## Monday, July 06, 2015

### Rheology and Memristors

Viscoselasticity in polymers is often modeled with mechanical elements, specifically springs and dashpots (think of a dashpot as a being like those cylinders that prevent doors from closing too quickly so as to not pinch fingers).
These elements can be connected in series such as in a Maxwell fluid shown on the left. This was proposed by James Maxwell of electromagnetic fame (more on that in a minute).
They can also be connected in parallel, such as in a Kelvin-Voight material.
Of course, there is no reason to limit any model to just two elements, and a Generalized Maxwell model will have a number of serially-connected springs and dashpots that are connected in parallel.

While knowing about these models is part of a good theoretical understanding of viscoelasticity (including knowing full well that these models, like all models, have limitations while still being useful), this can also be helpful when explaining viscoelasticity of people not trained in the subject. Mechanical/civil/aeronautical/... engineers and physicists catch on quickly, while electrical engineers can struggle. Fortunately, this struggle can easily be resolved by using mechanical-electrical analogies. A dashpot is analogous to a resister - energy is irrecoverably lost, while a spring is analogous to a capacitor as they both store energy. Viscoelasticity is simply an RC-circuit. I've made this analogy to electrical engineers many times and it works quite well.

But the analogy can be problematic at times. Electrical engineers also work with another basic circuit element, an inductor. Inductors resist the change of electrical flow by taking advantage of a nearby magnetic field. While mass/inertia/momentum can be considered analogs to inductance and we can describe a flowing polymer in unsteady conditions as an RCL circuit, the next element is where the analogy completely breaks down.

The next element? Beyond the resistor, capacitor and inductor? Maxwell (him again) first described those three as basic, circuit elements back in the 1800's, often using the electromechanical analogy described above. And for the longest time, those three elements were all that were known to exist, at least as passive devices. However, in 1971, Prof. Leon Chua, using symmetry arguments, proposed a forth basic circuit element - the memristor. This device, showing both a memory and resistance (hence the name), would show decreasing resistance as more current had flowed through the device, and it would also remember what its resistance was when the device was turned off. After restarting the element, the resistance would be the same as before. Changing the direction of current flow would reset the device to its initial resistance. As with the inductor, all of this is possible by a coupling the current to a nearby magnetic field.

Going from theory to practice took some 37 years. Or did it? Apparently a controversy exists. I'm not in any position to say who's right and who's wrong. Regardless of the physical existence of the device, it exists theoretically. But even trying to describe a mechanical analog of a memristor is complicated. HP Senior Fellow Stan Williams describes it as:
" An analogy for a memristor is an interesting kind of pipe that expands or shrinks when water flows through it. If water flows through the pipe in one direction, the diameter of the pipe increases, thus enabling the water to flow faster. If water flows through the pipe in the opposite direction, the diameter of the pipe decreases, thus slowing down the flow of water. If the water pressure is turned off, the pipe will retain it most recent diameter until the water is turned back on. Thus, the pipe does not store water like a bucket (or a capacitor) – it remembers how much water flowed through it."
And I further doubt that a polymeric analog of a memristor exists. While the whole idea of decreasing resistance with flow is quite analogous to shear thinning, the analogy fails on two fronts: polymers will lose the decreased viscosity upon cessation of flow (the rate of such loss being related to "the" relaxation time of the polymer), and switching flow directions will not reset the viscosity. Thank goodness, as dynamic mechanical analysis, where rheologists measure viscosity by imposing oscillatory shear on a sample, would not exist if it did.

Like all analogies, there are limits on their applications and if pushed too far, they break down. The analogy of a memristor to any part of a non-Newtonian fluid is just not going to happen, regardless of whether or not a memristor actually exists.

## Previous YearsJuly 6, 2011 - "Scientists Develop Bioplastic"July 6, 2010 - Ruminations on a Furnace Filter

Anonymous said...

It seems to me something mechanical vaguely akin to a memresistor could take the form of a set of hinged articulated plates distributed around the internal circumference of a pipe, so that flow in one direction would push the plates against the pipe wall (increasing pipe flow), and flow in the opposite direction would tend to push the plates toward the center, reducing opposite direction flow (although this would necessarily be dependent on the incoming flow rate). Assuming the hinges have *an ideal level of friction* the last state of the apparatus is more or less maintained until the next flow.

It's such a simple kind of device I'm sure it must have already been implemented somewhere.

John said...

Something along the lines of a reed valve?

Anonymous said...

Yes, that looks exactly like the sort of thing. Thank you.

Funny, isn't it, where lacking the name for something causes all kinds of difficulties in referring to it... I'm sure there's a name for this condition, but I forget what it is... ;) (I can imagine it now, "I was an "Aphasian Studies" major").

Something like this also came up in a recent Popular Mechanics, an article about "The 4D Nature of Materials" (at least this is close to the title) but it seems to me that pipes that vary circumference in response to water flow could be *extremely* problematic.

John said...

I would agree with you on being "extremely" problematic. For Newtonian fluids, flow rate is proportional to the fourth power of pipe diameter. Holey nonlinearities!

Joe Q. said...

I am a rank newbie when it comes to rheology, but when explaining things to those even more newboid than myself, I also find the electro-mechanical analogy to be helpful. Especially when comparing a rheology experiment with constant unidirectional rotation (like DC) to one with oscillating rotation (like AC). You can't really measure capacitance with DC.