Zhe
Wang
*^{abc},
Takuya
Iwashita
^{d},
Lionel
Porcar
^{e},
Yangyang
Wang
^{f},
Yun
Liu
^{g},
Luis E.
Sánchez-Díaz
^{c},
Bin
Wu
^{c},
Guan-Rong
Huang
^{h},
Takeshi
Egami
^{i} and
Wei-Ren
Chen
*^{c}
^{a}Department of Engineering Physics, Tsinghua University, Beijing 100084, China. E-mail: zwang2017@mail.tsinghua.edu.cn
^{b}Key Laboratory of Particle & Radiation Imaging (Tsinghua University), Ministry of Education, Beijing 100084, China
^{c}Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA. E-mail: chenw@ornl.gov
^{d}Department of Electrical and Electronic Engineering, Oita University, Oita 870-1192, Japan
^{e}Institut Laue-Langevin, B.P. 156, F-38042 Grenoble CEDEX 9, France
^{f}Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
^{g}Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899-6100, USA
^{h}Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan
^{i}Department of Materials Science and Engineering and Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996-1508, USA

Received
17th August 2018
, Accepted 27th September 2018

First published on 28th September 2018

The flow of colloidal suspensions is ubiquitous in nature and industry. Colloidal suspensions exhibit a wide range of rheological behavior, which should be closely related to the microscopic structure of the systems. With in situ small-angle neutron scattering complemented by rheological measurements, we investigated the deformation behavior of a charge-stabilized colloidal glass at particle level undergoing steady shear. A short-lived, localized elastic response at particle level, termed as the transient elasticity zone (TEZ), was identified from the neutron spectra. The existence of the TEZ, which could be promoted by the electrostatic interparticle potential, is a signature of deformation heterogeneity: the body of fluids under shear behaves like an elastic solid within the spatial range of the TEZ but like fluid outside the TEZ. The size of the TEZ shrinks as the shear rate increases in the shear thinning region, which shows that the shear thinning is accompanied by a diminishing deformation heterogeneity. More interestingly, the TEZ is found to be the structural unit that provides the resistance to the imposed shear, as evidenced by the quantitative agreement between the local elastic stress sustained by the TEZ and the macroscopic stress from rheological measurements at low and moderate shear rates. Our findings provide an understanding on the nonlinear rheology of interacting colloidal glasses from a micro-mechanical view.

In this work, we investigate the relation between the microscopic structure and rheology of a charge-stabilized colloidal glass as a model colloidal system with soft repulsive interactions. One reason for the current excitement stems from the description of flow based on the concept of “dynamical heterogeneity” – the spatial inhomogeneity in the relaxation dynamics or local configurational rearrangements.^{15,16} For example, Yamamoto and Onuki illustrated how the heterogeneity in bond breakage influences the nonlinear rheology of highly-supercooled liquids.^{17,18} Particularly, computer simulations suggest that the shear thinning phenomenon is a consequence of decreasing inhomogeneity of flow due to the increasingly frequent configurational fluctuations.^{17–20} In the past several decades, there were extensive theoretical and computational studies on the effects of local plasticity in developing a microscopic description of the flow of amorphous solids.^{16,21–26} The localized plastic arrangements and their spatial correlation were experimentally examined in colloidal glasses^{27–30} and their connection to the shear banding instability was further investigated.^{31,32} On the other hand, the role of local elasticity in determining the rheological behavior of soft matters has also been discussed.^{33,34} For many soft glasses, such as microgels, polymeric materials, foams and emulsions, the constitutive particles or molecules are deformable and possess significant elasticity. Therefore, the local elasticity can be clearly identified, and is found to deeply influence rheological behaviors.^{35} For example, polymeric molecules exhibit a strong entropic elasticity, which leads to the viscoelastic nature of these materials.^{36} For many colloidal suspensions, however, the constitutive particles are too hard to contribute any measurable elasticity at typical flow rates. In this case, the local elasticity should be due to the collective rearrangement of particles under deformation, as suggested by a recent computational study.^{37} Therefore, experimental identification of the local elasticity and its structural basis is crucial to understand the rheological behaviors, especially the viscoelasticity and the shear thinning phenomenon, of interacting colloidal glasses.

The aim of this work is to experimentally explore the origin of the nonlinear rheology of interacting colloidal glasses from the perspective of dynamical heterogeneity and local elasticity. Small-angle neutron scattering (SANS) technique is a powerful tool to study the microscopic structure of complex fluids at length scales from 1 to several hundreds of nanometers.^{38} It has been largely employed to investigate the structure of sheared colloidal suspensions.^{2,8–10,39} The analysis of our SANS data shows that the mechanical response of the charge-stabilized colloidal glass to the imposed shear is localized in a transient elasticity zone (TEZ), which could be promoted by the electrostatic interparticle potential. The correlation between the TEZ and the mechanical behavior of the sample is supported by the agreement between the microscopic stress revealed by scattering and the macroscopic stress measured by rheometry at low and moderate shear rates. Moreover, the size of the TEZ is found to shrink as the shear rate increases in the shear thinning region, which demonstrates that the shear thinning is accompanied by a diminishing heterogeneity of flow.

Fig. 2 Rheological measurements of the charge-stabilized colloidal suspension. (a) Frequency dependence of the storage and loss moduli G′ and G′′. (b) Shear viscosity η as a function of shear rate . |

(1) |

(2) |

(3) |

(4) |

In Fig. 3(a) to (d) we plot both of g_{2}^{−2}(r) and −rdg(r)/dr determined from the SANS experiment at = 3, 10, 30 and 100 s^{−1}, respectively. It is seen that the characteristic variations of these two functions are generally in phase within the shear thinning regime, which qualitatively agrees with the prediction of eqn (4). This observation suggests that the system is essentially elastically deformed at these shear rates, even when the system is flowing. Such deformation coherency is also observed in a simulation study on a model metallic liquid.^{37}

In a random stacking of particles, the local configurational environment is known to differ widely from one tagged particle to another. As a result, it is expected that the constant strain picture given by eqn (4) does not provide a complete description about the microscopic deformation. To further elucidate the structure of the flowing elasticity, we introduce the dependence of γ on the spatial range over which the elastic deformation is sustained:^{37}

(5) |

In Fig. 4(a) to (d) we present the γ(r) for the charge-stabilized colloidal suspension at = 3, 10, 30 and 100 s^{−1}, respectively. We would like to point out that the extraction of γ(r) does not involve any model fitting but only Bessel transforms and data binning. A region of effectively nonzero γ(r) with a spatial range of several particle diameter d is observed for all measured . We name this region transient elasticity zone (TEZ): Within the spatial range of this region, the local structure undergoes an elastic deformation with an average strain given by γ(r) when the system is under steady shear. Beyond this region, the particle motion is dominated by liquid-like random displacements. This localized elastic response survives only for a certain lifetime before it relaxes by flow and diffusion. The existence of TEZ suggests the dynamical heterogeneity in the mechanical response of the system to applied shear. γ(r) can be considered as a correlation function that describes a cooperative region characterized by mechanical coherency in the flow. Based on the previous simulation study,^{37} a Gaussian function is used to model the landscape of γ(r):

(6) |

Fig. 4 (a–d) γ(r) determined by eqn (5) at = 3, 10, 30 and 100 s^{−1}, respectively (yellow circles). The solid curves denote the fitting results by the Gaussian function eqn (6). Panel (e) summarizes the size of the TEZ 2ξ_{TEZ} as a function of . |

From Fig. 4 it is seen that at r ≳ 3d, the values of γ(r) are with large uncertainties and the dependence of γ(r) on r becomes irregular. Numerically this is due to the division between two small numbers. As evidenced by Fig. 3, at r ≳ 3d, both g_{2}^{−2}(r) and rdg(r)/dr are small, which suggest the loss of structural order and correlation at far distances. In this case, the local elastic coherency is no longer significant.

In the above mechanism, the interparticle electrostatic repulsion acts like a free energy barrier to resist the applied strain. Therefore, it is crucial in forming the local elasticity in the flow of charge-stabilized colloids. In fact, at the volume fraction of 0.4, it is known that the hard-sphere colloidal suspension is highly fluid, which suggests the absence of TEZ. The hard-sphere suspension exhibits evident elasticity only when the volume fraction is higher than about 0.58, in which case the excluded volume effect is significant.^{7}

Having established the picture of transient local elasticity from SANS experiment, we now proceed further to explore the role of the TEZ in the nonlinear rheological behavior of interacting colloidal suspensions. In the flowing charge-stabilized colloids, the elastic stress sustained by TEZ is estimated as σ_{TEZ} = G′γ_{M}, where G′ is the modulus of the local elasticity that is similar to the storage modulus given in Fig. 2(a).^{60} As given in Fig. 5(a), the microscopically determined stressσ_{TEZ} is seen to be in a quantitative agreement with the macroscopic shear stress σ_{CC} (σ_{CC} = η) determined from rheometry when ≤ 10 s^{−1}. This agreement is remarkable considering that the two approaches of measuring stress are completely different. It clearly reveals that the elasticity of the TEZ causes the high shear stress, or equivalently the viscosity, in the flow of the charged-stabilized sample. At higher shear rates ( ≫ 10 s^{−1}), σ_{TEZ} considerably deviates from σ_{CC}, manifesting the increasing fluidization.

It is known that Brownian and hydrodynamic effects contribute to the viscosity of colloidal suspensions.^{4,5} The Brownian viscosity contribution η_{B} can be calculated from g(r)^{5,61,62} and the result is shown in Fig. 5(b). The hydrodynamic viscosity contribution, estimated with the approximation given in ref. 61, is found to be well below 0.1 Pa s. This value is small since the electrostatic repulsion can suppress the hydrodynamic effect by reducing the near-contact lubrication.^{61} The viscosity contribution from the TEZ (η_{TEZ} = σ_{TEZ}/) is plotted in Fig. 5(b). The viscosity of the sample η_{CC} measured by rheometry is also shown in Fig. 5(b) for comparison. It is seen that for the flowing charge-stabilized colloids, η_{TEZ} is much larger than the Brownian viscosity contribution η_{B}. This result agrees with the theoretical prediction that in charge-stabilized colloids the potential viscosity contribution is much stronger than the Brownian viscosity contribution.^{61} Summarizing these results, we confirm that TEZ plays a key role in the nonlinear rheology of the sheared charge-stabilized colloids. This is very different from the hard-sphere colloids, in which the shear thinning is mainly attributed to the Brownian effect.^{4,11,61}

The concept of dynamical heterogeneity was introduced for the first time to explain the dramatic increase of the viscosity in the glass transition of supercooled liquids or the colloidal glass formation.^{63–65} Numerous computational and theoretical studies^{66,67} and recent experiments^{68} have shown that the drastic increase in viscosity is accompanied by the growth of cooperatively rearranging regions in fluids during the supercooling process. Yamamoto and Onuki generalized this thinking to the flowing supercooled and glassy liquids: With computer simulations, they demonstrated that the size of the region characterizing the collective bond breakage events shrinks during the shear thinning.^{17,18} Experimentally, we found that the size of the TEZ decreases with increasing the shear rate (Fig. 4(e)), implying the observed shear thinning behavior is accompanied by diminishing deformation heterogeneity. This result is consistent with the previous computational investigations.

(A1) |

(A2) |

(A3) |

Each function has a well-defined parity:

Y^{m}_{l}(θ,ϕ) → Y^{m}_{l}(π − θ,π + ϕ) = (−1)^{l}Y^{m}_{l}(θ,ϕ). | (A4) |

Within the accessed range of shear rate, the contribution from terms with l = 3,…,∞ to spectra should be small compared with the several leading terms. This approximation can be justified from the fact that the measured strain of the TEZ is only around 0.1 or even less. Therefore, eqn (A1) can be approximated as:

(A5) |

Under the geometry of shear flow, it is seen that S(Q) satisfies the conditions S(Q,θ,ϕ) = S(Q,π − θ,ϕ) and S(Q,θ,ϕ) = S(Q,θ,ϕ + π). These conditions can be justified from Fig. 1(b). Therefore, only terms with even l and m survive, which leads to the following expression:

(A6) |

In the 1–2 plane, we can define the following quantity:

(A7) |

Since S(Q,θ = π/2,ϕ) can be measured from the SANS experiment (see Fig. 1(b)), S^{xy}_{2,−2}(Q) can be obtained easily from the measured spectra in the 1–2 plane. Combining with eqn (A6), it is straightforward to show that:

(A8) |

With S_{2}^{−2}(Q), g_{2}^{−2}(r) can be obtained by spherical Bessel transformation:

(A9) |

The calculation of S^{0}_{0}(Q) from SANS spectra needs the information on the 1–3 plane. The detail can be found in ESI.†

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## Footnote |

† Electronic supplementary information (ESI) available: Sample preparation and characterization, SANS experiment, spherical harmonic expansion method, BD simulation, viscosity calculation details. See DOI: 10.1039/c8cp05247f |

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