Dynamic Light Scattering (aka QLS, PCS) Oriented particles create interference patterns, each bright spot being a speckle. The speckle pattern moves as the particle move, creating flickering.
All the motions and measurements are described by correlations functions •G2(!)- intensity correlation function describes particle motion •G1(!)- electric field correlation function describes measured fluctuations
Which are related to connect the measurement and motion
2 G (! ) ) B'1( * g (! ) $ 2 %& 1 #"
Analysis Techniques: • Treatment for monomodal distributions: linear and cumulant fits • Treatment for non-monomodal distributions: Contin fits
It is also possible to measure other motions, such as rotation. Particles behave like ‘slits’, the orientation of which generates interference patterns
Generates a ‘speckle’ pattern
Various points reflect different scattering angles
2 Movement of the particles cause fluctuations in the pattern
The pattern ‘fluctuates’
Movement is defined by the rate of fluctuation Measure the intensity of one speckle
Experimentally, the intensity of one ‘speckle’ is measured Order of magnitude for time-scale of fluctuations
fluctuations occur on the time-scale that particles move about one wavelength of light… -x / .
Assuming Brownian motion 2 of the particles… +-x, ) Dt
The time-scale is: Change on the . ~ 500 nm msec time frame 2 +,5x1005cm t / /100m sec 2.5x1008cm2 / s
D for ~ 200 nm particles How is the time scale of the fluctuations related to the particle movement?
Requires several steps:
1. Measure fluctuations an convert into an Intensity Correlation Function 2. Describe the correlated movement of the particles, as related to particle size into an Electric-Field Correlation Function.
3. Equate the correlation functions, with the Seigert Relationship
4. Analyze data using cumulants or CONTIN fitting routines • Math/Theory 2 texts:
‘Light scattering by Small Particles’ • Application/Optics by van de Hulst ‘Dynamic Light Scattering with applications to Chemistry, Biology • Data Analysis and Physics’ by Berne and Pecora First, the Intensity Correlation Function, G2(!)
Describes the rate of change in scattering intensity by comparing the intensity at time t to the intensity at a later time (t + !), providing a quantitative measurement of the flickering of the light
I(t) I(t+!) I(t+!’) Mathematically, the correlation function is written as an integral over the product of intensities at some time and with some delay time, !
1 T G (! ) ) 1 I( t )I+,t (! d! 2 T 0
Which can be visualized as taking the intensity at I(t) times the intensity at I(t+!)- red), followed by the same product at I(t+t’)- blue, and so on… The Intensity Correlation Function has the form of an exponential decay
plot linear in ! 1.8e+8 The correlation function typically
1.6e+8 exhibits an exponential decay
) 1.4e+8 ! ( 2 G 1.2e+8 plot logarithmic in ! 1.0e+8 1.8e+8
8.0e+7 1.6e+8 0 1000 2000 3000 4000 5000 Tau (2sec)
) 1.4e+8 ! ( 2 G 1.2e+8
1.0e+8
8.0e+7 1 10 100 1000 Tau (2sec) Second, Electric Field Correlation Function, G1(!)
It is Not Possible to Know How Each Particle Moves from the Flickering
Instead, we correlate the motion of the particles relative to each other Integrate the difference in distance between particles, assuming Brownian Motion
The electric field correlation function describes correlated particle movement, and is given as:
Constructive interference 1 T G (! ) ) 1 E( t )E+,t (! d! 1 T 0
G1(t) decays as and exponential with a decay constant 345for a system undergoing Brownian motion
03! G1(! ) ) exp Destructive interference The decay constant is re-written as a function of the particle size
The decay constant is related by Brownian Motion to the diffusivity by:
46n <= 9 3 ) 0Dq2 q ) sin: 7 . ; 2 8
with q2 reflecting the distance the particle travels … and the application of Stokes-Einstein equation
Boltzmann Constant
temperature kT thermodynamic D ) ) 662r hydrodynamic
viscosity particle radius Rate of decay depends on the particle size
1.0
0.9 0.8 large particles diffuse slower than 0.7 small particles, and the correlation 0.6 )
! function decays at a slower rate. (
2 0.5 G 0.4
0.3
0.2
0.1
0.0 1.0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.9 Tau 0.8 large particle 0.7
0.6 )
and the rate of ! (
2 0.5 G other motions 0.4 small particle 0.3 (internal, rotation…) 0.2 0.1
0.0
1e-6 1e-5 1e-4 1e-3 1e-2 1e-1
Tau Finally, the two correlation function can be equated using the Seigert Relationship Based on the principle of Guassian random processes – which the scattering light usually is 2 * The Seigert Relationship is expressed as: Intensity I ) E ) E > E
2 G (! ) ) B'1( * g (! ) $ 2 %& 1 #"
Intensity Correlation Electric Field Correlation Function Function (recall: this is measured) (recall: this is what the particles are doing) where B is the baseline and * is an instrumental response, both of which are constants •G2(!) intensity correlation function measures change in the scattering intensity
•G1(!) electric field correlation function describes correlated particle movements
• The Seigert Relationship equates the functions connecting the measurable to the motions
2 G (! ) ) B'1( * g (! ) $ 2 %& 1 #" • Math/Theory
• Application/Optics
• Data Analysis So, consider a simple example of the process
Measure the intensity fluctuations from a dispersion of particles. Commercial Equipment
• Need laser, optics, correlator, etc…
• Commercial Sources – Brookhaven Instruments – Malvern Instruments – Wyatt Instruments (multiangle measurements, HPLC detectors) – ALV (what we have) • Costs range $50K to $100K Instrumental Considerations
• Light Source – Monochromatic, polarized and continuous (laser) – Static light scattering goes as 1/!4, suggests shorter wavelengths give more signal • typical Ar+ ion laser at 488 nm – Dynamic light scattering S/N goes as !, while detector sensitivity goes as 1/!, so wavelength is not too critical. HeNe lasers are cheap and compact, but weaker (! = 633 nm) – Power needed depends on sample (but there can be heating!) – Calculation of G(") depends on two photons, and so on the power/area in the cell. Typically focus the beam to about 200 #m – Sample can be as small as 1 mm in diameter and 1 mm high. Typical volumes 3-5 ml. Instrument Considerations
• Need to avoid noise in the correlation functions – Dust! • Usually adds an unwanted (slow) component • See in analysis – some software help • AVOID by proper sample preparation when possible
– Poisson Noise • counting noise, decreases with added counts, important to have enough counts; typically 107 over all with 106 at baseline
– Stray light • adds an unwanted heterodyne component (exp (-$) instead of exp (-2$). Avoid with proper design Correlators
1 T • Need to calculate G2(! ) ) 1 I( t )I+,t (! d! T 0 1 N (l arge) which is approximated by G2 (! ) / @ I(ti )I(ti (! ) N i)1 so calculate by recording I(t) and sequentially multiplying and adding the result. To do in real time requires about ns calculations thus specialized hardware
• Pike – 1970s (Royal Signals and Radar Establishment, Malvern, England) • Langley and Ford (UMASS) ? Brookhaven • 1980’s Klaus Schatzel, Kiel University ? ALV Autocorrelation function is collected
The auto-correlator collects and integrates the intensity at the different delay times, !, all in real time
! (2sec) G2+!,
Each point is a different !. 2.000000000E+000 1.593461120E+008 2.400000095E+000 1.590897440E+008 5.000000000E+000 1.582029760E+008 1.8e+8 1.000000000E+001 1.564198880E+008 1.500000000E+001 1.546673760E+008 1.6e+8 2.000000000E+001 1.529991520E+008 2.500000000E+001 1.513296000E+008
) 1.4e+8 ! ( 3.000000000E+001 1.497655360E+008 2 G 1.2e+8 3.500000000E+001 1.482144000E+008 4.000000000E+001 1.466891040E+008 1.0e+8 4.500000000E+001 1.452316800E+008 5.000000000E+001 1.438225120E+008 8.0e+7 1 10 100 1000 Tau (2sec) …
6.000000000E+00 4 9.100139200E+007 … then, create the raw correlation function
Evaluate the autocorrelation function from the intensity data
1.8e+8
1.6e+8
) 1.4e+8 ! ( 2 G 1.2e+8
1.0e+8
8.0e+7 1 10 100 1000 Tau (2sec) … then, normalize the raw correlation function through some simple re-arrangements
G (! )0 B C+,! ) 2 ) *e023! B
0.8
0.6
0.4 * c(t) * is usually less than unity, from B should 0.2 measuring more go to zero than one speckle B 0.0
1 10 100 1000
! (2sec) General principle: the measured decay is the intensity-weighted sum of the decay of the individual particles
1.0 200 nm
100 nm 300 nm 0.8 400 nm
500 nm 0.6 ) t c(
0.4
0.2
0.0
1 10 100 1000 10000 tau (2sec)
Recall that different size particles exhibit different decay rates. Expressed in mathematical terms
g1(!) can be described as the movements from individual particles; where G(3) is the intensity-weighted coefficient associated with the amount of each particle.
03i! g1+! ,) @Gi +3,e i
For example, consider a mixture of particles:
0.30 intensity-weighted of 100 nm particles, 0.25 intensity-weighted of 200 nm particles, 0.20 intensity-weighted of 300 nm particles, 0.15 intensity-weighted of 400 nm particles, 0.10 intensity-weighted of 500 nm particles. A sample correlation function would look something like this…
Short times emphasize the intensity weighted- Recall sizes average 0.30 (100 nm)
1.0 0.25 (200 nm) wieghted sum of the 0.20 (300 nm) individual decay 0.15 (400 nm) 0.8 0.10 (500 nm)
0.6 ) t c(
0.4 100 nm 200 nm 0.2 300 nm 400 nm 500 nm 0.0
1 10 100 1000 10000 tau (2sec) long times reflected the larger particles • Math/Theory
• Application/Optics
• Data Analysis Finally, calculate the size from the decay constant
3 = ??? in sec-1 (experimentally determined)
3 Diffusivity is determined… D ) 2 need refractive index, wavelength and angle D = ??? in cm2 sec-1 q 46n <= 9 q ) sin: 7 . ; 2 8
r = ??? in cm kT Calculate the radius, but r ) need the Boltzmann 662D constant, temperature and viscosity What is left? Need a systematic way to determine 3’s
the distribution of particle sizes defines the approach to fitting the decay constant
Monomodal Distribution Linear Fit
Cumulant Expansion
Non-Monomodal Distribution Exponential Sampling
CONTIN regularization What is left? Need a systematic way to determine 3’s
First, consider the monomodal distribution, where the particles have an average mean with a distribution about the mean (red box, first)
Monomodal Distribution Linear Fit
Cumulant Expansion
Non-Monomodal Distribution Exponential Sampling
CONTIN regularization Simplest- the ‘basic’ linear fit
Assumes that all the particles fall about a relatively tight mean
C(!) ln C(!) ln * 0.8 Slope = -2Dq2 0.6 1e-1 ) ) ! 0.4 ! 1e-2 C( … but, need Long ! for C( 0.2 a good B 1e-3 0.0 1e-4 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 Tau Tau Long ! there is just noise… Cumulant expansion Assumes that the particles distribution is centered on a mean, with a Gaussian-like distribution about the mean. Where to start… A 03! Integral sum of decay curves g1+! ,) 1G 3 +e , d3 0 Larger particles are ‘seen’ more… Probability Density Function (Coefficients of Expansion) G+3,) M 2 P( q )S( q ) Intraparticle Form Factor G+,3 / N(R)R6 : solid And Interparticle Form Factor that both DEPEND ON q G+,3 / N(R)R4 : hollow shell (vesicle) Then, re-arrange the Seigert Relationship in terms of a cumulant expansion Recall that the correlation function can be expressed as 'G2+! ,0 B$ lnc+,! ) ln ) ln * 0 2ln g1+,! &% B #" Cumulant expansion is a rigorous defined tool of re-writing a sum of exponential decay functions as a power series expansion… so, that the sum from the previous page is replaced by the expansion (GET BACK HERE IN A FEW MINUTES) A n A n i3! A +i! , +i! , g1+! ,) 1 e P 3 +d3 , ln g1+,! B @ kn ) 1 kn d! 0 n)0 n! 0 n! http://mathworld.wolfram.com/cumulant.html … need to carry through some mathematics First, define a mean value 03! 0+303,! g1+! ,) e e 3 is the mean ‘gamma’ 2 3 0x x x Note: power series expansion e /10 x ( 0 ( ... 2! 3! Second, substitute the power series for the difference term (second term) A A 2 03! 03! ' +,3 0 3 $ g1+! ,) 1G 3 +e ,d3 ) 1G 3 e +% ,10 +,3 0 3 ! ( ! 0 ..."d3 0 0 &% 2! #" Cumulant Expansion (more) A A 2 03! 03! ' +,3 0 3 $ g1+! ,) 1G 3 +e , d3 ) e 1G+,3 %10 +,3 0 3 ! ( ! 0 ..."d3 0 0 %& 2! #" Working through the integrals… < k 2 k 3 9 g +,! ) e03! :10 0 ( 2 ! 2 0 3 ! 3 ( ...7 1 : 7 ; 2! 3! 8 Such that k2 is the second moment, k3 is the third moment, … A A 2 3 k2 ) 1G+,3 +,3 0 3 d3 k3 ) 1G+,3 +,3 0 3 d3 0 0 Cumulant Expansion (even more) 2 3 1 G +,! 0 B$ 1 ' < k k 9$ ln' 2 ) ln * ( ln e03! :1( 2 ! 2 0 3 ! 3 ( ...7 % " % : 7" 2 & B # 2 &% ; 2! 3! 8#" a b x Note: ln of products ln( ab ) ) lna ( lnb 1 Note: power series expansion ln(1( x ) / x 0 x2 ( ... 2 Note that x terms >> x2 terms, so that x2 are negligible Cumulant Expansion (more…) 'G2+! ,0 B$ 2 2 Note: ln% " ) ln * 0 23! ( K2 ! 0 ... Multiplied & B # by 2 intercept polydispersity average decay k2 Polydispersity index C ) 32 … and indicates the width of the distribution C ) 0.005 is mono-dispersed Sample of Cumulant Expansion 390 nm Beads 'G +!, 0 B$ Gamma linear ln 2 ) ln * 0 23! &% B #" 'G +!,0 B$ 2 ~ Poly quadratic ln 2 ) ln* 0 23! ( K 2! &% B #" 2 ~ Skew 'G +,! 0 B$ K 3 cubic ln 2 ) ln* 0 23! ( K 2!2 0 3 !3 &% B #" 2 3 ~ Kurtosis 'G +,! 0 B$ K 3 K 4 quartic ln 2 ) ln* 0 23! ( K 2!2 0 3 !3 ( 4 !4 &% B #" 2 3 12 Examine residuals to the fit uncorrelated correlated What is left? Need a systematic way to determine 3’s Second, consider the different non-monomodal distribution, where the particles have a distribution no longer centered about the mean (red box, next) Monomodal Distribution Linear Fit Cumulant Expansion Non-Monomodal Distribution Exponential Sampling CONTIN regularization Multiple modes because of polydispersity, internal modes, interactions… all of what make the sample interesting! Exponential Sampling for Bimodal Distribution 03i! Assume a finite number g1+,! ) @aie of particles, each with their own decay e.g. bimodal distribution To be reliable the sizes must be ~5X different Pitfalls • Correlation functions need to be measured properly a) Good measurements with appropriate delay times b) Incomplete, missing the early (fast) decays c) Incomplete, missing the long time (slow) decays CONTIN Fit for Random Distribution Laplace Transform of f(t) Note: Fourier Transform A A 0st 0iFt F+s ,) LDEf t +) , 1 e f +,t dt G+F ,) GDEf t + ,) 1 e f+,t dt 0 0A In light scattering regime. size distribution function A 03! g1+! ,) LDEG 3 +) ,1 e G+,3 d3 0 So, to find the distribution function, apply the inverse transformation which is done by numerical methods, with a combination of minimization of variance and regularization (smoothing). 01 G+3 ,) L DEg1 ! + , CONTIN • Developed by Steve Provencher in 1980’s A 03! • Recognize that g1+! ,) LDEG 3 +) ,1 e G+,3 d3 0 is an example of a “Fredholm Integral” where F(r) ) 1 K(r, s)A(s)ds measured object of desire defines experiment This is a classic ill-posed problem – which means that in the presence of noise many DIFFERENT sets of A(s) exist that satisfy the equation CONTIN (cont.) So how to proceed? 1. Limit information – i.e., be satisfied with the mean value (like in the cumulant analysis) 2. Use a priori information – Non-negative G($) (negative values are not physical) – Assume a form for G($) (like exponential sampling) – Assume a shape 3. Parsimony or regularization – Take the smoothest or simplest solution – Regularization (CONTIN) ERROR = (error of fit) +function of smoothness (usually minimization of second derivative) – Maximum entropy methods (+ p log (p) terms) Analysis of Decay Times First question: How do decay times vary with q? Finite Rotational Diffusion Diffusion (translation) Slope = Dapp Slope = Dapp 3 3 2 0+Dq 06Dr ,! g1+,! / e Dr q2 q2 2 $= Dappq where Dapp is a Rotational diffusion can collective diffusion coefficient change the offset of the that depends on interactions decay – can also observe with and concentration depolarized light Not spheres… but still dilute, so D = kT/f 1 Cylinders D ) +,D1 ( 2D2 3 D1 D1 KT D ) ln+,L Worms D 36HoL kT D ) Shape factor: A hydrodynamic term f that depends on shape 2 1/ 2 Prolate '10 +b ,$ % a " f ) & # 1/ 2 b < 2 9 :1( '10 +,b $ 7 2/3 % a " +,b ln: & # 7 a : b 7 a : a 7 ; 8 Concentration Dependence • In more concentrated dispersions (and can only find the definition of ‘concentrated’ generally by experiment’), measure a proper Dapp, but because of interactions Dapp (c) • Again, D = kT(1 + f(B) + …)/fo(1 + kfc + …) So Dapp= D0 (1 + kDc + …) like a second virial coefficient for diffusion with kD = 2B –kf – %2 partial molar volume of solute (polymer or micellar colloid) Virial Coefficient < IJ 9 2 01 • Driving force = : 7 ) kT[1( 46K1 drr (g(r) 01] ; IK 8T at low density / kT[1- 46K1 (g(r) -1)r 2dr (...] so for low density < IJ 9 : 7 / kT[1( KB2 (...] ; IK 8T where B ) 046 (g(r) 01)r 2dr 2 1 Multiple Scattering single scattering multiple scattering •Three approaches • Experimentally thin the sample or reduce contrast • Correct for the effects experimentally • Exploit it! Diffusing Wave Spectroscopy (DWS) • In an intensely scattering solution, the light is scattered so many times the progress of the light is essentially a random walk or diffusive process • Measure in transmission or backscattering mode • Probes faster times than QLS • See Pine et al. J. Phys. France 51 (1990) 2101-2127 Summary Oriented particles create interference patterns, each bright spot being a speckle. The speckle pattern moves as the particle move, creating flickering. All the motions and measurements are described by correlations functions •G2(!)- intensity correlation function describes particle motion •G1(!)- electric field correlation function describes measured fluctuations Which are related to connect the measurement and motion 2 G (! ) ) B'1( * g (! ) $ 2 &% 1 #" Analysis Techniques: • Treatment for monomodal distributions: linear and cumulant fits • Treatment for non-monomodal distributions: Contin fits • Interactions, polydispersity, require careful modeling to interpret Other motions, such as rotation, can be measured