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4
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M. R. Mackley, R. T. J. Marshall, J. B. A. F. Smeulders, and F. D. Zhao, Chem. Eng. Sci. 49, 2551 (1994).CESCAC
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Mackley, M.R.1
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Zhao, F.D.4
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8
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P. Panizza, D. Roux, V. Vuillaume, C. Y. D. Lu, and M. E. Cates, Langmuir 12, 248 (1996).LANGD5
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Langmuir
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Panizza, P.1
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Vuillaume, V.3
Lu, C.Y.D.4
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6244265764
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P. Sollich, F. Lequeux, P. Hébraud, and M. E. Cates, Phys. Rev. Lett. 78, 2020 (1997).PRLTAO
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Sollich, P.1
Lequeux, F.2
Hébraud, P.3
Cates, M.E.4
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18
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85036436655
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Soft systems may also be intrinsically metastable in a more drastic sense (for example, with respect to coalescence in emulsions)—we ignore this here
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Soft systems may also be intrinsically metastable in a more drastic sense (for example, with respect to coalescence in emulsions)—we ignore this here.
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20
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0001395913
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M. D. Lacasse, G. S. Grest, D. Levine, T. G. Mason, and D. A. Weitz, Phys. Rev. Lett. 76, 3448 (1996).PRLTAO
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Phys. Rev. Lett.
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Lacasse, M.D.1
Grest, G.S.2
Levine, D.3
Mason, T.G.4
Weitz, D.A.5
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31
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0003625787
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J. P. Hansen, D. Levesque, J. Zinn-Justin, North-Holland, Amsterdam
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W. Götze, in Liquids, Freezing and Glass Transition, edited by J. P. Hansen, D. Levesque, and J. Zinn-Justin (North-Holland, Amsterdam, 1991), pp. 287–503.
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(1991)
Liquids, Freezing and Glass Transition
, pp. 287-503
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Götze, W.1
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33
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85036433832
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(Structural) glasses are often classified into “strong” and “fragile” glass formers (see, e.g., c68). The former can be characterized, for example, by an Arrhenius dependence of the viscosity [Formula Presented] on [Formula Presented] temperature, while the latter show a much more dramatic variation of [Formula Presented] with [Formula Presented] near the glass transition. Mode-coupling theories generally predict fragile behavior, while trap models have been associated with strong glasses (see, e.g., c69). The latter is only true for approximately uniform trap depths, however; indeed, we shall find within the SGR model a strongly non-Arrhenius dependence of [Formula Presented] on the (noise) temperature [Formula Presented], due to a large spread of trap depths
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(Structural) glasses are often classified into “strong” and “fragile” glass formers (see, e.g., 68). The former can be characterized, for example, by an Arrhenius dependence of the viscosity η on T temperature, while the latter show a much more dramatic variation of η with T near the glass transition. Mode-coupling theories generally predict fragile behavior, while trap models have been associated with strong glasses (see, e.g., 69). The latter is only true for approximately uniform trap depths, however; indeed, we shall find within the SGR model a strongly non-Arrhenius dependence of η on the (noise) temperature x, due to a large spread of trap depths.
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34
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85036252879
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It is always understood in the following that trap depths are restricted to positive values, [Formula Presented]
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It is always understood in the following that trap depths are restricted to positive values, E>0.
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36
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85036259050
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Precisely this motion is predicted, on a global rather than mesoscopic scale, for perfectly ordered foams (see, e.g., c70
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Precisely this motion is predicted, on a global rather than mesoscopic scale, for perfectly ordered foams (see, e.g., 70).
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37
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85036300787
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This constitutes a major difference between the SGR model and theories for the motion of elastic manifolds in random media: While both approaches describe flow in the presence of disorder, for elastic manifolds the force [Formula Presented] driving the flow is assumed to be homogeneous throughout the sample (rather than the strain rate), and the distribution of trap depths has a cutoff whose strong dependence on [Formula Presented] entirely dominates the low-temperature behavior (see, e.g., c15
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This constitutes a major difference between the SGR model and theories for the motion of elastic manifolds in random media: While both approaches describe flow in the presence of disorder, for elastic manifolds the force F driving the flow is assumed to be homogeneous throughout the sample (rather than the strain rate), and the distribution of trap depths has a cutoff whose strong dependence on F entirely dominates the low-temperature behavior (see, e.g., 15).
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38
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85036285548
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This picture should not be taken too literally, however; the SGR model does not retain any memory of the properties of the old well and the hop back will in general be to a well with a different depth. Bistable elements that oscillate between two (or a small finite number of) equilibrium states are therefore not well described by the model. Correlations between old and new well depths could be introduced to account for such effects. However, an explicit description in terms of two-state elements might be more appropriate where such effects are important. [M. L. Falk and J. S. Langer (unpublished).]
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This picture should not be taken too literally, however; the SGR model does not retain any memory of the properties of the old well and the hop back will in general be to a well with a different depth. Bistable elements that oscillate between two (or a small finite number of) equilibrium states are therefore not well described by the model. Correlations between old and new well depths could be introduced to account for such effects. However, an explicit description in terms of two-state elements might be more appropriate where such effects are important. [M. L. Falk and J. S. Langer (unpublished).]
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39
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85036370673
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Within mode-coupling theory, the fact that [Formula Presented] relaxations correspond to localized motion—loosely describable as the rattling of particles in the cages formed by their neighbors (c32, p. 333)—has been deduced from the fact that correlation functions factorize into a time dependent and a separation (or wave vector) dependent part. Nevertheless, some degree of cooperative motion must be involved, at least on a local scale, because the motion of each particle within its cage affects the cages of its neighbors. The SGR model only captures the analog of the slower [Formula Presented] relaxations, corresponding to the breakup of particle cages and structural rearrangements
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Within mode-coupling theory, the fact that β relaxations correspond to localized motion—loosely describable as the rattling of particles in the cages formed by their neighbors (32, p. 333)—has been deduced from the fact that correlation functions factorize into a time dependent and a separation (or wave vector) dependent part. Nevertheless, some degree of cooperative motion must be involved, at least on a local scale, because the motion of each particle within its cage affects the cages of its neighbors. The SGR model only captures the analog of the slower α relaxations, corresponding to the breakup of particle cages and structural rearrangements.
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40
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85036437484
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This property of the model can be seen most clearly in the equilibrium state [Formula Presented] for [Formula Presented]. All elements are then unstrained, so both terms on the rhs of Eq. (6) are zero: The total elastic energy is constant (and equal to zero) and no energy is being dissipated. At the same time, however, rearrangements are predicted to occur at a finite rate [Formula Presented]
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This property of the model can be seen most clearly in the equilibrium state (γ̇=0) for x>1. All elements are then unstrained, so both terms on the rhs of Eq. (6) are zero: The total elastic energy is constant (and equal to zero) and no energy is being dissipated. At the same time, however, rearrangements are predicted to occur at a finite rate Γ.
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41
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85036320736
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This is true for a fixed distribution of yield energies [Formula Presented], such as the one that we use in all our numerical work [Formula Presented]. In general, our choice of [Formula Presented] fixes only the asymptotic behavior [Formula Presented], but not the details of the distribution of small yield energies [Formula Presented]. These do affect the quantitative predictions of the SGR model, but do not alter qualitative features such as the power-law behavior of the shear moduli and flow curves
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This is true for a fixed distribution of yield energies ρ(E), such as the one that we use in all our numerical work [ρ(E)=exp(-E)]. In general, our choice of xg=1 fixes only the asymptotic behavior ρ(E)∼exp(-E), but not the details of the distribution of small yield energies E. These do affect the quantitative predictions of the SGR model, but do not alter qualitative features such as the power-law behavior of the shear moduli and flow curves.
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42
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0000140335
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P. Hebraud, F. Lequeux, J. P. Munch, and D. J. Pine, Phys. Rev. Lett. 78, 4657 (1997).PRLTAO
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Phys. Rev. Lett.
, vol.78
, pp. 4657
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Hebraud, P.1
Lequeux, F.2
Munch, J.P.3
Pine, D.J.4
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46
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85036359160
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The lower limit in the response integral (17) is zero rather than [Formula Presented] because we assumed that the system is completely unstrained at [Formula Presented]
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The lower limit in the response integral (17) is zero rather than -∞ because we assumed that the system is completely unstrained at t=0.
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48
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85036187272
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the large shear rate regime, the steady yielding rate [Formula Presented] is found to be greater than the attempt frequency [Formula Presented]. This is unphysical because a significant fraction of elements must then be yielding in the “metastable” regime [Formula Presented], where the activation barrier for yielding is negative. For such negative barriers, the nominal yielding rate [Formula Presented] actually increases rather than decreases with decreasing [Formula Presented]; this causes a crossing of the flow curves at shear rates around [Formula Presented] (outside the range shown in Fig. 77). Such unphysical features could be avoided, for example, by fixing the yielding rate of an element to a constant [Formula Presented], say) in the metastable regime (negative activation barrier). Unfortunately, this makes the model significantly harder to solve
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In the large shear rate regime, the steady yielding rate Γ is found to be greater than the attempt frequency Γ0. This is unphysical because a significant fraction of elements must then be yielding in the “metastable” regime E-12l2<0, where the activation barrier for yielding is negative. For such negative barriers, the nominal yielding rate exp[-(E-12l2)/x] actually increases rather than decreases with decreasing x; this causes a crossing of the flow curves at shear rates around γ̇=1 (outside the range shown in Fig. 77). Such unphysical features could be avoided, for example, by fixing the yielding rate of an element to a constant (Γ0, say) in the metastable regime (negative activation barrier). Unfortunately, this makes the model significantly harder to solve.
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51
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0000950457
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A. J. Liu, S. Ramaswamy, T. G. Mason, H. Gang, and D. A. Weitz, Phys. Rev. Lett. 76, 3017 (1996).PRLTAO
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Phys. Rev. Lett.
, vol.76
, pp. 3017
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Liu, A.J.1
Ramaswamy, S.2
Mason, T.G.3
Gang, H.4
Weitz, D.A.5
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56
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85036258247
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For [Formula Presented] [Formula Presented] can actually be nonmonotonic, with a maximum for values of [Formula Presented] of order unity. This is due to the yielding of elements from “metastable” states with [Formula Presented]
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For γ̇≳x, P(ΔE) can actually be nonmonotonic, with a maximum for values of ΔE of order unity. This is due to the yielding of elements from “metastable” states with E-12l2<0.
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58
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85036344393
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The behavior in the glass phase [Formula Presented] could be investigated by introducing an energy cutoff [Formula Presented] as in Sec. IV B. However, the transient behavior [Formula Presented] turns out to be very sensitive to [Formula Presented] (whereas the steady state stress is not) and is therefore of questionable significance
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The behavior in the glass phase (x<1) could be investigated by introducing an energy cutoff Emax as in Sec. IV B. However, the transient behavior σ(t) turns out to be very sensitive to Emax (whereas the steady state stress is not) and is therefore of questionable significance.
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62
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0000954879
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T. G. Mason, M. D. Lacasse, G. S. Grest, D. Levine, J. Bibette, and D. A. Weitz, Phys. Rev. E 56, 3150 (1997).PLEEE8
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Phys. Rev. E
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, pp. 3150
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Mason, T.G.1
Lacasse, M.D.2
Grest, G.S.3
Levine, D.4
Bibette, J.5
Weitz, D.A.6
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64
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85036169467
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The observation of a decreasing [Formula Presented] as [Formula Presented] is nevertheless nontrivial. Naive arguments can easily be misleading: For example, one could argue incorrectly that, because the glass transition is caused by a strong increase in the fraction of elements with large yield energies [Formula Presented]—and therefore large yield strains—[Formula Presented] should actually increase for [Formula Presented]
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The observation of a decreasing γc as x→1 is nevertheless nontrivial. Naive arguments can easily be misleading: For example, one could argue incorrectly that, because the glass transition is caused by a strong increase in the fraction of elements with large yield energies E—and therefore large yield strains—γc should actually increase for x→1.
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65
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0000402686
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D. Weaire, F. Bolton, T. Herdtle, and H. Aref, Philos. Mag. Lett. 66, 293 (1992).PMLEEG
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Philos. Mag. Lett.
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Weaire, D.1
Bolton, F.2
Herdtle, T.3
Aref, H.4
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67
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85036329917
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(private communication)
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D. Roux (private communication).
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Roux, D.1
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68
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0009071257
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C. A. Angell, Science 267, 1924 (1995).SCIEAS
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Science
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Angell, C.A.1
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71
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85036177116
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The corresponding curves in Ref. c17 (Fig. 22) contained a numerical error. However, this caused only slight changes in the quantitative details of the flow curves, without altering any of the qualitative conclusions (power laws, etc.)
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The corresponding curves in Ref. 17 (Fig. 22) contained a numerical error. However, this caused only slight changes in the quantitative details of the flow curves, without altering any of the qualitative conclusions (power laws, etc.).
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