Static and dynamic light scattering have evolved into powerful methods to investigate a variety of soft and biological matter systems with structures on the nanometer to micrometer scale. They can provide detailed quantitative information on the shape, internal structure, size, and polydispersity of the system as well as interparticle interactions. I will present their fundamentals from a physics and instrumental point of view and also comment on experimental data analysis. The opportunities they offer will be discussed as well as their limits. This will be illustrated by a selection of examples, ranging from colloidal suspensions, detergent and polymer solutions to proteins and include topics like contrast and absolute intensity, determination of molar mass, polydispersity and interparticle interactions.

1. LIGHT SCATTERING:
FUNDAMENTALS
Andrea Vaccaro, LS Instruments
Modern Light Scattering Technologies, Fribourg 2011

2. “SCATTERING” OF LIGHT
water
particle dispersion

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

4. PART I: STATIC LIGHT
SCATTERING (SLS)

5. PHYSICAL ORIGIN OF SCATTERED
LIGHT
k
Light
2⇡n
k ⌘ |k| = ⇠ wave momentum

6. PHYSICAL ORIGIN OF SCATTERED
LIGHT
k
Light
2⇡n
Electron
k ⌘ |k| = ⇠ wave momentum
Nucleus
•An electron in the atomic cloud is subject to a Atom
force due to the electric field

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

8. PHYSICAL ORIGIN OF SCATTERED
LIGHT
k
Light
2⇡n
Electron
k ⌘ |k| = ⇠ wave momentum
Nucleus
•An electron in the atomic cloud is subject to a Atom
force due to the electric field
•The cloud deforms and a dipole is induced
•As the field oscillates so does the dipole
moment –
+

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

10. PHYSICAL ORIGIN OF SCATTERED
LIGHT
k
Light
2⇡n
Electron
k ⌘ |k| = ⇠ wave momentum
Nucleus
•An electron in the atomic cloud is subject to a Atom
force due to the electric field
•The cloud deforms and a dipole is induced
•As the field oscillates so does the dipole
moment –
•The resulting charge movement radiates
(“scatters”) light +
•“Elastic” scattering: momentum is preserved, no
energy loss ki ' ks = k

11. A COLLECTION OF ATOMS
Dielectric lump
#
dV
ˆ
r
dEs

12. A COLLECTION OF ATOMS
Dielectric lump
#
dV
Field 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

13. A COLLECTION OF ATOMS
Dielectric lump
#
dV
Field 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
By definition of polarizability ↵

14. A COLLECTION OF ATOMS
Dielectric lump
#
dV
Field 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
By definition of polarizability ↵
For an object smaller than λ
n
m⌘
ns

15. A COLLECTION OF ATOMS
Dielectric lump
#
dV
ˆ
r
Piecing everything together:
dEs

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

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

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

19. ORIGIN OF THE SCATTERING
CONTRAST

20. ORIGIN OF THE SCATTERING
CONTRAST
•Interference
αs
+
α
=

21. ORIGIN OF THE SCATTERING
CONTRAST
•Interference
•For a larger object it is
possible to find a second α
lump that scatters out of
phase and with the same
amplitude
+
α
=

22. ORIGIN OF THE SCATTERING
CONTRAST
•Interference
•For a larger object it is
possible to find a second α
lump that scatters out of
phase and with the same
amplitude
+
•Completely destructive α
interference
=

23. ORIGIN OF THE SCATTERING
CONTRAST
•Interference
•For a larger object it is
possible to find a second α
lump that scatters out of
phase and with the same
amplitude
+
•Completely destructive α
interference
=
•For an infinite object it is
always possible to do this
No contrast

24. THE MEASURED QUANTITY:
THE SCATTERED INTENSITY

25. 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

26. 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

27. 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 intensity
fluctuates in time

28. 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 intensity
fluctuates in time
•In SLS experiment the average
intensity is measured

29. SLS THEORETICAL
APPROACHES

30. SLS THEORETICAL
APPROACHES
Theory Assumption
Rayleigh (Electrostatic
Approximation)
Rayleigh-Ganz-Debye (RGD,
Optically “Soft” Particles)
Mie Scattering None
Fraunhofer (Geometrical Optics)

31. SLS THEORETICAL
APPROACHES
Theory Assumption
Rayleigh (Electrostatic
Approximation)
Rayleigh-Ganz-Debye (RGD,
Optically “Soft” Particles)
Mie Scattering None
Fraunhofer (Geometrical Optics)

32. SLS THEORETICAL
APPROACHES
Theory Assumption
Rayleigh (Electrostatic
Approximation)
Rayleigh-Ganz-Debye (RGD,
Optically “Soft” Particles)
Mie Scattering None
Fraunhofer (Geometrical Optics)

33. SLS THEORETICAL
APPROACHES
Theory Assumption
Rayleigh (Electrostatic
Approximation)
Rayleigh-Ganz-Debye (RGD,
Optically “Soft” Particles)
Mie Scattering None
Fraunhofer (Geometrical Optics)

34. RAYLEIGH (ELECTROSTATIC
APPROXIMATION)

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

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

37. 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

38. 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 dipole
with momentum

39. 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 dipole
with momentum
•<I>~ λ-4 The sky is blue

40. 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 dipole
with momentum
•<I>~ λ-4 The sky is blue
Contrast

41. 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 dipole
with momentum
•<I>~ λ-4 The sky is blue
Contrast Spherical Wave

42. 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 dipole
with momentum
•<I>~ λ-4 The sky is blue
Scattering
Geometry
Contrast Spherical Wave

43. RGD THEORY

44. RGD THEORY
•Assumption: The field inside the particle is the incident
field

45. RGD THEORY
•Assumption: The field inside the particle is the incident
field
•To satisfy this assumption we must require for the
incident field:

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

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

48. RGD INTERFERENCE: THE SCATTERING VECTOR

49. RGD INTERFERENCE: THE SCATTERING VECTOR

50. RGD INTERFERENCE: THE SCATTERING VECTOR
Scattering
length

51. RGD INTERFERENCE: THE SCATTERING VECTOR
Scattering Planar Wave, RGD
length Approximation

52. RGD INTERFERENCE: THE SCATTERING VECTOR
Scattering Planar Wave, RGD Spherical
length Approximation Wave

53. RGD INTERFERENCE: THE SCATTERING VECTOR
Scattering Planar Wave, RGD Spherical
Scattering
length Approximation Wave
Geometry

54. RGD INTERFERENCE: THE SCATTERING VECTOR
Scattering Planar Wave, RGD Spherical
Scattering
length Approximation Wave
Geometry

55. RGD INTERFERENCE: THE SCATTERING VECTOR
Scattering Planar Wave, RGD Spherical
Scattering
length Approximation Wave
Geometry

56. RGD INTERFERENCE: THE SCATTERING VECTOR
Scattering Planar Wave, RGD Spherical
Scattering
length Approximation Wave
Geometry

57. RGD INTERFERENCE: THE SCATTERING VECTOR
Scattering Planar Wave, RGD Spherical
Scattering
length Approximation Wave
Geometry

58. RGD INTERFERENCE: THE SCATTERING VECTOR
Scattering Planar Wave, RGD Spherical
Scattering
length Approximation Wave
Geometry

59. THE MEANING OF THE SCATTERING VECTOR
The module of the scattering vector has dimensions of inverse of
length: 4⇡n
q= sin(✓/2)

60. THE MEANING OF THE SCATTERING VECTOR
The module of the scattering vector has dimensions of inverse of
length: 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

61. THE MEANING OF THE SCATTERING VECTOR
The module of the scattering vector has dimensions of inverse of
length: 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

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

63. 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

64. 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

65. 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
Z
Particle j scattering ⇥ ⇤
amplitude Fj (q) ⇥ dVj (r0 ) exp
j iq · r0
j
Vj

66. AN ENSEMBLE OF PARTICLES
We are able to sum each
contribution

67. AN ENSEMBLE OF PARTICLES
We are able to sum each
contribution
Taking a time average

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

69. AN ENSEMBLE OF PARTICLES
We are able to sum each
contribution
Taking a time average
Independent 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

70. RGD SCATTERED INTENSITY
Indeed the electric field in not an observable
But intensity is:

71. RGD SCATTERED INTENSITY
Indeed the electric field in not an observable
But intensity is:
Is (q) = hIs (q, t)it = hEs (q, t)Es (q, t)⇤ it

72. RGD SCATTERED INTENSITY
Indeed the electric field in not an observable
But 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

73. RGD SCATTERED INTENSITY
Indeed the electric field in not an observable
But 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
Assumption: identical particles, i.e. Fj (q, t) = Fk (q, t) = F (q, t)

74. RGD SCATTERED INTENSITY
Indeed the electric field in not an observable
But 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
Assumption: identical particles, i.e. Fj (q, t) = Fk (q, t) = F (q, t)
f ( )2 ⌦ ↵X
Is (q) = Ii 2 |F (q, t)|2 t ⇥exp [ iq · (rj (t) rk (t))]⇤t
R0
j,k

75. RGD SCATTERED INTENSITY
Indeed the electric field in not an observable
But 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
Assumption: identical particles, i.e. Fj (q, t) = Fk (q, t) = F (q, t)
f ( )2 ⌦ ↵X
Is (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

76. RGD SCATTERED INTENSITY
Indeed the electric field in not an observable
But 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
Assumption: identical particles, i.e. Fj (q, t) = Fk (q, t) = F (q, t)
f ( )2 ⌦ ↵X
Is (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

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

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

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

80. 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

81. 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

82. 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 of
different contributions

83. 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 of
different contributions
•Same factorization for polydisperse systems

84. 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

85. 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

86. FORM FACTOR
Non radially symmetric shapes
D⇥R ⇤2 E
⌦ ↵ Z 1
|F (q, t)| t
2
V
dV (r) exp [ iq · r] 2
P (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

87. PAIR DISTANCE DISTRIBUTION
FUNCTION

88. FORM FACTOR AT SMALL Q: THE
RADIUS OF GYRATION

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

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

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

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

93. 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)

94. FORM FACTOR
The asymmetry parameter
Intensity weighted cosine of the scattering angle:

95. FORM FACTOR
The asymmetry parameter
Intensity weighted cosine of the scattering angle:
Measures the ability of the particle to scatter in the
forward direction

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

97. THE ERGODIC HYPOTHESIS
One of the starting hypotheses od statistical mechanics is the so
called “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 of
all the feasible physical configurations.

98. THE ERGODIC HYPOTHESIS
One of the starting hypotheses od statistical mechanics is the so
called “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 of
all the feasible physical configurations.
Once we know how to construct such an ensemble
this hypothesis enables us to “calculate” observed time
averaged quantities

99. THE STRUCTURE FACTOR
Applying the Ergodic Hypothesis:

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

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

102. EXAMPLE: STRUCTURE FACTOR
DETERMINATION, SILICA SUSPENSION
de Kruif et al., Langmuir 4, 668

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

104. 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 2
Is = Ii 2 N ⇥ V P (q)S(q) de Kruif et al., Langmuir 4, 668
R0

105. 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

106. STRUCTURE FACTOR AT SMALL Q
Density fluctuations:
✓ ◆
NA
1
⇧
lim S(q) = kT Osmotic Compressibilty
q!0 M c c

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

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

109. STRUCTURE FACTOR AT SMALL Q
Density fluctuations:
✓ ◆
NA
1
⇧
lim S(q) = kT Osmotic Compressibilty
q!0 M c c
Actually also pure liquid do scatter... though very little
N
Virial expansion of the osmotic compressibilty: lim S(q) = 1 B2
q!0 V
Example: Hard Sphere
g(r)
1
r
2R

110. STRUCTURE FACTOR AT SMALL Q
Density fluctuations:
✓ ◆
NA
1
⇧
lim S(q) = kT Osmotic Compressibilty
q!0 M c c
Actually also pure liquid do scatter... though very little
N
Virial expansion of the osmotic compressibilty: lim S(q) = 1 B2
q!0 V
Example: Hard Sphere
Z
g(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

111. MEASURING THE OSMOTIC COMPRESSIBILITY

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

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

114. INSTRUMENTATION FOR STATIC
LIGHT SCATTERING

115. INSTRUMENTATION FOR STATIC
LIGHT SCATTERING

116. INSTRUMENTATION FOR STATIC
LIGHT SCATTERING
CCD

117. INSTRUMENTATION FOR STATIC
LIGHT SCATTERING
PM

118. INSTRUMENTATION FOR STATIC
LIGHT SCATTERING
can be replaced
by fibre
PM

119. INSTRUMENTATION FOR STATIC
LIGHT SCATTERING
can be replaced
by fibre
PM

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

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

122. THE RAYLEIGH RATIO
Scattered Intensity:

123. THE RAYLEIGH RATIO
Scattered Intensity:

124. THE RAYLEIGH RATIO
Scattered 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

125. THE RAYLEIGH RATIO
Scattered 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

126. ABSOLUTE MEASUREMENTS

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

128. 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

129. 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?

130. ABSOLUTE MEASUREMENTS: HOW

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

132. 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:

133. 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:

134. DATA TREATMENT AND ABSOLUTE
INTENSITY

135. DATA TREATMENT AND ABSOLUTE
INTENSITY
Laser
PM
=

136. DATA TREATMENT AND ABSOLUTE
INTENSITY
Laser
PM
=
Laser
PM

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

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

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

140. 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

141. 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)

142. 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?

143. 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:

144. 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:

145. 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:

146. 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

147. 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

148. EXAMPLE POLYMER MASS
DETERMINATION

149. 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 (°)

150. 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 (°)

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

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

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

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

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

156. 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)

157. CORRECTIONS: REFLECTANCE
Reflections:
100
reflectivity
(n s - ng
) 180° - θ
10-1 2
R = ⎯⎯⎯ glass
10-2 n s + ng index matching
I (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)

158. PART II: DYNAMIC LIGHT
SCATTERING (DLS)

159. INTENSITY FLUCTUATION
Detector
DLS
Sample
Laser
!
Transmission > 95%

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

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

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

163. BROWNIAN MOTION AND
INTENSITY 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

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

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

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

167. 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

168. FIELD CORRELATION FUNCTION
The electric field correlation function is important as it
is directly connected to colloid dynamics models

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

170. FIELD CORRELATION FUNCTION
The electric field correlation function is important as it
is 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

171. FIELD CORRELATION FUNCTION
The electric field correlation function is important as it
is 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

172. FIELD CORRELATION FUNCTION
The electric field correlation function is important as it
is 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)

173. FIELD CORRELATION FUNCTION
The electric field correlation function is important as it
is 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!

174. INTENSITY CORRELATION FUNCTION
As for SLS we can measure the intensity correlation function

175. INTENSITY CORRELATION FUNCTION
As 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

176. INTENSITY CORRELATION FUNCTION
As 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

177. INTENSITY CORRELATION FUNCTION
As 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

178. INTENSITY CORRELATION FUNCTION
As 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 N

179. INTENSITY CORRELATION FUNCTION
As 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 N
At 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

180. INTENSITY CORRELATION FUNCTION
As 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 N
At 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( )]}⌅|

181. INTENSITY CORRELATION FUNCTION
As 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 N
At 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

182. COHERENCE AREA, SIEGERT
RELATIONSHIP

183. COHERENCE AREA, SIEGERT
RELATIONSHIP
•In practical implementations the field is
not always a Gaussian stochastic variable

184. COHERENCE AREA, SIEGERT
RELATIONSHIP
•In practical implementations the field is
not always a Gaussian stochastic variable
•This happens since we sometimes image
more than one coherence area (“speckle”)

185. COHERENCE AREA, SIEGERT
RELATIONSHIP
•In practical implementations the field is
not always a Gaussian stochastic variable
•This happens since we sometimes image
more than one coherence area (“speckle”)
n o
g (2) (q, ⇥ ) = 1 + 2
|g (1) (q, ⇥ )|2 1

186. COHERENCE AREA, SIEGERT
RELATIONSHIP
•In practical implementations the field is
not always a Gaussian stochastic variable
•This happens since we sometimes image
more than one coherence area (“speckle”)
n o
g (2) (q, ⇥ ) = 1 + 2
|g (1) (q, ⇥ )|2 1
•The signal to noise ratio is lowered

187. 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

188. 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
0
Identical particles (Monodisperse):
⇢
q2 ⌦ 2 ↵
g (1) (q, ) = ⇥exp { iq · [r(0) r( )]}⇤ = exp r ( )
6

189. 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
0
Identical particles (Monodisperse):
⇢
q2 ⌦ 2 ↵
g (1) (q, ) = ⇥exp { iq · [r(0) r( )]}⇤ = exp r ( )
6
⌦ ↵
Brownian diffusion theory r ( ) = 6D0
2

190. 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
0
Identical 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

191. 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
0
Identical 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

192. MONODISPERSE DLS
MEASUREMENT EXAMPLE
colloidal 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)

193. 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

194. 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
=g (1)
(q, , R)

195. 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)

196. 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 function

197. 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 function
Cumulant Expansion:
h i
ln g (1) (q, ) = ¯ 0 q 2 + cv D0 q 2
D ¯ 2
+ ···
2

198. 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 function
Cumulant 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

199. 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 function
Cumulant 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

200. 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 function
Cumulant 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%)

201. POLYDISPERSE PARTICLES:
CUMULANT ANALYSIS
Two 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

202. 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

203. 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?

204. DLS ON UNIMODAL SPHERES

205. 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

206. 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
angles
Output 45o - - 232.4 177 214
Output 75o - - 64 70 75
Output 105o - - 159 94 145
Output 135o - - 75 74 97