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Rabea
Seyboldt
^{ab},
Dimitri
Merger
^{c},
Fabian
Coupette
^{a},
Miriam
Siebenbürger
^{d},
Matthias
Ballauff
^{d},
Manfred
Wilhelm
^{c} and
Matthias
Fuchs
*^{a}
^{a}Department of Physics, Universität Konstanz, 78464 Konstanz, Germany. E-mail: matthias.fuchs@uni-konstanz.de
^{b}Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
^{c}Institute for Chemical Technology and Polymer Chemistry, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
^{d}Institute Soft Matter and Functional Materials, Helmholtz-Zentrum Berlin, 14109 Berlin, Germany

Received
15th July 2016
, Accepted 5th October 2016

First published on 7th October 2016

The leading nonlinear stress response in a periodically strained concentrated colloidal dispersion is studied experimentally and by theory. A thermosensitive microgel dispersion serves as well-characterized glass-forming model, where the stress response at the first higher harmonic frequency (3ω for strain at frequency ω) is investigated in the limit of small amplitude. The intrinsic nonlinearity at the third harmonic exhibits a scaling behavior which has a maximum in an intermediate frequency window and diverges when approaching the glass transition. It captures the (in-) stability of the transient elastic structure. Elastic stresses in-phase with the third power of the strain dominate the scaling. Our results qualitatively differ from previously derived scaling behavior in dielectric spectroscopy of supercooled molecular liquids. This might indicate a dependence of the nonlinear response on the symmetry of the external driving under time reversal.

Applying oscillatory shear strain and increasing the amplitude of the deformation, a sharp transition to plastic flow was discovered in athermal particle systems.^{2} If thermal fluctuations are present and trigger local displacements of particles from their time-averaged sites,^{3} the yielding of colloidal glass requires the shear-induced breaking of cages formed by neighboring particles.^{4–7} Varying amplitude and frequency, studies using large amplitude oscillatory shear (LAOS) observe a rich mechanical and structural response, and have been performed in fluid^{8} and glass-forming states.^{9–11} Recently, the application of medium amplitude oscillatory shear (MAOS) has been advocated^{12–16} to determine the frequency-dependent spectra of the leading nonlinear response. MAOS can be considered a direct extension of the linear response approach.

Bouchaud and Biroli argued on fundamental grounds that the nonlinear response in glassy systems should be important in general.^{17} Alluding to the well-understood case of continuous phase transitions among equilibrium phases, they argued that nonlinear susceptibility spectra detect the long sought-after domains of cooperative motion in supercooled liquids and record their growth at the glass transition. The response at the third harmonic of the applied sinusoidal perturbation records the number of cooperatively coupled molecules, which is predicted to diverge as captured in detailed α- and β-scaling laws at the glass transition.^{18} These predictions were crucial for enabling nonlinear dielectric spectroscopy to test the underlying theories about cooperative dynamics.^{19–21}

In the present contribution, we study the mechanical nonlinear susceptibility arising at the third harmonic in the stress response of a glass-forming colloidal dispersion driven by sinusoidal shear strain. Combining high-sensitivity rheological measurements on a well-characterized model glass-former with numerical and theoretical calculations in mode coupling theory (MCT), we establish that the nonlinear response function measured at the third harmonic diverges when approaching the glass transition. However, we find fundamental differences to the nonlinear response scenario predicted by Bouchaud and Biroli, and show that for the present case another class of scaling laws holds. Our experimental measurements of higher harmonic distortions in harmonically sheared colloidal dispersions rest on technical developments in Fourier transform rheology,^{22} including increased sensitivity,^{23} and the development and characterization of a colloidal model dispersion.^{24} Our theoretical investigations take place in the framework of MCT.^{25} It was developed to describe the structural relaxation in quiescent glass-forming liquids and rationalizes many phenomena observed in colloidal dispersions close to their glass transition.^{26} It also enters the discussion by Bouchaud and Biroli of the nonlinear spectra in supercooled molecular liquids, and its generalization to shear-driven Brownian systems^{27} gives the starting point for our analysis of the third harmonic response.

γ(t) = γ_{0}sinωt, | (1) |

(2) |

(3) |

The in-phase (G_{n}′) and out-of-phase (G_{n}′′) moduli are the real and imaginary parts of the complex modulus G_{n}(ω) at the nth harmonic and arise as Fourier coefficients of the stress σ(t), which (after transient effects that are already neglected in (2)) is periodic in time with period 2π/ω. Several LAOS analysis frameworks have emerged over the years: Fourier decomposition,^{22} stress decomposition,^{28} Chebishev polynomials^{13} and the sequence of physical processes approach^{29} have been used to analyze the controlled sinusoidal strain experiment. These methods have also been adopted to stress driven experiments,^{13,30,31} and differences of both techniques in viscoelastic materials were discussed.^{32} We will focus on frequency dependent spectra because they contain important information on the competition between external driving and intrinsic viscoelastic response.^{17}

Viscoelasticity in the linear response rheological moduli is one of the hallmarks of glass-formation. In the intermediate frequency window of predominantly elastic behavior – MCT calls it β-process – the elastic modulus takes a finite value, and the loss modulus exhibits a broad minimum. MCT predicts an asymptotic scaling-law for the frequency-dependent linear moduli , where the parameter ε denotes the relative separation from the glass transition, which lies at ε = 0. The critical elastic constant G^{c}_{∞} measures the rigidity surviving in fluid states for high enough frequencies,^{3} the amplitude factor h_{σ} links stress to structure, and the β-scaling function g_{β} contains the universal critical variation. It exhibits two power laws in a fluid state (ε < 0): the so-called critical law (exponent a) and the von Schweidler law (exponent −b) where a and b are material dependent.^{25} For the present model, a = 0.32 and b = 0.63. The β-scaling time t_{ε} diverges as a power law when approaching the glass transition, t_{ε} ∝ |ε|^{−1/(2a)}; albeit more slowly than τ ∝ |ε|^{−γ}, with γ = (a + b)/(2ab). The β-scale can most easily be read off from the minimum in G_{eq}′′(ω). Only for frequencies below the β-minimum, the final (or α-) relaxation commences and captures the decay of the elasticity in the fluid and the establishment of viscous flow with a finite Newtonian viscosity. The α-process shows up as a broad maximum in the loss modulus around the frequency where both moduli cross.

In order to identify the relevant frequency windows for the later analysis of the nonlinear response, the linear viscoelastic moduli of the model colloidal glass-forming dispersion are displayed in Fig. 1 using the hydrodynamic radius R_{H}, the thermal energy k_{B}T, and the diffusion coefficient at infinite dilution D_{0} to set the scales. Following ref. 35 we estimate the volume fractions of the measured samples as ϕ = 0.614, 0.62, 0.631 and 0.637, which are high owing to the noticeable particle size polydispersity of 17%. With increasing packing fraction, the α- and β-scaling regimes in the moduli shift to lower frequencies.

Fig. 1 Symbols represent the experimentally measured equilibrium storage and loss moduli of a model glass-forming dispersion of colloidal (near-) hard spheres at four different packing fractions ϕ approaching the glass transition. Lines show fits of the MCT model to the data using the parameters in Table 1. Labeled arrows mark the crossing of G_{eq}′(ω) and G_{eq}′′(ω) and the minimum in G_{eq}′′(ω), both at ϕ = 0.614. Dashed lines indicate the two spectral power laws of MCT's β-scaling regime. |

The schematic MCT model captures these trends reasonably. It is fitted to the linear response data following the procedure developed by ref. 35 described in the Materials and methods section, which gives the model parameters displayed in Table 1 of that section. The fit parameters provide insights into the applicability of the asymptotic laws of MCT. Asymptotically close to the transition, all changes in the spectra should be captured by the separation parameter ε = (ϕ − ϕ_{c})/ϕ_{c}, where ϕ_{c} is the packing fraction at the glass transition. Yet, additional density dependences restrict an unambiguous application of the β-scaling law to the larger two densities.

G_{1}(ω,γ_{0}) = G_{eq}(ω) + γ_{0}^{2}[G_{1}(ω)] + O(γ_{0}^{4}) |

G_{3}(ω,γ_{0}) = γ_{0}^{2}[G_{3}(ω)] + O(γ_{0}^{4}), | (4) |

(5) |

Fig. 2 Intrinsic nonlinearity Q_{0}(ω) of the third harmonic versus rescaled frequency. Measured data are given as symbols connected by dashed lines as guides to the eye. Taylor approximation results obtained from MCT are given as solid lines with matching colors. The corresponding spectra of the linear response are given in Fig. 1, which were used to determine the model parameters (given in Table 1). Two arrows with labels mark the position of the β-process minimum and of the α-process maximum in G_{eq}′′(ω) at ϕ = 0.614, read-off in Fig. 1. Two power laws are indicated by straight lines with exponents b and −a; see text for discussion. |

Numerical results for Q_{0}(ω) from MCT obtained by Taylor expansion are included with the experimental data in Fig. 2 and are given as lines. As the model parameters were fixed already, there are no adjustable parameters in the comparison of experiment and theory, which agree on the qualitative trends. Except for some mismatch in the overall amplitude, theory captures the intrinsic nonlinearity semi-quantitatively for the two lower packing fractions. For the two higher packing fractions, theory predicts a stronger variation with packing fraction than seen experimentally. This deviation in the nonlinear spectra matches the deviation already noticed in the linear spectra in Fig. 1, and may indicate a smearing of the singularity of idealized MCT well familiar from many experiments very close to the glass transition.^{37}

The calculated Q_{0}(ω,ε) in Fig. 3 exhibit a maximum, which scales with ε. The inset shows that the maximum is a feature of [G_{3}], as the first harmonic |G_{eq}| is monotonous in the respective frequency region. The height of the maximum diverges with and its position shifts with 1/t_{ε}; it lies in the center of the β-process window close to the minimum of G′′. The scaling master curve (see Fig. 4a), exhibits two power laws (see ESI†). The right flank behaves as ω^{−a}, the left as ω^{b}. The scaling with ω^{b} gets more pronounced upon approaching the glass transition because corrections arising from the α-process get suppressed (recall that τ diverges more strongly than t_{ε}). For frequencies smaller than the maximum, the curves for different ε collapse onto a second scaling master function valid in the α-process. Collapse in Fig. 4b holds when the frequency is rescaled by ω/|ε|^{γ}, while the amplitude does not depend on the distance to the glass transition. Rather, the nonlinearity vanishes with decreasing frequency: the curves are proportional to ω^{2} for ωτ ≪ 1. This quasistatic limit, see ESI,† is included as dotted lines in Fig. 3 and 4.

Fig. 4 Left panel: Scaling of the maximum of Q_{0} in the β-regime: the colored lines are the same numerical data as shown in Fig. 3, the black lines indicate the slopes of the asymptotic power laws. The height of the maximum scales with |ε|^{1/2}, and lies in the center of the β-process window close to the minimum of G′′ (see arrow). The right flank behaves as ω^{−a}, the left as ω^{b}, with a = 0.32 and b = 0.69. Right panel: Scaling of Q_{0} in the window of the α-process: again, colored lines are the data shown originally in Fig. 3, the black lines indicate the asymptotic power law behaviors. The crossover is roughly a decade below the α-maximum of G_{eq}′′ (see arrow). |

The discovered divergence of the intrinsic nonlinearity in the β-process window upon approaching the glass transition indicates the sensitivity of the incipient glass structure to the external deformation. In this frequency window characterized by the time scale t_{ε}, the dominant linear response of the stress in the material is elastic. Yet, this elastic structure has a finite life-time and is not stable with respect to thermal fluctuations; it will relax during the final α-relaxation. The external periodic strain causes large nonlinear reactions of the fragile elastic structure while it is still metastable. At lower frequencies, where the elastic structure has started to relax already by equilibrium structural processes, additional driving provided by the external straining has little effect. Thus the intrinsic nonlinearity decreases during the final (α-) relaxation and has its maximum in the β-process window.

Our results can be compared with similar nonlinear quantities derived for dielectric systems.^{17,18} The susceptibility χ_{3}(ω) gives the (normalized) nonlinear response at the third harmonic frequency of the applied external electric field and thus plays an analogous role as the third-harmonic modulus [G_{3}(ω)] under strain. The scaling of their absolute values with frequency and separation ε to the glass transition is compared in Fig. 6. (The scaling of [G_{3}(ω)] closely follows the one of Q_{0} and can readily be obtained from Fig. 3.) The power-law at the critical point of MCT (at ε = 0) agrees in both response functions and is given by [G_{3}(ω)], χ_{3}(ω) ∝ ω^{−a}; it diverges for ω → 0. (The different spectra at larger frequencies depend on microscopic details.) In supercooled states, for ε < 0, both response functions exhibit the critical law for frequencies in the early β-process (, so-called ‘critical regime’). For still lower frequencies, the dielectric nonlinearity increases during the late-β-process (‘von Schweidler regime’), while the shear modulus decreases. The maximum in the dielectric nonlinearity signals qualitatively different scaling behavior of χ_{3} compared to [G_{3}] in the α-relaxation region. This has important implications for the understanding of nonlinear phenomena in driven glass-forming systems. Tarzia et al.^{18} consider the nonlinear response to an oscillatory field which, when applied statically, shifts the glass transition locus,^{39} and allows to use a generalized fluctuation dissipation relation. In the present case of shear-deformation, detailed balance does not hold and, in the case of steady shearing at constant rate, the nonergodic glass state of MCT gets melted for any shear rate.^{40} Thus we propose as an explanation for the differences in the observed scaling behavior that the two different scenarios of the nonlinear response under oscillatory driving show the sensitivity of the glass state to the kind of perturbing external field: whether it connects to an equilibrium state or to a nonequilibrium steady state. Future fundamental studies possibly along the lines of systematic higher order response theory^{41} would be very useful for broadening this insight. Especially, determining the nonlinear response under applied stress would be useful to investigate the universality of the different scaling laws found in the nonlinear oscillatory response of glass-forming systems.

Fig. 6 Scaling of the magnitude of the third-harmonic modulus |[G_{3}(ω)]| in sheared systems found in this paper (black solid line) in comparison to the scaling of |χ_{3}(ω)| in dielectric systems^{18} (dashed grey line). The dotted lines indicate how the maxima and the shoulders of the curves scale with the distance to the glass transition ε. Only scaling behavior is shown (labeled by power-law exponents), thus, matching curves do not indicate matching values. While both nonlinear responses show the same scaling behavior ∝ ω^{−a} at the critical point, they differ qualitatively in the α-regime. |

(6) |

m(t,s,t′) = h(t,t′)h(t,s)[Φ](t,s) | (7) |

[Φ](t,s) = v_{1}Φ(t,s) + v_{2}Φ^{2}(t,s) | (8) |

(9) |

G(t,t′) = v_{σ}Φ^{2}(t,t′) + η_{∞}δ(t − t′), | (10) |

Φ(t,t′) = Φ_{eq}(t − t′) + (γ_{0}/γ_{c})^{2}Φ_{ω}(t,t′) + O((γ_{0}/γ_{c})^{4}). | (11) |

Φ_{ω}(t,t′) = f_{0}(t − t′) + e^{iω(t+t′)}f_{1}(t − t′) + e^{−iω(t+t′)}f_{1}*(t − t′). | (12) |

Inserting the Taylor solution into (10) and then into the Fourier-modes of the shear stress, (2),

(13) |

[G_{1}(ω)] = (2v_{σ}iω/γ_{c}^{2}) (F{Φ_{eq}(t) f_{0}(t)}(ω) + F{Φ_{eq}(t) f_{1}(t)}(0)), | (14) |

[G_{3}(ω)] = (2v_{σ}iω/γ_{c}^{2}) F{Φ_{eq}(t) f_{1}(t)}(2ω). | (15) |

The rheological experiments were conducted on a ARES-G2 (TA Instruments) strain controlled rheometer using a Couette geometry with a Peltier temperature control system. The same dispersion was measured at four temperatures, corresponding to different volume fractions; for details about the shear-protocol see the ESI.†

In Fig. 7, we present strain-amplitude measurements of the storage and loss moduli at ϕ = 0.631 for three different angular frequencies. At the third harmonic, a quadratic dependence of |G_{3}|/|G_{1}| on γ_{0} is observed as predicted by the above Taylor expansion. Therefore we reduce |G_{3}|/|G_{1}| to Q_{0} as shown in Fig. 7. This material function has been introduced by Hyun et al.^{36} and since then has been applied to investigate the nonlinear behavior of emulsions^{47} and foams,^{23} as well as linear^{48} and branched polymer melts.^{36} A collection of analytical expressions for Q_{0}(ω) derived for continuum and microscopic models is available in ref. 49. Poulos et al.^{50} recently measured |G_{3}|/|G_{1}| of fluid and glassy suspensions of star polymer particles as well as sterically stabilized poly(methyl methacrylate) particles. They analyzed the frequency dependence of |G_{3}|/|G_{1}| at a fixed strain amplitude of γ_{0} = 1 and found increasing intensities for increasing frequencies in fluid samples, whereas |G_{3}|/|G_{1}| was shown to decrease with increasing frequency for glassy samples.

Fig. 8 Symbols represent the experimentally measured flow curves for four ϕ approaching the glass transition, lines show fits of the schematic model to the data using the parameters in Table 1. |

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6sm01616b |

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