Recently I tried to measured the dynamic modulus of water using a rotational rheometer (AR-G2, TA Instruments).

Water is a Newtonian fluid at room temperature (T = 25°C). Using a cone-and-plate geometry of 60 mm/1° made of hard aluminium, I found that the lowest torque measurable is 1e-7 N·m, ten times higher than the claimed 1e-8 N·m on the instrument’s specification. The lowest shear rate that can generate a measurable torque is about 1.8 s^-1 under steady flow test. The steady state viscosity was measured to be 1.02 mPa·s.

To measure the dynamic modulus, the strain amplitude was set to 500%, and the data were averaged over 5 cycles. Meaningful results can be obtained at the range of ω = 1.0 ~ 10.0 rad/s.

As a Newtonian fluid, the storage modulus of water G’ = 0, and the loss modulus G” = ηω. The solid line in the figure is a fit of such relationship and the viscosity η = 1.14 mPa·s.

The dynamic modulus of water can also be measured by particle tracking microrheology or optical tweezer microrheology (see arXiv:1102.3035 [cond-mat.soft]).

Boukany, P., Hemminger, O., Wang, S., & Lee, L. (2010). Molecular Imaging of Slip in Entangled DNA Solution Physical Review Letters, 105 (2) DOI: 10.1103/PhysRevLett.105.027802

About the paper

Prof. Shiqing Wang (王十庆) have long been interested in observing the event inside the flowing sample during rheometry. He is known for showing the “inconvenient truth” of the research of experimental rheology: we do not always know if the flow field is homogeneous or not though we always assume it to be so.

He proposed the use of video to track the particle movement inside the test sample during shear. He found that a large proportion of conventional rheological measurement involved non-uniform shear field, i.e. nonlinear velocity profile along the vortex axes, which in effect undermines the reliability of most of the established basic knowledge of rheology. Fortunately, recently increasing number of physicists help solving this crisis by providing solid proposal for shear banding and shear localization theories. And many major experimental rheologists also seem to focus on the corresponding experiments, temporarily giving up their other own interests.

The instruments developed by Wang himself also allow new observations of event occurring in a flow field. The paper cited here is an observation of stretching or DNA molecules under parallel plate rheometry by addition of a confocal microscopy to a rheometer, directly relating the structural transition to the rheology. The results are simple and self-explained. Given such instrumental possibility, the first several theoretical things to validated, i.e. “what to see”, is obvious. But the work is still original in the instrument design. I love this kind of research. Imaginably, the rheometers in Wang’s lab are all in a hignly flexible situation, ready for all kinds of fixes. In contrast, rheometers in many labs are the same as home appliances, operated (sometimes even partly) according to the user manual released by the companies.

Why and how to be a good experimental rheologist

The definition of “rheology” as a discipline includes the following points: 1) the study of the relationship of stress and deformation; 2) the study of non-Newtonian fluid, viscoelasticity of soft solids and all other abnormal mechanical or flow behavior; and 3) study of structure-property relationship. The length of the history of rheology is longer than any discipline can last remaining phenomenological. However until now there is still much unknown in purely phenomenological sense in this area. Observation of new rheological behaviors keeps appearing the fast development of new materials. The understanding of these observation in a continuum sense is valuable for industrial practice and also challenging enough for pure research interest.

There is a large number of researchers who stay in this conventional rheology circle, while I myself am interested in the part of physics of complex fluids which rheology can lay a finger on. However, seeing from the physicists’ angle, rheology is limited at the experimental ends. It even represents a very special experimental condition of a sample; the state of the sample system is driven away from equilibrium by externally forced deformation (e.g. shearing). To be helpful for the study of complex fluids, esp. in non-equilibrium conditions, only knowing the macroscopic stress-deformation relationship cannot complete any logics. Although in polymer rheology much structural and dynamic information can be inferred by rheometry alone, this owes much to the rigorous development of polymer physics, which very much cares for rheology due to the fact that the application of polymer physics is largely under shearing conditions (e.g. polymer processing, coating, food processing). Instead, for research interests on other complex fluids, it is not always necessary to derive a rheological prediction after proposal of a structural theory. Rheologists should be more active in building the connection with theoretical study than physicists.

It is impractical but easy to imagine that an ideal, “omnipotent” complex fluids researcher should simultaneously have enough mathematical-physical skill to propose theoretical model for any system of interest, to deduce the stress tensor of viscosity prediction ready for experimental validation (e.g. a predicted density correlation function that can directly checked by dynamic light scattering or a constitutive equation in the case of rheology), and finally to validate it, with sophisticated experimental skill (of DLS or rheology for example).

However, in reality only a few proportion of grad students are excellent enough and properly trained to meet this standard (and congratulations, folks from CalTech and MIT). Practically a line should be draw somewhere between the theoretical and the experimental part of research. More lines may be necessary in the experimental part of research alone. In the narrow field of colloid rheology, some groups can achieve fruitful results by DLS experiments along. The DLS experiment is indeed very informative in the physical sense, directly measuring the local dynamic of structural transition with time, needless to mention that the prevailing mode-coupling theory (MCT) of colloidal glass directly gives an density correlation function, a convenience for DLS experimentalists. In the case of rheology however, only general, gross information of relaxation time is available without the support of existing structural theory. So while DLS can often provide new observations, rheological experiments are always limited in the hypothesis-testing ends.

I am particular interested in the imperfect case of me, who lack formal, solid training of nearly anything. I have trouble in understanding theory proposed by other in mathematical details; I have more in deducing experimentally checkable variables specifically for my own research project. Because of this limitation, for any new interesting research area I have to wait for the development of theoretical study until a refined and simple solution has come out, and even have to wait for the first experimental work which does the deduction toward the workable experimental parameter. My room to play is so small and immune of creativity if I remain theoretically handicap and always stay in the hypothesis testing ends in the line of scientific reasoning.

But I can still be more creative in another end — making new observations which can keep quite a few theorists busy for a while. Not every observations can attract the attention of physics, however, there are several characteristics of a highly cited physical observation.

Firstly, the observation should be very new and, of course, it should belong to you. Everyday there are billions of people doing all kinds of experiments of all kinds of samples on all kinds of instruments. A promising tip for people of low IQ and low creativity like me to have new and high impact observation is to develop an instrument that is completely unique all over the world, i.e. designed and DIY according to my own research interest. “Secondly”, “thirdly” and “fourthly” is common personal quality of scientific research research and will not repeat here.

Therefore, I should pay my limited intelligence to enhance my observability, esp. by designing ad hoc measuring devices for unconventional observational purposes.

Particularly, the conventional rotaional rheometry should combined with structural measurements. This idea has long be practiced but today it appears as a must. For example, in conventional DLS test the samples are filtered in to a cuvette and allow to stand at rest. To observe the behavior under shear condition it is now common that a shear cell is used instead of a cuvette in a DLS measurement. Shear cells is also widely used in various small-angle radiation measurements (e.g. SANS). However, these are not all the possibility. By such combination rheologists can not only say about the stresses and strains but also about the structures, the two parts mutually supported. However due to the limited performance of commercial shear cell, in most labs shear cells are still DIYed.

I have also seen many one-hit wonders of other rheometer fixes that serve very ad hoc purposes. These appear only a short while not because they are inefficient. Instead most of these designs are very clever. The difficulty for other to follow these research lies in maintaining the accuracy of data on a DIY setting. In many cases, once the student leave the lab, the setup he/she built is left unused, and the ideas behind is also dead. But I still love this kind of research. When I read one of these papers, I am not implied to following the authors citing their works (as is the case of most paper published), but refreshed by the extreme originality of it, reminding me to independently think of my own weird science and instruments.

In fact, the discipline of rheology bear this spirit of DIY. The early great experimental rheologists also made major contribution to the design of rheometer themselves. As far as I know, it was W. Philippoff who proposed oscillatory measurement. He was an electrical engineer, and got the idea at that time from the characterization of loss in oversea telephone signal by tanδ — the dielectric loss in this case. As his close colleague, A. Weissenberg develop a rheometer for this purpose, widely known as the “Weissenberg rheogoniometer”, e.g. the famous model R18, using a light beam to read the displacement analogous to the analytical balance. The accuracy of torque and displacement data is further enhanced by J. Ferry, who introduced electromagnetic transducers and obtained a rheometer that could output analytic signals. For a long period in a rheological experiment obtaining the data required knowledge of low-frequency circuit design. With the development of computer there was also once a time when a rheometer was connected to a “room”, the smallest size of computer at that time. It is intriguing that from the development of rheometer we actually see the history of electronics. These pioneers have set the excellent example of how to be a creative experimentalist (as they appeared to be more famous for their experimental work than theorectical ones): keep a close attention to the development of the vast field of technology and dare to apply new trick on the rheometers or other experimental instruments. On a uniquely designed instruments, all observation must be new, the rest lying on the physical meaning you want to address.

Abstract In this post an observation of negative G” values in a dynamic test of a sample is describes. A preliminary explanation for the result is provided by the author.

A negative value of G” is rheologically meaningless

It is well known that in conventional “dynamic rheological experiments” we apply a sinusoidal deformation to the sample and record the stress response. If the applied amplitude is small enough (within the linear viscoelastic regime) the responsive stress signal will be also a sinusoidal function with the same value of period, but different amplitude and phase. The test can be expressed mathematically as:

An example with δ=30° is illustrated in the following image (blue: strain; red: stress):

We utilize the difference in amplitude and phase to calculate G’, G”, tanδ, etc., which all rheologists must be very familiar with. Note, that although mathematically the value of the phase angle, δ, of two arbitrary sinusoidal curves of the same period can take any value in [0, 2π), I was told by an experienced rheologist that rheologically speaking the loss angle can only take the value between [0, 0.5π], the left limit being Hookian elasticity (δ=0 rad) while the right limit being Newtonian viscosity (δ=0.5π rad). In another word, the value of loss angle will never lie on any but the first quadrant, i.e. tanδ and G” never take negative values.

Observation of negative G” values

However, when I conducted dynamic tests on aged Laponite/PEG suspensions on a TA ARES strain-controlled rheometer, negative values of G” reproducibly appear in an angular frequency region of ω=10 ~ 100 rad/s.

A repeated scan in the range where G” take negative values exhibit good reproducibility. And the corresponding least value of δ is ca. 2.3°. See figure below:

It is expectable and indeed what actually turned out that the value of G” in this range decreases as the Laponite/PEG suspension ages, from a liquid like (G”>>G’) behavior to a gel like (G”<<G’) behavior. So the dipping into the negative regime also occurs during dynamic time tracing of the aging process at the correspondent test frequencies. As shown below, the decrease of G” with time at ω=8~160 rad/s is continuous from above to below zero. And the decrease rate first increase then decrease with increasing ω, following the trends of ω dependence of G” of the aged samples.

All of the above result consolidate my observation of negative G” values.

G” may not be reliable when the test exceed the instrument’s limit

A dynamic test can provides for the tester a bundle of materials functions including G’, G”, η’, η”, tanδ, etc. However, many of them are correlated. As the rheometer applies a fixed sinusoidal strain on the sample and records the responsive stress waveform, only two data are needed to calculate all these materials functions: the amplitude of the stress waveform, σ₀, and the time lag between the stress and strain waveforms, Δt (indicated in the first figure of this post). The resolution of the first data relies on the torque transducer of the rheometer, and the second relies on the highest sampling rate of the rheometer (assuming digital data collection). The relationship between the loss angle and the time lag is simple and geometry-independent: . Therefore, in the case where δ is very small (near Hookian behavior) and the dynamic test is conducted at a large value of ω, the time lag the rheometer needs to resolve will be extremely small. To get a quantitative sense, consider the case in which δ=0.8° and ω=126 rad/s (the least positive point of G”), the rheometer will need to resolve a Δt as small as 1.1E-04 s! Further decrease in δ with increasing ω (in a logarithmic way) will need even smaller value of Δt resolved. Right now I don’t know the exact lower limit of sampling rate of ARES rheometer but it is reasonable to suspect that the abnormal negative G” values when a sample exhibit near Hookian behavior might origin from instrumental error.

The results may not be explained by instrumental error

If the negative values of G” observed in the above listed results originated from the instrumental limit, the pattern of the values should be random, fluctuating with normally probability distribution around a position lightly above zero. After all, the “real” value of δ must be positive regardless of how small it is. However, the results show that G” decreases systematically with ω. In the time sweep test, especially, G” at some ω remain negative and continuously decrease with time. This is not statistically valid if we still assume the reason is the instrumental error. Alternatively I am forced to conclude that the instrument gave experimentally reliable whereas rheologically meaningless values of δ‘s which lie in the fourth quadrant. Or it is wrong to claim that the loss angle cannot lie on other quadrants than the first.

The interpretation of the G” minimum

Putting aside the above problem of negative G”, a deep minimum in G” curve in a dynamic frequency sweep test is commonly observed in my experiment on concentrated colloid samples, some may not go so far to give a negative G”‘s. For example, another Laponite/PEG sample with less extent of aging show systematic decrease in the G” minima with wating time, t_w (the results were obtained by time-resolved rheometry method proposed by H. Winter et al., Rheol. Acta 1994, 33, 385-397):

Another example is the dynamic frequency sweep of Carbopol pastes, a soft colloidal glass. A minimum in G” is again observed, though not so small as becoming negative:

The above examples indicate that the minimum of G” in a dynamic frequency sweep test in the range of 10~100 rad/s (λ=ω^-1=0.01~0.1s) is a common phenomena shared by the concentrated colloidal systems I used in my experiment. However, all the previous authors have reported no such minima in G” for colloidal samples. Typical result of a dynamic frequency sweep of a jammed colloidal sample should be like, for instance, those reported in Phys. Rev. Lett. 1995, 75, 2051, or what the soft glassy rheology (SGR) theory predicts. Are the minima of G” observed in my experiment merely an artifact?

Plans

Although I cannot figure out the above questions, there are still work can be done to narrow up the possibility. First, find another rheometer (maybe TA AR-G2, a stress-controlled rheometer) to conduct the same test. Second, contact TA’s engineer for inquiry of the detailed data recording method of their rheometer products. Third, ask experts for help (I know a few research groups that concentrate on soft colloid rheology). I hope I can clarify this problem soon.