Correlation Spectroscopy in Polymer Physics Methodenseminar im Wahlpflichtfach Basics diffusion and brownian motion correlations functions

Transcription

1 Correlation Spectroscopy in Polymer Physics Methodenseminar 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 2 1. remove wall 2. particles (gas) expand to the right Fick s first law: 1 particles passing through the plane per area x diffusion coefficient gradient of number density no net flow when particles are evenly distributed! D ~ average speed of particles, determined by thermal motion in a gas: in a liquid: Stokes-Einstein law avg. velocity mean free path between collisions (Maxwell) drag/friction coefficient solvent viscosity hydrodynamic radius (size determination!)

2 1. Basics 1.1 Macroscopic diffusion relation between flux and concentration/density change volume V = A Δx density Δx A between t and t +Δt: x ΔV x+δx gain by flux through A at x loss by flux through A at x+δx change of particle number in ΔV let (density change), derivative! 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 t 1 t2 σ x

3 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 1. Basics 1.2 Brownian motion simple 1D derivation for end distance: statistics (many walks) x l +l +l +l r random step, uniform length per time Δt: distance traveled: average but: 0 (uncorrelated random numbers!) compare with Einstein-Smoluchowski equation (full statistical/kinetical description of singleparticle motion):

4 1. Basics 1.3 Correlation functions χ: general property, e.g., density, intensity a: variable, e.g., r (space), t (time), define autocorrelation function (ACF) of χ(a): product-average, evaluated at constant difference a = c b in a system, where χ(a) varies randomly, C χ (a) is often a simple, analytical function that describes the basic statistical properties e.g. for - instantaneous positions of molecules in liquids - random location or velocity of Brownian particles C χ (a) is often directly measured, e.g., in scattering experiments, NMR, or 1. Basics 1.3 Correlation functions example: random hopping (Markov process) χ <χ> χ <χ> a 1 a 2 >a 1 b or c autocorrelate b or c C χ (a) <χ> 2 C χ (a) <χ> 2 τ τ long times: different sign as likely as equal sign cancellation! very often: C χ (a) ~ e a/τ a τ: decay constant, correlation time a so you can correlate the differences from the average! white noise ( most random variation of ν):

5 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 z φ z y with a molecule (or a fragment) with polarizability α induced dipole moment φ x =θ α x φ y oscillating dipole emits a scattered spherical wave according 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 (δω D ~ ω<v z >/c for absorption lines)

6 2. Dynamic light scattering (DLS) 2.3 Interference and scattering vector θ/2 1 b θ/2 scattering length =efficiency in solution: depends on refractive index mismatch! ( n=f(α) ) θ 2 a phase shift Δφ depends on path length difference δ=a b (simple trigonometry, k =2π/λ 0 ): includes the definition of the scattering vector (quantifies momentum transfer) note: for constructive interference, Δφ = n 2π interference = superposition for many particles, independent of origin (e.g., r 1 =0) 2. Dynamic light scattering (DLS) 2.4 Basic phenomenon and spectroscopic approach Doppler effect of diffusing scattering centers, r i (t): time-dependent scattered intensity (damped)! line width from FT{i det (t)}: δω ~ q 2 D c D c : collective diffusion coefficient (particles relative to each other) = D self for c 0 energy change δω ~ q 2 D c related to Doppler broadening: ~ 10 mev in a liquid a room temperature typical vis. photon energy: ev very small effect!, measurable only with narrow-banded lasers Fabry-Perot interferometers (~ since 1960) Brillouin doublet (phonon modes) or Raman lines ~q 2 D c ω

7 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!) i det <i det > t C i (t) autocorrelate <i det > 2 1/2q 2 Dc (autocorrelation done in real time on a fast computer card, Δt~100ns) t DLS terminology: intensity ACF Siegert relation for single species: simple exponential! ACF of the scattered electric field Γ: first cumulant 2. DLS 2.5 Time correlation in practice for polymers (large molecules, many scattering centers): Γ = q 2 D app 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

8 2. DLS 2.6 DLS on cellulose solutions in solution: random coil cellulose (cotton) = R h ~ molecular weight in bulk: semicrystalline polymer - hardly soluble -chainlength? Cuoxam: Schweizers reagent, copper sulfate - ammonia solution N solvent aggregates! bigger coil! N Cd N N Cd-tren: new high-tech coordinating solvent N Cd N N N N Cd N N N K. Saalwächter, W. Burchard, et al., Macromolecules 33 (2000), 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

9 3. Fluorescence correlation spectroscopy 3.1 Basic concepts fluorescence: red-shift of absorbed light term energy E j hν E i internal conversion (fast, radiationless) E k weakly allowed transition (lifetime ~10 9 s) hν confocal detection/imaging pinhole or fibre aperture tube lens Laser objective ~3 μm single-photon detection ~0.5 μm dichroic mirror focal/detection volume 3. Fluorescence correlation spectroscopy 3.2 Apparatus excitation wavelength mirror transmission Dichroic mirror Emissions filter

10 3. Fluorescence correlation spectroscopy 3.3 Intensity fluctuations and normalized autocorrelation function high concentration low concentration focal volume ~2 fl (2E-15) 1 molecule/2fl 1 nmol/l intensity [counts per ms] intensity [counts per ms] extremely high sensitivity!! time [s] also single-molecule spectroscopy (note: one molecule, but many photon cycles) <i> <N> ~ number of molecules in focus <i> C i (t)/<i> 2 [normalized] C i (t)/<i> 2 [normalized] autocorrelate /<N> 1/<N> t (ms) C i (t)/<i> E E+4 t (ms) 0.0 1E E+4 time [ms] 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 1/e 2 -widths of detection volume lateral diffusion time w z average particle number in focus w xy including triplet dynamics (triplet fraction f T, triplet lifetime τ T ) particles in triplet state are invisible for a certain time τ T

11 3. FCS 3.3 Autocorrelation function 3. FCS 3.4 Hybridization dynamics of RNA with DNA probes 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! RNA asdna rev. transcr. P. Schwille et al., Biochemistry 35 (1996), 10182

12 3. FCS 3.4 Hybridization dynamics of RNA with DNA probes 2-component fit: known: τ 1 (free DNA) < τ 2 (hybr. DNA), determined: rel. population Y 1,2 = N 1,2 /(N 1 +N 2 ) details on binding kinetics! P. Schwille et al., Biochemistry 35 (1996), 10182