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