrolliert wird.
Kapitel 3 zeigte, dass eine dreidimensionale (3d) Berechnung der MCT-ITT mittels moderner Parallelrechner möglich ist und zufriedenstellende Ergebnisse liefert, obwohl eine grobe Diskretisierung der Numerik noch in Kauf genommen werden muss. Voll-ständig anisotrope Dichtekorrelatoren, Spannungstensoren und Verformungen des Struk-turfaktors (SF) wurden jenseits von linearem Antwortverhalten berechnet. Die dreidi-mensionalen Ergebnisse sind konsistent mit früheren zweididreidi-mensionalen (2d) Rechnungen [42, 55, 64], was anhand vieler Aspekte betont wurde. Dazu zählen u.a. die Master-funktion für den (über-)exponentiellen scherinduzierten Zerfall der Korrelatoren, wie für den unter-exponentiellen, strukturellen Zerfall der Flüssigphase bei kleinen Weissenberg-Zahlen und außerdem das qualitative und quantitative Verhalten des verallgemeinerten Schermoduls und des Spannungsüberschwingers als Funktion von Packungsbruch und Péclet Zahl. Die MCT-ITT beschrieb darüber hinaus, dass Teilchen in 3d leichter einan-der ausweichen, als in 2d, denn Korrelatoroszillationen beim finalenα-Zerfall fallen in 3d schwächer aus.
Vergleiche des MCT-Dichtekorrelators mit MD-Simulationen [19] und PMMA-Expe-rimenten [11] zeigten eine hohe qualitative Übereinstimmung. Die α-Zerfallsskala (der Gleitung) wird allerdings von der MCT-ITT um einen Faktor 10 – 30 überschätzt. Auch die Spannungsspitzendeformation γ∗ wird um einen Faktor 3 – 5 überschätzt, wie der Vergleich mit Kolloidexperimenten in Kapitel 5 zeigte [42]. Die Spannungsspitze ist der Punkt maximaler Spannung in einem Spannungsüberschwinger, der im transienten Regime einer Deformation auftreten kann. Die MCT-ITT beschreibt dieses Phänomen ebenso wie das elastische und das stationäre Regime, welches sich bei konstanter Defor-mationsrate letztlich einstellt.
Die Auswertung der Scherspannung lieferte die Erkenntnis, dassγ∗ und die konstante Langzeitspannung (im stationären Regime) über die MCT-ITT Gleichungen monoton
gekoppelt sind. Das heißt, dass γ∗ mit dem Packungsbruch einer Dispersion und der Péclet-Zahl monoton ansteigt, was sich als Vorhersage experimentell prüfen ließe. In der Flüssigphase verschwindet der Spannungsüberschwinger für kleine Weissenberg-Zahlen.
Die Berechnung der Normalspannungen veranschaulichte den tensoriellen Charakter der MCT-ITT und bestätigte die Lodge-Meissner-Gleichung im elastischen Regime. Eine negative zweite Normalspannung wurde berechnet, deren Verhältnis zur ersten Nor-malspannung dem bei Polymerschmelzen ähnelt [85].
Eine unphysikalische Rückkopplung in den MCT-ITT EQMs für kleine q-Moden der Dichtekorrelatoren wurde entdeckt. Dabei koppeln größere Korrelatormoden so an die kleineren, dass letztere größer als 1 werden können. Die größeren Moden werden jedoch kaum beeinflußt, was daher lediglich eine klein-q Analyse mit den aktuellen Ergebnissen verhindert. Es ist noch unklar, ob diese Rückkopplung inhärent in den MCT-ITT Glei-chungen ist, oder durch eine sorgsamere Implementierung der Scheradvektion bei der numerischen Evaluation behoben werden muss.
Scherdeformierte Strukturfaktoren wurden dreidimensional dargestellt. Der Deforma-tionsfluß wird dem Strukturfaktor dabei im Fourier-Raum invers aufgeprägt. Dadurch wird die quadrupolare Deformation des elastischen Regimes im stationären Zustand hexa-decapolarer, was wiederum den Spannungsüberschwinger erzeugt, denn nur quadrupolare Deformationen erhöhen die Spannung (in der MCT-ITT). Diese Symmetriebetrachtun-gen motivierten einen heuristischen Ansatz, das Lindemannkriterium im Rahmen der MCT abzuleiten, mit dem Ergebnis, dass Gleitungen von der Größenordnung 20% ein Scherschmelzen induzieren. Die Größenordnung stimmt mit Experimenten und Simula-tionen überein (diese liefern etwa 10%, numerisch berechnete MCT-ITT liefert etwa 30%).
Ein Vergleich mit Röntgenstreuexperimenten an Hartkugeldispersionen [15] lieferte gute qualitative Übereinstimmung. Auch deren Strukturfaktor wird im Scherfluß entlang der Fließrichtung kontrahiert. Die MCT-ITT überschätzt die Deformation des Strukturfak-tors um einen Faktor 3, was gut zu der Überschätzung von γ∗ passt. Dadurch wird, im Rahmen der MCT-ITT, mehr Spannung und Strukturfaktordeformation akkumuliert.
In Kapitel4wurde eine anisotrope Korrektur zumβ-Korrelator hergeleitet und anhand der 2d und 3d MCT-ITT verifiziert. Der β-Korrelator beschreibt das asymptotische Verhalten der Dichtekorrelatoren nahe deren transienten Plateaus [41]. Die Korrektur hat eine quadrupolare Symmetrie, entspricht also dem elastischen Regime und kann besonders im Bereich um das erste Maximum des Strukturfaktors nicht vernachlässigt werden.
In Kapitel5wurde schließlich das schematischeF12( ˙γ)-Modell beschrieben und weiterent-wickelt um den Spannungsüberschwinger in beliebiger Deformationsgeometrie (zeitlich konstant und homogen) zu beschreiben. Damit beschreibt auch das F12( ˙γ)-Modell das
In Kapitel5wurde schließlich das schematischeF12( ˙γ)-Modell beschrieben und weiterent-wickelt um den Spannungsüberschwinger in beliebiger Deformationsgeometrie (zeitlich konstant und homogen) zu beschreiben. Damit beschreibt auch das F12( ˙γ)-Modell das