5 Schematic mode coupling theory
5.4 Metallic glasses under compressional flow
adjusted to be on the same linear trend, indicated by straight lines in Fig.5.8. For the latter, not γ∗ but its relation toγ∗∗ is set, and only γ∗ is fitted.
0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.1
Fig. 5.8:Strains γ∗ of the stress maximum and strains γ∗∗ marking the approach to the sta-tionary flow curve used in the fits of Fig.5.7. Circles denote the fluid phase anddiamondsthe glass phase. Theblack symbolsrefer to measuredγ∗s which suggest a linear relation betweenγ∗
andγ∗∗. Grey symbolsare extrapolated with this linear trend. The insetshows the position of the stress maximumγ∗as function of Pe0, with a guide to the eye (cubic polynomial inlogPe0).
Find the accordingF12( ˙γ)parameters in Tab.5.2.
5.4 Metallic glasses under compressional flow
In this section, the new time-dependent vertex vσ(I(t)), Eq. (5.10), is tested in an en-vironment of compressional flow, with compression rate γ˙u = ˙γ/√
3, see Eq. (5.11).
The F12( ˙γ)model is compared to experimental data measured by Jun Lu [7, 97] on a Zr41.2T i13.8Cu12.5N i10Be22.5 alloy, called Vitreloy 1. John Harmon [6, 98] performed experiments on an alloy of P d43N i10Cu27P20, which is used to broaden the basis of comparison. Both metallic glasses are regarded in a regime close to the glass transition temperature TG, which is experimentally defined as that temperature at which the vis-cosity exceeds a certain (arbitrary) value. It is taken from the above works [97,98]. It is one strength of schematic MCT (and MCT-ITT), not to need this specific temperature, but to provide and describe a glass close to its transition point by its asymptotic be-havior, i.e. the divergence of the quiescent α process and its behavior under strain flow.
This is in the following an important point in the considerations on metallic glasses. A new and interesting aspect at the end of this section is a comparison between metallic glasses and PNiPAM colloids, which means in the present context also a comparison of
5.4 Metallic glasses under compressional flow
shear and compressional flow in the framework of MCT.
5.4.1 Experimental aspects
The specific characteristics of the investigated alloys (viscosities, shear moduli, etc.) can be found in Refs. [7, 97] for Vitreloy 1 and Refs. [6, 98] for P d43N i10Cu27P20. The following investigation and the comparison in Sec. 5.4.3 shows the universality of the α process under shear in media of most different nature, viz. colloidal dispersions and metallic glasses. Hence, only those characteristics of the alloys will be mentioned here, which are important for this comparison, first of all the glass transient temperatures of TG = 623K of Vitreloy 1 andTG= 569K of P d43N i10Cu27P20.
Both experiments handled the compression of an alloy probe, where the strain rateγ˙u
was controlled, which is also denotedtrue strain rate. It must be distinguished from the often used engineering strainǫ, which is the time integral of the true strain rate and reads for compression ǫ = exp(Rt
t0dt′γ˙u)−1. The experiments measured engineering stress, which is not the force per area in the probe, due to the deformation of the probe. The latter is thetrue stress, which was accounted for numerically in Refs. [97,98] already. The true stress is the quantity described by the actual MCT-ITT. Unfortunately, the metallic alloys have not been well aged prior to measurement, but where rather compressed in the as-caststate, i.e. immediately after reaching the desired temperature. In the following, the argumentation of Ref. [98] is adopted, stating that steady state quantities are not subject to ageing. The elastic moduli and the extent of the stress overshoot decrease without ageing; see discussion on Fig. 5.13 below. The F12( ˙γ)model allows nonetheless for a comparison. However, because microscopic MCT describes only well aged systems, Sec. 2.3.2, the F12( ˙γ)parameters γ∗∗, γc, and v∗σ must be considered with caution when trying to draw conclusions for the microscopic theory.
dγ=dxx
Fig. 5.9:Sketch of compressional strain on a metallic glass, with according normal stressσxx; cf. to simple-shear flow in a colloidal dispersion Fig.1.2.
Figure 5.9 shows a sketch of compressional flow, together with a coordinate system, which is used for the indices of the stress tensor components σxx.
5.4 Metallic glasses under compressional flow
For a normalization of the F12( ˙γ)parameters, the covalent radii of the biggest elements in the alloys where chosen asR, which means RP d = 139pm for P d43N i10Cu27P20 and RZr = 175pm for Vitreloy 1, cf. the colloidal particles of Sec. 5.3.1 have RH ≃90nm.
The unit of stress is then[σxx] =kBT /R3. The rescaling of times (and thus the definition of Pe0 and Wi) is handled extra in Sec. 5.4.3 for the comparison with the PNiPAM colloidal experiments.
5.4.2 Comparison of schematic MCT and metallic glasses
The important schematic Eqs. (5.12) and (5.13) for a fit of compressional flow shall be recalled first, they read
˙ γ =√
3 ˙γu, σxx(t,γ˙) =−2 ˙γu
Z t 0
dt′e−2 ˙γut′g(t′,γ˙), (5.20) g(t,γ˙) =vσ(t,γ)Φ˙ 2(t,[I]), and (5.21) vσ(t,γ˙) =vσ∗(1−( ˙γt/γ∗)a) exph
−( ˙γt/γ∗∗)bi
. (5.22)
In the following, the minus sign ofσxx is dropped for simplicity. Compared to Secs. 5.2, 5.3, and 5.5, the raw data was not available for investigation, thus a comparison was made to the data extrapolated from figures in Refs. [97, 98] via the software package g3data.
Fitting procedure
There is a pronounced structural (fluid-like)αdecay in all measurement data even below the glass transition temperature ofTG, cf. Figs.5.10and5.12. For MCT, this is a problem addressed to as hopping processes, which is much more dominant in the metallic alloys than in colloidal dispersions, cf. Fig. 5.6. Within the metallic glass community this is addressed to as slowβ process [99], a final structural decay of the metallic melts, which must be distinguished from MCT’sβ decay onto a structural intermediate plateau in the density correlators. In Ref. [40], it was investigated, if this slow β decay is induced by diffusion of the small constituents in the alloy, while the α decay might be induced by the structural decay of the big constituents. However, MCT does not distinguishα, slow β, or hopping decay. As the structural decay is so pronounced in the alloys, a negative εis used to fit the metallic flow curves also below their glass transition temperatureTG. Deviating from Sec. 5.3.2, γ∗ and γ∗∗ are not fitted to each strain rate in order keep the amount of fitting limited, but also because the metallic glass data does not cover the regime in which the β decay influences γ∗ a lot (i.e. Pe0 ≪1). Also, a drift ofγ∗ cannot clearly be extrapolated from the experiments and should thus not be figured out.
Frequency dependent linear response moduli were not available, thereforevσ∗was gained in fitting the linear elastic increase ofσxx(t) for small strains. In the elastic regime holds σxx(t) = ˙γut2vσ∗fqc2 = ˙γut2Gc∞ (derived from Eq. (5.20)–(5.22)). Recall, for simple shear,σxy(t) = ˙γst v∗σfqc2 is valid for elastic response, so the theory of linear elasticity for
5.4 Metallic glasses under compressional flow
isotropic, homogeneous, incompressible media [9] is described correctly by theF12( ˙γ)model (with Lamé parameters µ=v∗σfq2 and λ=∞).
The increase of the flow curves from their intermediate plateau (due to the β decay of MCT) was fitted with Γ. This also influences the splay of the linear-regime slope of the stress-strain curves, because the idealized schematic elastic modulusG∞=v∗σf2depends on how fast the Φ(t)→f plateau is reached and thus on the time scaleΓ.
The vertex parameter γ∗ can be taken as stress-peak position of γ˙ut times √
3 in the compressional stress-strain curves, because of γ˙ =√
3 ˙γu, which is a consequence of the compressional-flow geometry in the invariants I1 and I2, cf. App.A.4.3. Together with γ∗∗,γcis used to tune the decay on the steady-state plateau in the stress vs strain curves, which gives the flow-curve ordinates. Albeit γ∗∗ sets the decay of the vertex vσ(t,γ),˙ Eq. (5.22), and γc the decay of the correlator Φ(t), Eq. (5.2), both decay processes are coupled by factorization of vertex and weights Φ2(t) in the generalized shear modulus, Eq. (5.21). Admittedly, with γ∗∗ and γc, theF12( ˙γ)model is somewhat overparameterized in this consideration without linear moduli G′ and G′′. Hence it is convenient to fix all strain scales γ∗,γ∗∗, and γc, in order to decrease the degrees of freedom to a minimum.
However, with these strain parameters fixed, the flow curves become nonmonotonic and unphysical for too high shear rates, cf. Sec.5.2.1, which is fortunately out of theγurange of the following considerations.
The instant viscosity was setη∞= 0, because there is no solvent in the metallic glass.
Such a viscosity would cause an offset in the stress-strain curves and yield an increase in the flow curves for very high shear rates. For both there is no evidence in the measured data (or now data available).
The vertex parameters in Eq. (5.22) were chosena=b= 2, because it fits the overshoot shapes much better than thea=b= 4used for the PNiPAM particles in Sec.5.3.2. This is also in good agreement with microscopic MCT, cf. Fig. 5.3. However, the metallic alloys were not aged compared to the PNiPAM colloids. The parameters a andb might therefore also express a more pronounced overshoot in a well aged system, implying that aand bincrease with ageing time.
Illustration of the comparison
Figures 5.10 and 5.11 show flow- and stress-strain curves of Vitreloy 1 [97] together with the corresponding schematic MCT fits. It can be verified that the low frequency decay (low strain rate decay) of the metallic glass can be fitted with theα decay of the F12( ˙γ)model. The vanishing of the stress overshoot with vanishing shear rate is explained correct by MCT. A plateau in the flow curves is not apparent in Fig. 5.10 and thus the β decay andΓcan only be fitted with the splay of the small strain slopes in Fig. 5.11. In consequence,Γpossesses a big uncertainty in the fits to Vitreloy 1 and must be considered as not significant.
Table 5.3 summarizes all F12( ˙γ)parameters of the fits to Vitreloy 1. Shear moduli of Gc∞ = 22.0GPa for T = 623K and Gc∞ = 21.9GPa for T = 613K can be derived from the F12( ˙γ)parameters. That the latter shear modulus is lower even though being
5.4 Metallic glasses under compressional flow TG= 623KandT = 613K(symbols, cf. legends) with corresponding schematic MCT fits(solid lines). All curves are fitted withε <0, i.e. below the critical MCT glass transition point. MCT fitting parameters are given in Tab.5.3.
deeper in the glass indicates that the error of the fitting is at least higher than±1% (as fitting was done ‘by eye’, an error of at least±5% should be assumed). The acoustically
0
Fig. 5.11: Stress vs strain curves for different strain rates according to Fig. 5.10 (data from Ref. [97]), with corresponding schematic MCT fits(solid lines). Both panels share their strain rate legends. Theleft panelshows stress-strain curves according to TG= 623K, while theright panelillustrates aT = 613K set. Find the accordingF12( ˙γ)parameters in Tab.5.3.
5.4 Metallic glasses under compressional flow
measured shear modulus at room temperature (20◦C) is G∞ = 34GPa [97]. Regarded that G∞ increases with decreasing temperature and is lower in the as-cast state under consideration [98], the trend and the order of magnitude fit well.
T[K] a;b ε vσ∗[kbT /R3Zr] Γ s−1
η∞ γc/√
3 γ∗/√
3 γ∗∗/√ 3
623 2; 2 −1.25·10−2 160 105 0 0.38 0.044 0.0665
613 2; 2 −5.1·10−3 162 105 0 0.38 0.0516 0.0705
Tab. 5.3:F12( ˙γ)parameters used to fit the experiments of Ref. [97] atTG = 623KandT = 613K shown in Figs.5.10and5.11.
0.1 0.12 0.15 0.18 0.2
10-5 10-4 10-3
Experiment: T= 548K ε= 2·10-4 ε= -2·10-3
˙ γu
s−1 σst xx kBT/R3 Pd
F12( ˙γ)fit:
Fig. 5.12: Flow curve ofP d43N i10Cu27P20from Ref. [98] atT = 548K < TG(red crosses) with correspondingF12( ˙γ)fits(solid lines). Thered curveshows a fit withε <0, while theblueone uses ε >0. F12( ˙γ)parameters are given in Tab.5.4.
Figures 5.12and 5.13 show flow- and stress vs strain curves ofP d43N i10Cu27P20 [98]
together with the corresponding schematic MCT fits. These curves, taken below the glass transition temperature of TG = 569K, serve especially to illustrate the apparent glassy structural decay in comparison to MCT. As the glass transition temperature TC
of MCT is larger than TG (cf. also φc < φg, Fig. 1.1), low frequency decay processes are apparent in the glass state of the metallic alloy. This can be seen in the low strain rate decrease of the flow curve, as well as in the vanishing of the stress overshoot with decreasing strain rate. The latter is the case, because the density correlator has been decayed already, when the strain driven overshoot takes place via vσ(t,γ˙) at a certain value ofγ∗, cf. Eq. (5.21). However, there is no low frequency correlator decay apparent
5.4 Metallic glasses under compressional flow
in MCT for ε> 0 (ideal MCT glass). Therefore this decay is fitted with ε < 0, which means to fit the structural decay in the metallic glass with a MCT fluid-likeαdecay. The hypothesis is that for the from of the structural decay it is not important which cause it really has. This shows the universality of the MCTα decay compared to all kinds of structural decay and its behavior under shear. Figures5.12and5.13right panel show as well the unsatisfactory result that follows from ε >0, i.e. an ideal MCT glass.
0
Fig. 5.13: Stress vs strain curves according to Fig. 5.12 from Ref. [98] at T = 548K < TG
(symbols), with corresponding schematic MCT fits (solid lines). Strain rates are given in the legends. Theleft panelshowsF12( ˙γ)fits withε <0, i.e. below the critical packing fractionφc. The right panelshowsF12( ˙γ)fits withε >0, i.e. above the critical packing fraction. Green crossesshow a 50h aged curve in comparison to thegreen circles, which correspond to the as-cast state (like all other curves). F12( ˙γ)parameters are given in Tab.5.4.
Figure 5.13shows additionally a well aged (50h) stress-strain curve for γ˙u = 10−4s−1 (the only one available in Ref. [98]). It can be verified that this has a major impact on the height of the stress overshoot, while having a small impact on the peak strainγ∗ and the flow curve valueσstxx. According to Ref. [98], the ageing increases the elastic moduli of the system. Thus, theF12( ˙γ)parametervσ∗ (G∞=vσ∗f2) is expected to be underestimated for all curves. The decay strain scales γc and γ∗∗, which determine the extent of the stress overshoot, will also be underestimated. However, a = b = 4 also increased the extent and height of the overshoot, cf. Fig. 5.1. One curve is not enough to produce a convenient fit here. The peak strain γ∗ is the most convenient quantity, assumed that ageing does not change the peak position a lot. However, as all other curves have the same ageing time, which is zero, theF12( ˙γ)parameters of the fits to the metallic glasses are consistent.
Table 5.4 summarizes all F12( ˙γ)parameters of the fits to P d43N i10Cu27P20. A shear modulus of Gc∞ = 23.9GPa (for ε < 0) can be derived from the F12( ˙γ)parameters. The acoustically measured shear modulus at room temperature (20◦C) isG∞= 31.5GPa [98].
Regarded thatG∞increases with decreasing temperature and is lower in the as-cast state, the trend and the order of magnitude fit well, like for Vitreloy 1.
5.4 Metallic glasses under compressional flow
T[K] a;b ε vσ∗[kbT /R3P d] Γ [1/s] η∞ γc/√
3 γ∗/√
3 γ∗∗/√ 3
548 2; 2 −2·10−3 99 105 0 0.43 0.0348 0.051
548 2; 2 2·10−4 80 25 0 0.3 0.0348 0.0517
Tab. 5.4:F12( ˙γ)parameters used to fit the P d43N i10Cu27P20 experiments of Ref. [98] at T = 548Kshown in Figs.5.12and5.13.
5.4.3 Comparison of metallic and colloidal glasses via schematic MCT In this section, the F12( ˙γ)model parameters of the fits to the PNiPAM colloids at 18◦C, Sec.5.3, are compared to the rheology experiments on the metallic alloy Vitreloy 1 of the preceding Sec. 5.4.2. Figure 5.14 and Tab. 5.5 illustrate this comparison. The systems show a vast difference on all scales: size of the constituents, shear moduli (energy scale per volume), and temperature, cf. Tab.5.5. Metallic glasses are used to fabricate highly elastic materials, e.g. golf clubs, while colloidal dispersions are used to fabricate very soft materials, like dispersion paints. Finding that these materials can after all be compared sensibly with the F12( ˙γ)model and thus with MCT means that their structural melting behaviour under straining has similarities. This implies on the other hand that MCT-ITT incorporates and describes a universal structural behavior under straining in a universal straining geometry of hard-sphere-like mixtures, no matter in which physical domain the microscopic constituents live.
The measure of fluctuation strength, viz. vσ∗[kBT /R3], is for both systems normalized by particle radiiR, yielding that they are of the same order of magnitude in both systems.
Recall, because the metallic glass system is not well aged, a further comparison than just the order of magnitude is not sensible. After geometric accounting (all strain scales of uniaxial compression rescaled by√
3), both,γ∗ andγ∗∗are of the same size, as long asγ∗ does not deviate for too highP e0 in the colloidal dispersion, i.e. as long as the asymptotic regime Pe0 ≪ 1 is regarded. A positive drift of γ∗ for higher Pe0 can be observed for the PNiPAM colloids. This should not create the impression that colloidal dispersions are more likely to drift in their peak position than metallic glasses. The Weissenberg numbers in Fig.5.14and Tab.5.6show a small overlap for the two experiments. Metallic glasses are regarded in a regime of the flow curve, where only the α decay matters, i.e.
γ∗ stays constant or even decreases with the flow curve as the overshoot vanishes. This holds only for the smallest Wi for the PNiPAM colloids, for the higher Wi, theβ decay already influences γ∗ and shifts it to higher values. The range of (normalized) stresses is still comparable, because the larger vσ∗ and the flow geometry, cf. Eq. (5.12), of the metal compression yield higher stresses for Vitreloy 1 by about a factor of 4 when fixing all otherF12( ˙γ)parameters.
Because a = b = 2 was used for Vitreloy 1 and a = b = 4 for PNiPAM, another fit using a= b= 2 was performed on the PNiPAM data, yielding similar F12( ˙γ)parameters.
However, γc is increased by a factor of about 2 compared to Sec. 5.3, cf. also Fig. 5.1.
This qualifies γc as being a rather sensitive parameter regarding the change of a andb.
5.4 Metallic glasses under compressional flow
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6
Wi = 0.647 0.207 0.065 0.021 0.007 Wi = 3.6·103
362 36.2
3.62 0.362
metal
colloids
˙ γt=√
3 ˙γut=γ˙st σxx[kBT/RZr],σxy[kBT/RH]
Fig. 5.14: Comparison of stress vs strain measurements for varying strain rates in Vitreloy 1 (blue triangles) and the PNiPAM dispersion (red circles). The metallic glass data is the same as in Fig.5.11 right panel, with T = 613K < TG, and the colloidal data of the fluid state is the same as Fig. 5.7a), with T = 291K. Strain rates, converted to and given in the legend as Weissenberg numbers, increase from bottom to top. Solid linesare MCT fits with the schematic model for the parameters in Tab.5.5.
As long as no transient density correlators are available, fitting γc from rheological data yields little information and is rather arbitrary (because γc is the decay scale of Φ(t), Eq. (5.2), and dominated by γ∗∗ ing(t,γ˙), Eq. (5.21)). When just regarding the aspect of fitting, γc should be coupled to γ∗∗ due to fit parameter reduction; a refined version of the F12( ˙γ)model could account for that.
The comparability of the short-time scale Γ is hardly possible. The fitting procedure of the Sec.5.4.2has leftΓwith a big uncertainty, because measurement data was lacking in the higher strain rate regime. It is moreover unclear, which relaxation process is fitted with Γ in the metallic glasses. An identification with short-time relaxation of metallic atoms, like with the Brownian relaxation of the colloids, is invalid, because the Debye frequency is much smaller. The comparability of colloids and metallic glasses is limited to theα process under strain. This is however a big step. It means thatstructural decay under strain in systems of completely different size and energy scales and flow geometries is similar in (normalized) fluctuation strength and appearing strain scales. Moreover this is incorporated in MCT-ITT, which implies that MCT-ITT describes a universal structural behavior under strain. This strengthens MCT-ITT and refers the hopping problematic as to be possibly specific to each system, meaning dependent on individual, chemical activation energies.
5.4 Metallic glasses under compressional flow
Metallic glass(Vitreloy 1) – T = 613K,RZr= 175pm
ε G∞[Pa] vσ∗[kbT /R3Zr] Γ [1/s] γc γ∗ γ∗∗
−5.1·10−3 ≃22·109 162 105 0.66 0.0894 0.122 Colloidal dispersion(PNiPAM, 18◦C) –T = 291K,RH ≃90nm
ε G∞[Pa] v∗σ[kbT /R3H] Γ [D0/R2H] γc hγ∗i hγ∗∗i
−1.8·10−4 ≃43 90 110 1.05 0.10 0.129
γ∗∗= 1.53γ∗−0.0240
Tab. 5.5:Parameters of theF12( ˙γ)model in the fits to the experiments on Vitreloy 1(upper table) and the PNiPAM dispersions(lower table)used for Fig. 5.14. For the metallic glass,γ∗ andγ∗∗
were chosen constant, for PNiPAM, the mean valuehγ∗iof the γ∗s from Tab.5.6is taken. The PNiPAM fit was performed witha=b= 2. For all fits it is η∞= 0.
To allow for a comparison of bare Péclet and Weissenberg numbers, theF12( ˙γ)parameters were used to normalize strain rates and define Pe0 = ˙γ/Γ and Wi = ˙γ η0/Gc∞, which is the only way to circumvent the unknown short-time dynamics of the metallic glass. The η0denotes the linear-response, long-time shear viscosityη0 = limγ˙→0σstxy/γ˙, cf. Fig.3.11, which was calculated with the F12( ˙γ)parameters from compressional flow (but then used in shear flow). The Gc∞=v∗σfc2 is theF12( ˙γ)shear modulus of the fluid phase.
Vitreloy 1, 613K PNiPAM,291K
˙ γu√
3/Γ Wi γ˙s/Γ Wi γ∗
3.84·10−5 3.62·103 6.34·10−2 3.84·10−6 362 7.62·10−2 3.84·10−7 36.2 9.13·10−2 1.65·10−5 0.647 3.84·10−8 3.62 0.117 5.28·10−6 0.207 3.84·10−9 0.362 0.153 1.65·10−6 6.47·10−2
5.28·10−7 2.07·10−2 1.65·10−7 6.47·10−3
Tab. 5.6:Strain rates and Weissenberg numbers of Vitreloy 1(columns 1 and 2)and PNiPAM dispersion(columns 3 and 4)according to the stress-strain curves in Fig.5.14. The Weissenberg numbers are calculated as Wi = ˙γGηc0
∞
. Column 5 shows the stress peak positions γ∗ used in the F12( ˙γ)fits to the PNiPAM curves with a = b = 2. The according parameters γ∗∗ were approximated viaγ∗∗ = 1.53γ∗−0.0240. The smallest γ∗ differs from Tab.5.2, because of the
. Column 5 shows the stress peak positions γ∗ used in the F12( ˙γ)fits to the PNiPAM curves with a = b = 2. The according parameters γ∗∗ were approximated viaγ∗∗ = 1.53γ∗−0.0240. The smallest γ∗ differs from Tab.5.2, because of the