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## Dynamic Light Scattering

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**Dynamic Light Scattering**ZetaPALS w/ 90Plus particle size analyzer Also equipped w/ BI-FOQELS & Otsuka DLS-700 (Rm CCR230)**Dynamic Light Scattering (DLS)**Photon Correlation Spectroscopy (PCS) Quasi-Elastic Light Scattering (QELS) • Measure Brownian motion by … • Collect scattered light from suspended particles to … • Obtain diffusion rate to … • Calculate particle size**Brownian motion**• Velocity of the Brownian motion is defined by the Translational Diffusion Coefficient (D) • Brownian motion is indirectly proportional to size • Larger particles diffuse slower than smaller particles • Temperature and viscosity must be known • Temperature stability is necessary • Convection currents induce particle movement that interferes with size determination • Temperature is proportional to diffusion rate • Increasing temperature increases Brownian motion**Brownian motion**Random movement of particles due to bombardment of solvent molecules**Stokes-Einstein Equation**dH = hydrodynamic diameter (m) k = Boltzmann constant (J/K=kg·m2/s2·K) T = temperature (K) η = solvent viscosity (kg/m·s) D = diffusion coefficient (m2/s)**Hydrodynamic diameter**Particle diameter • The diameter measured by DLS correlates to the effective particle movement within a liquid • Particle diameter + electrical double layer • Affected by surface bound species which slows diffusion Hydrodynamic diameter**Nonspherical particles**= Rapid Equivalent sphere Slow Hydrodynamic diameter is calculated based on the equivalent sphere with the same diffusion coefficient**Experimental DLS**• Measure the Brownian motion of particles and calculate size • DLS measures the intensity fluctuations of scattered light arising from Brownian motion • How do these fluctuations in scattered light intensity arise?**What causes light scattering from (small) particles?**• Explained by JW Strutt (Lord Rayleigh) • Electromagnetic wave (light) induces oscillations of electrons in a particle • This interaction causes a deviation in the light path through an angle calculated using vector analysis • Scattering coefficient varies inversely with the fourth power of the wavelength**Interaction of light with matterRayleigh approximation**• For small particles (d ≤ λ/10), scattering is isotropic • Rayleigh approximation tells us that I α d6 I α 1/λ4 where I = intensity of scattered light d = particle diameter λ = laser wavelength**Mie scattering from large particles**• Used for particles where d ~ λ0 • Complete analytical solution of Maxwell’s equations for scattering of electromagnetic radiation from spherical particles • Assumes homogeneous, isotropic and optically linear material Stratton, A. Electromagnetic theory, McGraw-Hill, New York (1941) www.lightscattering.de/MieCalc**Brownian motion and scattering**Constructive interference Destructive interference**Intensity fluctuations**• Apply the autocorrelation function to determine diffusion coefficient • Large particles – smooth curve • Small particles – noisy curve**Determining particle size**• Determine autocorrelation function • Fit measured function to G(τ) to calculate Γ • Calculate D, given n*, θ, and Γ • Calculate dH, given T* and η* *User defined values.**How a correlator works**• Random motion of small particles in a liquid gives rise to fluctuations in the time intensity of the scattered light • Fluctuating signal is processed by forming the autocorrelation function • Calculates diffusion**How a correlator works**• Large particles – the signal will be changing slowly and the correlation will persist for a long time • Small, rapidly moving particles – the correlation will disappear quickly**The correlation function**• For monodisperse particles the correlation function is • Where • A= baseline of the correlation function • B=intercept of the correlation function • Γ=Dq2 • D=translational diffusion coefficient • q=(4πn/λ0)sin(θ/2) • n=refractive index of solution • λ0=wavelength of laser • θ=scattering angle**The correlation function**• For polydisperse particles the correlation function becomes where g1(τ) is the sum of all exponential decays contained in the correlation function**Broad particle size distribution**• Correlation function becomes nontrivial • Measurement noise, baseline drifts, and dust make the function difficult to solve accurately • Cumulants analysis • Convert exponential to Taylor series • First two cumulants are used to describe data • Γ = Dq2 • μ2 = (D2*-D*2)q4 • Polydispersity = μ2/ Γ2**Cumulants analysis**• The decay in the correlation function is exponential • Simplest way to obtain size is to use cumulants analysis1 • A 3rd order fit to a semi-log plot of the correlation function • If the distribution is polydisperse, the semi-log plot will be curved • Fit error of less than 0.005 is good. 1ISO 13321:1996 Particle size analysis -- Photon correlation spectroscopy**Cumulants analysis**• Third order fit to correlation function • b = z-average diffusion coefficient • 2c/b2 = polydispersity index • This method only calculates a mean and width • Intensity mean size • Only good for narrow, monomodal samples • Use NNLS for multimodal samples**Polydispersity index**• 0 to 0.05 – only normally encountered w/ latex standards or particles made to be monodisperse • 0.05 to 0.08 – nearly monodisperse sample • 0.08 to 0.7 – This is a mid-range polydispersity • >0.7 – Very polydisperse. Care should be taken in interpreting results as the sample may not be suitable for the technique (e.g., a sedimenting high size tail may be present)**Non-Negatively constrained Least Squares (NNLS) algorithm**• Used for Multimodal size distribution (MSD) • Only positive contributions to the intensity-weighted distribution are allowed • Ratio between any two successive diameters is constant • Least squares criterion for judging each criterion is used • Iteration terminates on its own**Correlation functionCorrelograms**Correlograms show the correlation data providing information about the sample The shape of the curve provides clues related to sample quality • Decay is a function of the particle diffusion coefficient (D) • Stokes-Einstein relates D to dH • z-average diameter is obtained from an exponential fit • Distributions are obtained from multi-exponential fitting algorithms Noisy data can result from • Low count rate • Sample instability • Vibration or interference from external source**Data interpretationCorrelograms**• Very small particles • Medium range polydispersity • No large particles/aggregrates present (flat baseline)**Data interpretationCorrelograms**• Large particles • Medium range polydispersity • Presence of large particles/agglomerates (noisy baseline)**Data interpretationCorrelograms**• Very large particles • High polydispersity • Presence of large particles/agglomerates (noisy baseline)**Upper size limit of DLS**• DLS will have an upper limit wrt size and density • When particle motion is not random (sedimentation or creaming), DLS is not the correct technique to use • Upper limit is set by the onset of sedimentation • Upper size limit is therefore sample dependent • No advantage in suspending particles in a more viscous medium to prevent sedimentation because Brownian motion will be slowed down to the same extent making measurement time longer**Upper size limit of DLS**• Need to consider the number of particles in the detection volume • Amount of scattered light from large particles is sufficient to make successful measurements, but … • Number of particles in scattering volume may be too low • Number fluctuations – severe fluctuations of the number of particles in a time step can lead to problems defining the baseline of the correlation function • Increase particle concentration, but not too high or multiple scattering events might arise**Detection volume**Detector Laser**Lower particle size limit of DLS**• Lower size limit depends on • Sample concentration • Refractive index of sample compared to diluent • Laser power and wavelength • Detector sensitivity • Optical configuration of instrument Lower limit is typically ~ 2 nm**Sample preparation**• Measurements can be made on any sample in which the particles are mobile • Each sample material has an optimal concentration for DLS analysis • Low concentration → not enough scattering • High concentration → multiple scattering events affect particle size**Sample preparation**• Upper limit governed by onset of particle/particle interactions • Affects diffusion speed • Affects apparent size • Multiple scattering events and particle/particle interactions must be considered • Determining the correct particle concentration may require several measurements at different concentrations**Sample preparation**An important factor determining the maximum concentration for accurate measurements is the particle size**Sample concentrationSmall particles**• For particle sizes <10 nm, one must determine the minimum concentration to generate enough scattered light • Particles should generate ~ 10 kcps (count rate) in excess of the scattering from the solvent • Maximum concentration determined by the physical properties of the particles • Avoid particle/particle interactions • Should be at least 1000 particles in the scattering volume**Sample preparation**• When possible, perform DLS on as prepared sample • Dilute aqueous or organic suspensions • Alcohol and aggressive solvents require a glass/quartz cell • 0.0001 to 1%(v/v) • Dilution media (1) should be the same (or as close as possible) as the synthesis media, (2) HPLC grade and (3) filtered before use • Chemical equilibrium will be established if diluent is taken from the original sample • Suspension should be sonicated prior to analysis**Checking instrument operation**• DLS instruments are not calibrated • Measurement based on first principles • Verification of accuracy can be checked using standards • Duke Scientific (based on TEM) • Polysciences**Count rate and z-average diameter Repeatability**• Perform at least 3 repeat measurements on the same sample • Count rates should fall within a few percent of one another • z-average diameter should also be with 1-2% of one another**Count rateRepeatability problems**• Count rate DECREASES with successive measurements • Particle sedimentation • Particle creaming • Particle dissolution or breaking up • Resolution • Prepare a better, stabilized dispersion • Get rid of large particles • Coulter**Count rateRepeatability problems**• Count rate is RANDOM with successive measurements • Dispersion instability • Sample contains large particles • Bubbles • Resolution • Prepare a better, stabilized dispersion • Remove large particles • De-gas sample**Z-average diameterRepeatability problems**• Size DECREASES with successive measurements • Temperature not stable • Sample unstable • Resolution • Allow plenty of time for temperature equilibration • Prepare a better, stabilized dispersion**Repeatability of size distributions**• The sized distributions are derived from a NNLS analysis and should be checked for repeatability as well • If distributions are not repeatable, repeat measurements with longer measurement duration**References**• http://www.bic.com/90Plus.html • http://www.brainshark.com/brainshark/vu/view.asp?text=M913802&pi=62212 • http://www.malvern.co.uk/malvern/ondemand.nsf/frmondemandview • http://www.brainshark.com/brainshark/vu/view.asp?text=M913802&pi=96389 • http://www.brainshark.com/brainshark/vu/view.asp?text=M913802&pi=73504 • http://physics.ucsd.edu/neurophysics/courses/physics_173_273/dynamic_light_scattering_03.pdf • http://www.brookhaven.co.uk/dynamic-light-scattering.html • Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics, Bruce J. Berne and Robert Pecora, DOVER PUBLICATIONS, INC. Mineola. New York • Scattering of Light & Other Electromagnetic Radiation, Milton Kerker, Academic Press (1969)**Evaluating the correlation function**• If the intensity distribution is a fairly smooth peak, there is little point in conversion to a volume distribution using Mie theory • However, if the intensity plot shows a substantial tail or more than one peak, then a volume distribution will give a more realistic view of the importance of the tail or second peak • Number distributions are of little use because small error in data acquisition can lead to huge error in the distribution by number and are not displayed