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Rodrigo
Rivas-Barbosa
^{a},
Manuel A.
Escobedo-Sánchez
^{b},
Manlio
Tassieri
^{c} and
Marco
Laurati
*^{ad}
^{a}División de Ciencias e Ingenierías, Universidad de Guanajuato, Lomas del Bosque 103, 37150 León, Mexico. E-mail: mlaurati@fisica.ugto.mx
^{b}Soft Matter Laboratory, Heinrich-Heine University, Universitätsstrasse 1, 42150 Düsseldorf, Germany
^{c}Division of Biomedical Engineering, James Watt School of Engineering, University of Glasgow, Glasgow G12 8LT, UK. E-mail: Manlio.Tassieri@glasgow.ac.uk
^{d}Department of Chemistry “Ugo Schiff” and CSGI, University of Florence, Sesto Fiorentino, Florence I-50019, Italy

Received
15th November 2019
, Accepted 23rd January 2020

First published on 24th January 2020

We report on the application of a Fourier transform-based method, ‘i-Rheo’, to evaluate the linear viscoelastic moduli of hard-sphere colloidal dispersions, both in the fluid and glass states, from a direct analysis of raw step-stress (creep) experimental data. We corroborate the efficacy of i-Rheo by comparing the outputs of creep tests performed on homogenous complex fluids to conventional dynamic frequency sweeps. A similar approach is adopted for a number of colloidal suspensions over a broad range of volume fractions. For these systems, we test the limits of the method by varying the applied stress across the materials' linear and non-linear viscoelastic regimes, and we show that the best results are achieved for stress values close to the upper limit of the materials' linear viscoelastic regime, where the signal-to-noise ratio is at its highest and the non-linear phenomena have not appeared yet. We record that, the range of accessible frequencies is controlled at the higher end by the relative weight between the inertia of the instrument and the elasticity of the complex material under investigation; whereas, the lowest accessible frequency is dictated by the extent of the materials' linear viscoelastic regime. Nonetheless, despite these constrains, we confirm the effectiveness of i-Rheo for gaining valuable information on the materials' linear viscoelastic properties even from ‘creep ringing’ data, confirming its potency and general validity as an accurate method for determining the material's rheological behaviour for a variety of complex systems.

(1) |

The standard method of measuring G*(ω) over a finite set of frequencies is based on the imposition of an oscillatory stress σ(ω,t) = σ_{0}sin(ωt) (where σ_{0} is the amplitude of the stress function) and the measurement of the resulting oscillatory strain, which would have a form of γ(ω,t) = γ_{0}(ω)sin(ωt + φ(ω)), where γ_{0}(ω) is the strain amplitude and φ(ω) is the phase-shift between the stress and the strain. From eqn (1) it follows that:

(2) |

(3) |

Fig. 1 A schematic representation of half square wave u(t) and its approximation obtained by considering only the first four terms of the Fourier series shown in eqn (3). |

(4) |

(5) |

γ(t) = σ_{0}J(t) | (6) |

Eqn (6) has been accepted by the rheology community because the data acquisition rates of rheometers were (until few years ago) slower than ε. However, nowadays, thanks to the development of sensitive and fast transducers, it is possible to acquire a significant number of experimental data within the time window defined by ε. Nevertheless, these data are commonly discarded because of the condition on which eqn (6) relies on, i.e. t ≫ ε.

(7) |

This method has been improved by Tassieri et al.,^{38} while analysing microrheology measurements performed with optical tweezers. The authors found that a substantial reduction in the size of the high-frequency artefacts, from which some high-frequency noise tends to spill over into the top of the experimental frequency range, can be achieved by an over-sampling technique. The technique involves first numerically interpolating between data points using a standard non-overshooting cubic spline, and then generating a new, over-sampled data set, by sampling the interpolating function not only at the exact data points but also at a number of equally-spaced points in between. Notice that, over-sampling is a common procedure in signal processing and it consists of sampling a signal with a sampling frequency f_{s} much higher than the Nyquist rate 2B, where B is the highest frequency contained in the original signal. A signal is said to be oversampled by a factor of β ≡ f_{s}/(2B).^{39}

The advantages of an analytical tool, such as i-Rheo, which allows the evaluation of the Fourier transform of raw experimental data, become apparent when both eqn (4) and the relationship between the material's complex shear modulus and the creep compliance are considered together in the frequency domain:

(8) |

3.1.1 Homogeneous complex fluids.
In order to validate the efficacy of i-Rheo for analysing creep measurements, we prepared two homogeneous complex fluids with significantly different LVE properties: (I) a polyacrylamide (PAM, from Polysciences Inc.) with a nominal molecular weight of M_{w} = 18 MDa was used to prepare an aqueous solution at a mass concentration of 1% w/w. The solutions were mixed with a magnetic stirrer at 200 rpm for 48 h at room temperature. (II) A standard calibration fluid was provided with the purchase of the rotational rheometer MCR 302 (Anton Paar). This fluid is a polydimethylsiloxane (PDMS) sample that is expected to show a low frequency crossover of the moduli at ω = 100 rad s^{−1} and T = 25 °C.

3.1.2 Colloidal suspensions.
We prepared dispersions of polymethylmethacrylate (PMMA) spheres of radius R = 150 nm (polydispersity 12%), sterically stabilized with a layer of polyhydroxystearic acid (PHSA), in cis–trans-decalin.^{40} In this solvent the particles behave like hard spheres.^{41} Dispersions with different colloidal volume fractions ϕ were prepared by diluting a sediment obtained by centrifugation of a dilute suspension. The volume fraction of the sediment was estimated by means of simulation and experimental results to be ϕ ≈0.66.^{42–44} We prepared dispersions with ϕ = 0.40, 0.48, 0.50, 0.52, 0.54, 0.56, 0.58, and 0.60, going from the fluid to the glass state. After dilution, the samples were mixed in a rotating wheel for a least 12 h before the experiments were started.

Measurements on the colloidal suspensions were performed with a DHR3 stress-controlled rotational rheometer (TA instruments), using a plate-plate geometry of diameter D = 30 mm and a gap d = 0.5 mm. We used a solvent trap to minimize solvent evaporation. The temperature was set to T = 20 °C and controlled within 0.1 °C via a Peltier system. For all samples, a dynamic strain sweep (DSS) at frequency ω = 1 rad s^{−1} was performed directly after loading to determine the linear response regime and, for glassy samples, the strain amplitude at which fluidization occurs. For these last samples, corresponding to dispersions with ϕ = 0.58 and 0.60, the effects of sample loading and ageing were reduced by performing the following rejuvenation procedure before each test. First, a dynamic time sweep with a large strain amplitude, typically γ = 500%, well above the yield strain, with a duration of 200 s and a frequency ω = 1 rad s^{−1} was performed in order to fluidize the sample. After that, another dynamic time sweep with the same frequency but strain amplitude within the linear viscoelastic regime, typically γ = 1% was applied until a steady-state response, i.e. a time-independent storage G′(ω) and loss modulus G′′(ω), was observed, which typically took about 150 s. This indicated that no further structural changes occurred and hence a reproducible state of the sample was obtained. The linear viscoelastic moduli where measured using a conventional DFS with a strain amplitude of γ = 0.1–1%, depending on the sample, in the frequency range 10^{−1} < ω < 10^{2} rad s^{−1}. Creep experiments for different applied stress σ values were measured for all the samples for a duration of 10^{3} s using the fast sampling option, which provides a sampling time t_{sam} ≈ 1.5 × 10^{−3} s.

It is important to highlight that the values of the two parameters g(0) and ġ_{∞} required by eqn (7) have been set equal to zero for γ(0) = 0, and ; whereas, σ(0) and _{∞} are obtained from the experimental data as follows: (i) σ(0) is the first experimental value of the stress (e.g., in Fig. 3σ(0) = 426 Pa); (ii) _{∞} is obtained from a linear regression of the long-time behaviour of γ(t). Notably, the assumptions made for γ(0) and are actually exact, because the sample is initially undeformed and when the stress is constant its time derivative is null. In the cases of σ(0) and _{∞}, they are sensible approximations. Indeed, for viscoelastic fluids, whose linear response is characterised by the existence of the terminal region at low frequencies (where G′(ω) ∝ ω^{2} and G′′(ω) ∝ ω for ω → 0), the long-time behaviour of the strain is expected to be: γ(t) ∝ t/η for t → ∞ and therefore _{∞} ∝ 1/η for t → ∞, where η is the fluid's steady state shear viscosity. A similar conclusion is achieved in the case of viscoelastic solids (e.g., gels and rubbers), for which J(t) tends to a finite value J_{0} at long times and for which _{∞} ≅ 0. The effectiveness of i-Rheo to evaluate the LVE properties of complex materials from conventional bulk rheology step-stress measurements has been initially corroborated by investigating the rheological properties of two homogeneous complex fluids (Section 4.1), then applied to the rheological study of concentrated suspensions of colloidal particles, including glasses (Section 4.2).

Moreover, with regards to the lowest accessible frequencies, typically the limiting factor is the signal-to-noise (SNR) ratio of the data measured after a long time. In a creep experiment though, large strain γ values are measured at long times, meaning that the SNR value should be large. This is different from the use of i-Rheo to determine viscoelastic moduli from stress relaxation after a step strain. Indeed, in that case the stress tends to zero at long times and therefore SNR is small. Thus creep tests are in principle a better choice to determine viscoelastic moduli at small frequencies. However, as shown hereafter, the sample behaviour at long times might impose additional constraints.

Hereafter we report the results for a selected sample with ϕ = 0.54 to illustrate the procedure used to determine the linear viscoelastic moduli by means of i-Rheo and investigate the dependence on the applied stress. Then, we report the linear viscoelastic moduli obtained via i-Rheo for all the investigated samples and their comparison with those of conventional DFS.

The results of a dynamic strain sweep (DSS) test performed on a sample with ϕ = 0.54 at ω = 1 rad s^{−1} are reported in Fig. 6. At this frequency, the sample behaves like a viscoelastic solid, with the storage modulus larger than the loss modulus within the material's linear response regime, which extends up to a strain amplitude of γ_{0} ≈ 1%, with a corresponding oscillatory stress amplitude of circa 0.3 Pa. In the non-linear regime of the material response the moduli cross at γ_{0} ≈ 12%, which corresponds to a stress amplitude of approximately 1.5 Pa, after which a fluid-like response is observed. This is consistent with recent studies, which indicate the crossing point of the frequency dependent viscoelastic moduli as the yielding point of soft glassy materials.^{47} We have used the DSS tests to identify for each sample the stress values that fall within the linear and non-linear regimes of the materials. We then selected a series of stresses across the two regimes to investigate the effects of the applied stresses on the frequency-dependent moduli evaluated by means of i-Rheo. Fig. 7 shows the creep measurements performed at different applied stresses (i.e., σ = 0.1, 0.2, 0.3, 0.4, 0.5 and 1 Pa) on a sample having a volume fraction of ϕ = 0.54. As for the polyacrylamide solution, also the response of this system shows inertia effects in the time interval 10^{−3} < t < 10^{−1} s, followed by a ‘creep ringing’ regime for 10^{−1} < t < 1 s. At longer times, all curves show a sub-linear increase of the strain γ(t) with time, which is characteristic of solid-like materials for applied stresses lower than their yield stress σ_{y}, which for this system can be estimated from Fig. 6 to be σ_{y} ≅ 1.5 Pa. Moreover, from Fig. 7 it can be seen that the measured strain increases with the increase of the applied stress, as expected.

Fig. 7 Strain γ(t) as a function of time t obtained from creep measurements performed with different applied stresses on a sample with ϕ = 0.54. Inset: t/γ as a function of t. |

Fig. 10 Strain γ(t) as a function of time t obtained in creep experiments for different colloidal volume fractions ϕ, as indicated. The applied stress depends on the sample and corresponds to the value used to obtain the viscoelastic moduli in Fig. 9. Arrows indicate the limit of the response affected by tool inertia effects. |

In terms of the lowest accessible frequency, we can argue that it is affected by the physics of the suspension rather than by the technical limitations of the rheometer. Indeed, concentrated colloidal suspensions are often characterised by flow-induced inhomogeneities, such as shear banding, which can affect the response of colloidal dispersions during creep measurements, especially after a long time,^{30,32} when approaching and/or crossing the glass transition.^{30–32,51} This was not a problem for the lowest volume fractions explored in this work, i.e. ϕ = 0.40, 0.48 and 0.50, which did not manifest flow inhomogeneities within the measurement time-window (Fig. 10) and therefore the viscoelastic moduli were evaluated down to ω ≈ 4 × 10^{−3} rad s^{−1}. In the cases of ϕ = 0.40 and 0.48 some oscillations observed in G′′(ω) could be attributed to a low signal and noise of the measurements. Moreover, for all ϕ ≥ 0.52 the samples showed an anomalous behaviour of the strain γ(t) for t ≳ 10^{2} s (Fig. 10), which was limiting the range of the lowest accessible frequencies by means of i-Rheo.

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