Correlation Spectroscopy in Polymer PhysicsMethodenseminar im Wahlpflichtfach 3

1. Basics• diffusion and brownian motion• correlations functions

2. Dynamic light scattering (DLS)• DLS on cellulose solutions

3. Fluorescence correlation spectroscopy (FCS)• hybridization dynamics of RNA/DNA

Literature:• B. J. Berne, R. Pecora, “Dynamic Light Scattering: With Applications

to Chemistry, Biology, and Physics”, Dover Publications • C. Zander, J. Enderlein, R. A. Keller (Eds.), “Single-Molecule

Detection in Solution - Methods and Applications”, VCH-Wiley

1. Basics

1.1 Macroscopic diffusion

1

2

1. remove wall2. particles (gas) expand to the right

x

• Fick‘s first law:

particles passingthrough the plane

per area

diffusioncoefficient

gradient of number density

no net flow when particlesare evenly distributed!

D ~ average “speed” of particles, determined by thermal motion

in a gas:

mean free path between collisions

avg. velocity

(Maxwell)

in a liquid: Stokes-Einstein law

drag/friction coefficient

solvent viscosity

hydrodynamic radius (size determination!)

1. Basics 1.1 Macroscopic diffusion

• relation between flux and concentration/density changeΔx

A

x x+ΔxΔV

volume V = A Δxdensity

between t and t + Δt:

gain by flux through A at x

loss by flux through A at x+Δx

change of particle number in ΔV

≡ derivative! let (density change),

continuity equation

1. Basics 1.1 Macroscopic diffusion

• combine Fick‘s first law and continuity equation:

Fick’s second law, diffusion equation

- second-order differential equation!- solution with appropriate initial and boundary conditions

- is a function of space and time- replace by c(x,t) in a solution (= concentration)

• point source, 1D solution:

initial condition:boundary conditions:

t=0

σx

t1

t2

1. Basics

1.2 Brownian motion

• describe the random motion of individual particles(“self motion” or “self diffusion”)

• first observed by Brown on cell organellae

r

simple 1D derivation for end distance:

1. Basics 1.2 Brownian motion

+l +l +l−l

random step, uniform length per time Δt:

x

statistics (many walks)

average

distance traveled:

0 (uncorrelated random numbers!)

compare with Einstein-Smoluchowski equation (full statistical/kinetical description of single-particle motion):

but:

1. Basics

1.3 Correlation functions

χ: general property, e.g., density, intensitya: variable, e.g., r (space), t (time), …

• define autocorrelation function (ACF) of χ(a):

product-average, evaluated at constant differencea = c−b

• in a system, where χ(a) varies randomly, Cχ(a) is often a simple, analytical function that describes the basic statistical propertiese.g. for - instantaneous positions of molecules in liquids

- random location or velocity of Brownian particles

• Cχ(a) is often directly measured, e.g., in scattering experiments, NMR, or…

1. Basics 1.3 Correlation functions

• example: random hopping (Markov process)

χ

a1

b or c

<χ>

a

Cχ(a)

<χ>2

τ

auto-correlate

Cχ(a)χ

<χ>

b or c a<χ>2

τ

very often: Cχχχχ(a) ~ e−−−−a/ττττ

ττττ: decay constant, correlation time

a2>a1

so you can correlate the differences from the average!

long times: different signas likely as equal sign

⇒ cancellation!

white noise (“most random” variation of ν):

2. Dynamic light scattering (DLS)

2.1 The light scattering experiment

2. Dynamic light scattering (DLS)

2.2 Rayleigh scattering

• interaction of an incoming (vertically polarized) wave

with a molecule (or a fragment) with polarizability α

⇒ induced dipole moment x

z

y

α

φz

φy

φx=θ

• oscillating dipole emits a scattered spherical waveaccording to the Maxwell eqs. (accelerated charge!):

⇒ intensity ratio (scattered vs. incoming), use

(!! blue sky, red sunset…)

• modifications when scattering center moves:⇒ internal dynamics (vibrations etc.): polarizability changes, α = f(t), Raman effect!⇒ molecule moves: Doppler shift/broadening (δωD ~ ω<vz>/c for absorption lines)

2. Dynamic light scattering (DLS)

2.3 Interference and scattering vector

1

2

θ/2

θ

b

a

θ/2

· ·

• note: for constructive interference, Δφ = n 2π

scattering length =efficiencyin solution: depends on

refractive index mismatch!( n=f(α) )

• phase shift Δφ depends on path length difference δ=a−b (simple trigonometry, |k|=2π/λ0):

includes the definition of the scattering vector (quantifies momentum transfer)

• interference = superposition

for many particles, independent of origin (e.g., r1=0)

2. Dynamic light scattering (DLS)

2.4 Basic phenomenon and spectroscopic approach• Doppler effect of diffusing scattering centers, ri(t):

• line width from FT{idet(t)}: δω ~ q2Dc

Dc: collective diffusion coefficient (particles relative to each other)= Dself for c → 0

time-dependent scattered intensity (damped)!

~q2Dc

ω

Brillouin doublet (phonon modes)or Raman lines

• energy change δω ~ q2Dc related to Doppler broadening:~ 10 meV in a liquid a room temperature

• typical vis. photon energy: 1.7−3 eV

⇒ very small effect!, measurable only with • narrow-banded lasers• Fabry-Perot interferometers

(~ since 1960)

2. Dynamic light scattering (DLS)

2.5 Time correlation• idea: direct observation of intensity fluctuations,

possible for small scattering volume (⇔ static LS: large volume, fluctuations are averaged!)

Ci(t)idet

<idet>

t t<idet>2

1/2q2Dc

auto-correlate

(autocorrelation done in real time on a fast

computer card, Δt~100ns)

• DLS terminology:

intensity ACF

Siegert relation

ACF of the scattered

electric field

Γ: “first cumulant”

for single species: simple exponential!

2. DLS 2.5 Time correlation

• in practice for polymers (large molecules, many scattering centers):

Γ = q2Dapp

apparent diffusion coefficent, depends on

• intramolecular interferences, intramolecular motions ⇒ size effect, q dependence • hydrodynamic interactions between particles ⇒ concentration (c) dependence

series expansion in c and q:

⇒ do extrapolation for c → 0, q → 0

cellulose (cotton) =

in solution:random coil

in bulk: semicrystallinepolymer- hardly soluble- chain length?

Rh ~ molecular weight

2. DLS

K. Saalwächter, W. Burchard, et al., Macromolecules 33 (2000), 4094

CdNN N

N

CdNN N

N

Cd NNN

N

Cuoxam: “Schweizers reagent”, copper sulfate - ammonia solution

solvent aggregates!

Cd-tren: new high-tech coordinating solvent

bigger coil!

2.6 DLS on cellulose solutions

2. DLS 2.6 DLS on Cellulose solutions

“dynamic Zimm plots“to remove c (concentration) and q (angle) dependencies:

K. Saalwächter, W. Burchard, et al., Macromolecules 33 (2000), 4094

3. Fluorescence correlation spectroscopy

3.1 Basic concepts

Ei

EjEk

internal conversion (fast, radiationless)

hν

hν’

weakly allowed transition(lifetime ~10−9 s)

termenergy

• fluorescence: red-shift of absorbed light

~0.5 μm

• confocal detection/imagingLaser

dichroic mirror

single-photondetection

pinhole or fibre aperture

objectivetube lens

~3 μm

focal/detection volume

3. Fluorescence correlation spectroscopy

Emissions filter

Dichroic mirror

excitation wavelength

mirrortransmission

3.2 Apparatus

3. Fluorescence correlation spectroscopy

3.3 Intensity fluctuations and normalized autocorrelation function

5.6 5.8 6.0 6.2 6.40

50

100

150

200

250

300

inte

nsity

[cou

nts

per

ms]

time [s]

5.3 5.5 5.7 5.9 6.10

100

200

300

400

500

600

inte

nsity

[cou

nts

per

ms]

<i>

<i>

highconcentration

lowconcentration

focal volume~2 fL (2E-15)

1 molecule/2fL≈ 1 nmol/l

extremely high sensitivity!!⇒ also “single-molecule” spectroscopy (note: one molecule, but many photon cycles)

t (ms)

1E-4 0.01 1 100 1E+40.0

4.0

8.0

10.0

12.0

14.0

time [ms]

1/<N>

t (ms)

1E-4 0.01 1 100 1E+40.0

0.4

0.8

1.2

1.6

2.0

auto-correlate

Ci(t

)/<

i>2

[nor

mal

ized

]C

i(t)/

<i>

2 [n

orm

aliz

ed]

1/<N>

Ci(t)/<i>2 → 1

<N> ~ number of molecules

in focus

3. FCS 3.3 Autocorrelation function

• remove offset: normalized autocorrelation function

inverse number of particles in the focus

• theory assuming Brownian motion (free diffusion) and Gaussian detection volume

wz

wxy

1/e2-widths of detection volume

lateral diffusion time

average particle number in focus

• including triplet dynamics (triplet fraction fT, triplet lifetime τT) particles in triplet state are invisible

for a certain time τT

3. FCS 3.3 Autocorrelation function

• RNA: mediates between genetic code (DNA) and protein synthesis• virus replication (e.g., AIDS) is based on certain RNAs• replication (binding of reverse transcriptase) can be hindered by hybridization

(=binding and blocking) with „anti-sense“-DNA • understanding of hybridization dynamics (accompanied by unfolding of RNA)

⇒ powerful drugs on the basis of DNA!

asDNA

RNA

rev. transcr.

3. FCS 3.4 Hybridization dynamics of RNA with DNA probes

P. Schwille et al., Biochemistry35 (1996), 10182

2-component fit:

⇒ details on binding kinetics!

P. Schwille et al., Biochemistry35 (1996), 10182

3. FCS 3.4 Hybridization dynamics of RNA with DNA probes

known: τ1 (free DNA) < τ2 (hybr. DNA), determined: rel. population Y1,2 = N1,2/(N1+N2)