# Rheology

Rheology is the study of a fluid's mechanical properties: its viscosity, elasticity etc. Traditionally, these properties are measured in a rheometer - a standard piece of equipment in any lab developing complex fluids such as plastics, foods, cosmetics, paints, lubricants, detergents, liquid crystals. A rheometer typically involves two metal plates separated by a small gap, into which the fluid is introduced. The machine then measures the rate at which one plate slides past the other when a time-dependent stress is applied, to find the time-dependent response of the fluid.

Usually, oscillatory stresses are applied at a range of frequencies, to find the fluid's frequency-dependent response spectrum, expressed in the frequency-dependent complex modulus G*(w). Other possible protocols include suddenly switching on a constant stress sigma and measuring the resultant strain gamma as a function of time t, the ratio of the two being the fluid's time-dependent compliance J(t)=gamma(t)/sigma which vanishes for negative t (by causality) and grows without limit as t increases (because fluids flow).

Another method, invented in the 1990s and still a topic of research and development, is microrheology. This requires only tiny fluid volumes, and so is useful for rare or expensive fluids, or those in situ within biological cells, foods etc. It involves introducing nano-particles (colloids) into the fluid and measuring their movement (either by particle-tracking video microscopy, or by scattering of laser light) under the action of forces, thus inferring the properties of the fluid around them. Those forces may be naturally arising thermal forces causing Brownian motion of the probe particles, or applied by optical tweezers or magnetic field gradients (if the probes are magnetic) using an oscillatory or step protocol as for a rheometer.

If the forces are only thermal (i.e. random) then the probe particles perform random walks with time- and position-dependent correlations due to the non-trivial properties of the fluid. Mason et al showed, in their seminal paper [3], that G*(w) could be derived from the particle motions using a method of data analysis and fitting, which became the standard, and was therefore reproduced by many labs independently, always with a significant time investment and often with bugs.

Their method requires the mean-square displacements of particle positions <r^2(t)> to be measured as a function of time, and its Laplace transform <r^2(s)> to be found (usually involving some fitting and approximation by black-box algorithms that contain hidden, model-dependent assumptions). This is related to the Laplace-transformed fluid modulus G(s), from which the complex modulus G*(w) can be found by analytic continuation using the substitution s=iw. The time-dependent data were not converted directly to the frequency domain because <r^2(t)>, like J(t), grows without limit with time, so its Fourier transform does not converge. In fact, <r^2(t)> is proportional to J(t), with a constant of proportionality given by probe particle size.

In [1], I derived a one-line formula (tested by my experimentalists co-authors) to perform the conversion directly, from a discrete, finite data set to G*(w) without any hidden assumptions. This has saved large amounts of time and mistakes for microrheologists, and provided a quick and easy protocol for traditional rheologists to find G*(w) from a measurement of J(t), which tends to be faster than performing a whole frequency sweep.

Performing the conversion directly not only requires (i) working around the problem of the divergent integral (which I solved by double differentiation followed by double integration, making use of the causality property), but also (ii) the problem that J(t) is not fully known, only finitely sampled, and (iii) contains noise. Many people have instinctively conflated issues (ii) and (iii), addressing them both by replacing the data with a preconceived fitted model. This yields results that neither have the status of experiment nor theory, but some subjective compromise between the two. In fact, (iii) is not a problem but simply a scientific fact that experimental data contain noise. In [1], I avoided any model assumptions, and addressed (ii) by simply interpolating and extrapolating the data by a piecewise-linear function that visits every data point, so that both the signal and noise are faithfully reproduced in the output G*(w).

More on the topics of rheology and microrheology can be found in [2].

Details:

Direct conversion of rheological compliance measurements into storage and loss moduli

The procedure that has become established, for obtaining frequency-dependent dynamic moduli from non-oscillatory rheometry, is to fit the experimental data to a particular model (often the generalized Maxwell model is used), and subsequently to calculate the resulting complex viscoelastic modulus,

G*(w)= G'(w) + i G''(w),

for that parametrization of the model.

That procedure can be somewhat restrictive, as it may force the user to approximate their data into the prescribed form, or to use a very large number of fitting parameters. It also artificially hides experimental noise, making the uncertainties in the final results difficult to quantify. Equivalently, one can find an approximate Laplace transform of the time-dependent data, then derive the Laplace transform of the stress relaxation modulus, and subsequently transform from a Laplace to a Fourier description (either numerically or, for certain functional forms, analytically). In either case, the procedure limits the user's freedom in the types of formulae that can easily be fitted and manipulated, and it can be somewhat laborious. Furthermore, in cases where the experimental data are imperfectly fitted by the pre-conceived functions, or other approximations are introduced (such as an approximate Laplace transform or its inverse), the accuracy of the derived moduli becomes vague.

A more direct, straightforward and accurate treatment of rheometric data is possible. We have derived a formula for G*(w) in terms of the experimental data points themselves. The figure shows the results (red and blue lines) of substituting data from compliance meaurements of a polyisoprene melt (Mw=152kg/mol, Mw/Mn=1.03), at 0 Celsius, directly into the formula. Data points from much more time-consuming oscillatory measurements are shown for comparison. It can be implemented in just three lines of Mathematica, as follows:

{t,J} = Transpose[Import["filename.txt","Table"]];

G[w_, J0_, eta_, {t_, J_}] := I w /(I w J0 + (1-Exp[-I w t[[1]]])(J[[1]]-J0)/t[[1]] + Exp[-I w t[[Length[t]]] ] / eta + Sum[ (Exp[-I w t[[k-1]]]-Exp[-I w t[[k]]]) (J[[k]]-J[[k-1]])/(t[[k]]-t[[k-1]]), {k,2,Length[t]} ] );

LogLogPlot[ {Re[G[w,0,22151,{t,J}]], Im[G[w,0,22151,{t,J}]]}, {w,0.001,1000} ]

(Some installations of Mathematica first require a library to be loaded, using << Graphics`Graphics` and that LogLogPlot executes with low precision in some versions, so that a linear plot of Log[G[10^s]] versus s (where w=10^s) may produce a better graph.)

The first line imports the experimental data (a list of pairs of numbers) from a file (here named filename.txt). The second line defines the function G, expressing the complex modulus in terms of the data, and the third displays the resulting real and imaginary parts of the complex modulus.

To test the routine, we provide the compliance data published in the paper ( Phys. Rev. E 80, 012501 (2009) ) as a text file containing a list of pairs of numbers (t,J) where t is time measured in seconds and J is compliance in reciprocal Pascals here: ComplianceData.txt. Applying the conversion formula to this file should reproduce the continuous lines in Fig.4 of the paper.

An implementation for Maple (slightly inelegant, to work around some Maple bugs) can be downloaded as a Maple worksheet: G'&G'' from J.mw or alternatively as plain text: G'&G'' from J.txt

A LabVIEW executable code and accompanying documentation can be dowloaded fromftp://userweb.elec.gla.ac.uk/pub/andrewg/manlio/

and a discussion of the scientific philosophy behind the data analysis here: BSR Bulletin 50, 76 (2009).

References

[1] Direct conversion of rheological compliance measurements into storage and loss moduli, R M L Evans, Manlio Tassieri, Dietmar Auhl, and Thomas A Waigh, Phys. Rev E, 012501 (2009).

[2] ``Mirorheology with optical tweezers: Principles and applications", edited by Manlio Tassieri. (Pan Stanford, 2016). ISBN 9789814669184.

[3] T G Mason, K Ganesan, J H van Zanten, D Wirtz and S C Kuo, Phys Rev Lett. 79, 3282 (1997).