Slide 1

Microrheology. based on review by Gardel, Valentine & Weitz 1

Slide 2

Microrheology may replace normal rheology for many kinds of soft material. At the very least, deserves the attention of anyone who studies: Gels Liquid Crystals Viscous fluids Probably not for people who need: Stress-strain curves on composites, rubbers, fibers. 2

Slide 3

Normal DLS and FPR can be forms of microrheology, as can FCS. Stokes-Einstein Equation: Invert that equation to get : 3 But D = D(q) = /q 2 .or maybe not! What are the time and distance scales associated with that microviscosity?

Slide 4

Here are some golden oldies from an obscure group down in Louisiana. 4

Slide 5

Here, ZADS is used to follow gelation of poly(acrylamide). 5 Fraction of probes that are frozen During gelation of poly(acrylamide), Langmuir 1994, 10, 4053-4059.

Slide 6

Not only obscure Louisiana groups but also others studied failures of Stokes-Einstein. Jim Martin George Phillies Tim Lodge 6

Slide 7

Failure is such a harsh word. Maybe the Stokes-Einstein is not failing. Perhaps we just are not looking at it correctly. Maybe there is something like shear thinning or thickening going on, to explain the failures. Following empowerment educational practice, maybe we should say that Stokes-Einstein (SE) hasnt failed, but is challenged. Maybe if we look at success more generally, SE is working! We may need a way to generalize the Stokes-Einstein law. But later on that. 7

Slide 8

is a long way from G( ) and G( ) You remember why we want these, right? *Some people say when G = G you have a gel. *Thats actually bogus: when G and G obey power laws with , thats when you have a gel. 8 Its a long way to rhe-o-lo-gy http://www.bing.com/videos/search?q=it's+a+long+way+to+tipperary&mid=2E91F80A34CCAE2E36262E9 1F80A34CCAE2E3626&view=detail&FORM=VIRE3 Martin & Adolph >3 orders of magnitude power law at gel point.

Slide 9

Better (maybe) microrheology strategies are in two particular (particle-erget it?) varieties: Active: you push or pull particles about, measuring how much force this requires, phase lag, etc. *Magnetic tweezing (MT) *Optical tweezing (OT) *Atomic Force Microscopy Passive: particles do their thingyou follow that as best you can. *Regular or Depolarized DLSwith estimation *Particle Tracking (PT) *Diffusing Wave Spectroscopy (DWS) 9

Slide 10

Lets begin with magnetic tweezing.* 10 *Our laboratory is committed to do this type of microrheology. We have begun building little amplifiers we think can drive the coils that generate the magnetic field. The particle is magnetic or superparamagnetic. An oscillating field is applied to make it move. Various ways are used to determine its position. We realize from our studies of driven harmonic oscillators that the amplitude of the motion and the phase with respect to driving force will vary with frequency. We will vary that frequency deliberately. The fluid may be simple or complex (latter case showni.e., entangled polymer matrix).

Slide 11

Moving the particles should be simple. 11 Chip Amp in Dynaco ST 120 case at Camp Chaosalmost assembled. Six amplifiers like this will drive the coilswe hope.

Slide 12

Next, you have to calibrate the magnetic force and observe the particle motion. One way to calibrate: pump a fluid at a known speed while holding a particle of known size steady in the tweezers. The friction force is f= 6 Rand that is the force the tweezers supply if there is no motion. 12

Slide 13

Now, how do we track the probe particles motion? Video camera like our Dage 66: 30 frames/s for $0 CCD interline camera we requested from British Petroleum: 30 frames/s for $5,000 to $15,000 sCMOS camera we are drooling over: 100 frames/s for $16,000 13

Slide 14

So, that gives imageswhat are the particle coordinates, though? 14 http://www.youtube.com/watch?v=EzhMFPwW0rA http://www.youtube.com/watch?v=YUr_8sIewTU Plan A: Able Particle Tracker = about $600 Plan B: Donovan at Kent State LCI = about $7000 Plan C (for correct): Both for about $7600

Slide 15

Heres an MT example, cast in typical rheological terms like creep. 15

Slide 16

There are other ways to follow the particlesfaster ways. For example, a laser beam can illuminate the particle. As particle passes in front of beam, scattering goes up (or transmission goes down). You can follow these light signals with PMT or photodiode. You can go to much higher frequencies this waymaybe 50kHz. Try that with a conventional rheometertypically limited to ~100Hz. This extension is important for soft materials limited opportunities for time-temperature superposition. WLF cannot swim! 16

Slide 17

Optical tweezing, an alternative to magnetic tweezers, relies on the Ashkin effect. 17 Particles with higher refractive index than the surroundings are drawn towards the beam. Too high in the trap, the particles can feel the axial force and get expelled in the vertical direction. If the light is very tightly focused, typically by using a high-NA microscope objective lens, the dominant force tends to draw the particle towards the waist of the intensity profile.

Slide 18

Optical tweezers apply exquisitely localized forces that can measure, for example, the entropic force associated with stretching a single molecule. 18 http://www.youtube.com/watch?v=U88iBwc2qIE&feature=player_embedded This kind of single-molecule extensional rheology is not considered further here, but see this video from Finland (apparently celebrating National Instruments Labview) http://www.youtube.com/watch?v=OmfW2CuBm1g&feature=relatedhttp://www.youtube.com/watch?v=OmfW2CuBm1g&feature=related This kind of single-molecule extensional rheology is not considered further here, but see this video from Finland (apparently celebrating National Instruments Labview) http://www.youtube.com/watch?v=OmfW2CuBm1g&feature=relatedhttp://www.youtube.com/watch?v=OmfW2CuBm1g&feature=related

Slide 19

For us, OT are just another way to wiggle a bead; you again follow its position vs. force at various frequencies. 19 The forces exerted on the particle are, as always, the gradient of some potential. Near the trap center, they are radial forces. Heres the force Depends on refractive index of particle & surroundings.

Slide 20

I cant say it better than this. .if the laser beam center is offset from the center of the particle, the particle experiences a restoring force toward the center of the trap. By moving the trap with respect to the position of the bead, stress can be applied locally to the sample, and the resultant particle displacement reports the strain, from which rheological information can be obtained. 20

Slide 21

Force calibration and particle location can be similar to MT, but how do we create and move the tightly focused beam? 21

Slide 22

With Sketchup, you can fly like a photon through a tweezing instrument. 22 http://www.youtube.com/watch?v=TeJAgNgKsX8 http://www.youtube.com/watch?v=_7HZiDb3NHY&feature=topics

Slide 23

These days, you can just buy your OT. 23 http://www.youtube.com/watch?v=ju6wENPtXu8 http://www.elliotscientific.com/ I cant help but think the best home-brew systems are still better. It was a long time until commercial DLS systems came even close to the home- built ones. On the other hand, there is big money in OT because the user base is heavily biological.

Slide 24

Once again, the relevant equations are familiar to us: a driven, damped harmonic oscillator. 24 Force balance. Displacement as a function of requency: maximal at resonance.

Slide 25

We will skip AFM, but its in the same category as OT and MTjust another way of doing conventional rheology on a tiny scale. The technology which enables these active methods is remarkable especially OT and AFMbut ultimately theyre just micron-scale rheometers. 25

Slide 26

Ironically, it is the passive microrheology schemeswhere stuff just happens that you can interpret as rheologywhich are complex. Ease of Experiment Ease of Interpretation = Constant 26

Slide 27

Several factors make the mathematical baggage worth carrying. Passive is gentler than active. Passive is cheaper. Passive usually explores shorter distance scales. Passive usually explores a shorter time scale. 27 This equation reads: energy density is equal to energy of deformation (modulus times strain, where strain = L/a). Typical: 1 to 100 distance scale and 10 to 500 Pascals

Slide 28

To understand passive methods, we must generalize the Stokes-Einstein relation. And that will have to wait until after July 4 th ! Then we will have a sance with Messrs. Fourier, Langevin and Laplace. But I do want to give a preview.

Slide 29

Regular DLS can be re-interpreted. 29 The distance scale in traditional DLS is 2 /q which is on the order of 1000 to 5000 for typical wavelengths and scattering angles. The DLS data contain info on where the particle has been and, therefore, how it moves on those distance and time scales. For regular DLS, the direction is also specific and encoded in the data. The distance vs. time data is more specific than just a diffusion coefficient. This specific data permits the Stokes-Einstein relation to be generalized from one that involves diffusion, particle radius and viscosity to one that involves mean squared displacements, particle radius and frequency- dependent viscosity. q where q = 2 /d d

Slide 30

So, now you can plot distance vs time. 30 What DLS measures A new way to look at what it means, with a handlestill to be fleshed outto rheology.. And now.one last thing. If you toss away the directional information by forcing the light to be scattered multiple times as it goes through the medium, you greatly shorten the distance scale. In that case, you can measure displacements on the order of 1 .sub-nanoscale rheology.

Slide 31

One last thing today: this kind of thinking is often associated with something called diffusing wave spectroscopy (DWS). 31 If you toss away the directional information by forcing the light to be scattered multiple times as it goes through the medium, you greatly shorten the distance scale of particle displacements required to dephase the light to make it fluctuate and do DLS. In that case, you can measure displacements on the order of 1 .sub-nanoscale rheology. The light sort of diffuses through the medium, scattering off many different particles. Directional information is totally lost. Distances shortened greatly. Time scales reduced greatly.

Slide 32

Day two: let us recap. We can apply forces to particles and follow their motion; a given response for a given force is just like conventional rheology. This is active microrheology. 32 Passive forms of microrheology rely on the particles innate tendency to wiggle about by thermal motion. No setup is needed to make that happen, just to observe it. There is no free lunch (except in polymer science) so whatever you gain in simplicity of measurement is paid for by complexity of analysis. Today it is our grim duty to do what we can to understand something of that analysis. This wont be pretty: http://www.oddtidings.com/http://www.oddtidings.com/ This wont be pretty: http://www.oddtidings.com/http://www.oddtidings.com/

Slide 33

This is going to involve Langevin, Laplace and Fourier. 33 http://en.wikipedia.org/wiki/Paul_Langevin http://www.encyclopedia.com/topic/Paul_Langevin.aspxhttp://www.encyclopedia.com/topic/Paul_Langevin.aspx (much better) Son of appraiser; student of P. Curie; may have had affair with M. Curie; trained de Broglie; research from anti-submarine warfare to diffusion theory. Wife did not understand him. Langevin: 1800s - 1900s http://en.wikipedia.org/wiki/Pierre-Simon_Laplace According to his great-great-grandson, [3] d'Alembert received him rather poorly, and to get rid of him gave him a thick mathematics book, saying to come back when he had read it. When Laplace came back a few days later, d'Alembert was even less friendly and did not hide his opinion that it was impossible that Laplace could have read and understood the book. But upon questioning him, he realized that it was true, and from that time he took Laplace under his care. [3] Laplace: 1700s 1800s http://en.wikipedia.org/wiki/Joseph_Fourier Orphan son of a tailor. Fourier: 1700s 1800x

Slide 34

34 The Laplace operator is an INTEGRAL operator. We are more familiar with differential operators, like this one: Space to write: Factoid: the little triangle is called the Laplacian, so our guy did differential operators, too.

Slide 35

Space to write: Notice: finite vs infinite limits; only positive; no oscillations, and s both have units of inverse time, but one is an oscillation frequency and the other a decay rate. Compare Laplace and Fourier integral operators. 35

Slide 36

WHY do we have these integral operators? What good do they do? 36 Chemists usually know Fourier synthesis: how much of each wave it takes to make a given signal. We even saw AND HEARD this stuff SquareWavesAndFT.vi SquareWavesAndFT.vi (local link only) Chemists know the FID signals of NMR FID_TwoDampedCosines.vi FID_TwoDampedCosines.vi (local link only) The NMR measures in the time domain; we convert this to frequency domain by FT.

Slide 37

DLS people know the same kind of synthesis of exponentials (not sine or cosine waves) is what makes up their measured signal, g (1) (t). 37 Space to write By convention, the decay rate in DLS is called , not s. Otherwise, its exactly the form of a Laplace transform. One could say the DLS experiment transforms a decay rate distribution into a correlation function g (1) (t). Happens because each diffuser contributes an exponential term.

Slide 38

Laplace inversion is a momentous development.* In DLS, you try to invert the measured g (1) (t) data to obtain how importantor momentouseach term A( ) is. It is moment analysis. Space to write *LI returns the moments of a function, in our case exp(-gt). The order of the moments does not concern us here. Laplace spent the last part of his career on statistics, and his transform has a role there, too. By contrast, Fourier transformation resolves a signal into its sine or cosine waves of varying importance. 38

Slide 39

Methods to go backward and forward are a lot better defined for Fourier transform than for Laplace transform. Either is subject to issues like: * Noise *Windowing (how much signal to use) *Closeness of data in measured space. *Evenness of data in measured space. 39

Slide 40

Math, Engineering & Physics people learn a lot more than most chemists about Messrs. Laplace and Fourier. Fourier transformation and Laplace transformation are valuable tools for solving mathematical problems, especially differential equations. We cover a Fourier example in Chem4011, where the FT is used to solve Ficks equations for diffusion (see Cantor & Schimmel, Biophysical Chemistry on diffusion or Chem4010 Virtual Book, Ch. 15) Chem4010 Virtual Book, Ch. 15 Here we will focus on some brilliant successes of the Laplace transform. 40

Slide 41

Its easy to get L {f(t)} for some simple f(t). Space to write So.if t is time in seconds, so that s is a rate in inverse seconds, the simple constant 1 has somehow acquired units due to the Laplace transform. I dont know what to think about this. 41

Slide 42

Any constant will factor out of the integral, so L {F(t)=a} = a/s, where a = constant. So.if f(s) = a/s we know that F(t) = a. There.thats our first LT transform pair. a a/s t-space s-space unitless seconds -1 42 LT

Slide 43

Lets do another transform pair, this time for exponential growth. Space to write 43

Slide 44

Exponential growth plugs into the hyperbolic sine and cosine functions, sinh and cosh. 44

Slide 45

This gets us a route to the cosine and sine functions. 45 This kinda has the form of those Lorentzian lineshapes we saw back when we were doing harmonic oscillator/spectroscopy/electronic circuits/rheology. Its a clue that Laplace transform will be useful in all these!

Slide 46

Laplace transform pairs have been worked out for many functions. 46 http://en.wikipedia.org/wiki/Laplace_transform http://mathworld.wolfram.com/LaplaceTransform.html Of course, its in Mathematica, too. Scroll down and look at exponentially decaying sine functions, for example.

Slide 47

How does all this make our lives easier? 47 Written in Windows Journal File: Microrheo.jnt

Slide 48

Apply L to the both sides. Space to write 48

Slide 49

That integral is do-able and look-upable. 49 Space to write So the Laplace transform has helped us solve the integral, but that is not all.

Slide 50

Because we know f(s) = L (F(t)) we can look up what F(t) must be. In fact, we dont have to look it upit was the first thing we derived. Space to write 50

Slide 51

Nowlets explore a similar integral numerically, using LabView. 51 Rod Form Factor VIRod Form Factor VI (local link only) *Integral being computed has t=1 and *Upper limit of u not . So.whats /2? 1.507 Now, what do we find numerically for u = Result: Laplace transform helps us check our numerical integral is working.

Slide 52

OK, so LT helped us solve an integral. Its even better at solving differential equations. 52 So, if you know L {F(t))} all you need to do to get L {F(t)} is multiply by s and subtract F(0). It could be easier to transform to s space, do the multiplication/subtraction, then transform back to t space to get F(t) than it is to compute F(t) directly. So, if you know L {F(t))} all you need to do to get L {F(t)} is multiply by s and subtract F(0). It could be easier to transform to s space, do the multiplication/subtraction, then transform back to t space to get F(t) than it is to compute F(t) directly.