Light Scattering: Fundamentals

LIGHT SCATTERING: FUNDAMENTALS Andrea Vaccaro, LS InstrumentsModern Light Scattering Technologies, Fribourg 2011

“SCATTERING” OF LIGHT waterparticle dispersion

LIGHT SCATTERING AS SIZING TOOL Nanoparticles: 1-100 nm Polymers, Macromolecules Clays, Oxides, Proteins etc.

PART I: STATIC LIGHT SCATTERING (SLS)

PHYSICAL ORIGIN OF SCATTERED LIGHT kLight 2⇡nk ⌘ |k| = ⇠ wave momentum

PHYSICAL ORIGIN OF SCATTERED LIGHT kLight 2⇡n Electronk ⌘ |k| = ⇠ wave momentum Nucleus•An electron in the atomic cloud is subject to a Atomforce due to the electric ﬁeld

PHYSICAL ORIGIN OF SCATTERED LIGHT kLight 2⇡n Electronk ⌘ |k| = ⇠ wave momentum Nucleus•An electron in the atomic cloud is subject to a Atomforce due to the electric ﬁeld•The cloud deforms and a dipole is induced – +

PHYSICAL ORIGIN OF SCATTERED LIGHT kLight 2⇡n Electronk ⌘ |k| = ⇠ wave momentum Nucleus•An electron in the atomic cloud is subject to a Atomforce due to the electric ﬁeld•The cloud deforms and a dipole is induced•As the ﬁeld oscillates so does the dipolemoment – +

PHYSICAL ORIGIN OF SCATTERED LIGHT kLight 2⇡n Electronk ⌘ |k| = ⇠ wave momentum Nucleus•An electron in the atomic cloud is subject to a Atomforce due to the electric ﬁeld•The cloud deforms and a dipole is induced•As the ﬁeld oscillates so does the dipolemoment –•The resulting charge movement radiates(“scatters”) light +

PHYSICAL ORIGIN OF SCATTERED LIGHT kLight 2⇡n Electronk ⌘ |k| = ⇠ wave momentum Nucleus•An electron in the atomic cloud is subject to a Atomforce due to the electric ﬁeld•The cloud deforms and a dipole is induced•As the ﬁeld oscillates so does the dipolemoment –•The resulting charge movement radiates(“scatters”) light +•“Elastic” scattering: momentum is preserved, noenergy loss ki ks = k

A COLLECTION OF ATOMS Dielectric lump # dV ˆ r dEs

A COLLECTION OF ATOMS Dielectric lump # dVField scattered by a dipole of momentum p ˆ r k 2 exp [ikr] dEs dEs (t) = 2 ˆ ⇥ p(t) ⇥ ˆ r r ns 4 r

A COLLECTION OF ATOMS Dielectric lump # dVField scattered by a dipole of momentum p ˆ r k 2 exp [ikr] dEs dEs (t) = 2 ˆ ⇥ p(t) ⇥ ˆ r r ns 4 rBy deﬁnition of polarizability ↵

A COLLECTION OF ATOMS Dielectric lump # dVField scattered by a dipole of momentum p ˆ r k 2 exp [ikr] dEs dEs (t) = 2 ˆ ⇥ p(t) ⇥ ˆ r r ns 4 rBy deﬁnition of polarizability ↵For an object smaller than λ n m⌘ ns

A COLLECTION OF ATOMS Dielectric lump # dV ˆ rPiecing everything together: dEs

A COLLECTION OF ATOMS Dielectric lump # dV ˆ rPiecing everything together: dEs ⇢ Scattering length density: Ability to scatter of the material

A COLLECTION OF ATOMS Dielectric lump # dV ˆ rPiecing everything together: dEs ⇢ Scattering length Spherical density: Wave Ability to scatter of the material

A COLLECTION OF ATOMS Dielectric lump # dV ˆ rPiecing everything together: dEs ⇢ Scattering length Spherical Scattering density: Wave Geometry Ability to scatter of the material

ORIGIN OF THE SCATTERING CONTRAST

ORIGIN OF THE SCATTERING CONTRAST•Interference αs + α =

ORIGIN OF THE SCATTERING CONTRAST•Interference•For a larger object it ispossible to ﬁnd a second αlump that scatters out ofphase and with the sameamplitude + α =

ORIGIN OF THE SCATTERING CONTRAST•Interference•For a larger object it ispossible to ﬁnd a second αlump that scatters out ofphase and with the sameamplitude +•Completely destructive αinterference =

ORIGIN OF THE SCATTERING CONTRAST•Interference•For a larger object it ispossible to ﬁnd a second αlump that scatters out ofphase and with the sameamplitude +•Completely destructive αinterference =•For an inﬁnite object it isalways possible to do this No contrast

THE MEASURED QUANTITY:THE SCATTERED INTENSITY

THE MEASURED QUANTITY: THE SCATTERED INTENSITY• Whatever the detection technology, the observable quantity is not the electric ﬁeld but the ﬂux of energy, the so-called light intensity

THE MEASURED QUANTITY: THE SCATTERED INTENSITY• Whatever the detection technology, the observable quantity is not the electric ﬁeld but the ﬂux of energy, the so-called light intensity• It can be shown that in most conditions

THE MEASURED QUANTITY: THE SCATTERED INTENSITY• Whatever the detection technology, the observable quantity is not the electric field but the flux of energy, the so-called light intensity• It can be shown that in most conditions•In practice the intensityfluctuates in time

THE MEASURED QUANTITY: THE SCATTERED INTENSITY• Whatever the detection technology, the observable quantity is not the electric field but the flux of energy, the so-called light intensity• It can be shown that in most conditions•In practice the intensityfluctuates in time•In SLS experiment the averageintensity is measured

SLS THEORETICAL APPROACHES

SLS THEORETICAL APPROACHES Theory Assumption Rayleigh (Electrostatic Approximation) Rayleigh-Ganz-Debye (RGD, Optically “Soft” Particles) Mie Scattering NoneFraunhofer (Geometrical Optics)

RAYLEIGH (ELECTROSTATIC APPROXIMATION)

RAYLEIGH (ELECTROSTATIC APPROXIMATION) The scattering object sees in every point the same incident electric ﬁeld at any time

RAYLEIGH (ELECTROSTATIC APPROXIMATION) The scattering object sees in every point the same incident electric ﬁeld at any time•The equations of electrostatics theory apply

RAYLEIGH (ELECTROSTATIC APPROXIMATION) The scattering object sees in every point the same incident electric ﬁeld at any time•The equations of electrostatics theory apply•Their solution gives the polarizability

RAYLEIGH (ELECTROSTATIC APPROXIMATION) The scattering object sees in every point the same incident electric ﬁeld at any time•The equations of electrostatics theory apply•Their solution gives the polarizability•Every particle scatters (radiates) as an ideal dipolewith momentum

RAYLEIGH (ELECTROSTATIC APPROXIMATION) The scattering object sees in every point the same incident electric ﬁeld at any time•The equations of electrostatics theory apply•Their solution gives the polarizability•Every particle scatters (radiates) as an ideal dipolewith momentum•<I>~ λ-4 The sky is blue

RAYLEIGH (ELECTROSTATIC APPROXIMATION) The scattering object sees in every point the same incident electric ﬁeld at any time•The equations of electrostatics theory apply•Their solution gives the polarizability•Every particle scatters (radiates) as an ideal dipolewith momentum•<I>~ λ-4 The sky is blue Contrast

RAYLEIGH (ELECTROSTATIC APPROXIMATION) The scattering object sees in every point the same incident electric ﬁeld at any time•The equations of electrostatics theory apply•Their solution gives the polarizability•Every particle scatters (radiates) as an ideal dipolewith momentum•<I>~ λ-4 The sky is blue Contrast Spherical Wave

RAYLEIGH (ELECTROSTATIC APPROXIMATION) The scattering object sees in every point the same incident electric ﬁeld at any time•The equations of electrostatics theory apply•Their solution gives the polarizability•Every particle scatters (radiates) as an ideal dipolewith momentum•<I>~ λ-4 The sky is blue Scattering Geometry Contrast Spherical Wave

RGD THEORY

RGD THEORY•Assumption: The ﬁeld inside the particle is the incidentﬁeld

RGD THEORY•Assumption: The field inside the particle is the incidentfield•To satisfy this assumption we must require for theincident field:

RGD THEORY•Assumption: The field inside the particle is the incidentfield•To satisfy this assumption we must require for theincident field: i) no reflection at the particle/solvent interface i)

RGD THEORY•Assumption: The field inside the particle is the incidentfield•To satisfy this assumption we must require for theincident field: i) no reflection at the particle/solvent interface ii) no phase change within the particle i) ii)

RGD INTERFERENCE: THE SCATTERING VECTOR

RGD INTERFERENCE: THE SCATTERING VECTORScattering length

RGD INTERFERENCE: THE SCATTERING VECTORScattering Planar Wave, RGD length Approximation

RGD INTERFERENCE: THE SCATTERING VECTORScattering Planar Wave, RGD Spherical length Approximation Wave

RGD INTERFERENCE: THE SCATTERING VECTORScattering Planar Wave, RGD Spherical Scattering length Approximation Wave Geometry

THE MEANING OF THE SCATTERING VECTORThe module of the scattering vector has dimensions of inverse oflength: 4⇡n q= sin(✓/2)

THE MEANING OF THE SCATTERING VECTORThe module of the scattering vector has dimensions of inverse oflength: 4⇡n q= sin(✓/2)q-1 is the length-scale of the interference phenomenon.Two material lumps farther than q-1 interfere destructively.Closer than q-1 interfere additively Destructive Interference, Smaller Intensities No internal Interference Maximum Intensity

THE MEANING OF THE SCATTERING VECTORThe module of the scattering vector has dimensions of inverse oflength: 4⇡n q= sin(✓/2)q-1 is the length-scale of the interference phenomenon.Two material lumps farther than q-1 interfere destructively.Closer than q-1 interfere additively Destructive Interference, Smaller q-1 can be Intensities interpreted as a rough measure of No internal the probed Interference length-scale Maximum Intensity

ONE PARTICLE: THE SCATTERING AMPLITUDEIntegrating previous equation over the whole particle: Z Ei Es = f (⇥) dV (r) exp [ iq · r] R0 V

ONE PARTICLE: THE SCATTERING AMPLITUDEIntegrating previous equation over the whole particle: Z Ei Es = f (⇥) dV (r) exp [ iq · r] R0 VLabeling the particle with the subscript j and factoring out its position by means the variable substitution

ONE PARTICLE: THE SCATTERING AMPLITUDEIntegrating previous equation over the whole particle: Z Ei Es = f (⇥) dV (r) exp [ iq · r] R0 VLabeling the particle with the subscript j and factoring out its position by means the variable substitution We obtain Ei Es = f ( ) exp [ iq · rj ] Fj (q) R0

ONE PARTICLE: THE SCATTERING AMPLITUDE Integrating previous equation over the whole particle: Z Ei Es = f (⇥) dV (r) exp [ iq · r] R0 V Labeling the particle with the subscript j and factoring out its position by means the variable substitution We obtain Ei Es = f ( ) exp [ iq · rj ] Fj (q) R0 ZParticle j scattering ⇥ ⇤ amplitude Fj (q) ⇥ dVj (r0 ) exp j iq · r0 j Vj

AN ENSEMBLE OF PARTICLESWe are able to sum each contribution

AN ENSEMBLE OF PARTICLESWe are able to sum each contribution Taking a time average

AN ENSEMBLE OF PARTICLES We are able to sum each contribution Taking a time averageIndependent position and orientation X⇤Es ⌅t ⇥ ⇤exp [ iq · rj ]⌅t ⇤Fj (q)⌅t j

AN ENSEMBLE OF PARTICLES We are able to sum each contribution Taking a time averageIndependent position and orientation X !⇤Es ⌅t ⇥ ⇤exp [ iq · rj ]⌅t ⇤Fj (q)⌅t j X⇥ ⇤ ⇥ ⇤cos(q · rj )⌅t i ⇤sin(q · rj )⌅t ⇤Fj (q)⌅t = 0 j =0 =0

RGD SCATTERED INTENSITYIndeed the electric ﬁeld in not an observableBut intensity is:

RGD SCATTERED INTENSITYIndeed the electric ﬁeld in not an observableBut intensity is: Is (q) = hIs (q, t)it = hEs (q, t)Es (q, t)⇤ it

RGD SCATTERED INTENSITYIndeed the electric ﬁeld in not an observableBut intensity is: Is (q) = hIs (q, t)it = hEs (q, t)Es (q, t)⇤ it f ( )2 X = E i Ei ⇤ 2 ⇥Fj (q, t)Fk (q, t)exp [ iq · (rj (t) ⇤ rk (t))]⇤t R0 j,k

RGD SCATTERED INTENSITYIndeed the electric ﬁeld in not an observableBut intensity is: Is (q) = hIs (q, t)it = hEs (q, t)Es (q, t)⇤ it f ( )2 X = E i Ei ⇤ 2 ⇥Fj (q, t)Fk (q, t)exp [ iq · (rj (t) ⇤ rk (t))]⇤t R0 j,kAssumption: identical particles, i.e. Fj (q, t) = Fk (q, t) = F (q, t)

RGD SCATTERED INTENSITYIndeed the electric ﬁeld in not an observableBut intensity is: Is (q) = hIs (q, t)it = hEs (q, t)Es (q, t)⇤ it f ( )2 X = E i Ei ⇤ 2 ⇥Fj (q, t)Fk (q, t)exp [ iq · (rj (t) ⇤ rk (t))]⇤t R0 j,kAssumption: identical particles, i.e. Fj (q, t) = Fk (q, t) = F (q, t) f ( )2 ⌦ ↵XIs (q) = Ii 2 |F (q, t)|2 t ⇥exp [ iq · (rj (t) rk (t))]⇤t R0 j,k

RGD SCATTERED INTENSITYIndeed the electric ﬁeld in not an observableBut intensity is: Is (q) = hIs (q, t)it = hEs (q, t)Es (q, t)⇤ it f ( )2 X = E i Ei ⇤ 2 ⇥Fj (q, t)Fk (q, t)exp [ iq · (rj (t) ⇤ rk (t))]⇤t R0 j,kAssumption: identical particles, i.e. Fj (q, t) = Fk (q, t) = F (q, t) f ( )2 ⌦ ↵XIs (q) = Ii 2 |F (q, t)|2 t ⇥exp [ iq · (rj (t) rk (t))]⇤t R0 j,k ⌦ ↵ f ( )2 ⌦ ↵ |F (q, t)| t 1 X 2 = Ii N |F (0, t)|2 t ⇥exp [ iq · (rj (t) rk (t))]⇤t R02 ⇥|F (0, t)| t 2⇤ N j,k

RGD SCATTERED INTENSITYIndeed the electric ﬁeld in not an observableBut intensity is: Is (q) = hIs (q, t)it = hEs (q, t)Es (q, t)⇤ it f ( )2 X = E i Ei ⇤ 2 ⇥Fj (q, t)Fk (q, t)exp [ iq · (rj (t) ⇤ rk (t))]⇤t R0 j,kAssumption: identical particles, i.e. Fj (q, t) = Fk (q, t) = F (q, t) f ( )2 ⌦ ↵XIs (q) = Ii 2 |F (q, t)|2 t ⇥exp [ iq · (rj (t) rk (t))]⇤t R0 j,k ⌦ ↵ f ( )2 ⌦ ↵ |F (q, t)| t 1 X 2 = Ii N |F (0, t)|2 t ⇥exp [ iq · (rj (t) rk (t))]⇤t R02 ⇥|F (0, t)| t 2⇤ N j,k f ( )2 2 2 = Ii 2 N ⇥ V P (q)S(q) R0

RGD SCATTERED INTENSITY f ( )2 2 2 Is (q) = Ii 2 N ⇥ V P (q)S(q) R0

RGD SCATTERED INTENSITY f ( )2 2 2 Is (q) = Ii 2 N ⇥ V P (q)S(q) R0 Scattering Geometry

RGD SCATTERED INTENSITY f ( )2 2 2 Is (q) = Ii 2 N ⇥ V P (q)S(q) R0 Scattering Geometry Total Scattering Length: Contrast

RGD SCATTERED INTENSITY f ( )2 2 2 Is (q) = Ii 2 N ⇥ V P (q)S(q) R0 Form Factor: Scattering Intraparticle Geometry Interference Total Scattering Length: Contrast

RGD SCATTERED INTENSITY f ( )2 2 2 Is (q) = Ii 2 N ⇥ V P (q)S(q) R0 Structure Form Factor: Factor: Scattering Intraparticle Interparticle Geometry Interference Interference Total Scattering Length: Contrast

RGD SCATTERED INTENSITY f ( )2 2 2 Is (q) = Ii 2 N ⇥ V P (q)S(q) R0 Structure Form Factor: Factor: Scattering Intraparticle Interparticle Geometry Interference Interference Total Scattering Length: Contrast•The RGD assumption results in the factorization ofdifferent contributions

RGD SCATTERED INTENSITY f ( )2 2 2 Is (q) = Ii 2 N ⇥ V P (q)S(q) R0 Structure Form Factor: Factor: Scattering Intraparticle Interparticle Geometry Interference Interference Total Scattering Length: Contrast•The RGD assumption results in the factorization ofdifferent contributions•Same factorization for polydisperse systems

FORM FACTOR Homogeneous Sphere with radius R ⌦ ↵ R 2 R 2 |F (q, t)| t 2 |F (q)| 2 dV (r) exp [ iq · r] dV exp [ iq · r]P (q) = = = V R = V R ⇥|F (0, t)| t 2⇤ |F (0)|2 V dV (r) V dV  2 3 = (sin(qR) qR cos(qR)) (qR)3 F (q)/F (0) 4.49 7.73 10.90" P (q) qR qR

FORM FACTOR Homogeneous Sphere with radius R ⌦ ↵ R 2 R 2 |F (q, t)| t 2 |F (q)| 2 dV (r) exp [ iq · r] dV exp [ iq · r]P (q) = = = V R = V R ⇥|F (0, t)| t 2⇤ |F (0)|2 V dV (r) V dV  2 3 = (sin(qR) qR cos(qR)) (qR)3 Polydisperse Homogeneous Sphere F (q)/F (0) 4.49 7.73 10.90" P (q) qR qR

FORM FACTOR Non radially symmetric shapes D⇥R ⇤2 E ⌦ ↵ Z 1 |F (q, t)| t 2 V dV (r) exp [ iq · r] 2P (q) = = D⇥R ⇤2 E t = g(r) sin(qr)/(qr)dr ⇥|F (0, t)| t 2⇤ dV (r) 0 V t Scattering length density weighted pair distance function: Z r g(r) = r 2 (r ) 0 (r r )d r 0 3 0 r0 V r r0

PAIR DISTANCE DISTRIBUTION FUNCTION

FORM FACTOR AT SMALL Q: THE RADIUS OF GYRATION

FORM FACTOR AT SMALL Q: THE RADIUS OF GYRATIONExpanding in series the interference factor... sin(qr) (qr)2 ⇥1 + ··· (qr) 6

FORM FACTOR AT SMALL Q: THE RADIUS OF GYRATIONExpanding in series the interference factor... sin(qr) (qr)2 ⇥1 + ··· (qr) 6The form factor becomes Z 1 2  Z 1 2 2 q Rg q 2 2P (q) = g(r) sin(qr)/(qr)dr 1 r2 g(r)dr 1 0 6 0 3

FORM FACTOR AT SMALL Q: THE RADIUS OF GYRATIONExpanding in series the interference factor... sin(qr) (qr)2 ⇥1 + ··· (qr) 6The form factor becomes Z 1 2  Z 1 2 q 2 Rg q 2 2P (q) = g(r) sin(qr)/(qr)dr 1 r2 g(r)dr 1 0 6 0 3 Z 1 2 2Radius of Gyration: Rg ⌘ r g(r)dr 0

FORM FACTOR AT SMALL Q: THE RADIUS OF GYRATIONExpanding in series the interference factor... sin(qr) (qr)2 ⇥1 + ··· (qr) 6The form factor becomes Z 1 2  Z 1 2 q 2 Rg q 2 2P (q) = g(r) sin(qr)/(qr)dr 1 r2 g(r)dr 1 0 6 0 3 Z 1 2 2Radius of Gyration: Rg ⌘ r g(r)dr 0In a plot of the intensity vs. q2 the extrapolation to zero yields a size parameter that is model independent

FORM FACTOR Polymers global structure Mw Q-1.66 (self avoiding) random walk log(Is) Q-2 cylinder Q-1 . cross section 1/Rg 2/lp 1/Rg,cs log(Q)

FORM FACTOR The asymmetry parameterIntensity weighted cosine of the scattering angle:

FORM FACTOR The asymmetry parameterIntensity weighted cosine of the scattering angle:Measures the ability of the particle to scatter in theforward direction

THE ERGODIC HYPOTHESISOne of the starting hypotheses od statistical mechanics is the socalled “Ergodic Hypothesis”: For any system at equilibrium inﬁnite time averages of observable quantities are equivalent to ensemble averages, i.e.:

THE ERGODIC HYPOTHESISOne of the starting hypotheses od statistical mechanics is the socalled “Ergodic Hypothesis”: For any system at equilibrium inﬁnite time averages of observable quantities are equivalent to ensemble averages, i.e.:An ensamble average, is an average over the ensable ofall the feasible physical conﬁgurations.

THE ERGODIC HYPOTHESISOne of the starting hypotheses od statistical mechanics is the socalled “Ergodic Hypothesis”: For any system at equilibrium inﬁnite time averages of observable quantities are equivalent to ensemble averages, i.e.:An ensamble average, is an average over the ensable ofall the feasible physical conﬁgurations.Once we know how to construct such an ensemblethis hypothesis enables us to “calculate” observed timeaveraged quantities

THE STRUCTURE FACTOR Applying the Ergodic Hypothesis:

THE STRUCTURE FACTOR Applying the Ergodic Hypothesis:In the same vein as the form factor it can be shown thataccounting only for pair interactions (“on the pair level”)

THE STRUCTURE FACTOR Applying the Ergodic Hypothesis:In the same vein as the form factor it can be shown thataccounting only for pair interactions (“on the pair level”) r

EXAMPLE: STRUCTURE FACTORDETERMINATION, SILICA SUSPENSION de Kruif et al., Langmuir 4, 668

EXAMPLE: STRUCTURE FACTOR DETERMINATION, SILICA SUSPENSION Raw Data Volume Fraction f ( )2 2 2Is = Ii 2 N ⇥ V P (q)S(q) de Kruif et al., Langmuir 4, 668 R0

EXAMPLE: STRUCTURE FACTOR DETERMINATION, SILICA SUSPENSION Raw Data Very Small Volume Volume Fraction Fraction f ( )2 2 2 Is = Ii 2 N ⇥ V P (q) R0 f ( )2 2 2Is = Ii 2 N ⇥ V P (q)S(q) de Kruif et al., Langmuir 4, 668 R0

EXAMPLE: STRUCTURE FACTOR DETERMINATION, SILICA SUSPENSION Raw Data Very Small Volume Fraction Volume Fraction Volume Fraction Divide f ( )2 2 2 Is = Ii 2 N ⇥ V P (q) R0 f ( )2 2 2 S(q)Is = Ii 2 N ⇥ V P (q)S(q) de Kruif et al., Langmuir 4, 668 R0

STRUCTURE FACTOR AT SMALL QDensity ﬂuctuations: ✓ ◆ NA 1 ⇧lim S(q) = kT Osmotic Compressibiltyq!0 M c c

STRUCTURE FACTOR AT SMALL QDensity ﬂuctuations: ✓ ◆ NA 1 ⇧lim S(q) = kT Osmotic Compressibiltyq!0 M c c Actually also pure liquid do scatter... though very little

STRUCTURE FACTOR AT SMALL QDensity ﬂuctuations: ✓ ◆ NA 1 ⇧lim S(q) = kT Osmotic Compressibiltyq!0 M c c Actually also pure liquid do scatter... though very little NVirial expansion of the osmotic compressibilty: lim S(q) = 1 B2 q!0 V

STRUCTURE FACTOR AT SMALL QDensity ﬂuctuations: ✓ ◆ NA 1 ⇧lim S(q) = kT Osmotic Compressibiltyq!0 M c c Actually also pure liquid do scatter... though very little NVirial expansion of the osmotic compressibilty: lim S(q) = 1 B2 q!0 VExample: Hard Sphereg(r) 1 r 2R

STRUCTURE FACTOR AT SMALL QDensity ﬂuctuations: ✓ ◆ NA 1 ⇧lim S(q) = kT Osmotic Compressibiltyq!0 M c c Actually also pure liquid do scatter... though very little NVirial expansion of the osmotic compressibilty: lim S(q) = 1 B2 q!0 VExample: Hard Sphere Zg(r) N 1 2 sin(qr) S(q) = 1 + 4 [g(r) 1]r dr V qr 1 0 Z 2 sin(qr) 2R N =1+4 ( 1)r dr V 0 qr 4 N r =1 (2R)3 = 1 8⇥ 3 V 2R

MEASURING THE OSMOTIC COMPRESSIBILITY

MEASURING THE OSMOTIC COMPRESSIBILITY 1/S(q)

MEASURING THE OSMOTIC COMPRESSIBILITY 1/S(q)1/S(q)

INSTRUMENTATION FOR STATIC LIGHT SCATTERING

INSTRUMENTATION FOR STATIC LIGHT SCATTERING CCD

INSTRUMENTATION FOR STATIC LIGHT SCATTERING PM

INSTRUMENTATION FOR STATIC LIGHT SCATTERING can be replaced by ﬁbre PM

INSTRUMENTATION FOR STATIC LIGHT SCATTERING DLS can be replaced by ﬁbre Laser PM

THE RAYLEIGH RATIOScattered Intensity:

THE RAYLEIGH RATIOScattered Intensity: The Rayleigh Ratio: Scattered intensity per unit incident intensity, unit solid angle, and unit scattering volume. Depends only on the thermodynamic state of the solvent not on the measuring apparatus

ABSOLUTE MEASUREMENTS

ABSOLUTE MEASUREMENTS• Knowledge of the constant A enables absolute intensity measurements

ABSOLUTE MEASUREMENTS• Knowledge of the constant A enables absolute intensity measurements• Absolute measurements allow for the determination of the radius of gyration and the second virial coefﬁcient but also of the molar mass M, or the particle concentration

ABSOLUTE MEASUREMENTS• Knowledge of the constant A enables absolute intensity measurements• Absolute measurements allow for the determination of the radius of gyration and the second virial coefﬁcient but also of the molar mass M, or the particle concentration• How do we do it?

ABSOLUTE MEASUREMENTS: HOW

ABSOLUTE MEASUREMENTS: HOW• Scientists have built special devices that allow the measurement of Rayleigh ratios, values for common reference solvents are available in literature

ABSOLUTE MEASUREMENTS: HOW• Scientistshave built special devices that allow the measurement of Rayleigh ratios, values for common reference solvents are available in literature• Ifwe measure the same reference solvent in the same thermodynamic conditions we have:

ABSOLUTE MEASUREMENTS: HOW• Scientistshave built special devices that allow the measurement of Rayleigh ratios, values for common reference solvents are available in literature• Ifwe measure the same reference solvent in the same thermodynamic conditions we have: Substituting back:

DATA TREATMENT AND ABSOLUTE INTENSITY

DATA TREATMENT AND ABSOLUTE INTENSITY Laser PM=

DATA TREATMENT AND ABSOLUTE INTENSITY Laser PM= Laser PM

DATA TREATMENT AND ABSOLUTE INTENSITY Laser Laser PM PM= - Laser Laser PM PM

DATA TREATMENT AND ABSOLUTE INTENSITY Laser Laser PM PM= - Laser Laser PM PM reference solvent e.g. ℜtoluene = 39.6×10-4 m-1

DATA TREATMENT AND ABSOLUTE INTENSITY Laser Laser PM PM= - Laser Laser PM PM reference solvent volume e.g. ℜtoluene = 39.6×10-4 m-1 correction

DATA TREATMENT AND ABSOLUTE INTENSITY M 18 ∆ℜ (10-3 m-1) R2g/3 16 ∆ℜ = 18.4×10-3 m-1(1 - 1.78×10-16 m2 Q2) 14 0 2 4 6 8 10 12 Q-2 (10-4 nm-2)

MACROMOLECULAR SYSTEMS• The treatment so far was focused on particulate systems• For macromolecular systems we cannot precisely deﬁne a refractive index, how do we obtain the contrast?

MACROMOLECULAR SYSTEMS• The treatment so far was focused on particulate systems• For macromolecular systems we cannot precisely deﬁne a refractive index, how do we obtain the contrast?Simple mixing rule for refractive indices:

MACROMOLECULAR SYSTEMS• The treatment so far was focused on particulate systems• For macromolecular systems we cannot precisely deﬁne a refractive index, how do we obtain the contrast?Simple mixing rule for refractive indices:We measure in dilute conditions:

MACROMOLECULAR SYSTEMS• The treatment so far was focused on particulate systems• For macromolecular systems we cannot precisely deﬁne a refractive index, how do we obtain the contrast?Simple mixing rule for refractive indices:We measure in dilute conditions: RGD Hypothesis: m close to 1

MACROMOLECULAR SYSTEMS• The treatment so far was focused on particulate systems• For macromolecular systems we cannot precisely deﬁne a refractive index, how do we obtain the contrast?Simple mixing rule for refractive indices:We measure in dilute conditions: Tabulated or RGD Hypothesis: m close to 1 Measured

EXAMPLE POLYMER MASS DETERMINATION

EXAMPLE POLYMER MASS DETERMINATION 300 250 200 I(θ) (kHz) sample 150 100 reference 50 solvent 0 0 20 40 60 80 100 120 140 160 θ (°)

CORRECTIONS: TURBIDITY = -Haller et al., Rev. Sci. Instr. (1983); Schurtenberger&Augusteyn, Biopolymers (1991)

CORRECTIONS: TURBIDITY = - replace 〈I(q)〉 by 〈I(q)〉/T with transmission T = 〈Is(q=0)〉 / 〈Ii〉 important in connection with 3D technology and turbid samplesHaller et al., Rev. Sci. Instr. (1983); Schurtenberger&Augusteyn, Biopolymers (1991)

CORRECTIONS: REFLECTANCE Reﬂections: glass index matching liquid sample θ Imeas(θ) = Is(θ) + R Is(180°- θ)

CORRECTIONS: REFLECTANCE Reﬂections: reﬂectivity (n s - ng ) 180° - θ 2 R = ⎯⎯⎯ glass n s + ng index matching liquid sample θ Imeas(θ) = Is(θ) + R Is(180°- θ)

CORRECTIONS: REFLECTANCE Reﬂections: 100 reﬂectivity (n s - ng ) 180° - θ 10-1 2 R = ⎯⎯⎯ glass 10-2 n s + ng index matchingI (a.u.) liquid sample 10-3 θ 10-4 10-5 0 45 90 135 180 θ (°) Imeas(θ) = Is(θ) + R Is(180°- θ)

PART II: DYNAMIC LIGHT SCATTERING (DLS)

INTENSITY FLUCTUATION Detector DLS Sample Laser ! Transmission > 95%

INTENSITY FLUCTUATION Detector DLS Sample Laser ! Transmission > 95% 0.12 µm 1.6 µm

PARTICLE DYNAMICS IN REAL AND RECIPROCAL SPACE Particle tracking with a microscope Dynamics in reciprocal (Fourier) space

BROWNIAN MOTION ANDINTENSITY FLUCTUATIONS l(t2 t1 ) Brownian motion • Particle diffusion due to thermal motion • Interference effects on scattered light • Stokes-Einstein equation

BROWNIAN MOTION ANDINTENSITY FLUCTUATIONS l(t2 t1 ) Brownian motion • Particle diffusion due to thermal motion • Interference effects on scattered light • Stokes-Einstein equation RANDOM FLUCTUATION IN SCATTERED INTENSITY

INTENSITY DECORRELATION Intensity correlation function small large I(t), I(t + ) Uncorrelated > CORRELATION DECORRELATI ON 9

INTENSITY DECORRELATION Intensity correlation function small large I(t), I(t + ) Uncorrelated > CORRELATION DECORRELATI ∗ ON 9

INTENSITY DECORRELATION Intensity correlation function small large I(t), I(t + ) Uncorrelated > CORRELATION DECORRELATI ∗ ON p l( ⇤ ) = D ⇤ =q 1 9

INTENSITY DECORRELATION Intensity correlation function small large I(t), I(t + ) Uncorrelated > CORRELATION DECORRELATI ∗ ON p l( ⇤ ) = D ⇤ =q 1 9 ⇤ = (q 2 D) 1

FIELD CORRELATION FUNCTIONThe electric ﬁeld correlation function is important as itis directly connected to colloid dynamics models

FIELD CORRELATION FUNCTIONThe electric ﬁeld correlation function is important as itis directly connected to colloid dynamics models X⇤E(q, 0)E ⇤ (q, )⌅ ⇥ ⇤Fj (q)Fk (q) exp { iq · [rj (0) ⇤ rk ( )]}⌅ j,k

FIELD CORRELATION FUNCTIONThe electric ﬁeld correlation function is important as itis directly connected to colloid dynamics models X⇤E(q, 0)E ⇤ (q, )⌅ ⇥ ⇤Fj (q)Fk (q) exp { iq · [rj (0) ⇤ rk ( )]}⌅ j,k X ⇥ Fj2 (q) ⇤exp { iq · [rj (0) rj ( )]}⌅ j Independent Particles

FIELD CORRELATION FUNCTIONThe electric ﬁeld correlation function is important as itis directly connected to colloid dynamics models X⇤E(q, 0)E ⇤ (q, )⌅ ⇥ ⇤Fj (q)Fk (q) exp { iq · [rj (0) ⇤ rk ( )]}⌅ j,k X ⇥ Fj2 (q) ⇤exp { iq · [rj (0) rj ( )]}⌅ ⇥ N F 2 (q) ⇤exp { iq · [r(0) r( )]}⌅ j Independent Particles Identical Particles

FIELD CORRELATION FUNCTIONThe electric ﬁeld correlation function is important as itis directly connected to colloid dynamics models X⇤E(q, 0)E ⇤ (q, )⌅ ⇥ ⇤Fj (q)Fk (q) exp { iq · [rj (0) ⇤ rk ( )]}⌅ j,k X ⇥ Fj2 (q) ⇤exp { iq · [rj (0) rj ( )]}⌅ ⇥ N F 2 (q) ⇤exp { iq · [r(0) r( )]}⌅ j Independent Particles Identical Particles Upon normalization: ⇥E(q, 0)E ⇤ (q, )⇤ g (1) (q, ) = = ⇥exp { iq · [r(0) r( )]}⇤ I(q)

FIELD CORRELATION FUNCTIONThe electric ﬁeld correlation function is important as itis directly connected to colloid dynamics models X⇤E(q, 0)E ⇤ (q, )⌅ ⇥ ⇤Fj (q)Fk (q) exp { iq · [rj (0) ⇤ rk ( )]}⌅ j,k X ⇥ Fj2 (q) ⇤exp { iq · [rj (0) rj ( )]}⌅ ⇥ N F 2 (q) ⇤exp { iq · [r(0) r( )]}⌅ j Independent Particles Identical Particles Upon normalization: ⇥E(q, 0)E ⇤ (q, )⇤ g (1) (q, ) = = ⇥exp { iq · [r(0) r( )]}⇤ I(q) But cannot be measured!

INTENSITY CORRELATION FUNCTIONAs for SLS we can measure the intensity correlation function

INTENSITY CORRELATION FUNCTIONAs for SLS we can measure the intensity correlation functionOmitting the scattering amplitudes Fj : * + X X⇤I(q, 0)I(q, )⌅ ⇥ exp { iq · [rj (0) rk (0)]} exp { iq · [rl ( ) rm ( )]} j,k l,m

INTENSITY CORRELATION FUNCTIONAs for SLS we can measure the intensity correlation functionOmitting the scattering amplitudes Fj : * + X X⇤I(q, 0)I(q, )⌅ ⇥ exp { iq · [rj (0) rk (0)]} exp { iq · [rl ( ) rm ( )]} j,k l,m X ⇥ ⇤exp { iq · [rj (0) rk (0) + rl ( ) rm ( )]}⌅ j,k,l,m

INTENSITY CORRELATION FUNCTIONAs for SLS we can measure the intensity correlation functionOmitting the scattering amplitudes Fj : * + X X⇤I(q, 0)I(q, )⌅ ⇥ exp { iq · [rj (0) rk (0)]} exp { iq · [rl ( ) rm ( )]} j,k l,m X ⇥ ⇤exp { iq · [rj (0) rk (0) + rl ( ) rm ( )]}⌅ j,k,l,mj=k=l=m!N

INTENSITY CORRELATION FUNCTIONAs for SLS we can measure the intensity correlation functionOmitting the scattering amplitudes Fj : * + X X⇤I(q, 0)I(q, )⌅ ⇥ exp { iq · [rj (0) rk (0)]} exp { iq · [rl ( ) rm ( )]} j,k l,m X ⇥ ⇤exp { iq · [rj (0) rk (0) + rl ( ) rm ( )]}⌅ j,k,l,mj=k=l=m!Nj = k 6= l = m ! N 2 N

INTENSITY CORRELATION FUNCTIONAs for SLS we can measure the intensity correlation function Omitting the scattering amplitudes Fj : * + X X ⇤I(q, 0)I(q, )⌅ ⇥ exp { iq · [rj (0) rk (0)]} exp { iq · [rl ( ) rm ( )]} j,k l,m X ⇥ ⇤exp { iq · [rj (0) rk (0) + rl ( ) rm ( )]}⌅ j,k,l,m j=k=l=m!N j = k 6= l = m ! N 2 NAt the pair level and assuming the ﬁeld be a Gaussian stochastic variable: X X j = m ⇤= l = k ⇥ ⌅exp { iq · [rj (0) rj ( )]}⇧ ⌅exp { iq · [rk (0) rk ( )]}⇧ j k

INTENSITY CORRELATION FUNCTIONAs for SLS we can measure the intensity correlation function Omitting the scattering amplitudes Fj : * + X X ⇤I(q, 0)I(q, )⌅ ⇥ exp { iq · [rj (0) rk (0)]} exp { iq · [rl ( ) rm ( )]} j,k l,m X ⇥ ⇤exp { iq · [rj (0) rk (0) + rl ( ) rm ( )]}⌅ j,k,l,m j=k=l=m!N j = k 6= l = m ! N 2 NAt the pair level and assuming the ﬁeld be a Gaussian stochastic variable: X X j = m ⇤= l = k ⇥ ⌅exp { iq · [rj (0) rj ( )]}⇧ ⌅exp { iq · [rk (0) rk ( )]}⇧ j k 2 2 2 ⇥ N 2 |⇤exp { iq · [r(0) r( )]}⌅| ⇥ ⇤I⌅ |⇤exp { iq · [r(0) r( )]}⌅|

INTENSITY CORRELATION FUNCTIONAs for SLS we can measure the intensity correlation function Omitting the scattering amplitudes Fj : * + X X ⇤I(q, 0)I(q, )⌅ ⇥ exp { iq · [rj (0) rk (0)]} exp { iq · [rl ( ) rm ( )]} j,k l,m X ⇥ ⇤exp { iq · [rj (0) rk (0) + rl ( ) rm ( )]}⌅ j,k,l,m j=k=l=m!N j = k 6= l = m ! N 2 NAt the pair level and assuming the ﬁeld be a Gaussian stochastic variable: X X j = m ⇤= l = k ⇥ ⌅exp { iq · [rj (0) rj ( )]}⇧ ⌅exp { iq · [rk (0) rk ( )]}⇧ j k 2 2 2 ⇥ N 2 |⇤exp { iq · [r(0) r( )]}⌅| ⇥ ⇤I⌅ |⇤exp { iq · [r(0) r( )]}⌅| n o 2 g (2) (q, ) = I(q, 0)I(q, )⇥ / I⇥ = 1 + |g (1) (q, )|2

COHERENCE AREA, SIEGERT RELATIONSHIP

COHERENCE AREA, SIEGERT RELATIONSHIP•In practical implementations the ﬁeld isnot always a Gaussian stochastic variable

COHERENCE AREA, SIEGERT RELATIONSHIP•In practical implementations the ﬁeld isnot always a Gaussian stochastic variable•This happens since we sometimes imagemore than one coherence area (“speckle”)

COHERENCE AREA, SIEGERT RELATIONSHIP•In practical implementations the ﬁeld isnot always a Gaussian stochastic variable•This happens since we sometimes imagemore than one coherence area (“speckle”) n o g (2) (q, ⇥ ) = 1 + 2 |g (1) (q, ⇥ )|2 1

COHERENCE AREA, SIEGERT RELATIONSHIP•In practical implementations the ﬁeld isnot always a Gaussian stochastic variable•This happens since we sometimes imagemore than one coherence area (“speckle”) n o g (2) (q, ⇥ ) = 1 + 2 |g (1) (q, ⇥ )|2 1 •The signal to noise ratio is lowered

DLS INSTRUMENTATION To image one Laser coherence area: ⇥L l1 l2 < ) =1 l1 l2 I(q,t) t 1010011131201100110100 Intensity correlation function (= photoncount correlation function)computed by purpose-built hardware (photon correlator) or software

DLS DATA TREATMENT g 2 q, g 1 q, 1+ 2 g 1 q, g 2 q, 1 1 2 I q,0 I q, with g q, 2 Iq 0Identical particles (Monodisperse): ⇢ q2 ⌦ 2 ↵ g (1) (q, ) = ⇥exp { iq · [r(0) r( )]}⇤ = exp r ( ) 6

DLS DATA TREATMENT g 2 q, g 1 q, 1+ 2 g 1 q, g 2 q, 1 1 2 I q,0 I q, with g q, 2 Iq 0Identical particles (Monodisperse): ⇢ q2 ⌦ 2 ↵ g (1) (q, ) = ⇥exp { iq · [r(0) r( )]}⇤ = exp r ( ) 6 ⌦ ↵ Brownian diffusion theory r ( ) = 6D0 2

DLS DATA TREATMENT g 2 q, g 1 q, 1+ 2 g 1 q, g 2 q, 1 1 2 I q,0 I q, with g q, 2 Iq 0Identical particles (Monodisperse): ⇢ q2 ⌦ 2 ↵ g (1) (q, ) = ⇥exp { iq · [r(0) r( )]}⇤ = exp r ( ) 6 ⌦ ↵ Brownian diffusion theory r ( ) = 6D0 2 g (1) (q, ) = exp 2 q D0

DLS DATA TREATMENT g 2 q, g 1 q, 1+ 2 g 1 q, g 2 q, 1 1 2 I q,0 I q, with g q, 2 Iq 0Identical particles (Monodisperse): ⇢ q2 ⌦ 2 ↵ g (1) (q, ) = ⇥exp { iq · [r(0) r( )]}⇤ = exp r ( ) 6 ⌦ ↵ Brownian diffusion theory r ( ) = 6D0 2 kT g (1) (q, ) = exp 2 q D0 R= 6⇥ D0

MONODISPERSE DLS MEASUREMENT EXAMPLEcolloidal polystyrene (R=25 nm) in water ( =10-3 Pa s, n=1.33) at 25°C: 0 = 488 nm q = 8.9 106 m-1 (30°), 24.2 106 m-1 (90°), 33.1 106 m-1 (150°) 0 1 30°, = 1.47 ms 30° c = 1.47 ms c -1 0.8 ln(g (1)( )) -2 0.6 90°, = 0.20 ms g(1)( ) c 90° = 0.20 ms c -3 0.4 -4 0.2 150° 150° = 0.11 ms = 0.11 ms -5 c 0 c 0 0.5 1 1.5 2 0.001 0.01 0.1 1 10 (ms) (ms)

POLYDISPERSE PARTICLES: CUMULANT ANALYSISFor a polydisperse sample R ⇥ ⇤ N (R)V (R) P (q, R) exp q D0 (R) 2 2 dR g (1) (q, ) = R N (R)V (R)2 P (q, R)dR

POLYDISPERSE PARTICLES: CUMULANT ANALYSISFor a polydisperse sample R ⇥ ⇤ N (R)V (R) P (q, R) exp q D0 (R) 2 2 dR g (1) (q, ) = R N (R)V (R)2 P (q, R)dR =g (1) (q, , R)

POLYDISPERSE PARTICLES: CUMULANT ANALYSISFor a polydisperse sample R ⇥ ⇤ N (R)V (R) P (q, R) exp q D0 (R) 2 2 dR g (1) (q, ) = R N (R)V (R)2 P (q, R)dR ⇠ I(R) =g (1) (q, , R)

POLYDISPERSE PARTICLES: CUMULANT ANALYSISFor a polydisperse sample R ⇥ ⇤ N (R)V (R) P (q, R) exp q D0 (R) 2 2 dR g (1) (q, ) = R N (R)V (R)2 P (q, R)dR ⇠ I(R) =g (1) (q, , R) Intensity weighted correlation function

POLYDISPERSE PARTICLES: CUMULANT ANALYSIS For a polydisperse sample R ⇥ ⇤ N (R)V (R) P (q, R) exp q D0 (R) 2 2 dR g (1) (q, ) = R N (R)V (R)2 P (q, R)dR ⇠ I(R) =g (1) (q, , R) Intensity weighted correlation functionCumulant Expansion: h i ln g (1) (q, ) = ¯ 0 q 2 + cv D0 q 2 D ¯ 2 + ··· 2

POLYDISPERSE PARTICLES: CUMULANT ANALYSIS For a polydisperse sample R ⇥ ⇤ N (R)V (R) P (q, R) exp q D0 (R) 2 2 dR g (1) (q, ) = R N (R)V (R)2 P (q, R)dR ⇠ I(R) =g (1) (q, , R) Intensity weighted correlation functionCumulant Expansion: h i ln g (1) (q, ) = ¯ 0 q 2 + cv D0 q 2 D ¯ 2 + ··· 2 R ¯ N (R)V (R)2 P (q, R)D0 (R)dR D0 = R N (R)V (R)2 P (q, R)dR

POLYDISPERSE PARTICLES: CUMULANT ANALYSIS For a polydisperse sample R ⇥ ⇤ N (R)V (R) P (q, R) exp q D0 (R) 2 2 dR g (1) (q, ) = R N (R)V (R)2 P (q, R)dR ⇠ I(R) =g (1) (q, , R) Intensity weighted correlation functionCumulant Expansion: h i ln g (1) (q, ) = ¯ 0 q 2 + cv D0 q 2 D ¯ 2 + ··· 2 R ¯ N (R)V (R)2 P (q, R)D0 (R)dR ¯ kT D0 = R R= ¯ N (R)V (R)2 P (q, R)dR 6⇥ D0

POLYDISPERSE PARTICLES: CUMULANT ANALYSIS For a polydisperse sample R ⇥ ⇤ N (R)V (R) P (q, R) exp q D0 (R) 2 2 dR g (1) (q, ) = R N (R)V (R)2 P (q, R)dR ⇠ I(R) =g (1) (q, , R) Intensity weighted correlation functionCumulant Expansion: h i ln g (1) (q, ) = ¯ 0 q 2 + cv D0 q 2 D ¯ 2 + ··· 2 R ¯ N (R)V (R)2 P (q, R)D0 (R)dR ¯ kT D0 = R R= ¯ N (R)V (R)2 P (q, R)dR 6⇥ D0 Typical uncertainty in cv is ± 0.02, i.e. it is hard to determine cv ≤ 0.2 (20%)

POLYDISPERSE PARTICLES: CUMULANT ANALYSISTwo species, differing in size by 50% ⇥ ⇤ln g (1) (q, ) = 1/2 exp ¯ 0.8D0 q 2 + exp ¯ 1.2D0 q 2 ln g (1) (q, ) Dq 2τ

A TEST EXPERIMENT: BIMODAL LATEX SPHERE DISPERSIONS Sample: Polystyrene spheres in water at 25 oC. Unimodal systems: nominal particle radii of 46 nm and 205 nm Bimodal systems: mixture of spheres (46 nm and 205 nm), volume ratio of 2:1 (number ratio 178:1). I(θ) x10 x10 X3.6 20 40 60 80 100 120 140 angle

A TEST EXPERIMENT: BIMODAL LATEX SPHERE DISPERSIONS Sample: Polystyrene spheres in water at 25 oC. Unimodal systems: nominal particle radii of 46 nm and 205 nm Bimodal systems: mixture of spheres (46 nm and 205 nm), volume ratio of 2:1 (number ratio 178:1). I(θ) x10 x10 X3.6 20 40 60 80 100 120 140 angle What will we measure at different angles?

DLS ON UNIMODAL SPHERES

DLS ON UNIMODAL SPHERES R1 (nm) R2 (nm) Nominal 46 205 value Output 75o 60.5 226 Output 90o 60 250 Output 135o 59 232

MULTI-ANGLE ANALYSIS R1 (nm) V (R1) R2 (nm) V (R2) (%) Cumulant Cumulant (%) (2 nd) theoret. Nominal 60 (46) 66 230 (205) 33 value Output 8 72 65.6 240 34.4 anglesOutput 45o - - 232.4 177 214Output 75o - - 64 70 75Output 105o - - 159 94 145Output 135o - - 75 74 97

Light Scattering: Fundamentals

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