UGC-DAE Consortium for Scientific Research, Mumbai Light Scattering studies on Colloids & Gels Goutam Ghosh

Elastic Light Scattering Brownian motion  UGC-DAE Consortium for Scientific Research, Mumbai  scattering angle q = Ik s – k i I = scattering vector Ik s I = Ik i I = 2  / q = - + I ~ (  p –  s ) V 2 P(q) S(q) Case 1 I ~ I(t) ~ n(t) ~  (t) Case 2 Dynamic Light Scattering (DLS) Static Light Scattering (SLS) kiki ksks Count scattered intensity as a function of q or time (t) d =  / q  t ~ 50 nSec I ~ (  p –  s ) V 2 P(q) kiki  kiki ksks q

UGC-DAE Consortium for Scientific Research, Mumbai   y x z Scattering vector, q kiki ksks Incident Scattered Basic scattering geometry  Angle between direction of polarization and scattering plane,   Scattering angle,   Scattering vector, q = k i - k s Source: Vertically polarized, monochromatic ( = 532 nm) laser light. Classical theory of Scattering

UGC-DAE Consortium for Scientific Research, Mumbai Oscillating electric field Induced dipole moment Scattered Electric field Scattered intensityRayleigh law True for particles whose size less than  /20 Classical theory of Scattering - +

UGC-DAE Consortium for Scientific Research, Mumbai Polarization dependence I For perpendicular polarization,  = 90, for all  II For parallel polarization,  = 0 when  = 90 and  = 90 when  = 0 and 180 III For unpolarized light Rayleigh ratio Angular distribution independent of size Classical theory of Scattering

UGC-DAE Consortium for Scientific Research, Mumbai LS P1 L S PD PM P2  C PMT Light Scattering Set up LS → Laser source (100 mW, He-Ne, 532 nm vertically polarized) P1 and P2 → Linear Polarizers L → Lens used to focus the incident beam at the sample PD → Photo-Diode PM → Power meter PMT → Photo-multiplier Tube detector C → Computer for data collection  → Scattering angle VVVH

Indirect (reciprocal space) Low resolution (DLS) Interference (dust) Average structure Interactions Liquid phase Fast and wide range (DLS) Non-destructive UGC-DAE Consortium for Scientific Research, Mumbai Light Scattering method

UGC-DAE Consortium for Scientific Research, Mumbai Light, neutron and X-ray scattering Size range of different scattering methods Dimension (nm) SANS, SAXS USANS SLS RGD MIE DLS Comparison (Static scattering) - Different length scales - scattering power light (refractive index) x-rays (electron density) neutrons (scattering length density) Works in a range where optical microscopy fails!

Systems studied using Light Scattering method  Colloidal solutions - Surfactants - Polymers - Drugs - Nanoparticles  Gels UGC-DAE Consortium for Scientific Research, Mumbai In a colloidal solution particles execute Brownian motion in the entire volume. In a gel a macroscopic network is formed and no Brownian motion exist.

UGC-DAE Consortium for Scientific Research, Mumbai Static Light Scattering I ~ (  p –  s ) V 2 P(q) S(q) Small limit, qr <<1, where Form factor for non-interacting particles, S(q) = 1 P(q) = F(q) 2 =, 50 nm I(q)= I(0) exp(-q 2 R g 2 /3) Guinier law For sphere For non-interacting particles, S(q) = 1 Non-aqueous solution Infinitely dilute solution in water Moderate concentration (vol. frac. < ) in water with salt concentration > 10 mM

UGC-DAE Consortium for Scientific Research, Mumbai P(q) for a large sphere & S(q) = nm450 nm Static Light Scattering

UGC-DAE Consortium for Scientific Research, Mumbai Interparticle Structure factor, S(q) Static Light Scattering When the positions are correlated S(q), Concentration dependence Vol. Fraction Charge Ionic strength g(r) represents the probability of finding another particle at a distance r and r+dr Small particle limit As concentration increases, peak develops at q ~  /d I(q) ~ (  p –  s ) V 2 P(q) S(q)

UGC-DAE Consortium for Scientific Research, Mumbai Dynamic Light Scattering (DLS) Also known as Photon Correlation Spectroscopy (PCS) and Quasi-Elastic Light Scattering (QELS)

UGC-DAE Consortium for Scientific Research, Mumbai Dynamic Light Scattering (colloidal system) Time Resolved Experiment Number density changes with time Net intensity changes with time Diffusion rate depends on particle size, medium viscosity, temperature Auto-correlation function “large” slow moving particles “small” fast moving particles I( t ) t (  S)  S  g (2) (  ) q r(t) I ~ I(t) ~ n(t) ~  (t)

UGC-DAE Consortium for Scientific Research, Mumbai Siegart’s relation: g 2 (  ) = 1 + |  g 1 (  )| 2, 0 <  ≤ 1 Polydispersity index = Method of Cumulant = Exponential fit Relaxation time dist. NNLS CONTIN  =1/T R = q 2 D I ~ 2 DLS on colloids

UGC-DAE Consortium for Scientific Research, Mumbai  Monodisperse spheres (single exponential decay) translational g 1 (  ) = A exp (- Dq 2  ) 1234 D o - diffusion coefficient, T - temperature  - viscosity, R h - hydrodynamic radius D o = kT / (  R h ) Stokes-Einstein relation 2.5 nm (1) 54 nm (2) 214 nm (3) 422 nm (4) DLS on colloids

For translational diffusion,  = Dq 2 SDS micelles in presence of additive D  (slope) UGC-DAE Consortium for Scientific Research, Mumbai DLS on colloids

Spherical particles A2A2 rotational A1A1 A translational   q2q2 D kT qL > 3 D = 3  L F(p) F(p) – shape factor Isotropic Anisotropic Non- (rotational diffusion) UGC-DAE Consortium for Scientific Research, Mumbai q2q2 DRDR  VV VH DLS on colloids

UGC-DAE Consortium for Scientific Research, Mumbai Sphere-to-Rod transition of SDS micelles with addition of TBABr DLS on colloids

S. No.Frictn. Dirn.Cylindrical (L>>r)Ellipsoid (b>>a)Sphere (r) 1 f parallel 2 f perpendicular 3 f rotational 4 f axial rotation UGC-DAE Consortium for Scientific Research, Mumbai Stokes-Einstein relation D = k B T / f DLS on colloids

UGC-DAE Consortium for Scientific Research, Mumbai Polymer solutions Semi-dilute regime Number density > 4R g 3 Fast mode – Diffusion Slow mode – stress relaxation DLS on colloids

DLS on Gel What is a gel ? A gel is a physically or chemically cross-linked three dimensional network which can hold liquid; therefore, visco-elastic in nature. Colloidal solution Polymer gel  UGC-DAE Consortium for Scientific Research, Mumbai

Characterization of the Sol-gel transition :  Gelation kinetics  Gelation mechanism Characterization of the gel phase :  Morphology  Dynamics What is measured on gel using DLS ? DLS on Gel

Gelation kinetics :- The time taken by the polymer solution to transform to a macroscopic gel phase is called the gelation time (t gel ). The inverse of t gel is the gelation rate (t gel -1 ), or the gelation kinetics. Measurement methods (gelation kinetics): ● Test tube tilt (TT) ● Light Scattering (LS) NO FLOW Scattered counts Measured time UGC-DAE Consortium for Scientific Research, Mumbai DLS on Gel

A gel is a non-ergodic system, as its dynamics are restricted by bonds. Therefore, a time- averaged measurement does not represent the complete structural and dynamical aspects of a gel system. UGC-DAE Consortium for Scientific Research, Mumbai DLS on Gel So, how to measure a Gel using DLS ? Method 1 (Pusey): T = E [S (q,t ) – S (q, ∞)] Non-ergodic ratio: Y  E / T where T E

Method 2 (Xue): In this method, the detection area has to be such that multiple speckles can be seen at a time. The sample (gel) is either rotated or translated to average over multiple orientation of the sample. UGC-DAE Consortium for Scientific Research, Mumbai DLS on Gel Therefore, directly measures g 1 (t) or S(q,t)

UGC-DAE Consortium for Scientific Research, Mumbai DLS on Gel Dynamic structure factor, S(q,t ) of a gel phase has two modes, namely, fast mode and slow mode, i.e., S(q,t ) = S f (q,t ) + S s (q,t ) Fast mode : The fast mode relaxation which gives rise to the initial exponential decay of S(q,t ) arises due to the diffusive mode of segmental dynamics in polymer chains between two cross-link points. D f = E /  Stokes-Einstein’s equation : D f = E /  = k B T / 6 

UGC-DAE Consortium for Scientific Research, Mumbai DLS on Gel Slow mode : The origin of the slow mode is not very clear. Two models are reported. (1)Gel mode plus inhomogeneity (GMPI) – gel is viewed as an elastic medium with overdamped modes describing the density fluctuations. Coupled with some static inhomogeneities this picture can qualitatively describe the initial decay of the correlation function (fast mode) and its saturation at long time (slow mode). (2)Harmonically bound Brownian particle (HBBP) – at short time the particles (chain segments) undergo simple diffusion, but at longer time they find that they are restricted to a maximum displacement when the elastic energy equals the thermal energy, i.e., kx 2 / 2 = k B T/ 2

UGC-DAE Consortium for Scientific Research, Mumbai DLS on Gel How to determine to dynamics of the slow mode ? The two models can be distinguished by studying S(q,t) at different wavevector (q). For example, (1) in GMPI model, the fast mode of S(q,t) is q dependent, but the slow mode is independent of q, and (2) in HBBP model, both modes are q dependent.

UGC-DAE Consortium for Scientific Research, Mumbai Thank you Reference: Dynamic Light Scattering: Application of Photon Correlation Spectroscopy, Ed. Robert Pecora, Plenum Press, New York and London, Goutam Ghosh Ph: UGC-DAE Consortium for Scientific Research