Power law viscoelasticity of a fractal colloidal gel

Abstract

Rheology is the science of deformation and flow, with a focus on materials that do not exhibit simple linear elastic or viscous Newtonian behaviours. Rheology plays an important role in the characterisation of soft viscoelastic materials commonly found in the food and cosmetics industries, as well as in biology and bioengineering. Empirical and theoretical approaches are commonly used to identify and quantify material behaviours based on experimental data. RHEOS (RHEology, Open-Source) is a software package designed to make the analysis of rheological data simpler, faster, and more reproducible. RHEOS is currently limited to the broad family of linear viscoelastic models. A particular strength of the library is its ability to handle rheological models containing fractional derivatives, which have demonstrable utility for the modelling of biological materials [1,2,3,4], but have hitherto remained in relative obscurity -possibly due to their mathematical and computational complexity. RHEOS is written in Julia [5], which provides excellent computational efficiency and approachable syntax. RHEOS is fully documented and has extensive testing coverage.To our knowledge, there is to this date no other software package that offers RHEOS' broad selection of rheology analysis tools and extensive library of both traditional and fractional models. It has been used to process data and validate a model in [3], and is currently in use for several ongoing projects.It should be noted that RHEOS is not an optimisation package. It builds on another optimisation package, NLopt [6], by adding a large number of abstractions and functionality specific to the exploration of viscoelastic data. Statement of NeedArbitrary stress-strain curves and broad relaxation spectra require advanced software Many scientists and engineers who undertake rheological experiments would fit their data with one or several viscoelastic models in order to classify materials, quantify their behaviour, and predict their response to external perturbations.Standard linear viscoelastic models take the form of an ordinary differential equation between stress σ and strain . Under simple perturbations (step or ramp in stress or strain, or frequency sweep), it is relatively straight-forward to extract time-scales and identify asymptotic behaviours required to identify parameter values. However, data often involves complex stress and strain signals, and materials whose behaviour involves a broad distribution of time-scales, including power law behaviours. Fitting models and predicting their response in the time domain then requires computing viscoelastic hereditary integrals such as:τ ) dτ dτ * J L Kaplan, email: jlk49@cam.ac.uk † A Bonfanti, email: ab2425@cam.ac.uk ‡ A Kabla,

Rheology of protein-stabilised emulsion gels envisioned as composite networks 1– Comparison of pure droplet gels and protein gels

“…The power law dependence of the elastic modulus G is a feature of fractal colloidal gels, as previously observed experimentally and numerically [5,8,49,50,51,52], that can be related to the fractal dimension D f . However, the large range of volume fractions for the gels presented here makes such analysis impractical, in the absence of additional characterisation of these networks..…”

Colloidal gelation, a means to study elasto-capillarity effects in foam

“…Though power law dependences of the elasticity on time are not unusual in colloidal gels, the exponents are generally found to be significantly lower, of the order of 1/3. [20][21][22][23][24] However, larger power law exponents are expected from percolation theory, 25 and we may conceive that the distinct behaviour observed for our gel is due to the reaction limited process governing the aggregation of our colloidal system, as opposed to the diffusion limited processes that govern the aggregation of most colloidal systems.…”

Fractional viscoelastic models for power-law materials

“…55 Modelling of the viscoelastic response of tight sandstone. 56 Modelling of both colloidal 64 and carbopol 129 gel rheology. Modelling of arterial tissue.…”

“…52,53 This led to the development of a new formalism for the modelling of viscoelastic materials known as fractional viscoelasticity. Fractional viscoelasticity has been applied to complex geological and construction materials such as bitumen (asphalt), 44,45 concrete, 42,43 rock mass, [54][55][56][57][58][59] waxy crude oil, 60,61 as well as polymers and gels, 41,[62][63][64][65] and food. 11 Numerous examples can also be found of fractional viscoelasticity applied to biological materials such as epithelial cells, 66 breast tissue cells, 67,68 lung parenchyma, 69 blood flow, 70,71 as well as red blood cell membranes.…”