seminar pcf “lightscattering”. 1. light scattering – theoretical background 1.1. introduction...
TRANSCRIPT
Seminar PCF
ldquoLightscatteringrdquo
0
2 2 cos
x tE x t E
c
1 Light Scattering ndash Theoretical Background
11 Introduction
Light-wave interacts with the charges constituting a given molecule in remodelling the spatial charge distribution
Molecule constitutes the emitter of an electromagnetic wave of the same wavelength as the incident one (ldquoelastic scatteringrdquo)
E
m
sE
Wave-equation of oscillating electic fieldof the incident light
Particles larger than 20 nm (right picture) - several oscillating dipoles created simultaneously within one given particle- interference leads to a non-isotropic angular dependence of the scattered light
intensity - particle form factor characteristic for size and shape of the scattering particle- scattered intensity I ~ NiMi
2Pi(q) (scattering vector q see below)
Particles smaller than l20 (left picture) - scattered intensity independent of scattering angle I ~ NiMi
2
Particles in solution show Brownian motion (D = kT(6phR) and ltDr(t)2gt=6Dt)THORN Interference pattern and resulting scattered intensity fluctuate with time
THORN Change in respective particle positions leads to changes in interparticular () interference and therefore temporal fluctuations in the scattered intensity detected at given scattering angle (s Static Structurefactor ltS(q)gt Dynamic Lightscattering S(qt) (DLS))
2 22
02 2 2
41exp 2 Ds
D D
EmE i t kr
t r c r c
2 Lichtstreuung ndash experimenteller Aufbau
Detector (photomultiplier photodiode) scattered intensity only 2
s s s sI E E E
detector
rD
I
sampleI0
Scattered light wave emitted by one oscillating dipole
Light source I0 = laser focussed monochromatic coherent
Coherent the light has a defined oscillation phase over a certain distance (05 ndash 1 m) and time so it can show interference Note that only laser light is coherent in time soNo laser =gt no dynamic light scattering
Sample cell cylindrical quartz cuvette embedded in toluene bath (T nD)
Light Scattering Setup of the F-Practical Course PhysChem Mainz
Scattering volume defined by intersection of incident beam and optical aperture of the detection opticsvaries with scattering angle
Important scattered intensity has to be normalized
Scattering from dilute solutions of very small particles (ldquopoint scatterersrdquo)(eg nanoparticles or polymer chains smaller than l20)
222 2
040
4 ( )DD
L
nb n K
cN
in cm2g-2Mol
22 ( ) D
solution solvent
rR b c M I I
V
std abssolution solvent
std
IR I I
I
contrast factor
Absolute scattered intensity of ideal solutions Rayleigh ratio ([cm-1])
For calibration of the setup one uses a scattering standard Istd Toluene ( Iabs = 14 e-5 cm-1 )
Reason of ldquoSky Bluerdquo (scattering from gas molecules of atmosphere)
Scattering from dilute solutions of larger particles
- scattered intensity dependent on scattering angle (interference)
0q k k
0k
k
4 sin( )2Dnq
0k
The scattering vector q (in [cm-1]) length scale of the light scattering experiment
q
q = inverse observational length scale of the light scattering experiment
q-scale resolution information comment
qR ltlt 1 whole coil mass radius of gyration eg Zimm plot
qR lt 1 topology cylinder sphere hellip
qR asymp 1 topology quantitative size of cylinder
qR gt 1 chain conformation helical stretched
qR gtgt 1 chain segments chain segment density
0 2 4 6 8 10 1210
-5
10-4
10-3
10-2
10-1
100
P(q
)
qR
For large (ca 500 nm) homogeneous spheres
26
9( ) sin cosP q qR qR qR
qR
Minimum bei qR = 449
Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
0
2 2 cos
x tE x t E
c
1 Light Scattering ndash Theoretical Background
11 Introduction
Light-wave interacts with the charges constituting a given molecule in remodelling the spatial charge distribution
Molecule constitutes the emitter of an electromagnetic wave of the same wavelength as the incident one (ldquoelastic scatteringrdquo)
E
m
sE
Wave-equation of oscillating electic fieldof the incident light
Particles larger than 20 nm (right picture) - several oscillating dipoles created simultaneously within one given particle- interference leads to a non-isotropic angular dependence of the scattered light
intensity - particle form factor characteristic for size and shape of the scattering particle- scattered intensity I ~ NiMi
2Pi(q) (scattering vector q see below)
Particles smaller than l20 (left picture) - scattered intensity independent of scattering angle I ~ NiMi
2
Particles in solution show Brownian motion (D = kT(6phR) and ltDr(t)2gt=6Dt)THORN Interference pattern and resulting scattered intensity fluctuate with time
THORN Change in respective particle positions leads to changes in interparticular () interference and therefore temporal fluctuations in the scattered intensity detected at given scattering angle (s Static Structurefactor ltS(q)gt Dynamic Lightscattering S(qt) (DLS))
2 22
02 2 2
41exp 2 Ds
D D
EmE i t kr
t r c r c
2 Lichtstreuung ndash experimenteller Aufbau
Detector (photomultiplier photodiode) scattered intensity only 2
s s s sI E E E
detector
rD
I
sampleI0
Scattered light wave emitted by one oscillating dipole
Light source I0 = laser focussed monochromatic coherent
Coherent the light has a defined oscillation phase over a certain distance (05 ndash 1 m) and time so it can show interference Note that only laser light is coherent in time soNo laser =gt no dynamic light scattering
Sample cell cylindrical quartz cuvette embedded in toluene bath (T nD)
Light Scattering Setup of the F-Practical Course PhysChem Mainz
Scattering volume defined by intersection of incident beam and optical aperture of the detection opticsvaries with scattering angle
Important scattered intensity has to be normalized
Scattering from dilute solutions of very small particles (ldquopoint scatterersrdquo)(eg nanoparticles or polymer chains smaller than l20)
222 2
040
4 ( )DD
L
nb n K
cN
in cm2g-2Mol
22 ( ) D
solution solvent
rR b c M I I
V
std abssolution solvent
std
IR I I
I
contrast factor
Absolute scattered intensity of ideal solutions Rayleigh ratio ([cm-1])
For calibration of the setup one uses a scattering standard Istd Toluene ( Iabs = 14 e-5 cm-1 )
Reason of ldquoSky Bluerdquo (scattering from gas molecules of atmosphere)
Scattering from dilute solutions of larger particles
- scattered intensity dependent on scattering angle (interference)
0q k k
0k
k
4 sin( )2Dnq
0k
The scattering vector q (in [cm-1]) length scale of the light scattering experiment
q
q = inverse observational length scale of the light scattering experiment
q-scale resolution information comment
qR ltlt 1 whole coil mass radius of gyration eg Zimm plot
qR lt 1 topology cylinder sphere hellip
qR asymp 1 topology quantitative size of cylinder
qR gt 1 chain conformation helical stretched
qR gtgt 1 chain segments chain segment density
0 2 4 6 8 10 1210
-5
10-4
10-3
10-2
10-1
100
P(q
)
qR
For large (ca 500 nm) homogeneous spheres
26
9( ) sin cosP q qR qR qR
qR
Minimum bei qR = 449
Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
Particles larger than 20 nm (right picture) - several oscillating dipoles created simultaneously within one given particle- interference leads to a non-isotropic angular dependence of the scattered light
intensity - particle form factor characteristic for size and shape of the scattering particle- scattered intensity I ~ NiMi
2Pi(q) (scattering vector q see below)
Particles smaller than l20 (left picture) - scattered intensity independent of scattering angle I ~ NiMi
2
Particles in solution show Brownian motion (D = kT(6phR) and ltDr(t)2gt=6Dt)THORN Interference pattern and resulting scattered intensity fluctuate with time
THORN Change in respective particle positions leads to changes in interparticular () interference and therefore temporal fluctuations in the scattered intensity detected at given scattering angle (s Static Structurefactor ltS(q)gt Dynamic Lightscattering S(qt) (DLS))
2 22
02 2 2
41exp 2 Ds
D D
EmE i t kr
t r c r c
2 Lichtstreuung ndash experimenteller Aufbau
Detector (photomultiplier photodiode) scattered intensity only 2
s s s sI E E E
detector
rD
I
sampleI0
Scattered light wave emitted by one oscillating dipole
Light source I0 = laser focussed monochromatic coherent
Coherent the light has a defined oscillation phase over a certain distance (05 ndash 1 m) and time so it can show interference Note that only laser light is coherent in time soNo laser =gt no dynamic light scattering
Sample cell cylindrical quartz cuvette embedded in toluene bath (T nD)
Light Scattering Setup of the F-Practical Course PhysChem Mainz
Scattering volume defined by intersection of incident beam and optical aperture of the detection opticsvaries with scattering angle
Important scattered intensity has to be normalized
Scattering from dilute solutions of very small particles (ldquopoint scatterersrdquo)(eg nanoparticles or polymer chains smaller than l20)
222 2
040
4 ( )DD
L
nb n K
cN
in cm2g-2Mol
22 ( ) D
solution solvent
rR b c M I I
V
std abssolution solvent
std
IR I I
I
contrast factor
Absolute scattered intensity of ideal solutions Rayleigh ratio ([cm-1])
For calibration of the setup one uses a scattering standard Istd Toluene ( Iabs = 14 e-5 cm-1 )
Reason of ldquoSky Bluerdquo (scattering from gas molecules of atmosphere)
Scattering from dilute solutions of larger particles
- scattered intensity dependent on scattering angle (interference)
0q k k
0k
k
4 sin( )2Dnq
0k
The scattering vector q (in [cm-1]) length scale of the light scattering experiment
q
q = inverse observational length scale of the light scattering experiment
q-scale resolution information comment
qR ltlt 1 whole coil mass radius of gyration eg Zimm plot
qR lt 1 topology cylinder sphere hellip
qR asymp 1 topology quantitative size of cylinder
qR gt 1 chain conformation helical stretched
qR gtgt 1 chain segments chain segment density
0 2 4 6 8 10 1210
-5
10-4
10-3
10-2
10-1
100
P(q
)
qR
For large (ca 500 nm) homogeneous spheres
26
9( ) sin cosP q qR qR qR
qR
Minimum bei qR = 449
Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
Particles in solution show Brownian motion (D = kT(6phR) and ltDr(t)2gt=6Dt)THORN Interference pattern and resulting scattered intensity fluctuate with time
THORN Change in respective particle positions leads to changes in interparticular () interference and therefore temporal fluctuations in the scattered intensity detected at given scattering angle (s Static Structurefactor ltS(q)gt Dynamic Lightscattering S(qt) (DLS))
2 22
02 2 2
41exp 2 Ds
D D
EmE i t kr
t r c r c
2 Lichtstreuung ndash experimenteller Aufbau
Detector (photomultiplier photodiode) scattered intensity only 2
s s s sI E E E
detector
rD
I
sampleI0
Scattered light wave emitted by one oscillating dipole
Light source I0 = laser focussed monochromatic coherent
Coherent the light has a defined oscillation phase over a certain distance (05 ndash 1 m) and time so it can show interference Note that only laser light is coherent in time soNo laser =gt no dynamic light scattering
Sample cell cylindrical quartz cuvette embedded in toluene bath (T nD)
Light Scattering Setup of the F-Practical Course PhysChem Mainz
Scattering volume defined by intersection of incident beam and optical aperture of the detection opticsvaries with scattering angle
Important scattered intensity has to be normalized
Scattering from dilute solutions of very small particles (ldquopoint scatterersrdquo)(eg nanoparticles or polymer chains smaller than l20)
222 2
040
4 ( )DD
L
nb n K
cN
in cm2g-2Mol
22 ( ) D
solution solvent
rR b c M I I
V
std abssolution solvent
std
IR I I
I
contrast factor
Absolute scattered intensity of ideal solutions Rayleigh ratio ([cm-1])
For calibration of the setup one uses a scattering standard Istd Toluene ( Iabs = 14 e-5 cm-1 )
Reason of ldquoSky Bluerdquo (scattering from gas molecules of atmosphere)
Scattering from dilute solutions of larger particles
- scattered intensity dependent on scattering angle (interference)
0q k k
0k
k
4 sin( )2Dnq
0k
The scattering vector q (in [cm-1]) length scale of the light scattering experiment
q
q = inverse observational length scale of the light scattering experiment
q-scale resolution information comment
qR ltlt 1 whole coil mass radius of gyration eg Zimm plot
qR lt 1 topology cylinder sphere hellip
qR asymp 1 topology quantitative size of cylinder
qR gt 1 chain conformation helical stretched
qR gtgt 1 chain segments chain segment density
0 2 4 6 8 10 1210
-5
10-4
10-3
10-2
10-1
100
P(q
)
qR
For large (ca 500 nm) homogeneous spheres
26
9( ) sin cosP q qR qR qR
qR
Minimum bei qR = 449
Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
2 22
02 2 2
41exp 2 Ds
D D
EmE i t kr
t r c r c
2 Lichtstreuung ndash experimenteller Aufbau
Detector (photomultiplier photodiode) scattered intensity only 2
s s s sI E E E
detector
rD
I
sampleI0
Scattered light wave emitted by one oscillating dipole
Light source I0 = laser focussed monochromatic coherent
Coherent the light has a defined oscillation phase over a certain distance (05 ndash 1 m) and time so it can show interference Note that only laser light is coherent in time soNo laser =gt no dynamic light scattering
Sample cell cylindrical quartz cuvette embedded in toluene bath (T nD)
Light Scattering Setup of the F-Practical Course PhysChem Mainz
Scattering volume defined by intersection of incident beam and optical aperture of the detection opticsvaries with scattering angle
Important scattered intensity has to be normalized
Scattering from dilute solutions of very small particles (ldquopoint scatterersrdquo)(eg nanoparticles or polymer chains smaller than l20)
222 2
040
4 ( )DD
L
nb n K
cN
in cm2g-2Mol
22 ( ) D
solution solvent
rR b c M I I
V
std abssolution solvent
std
IR I I
I
contrast factor
Absolute scattered intensity of ideal solutions Rayleigh ratio ([cm-1])
For calibration of the setup one uses a scattering standard Istd Toluene ( Iabs = 14 e-5 cm-1 )
Reason of ldquoSky Bluerdquo (scattering from gas molecules of atmosphere)
Scattering from dilute solutions of larger particles
- scattered intensity dependent on scattering angle (interference)
0q k k
0k
k
4 sin( )2Dnq
0k
The scattering vector q (in [cm-1]) length scale of the light scattering experiment
q
q = inverse observational length scale of the light scattering experiment
q-scale resolution information comment
qR ltlt 1 whole coil mass radius of gyration eg Zimm plot
qR lt 1 topology cylinder sphere hellip
qR asymp 1 topology quantitative size of cylinder
qR gt 1 chain conformation helical stretched
qR gtgt 1 chain segments chain segment density
0 2 4 6 8 10 1210
-5
10-4
10-3
10-2
10-1
100
P(q
)
qR
For large (ca 500 nm) homogeneous spheres
26
9( ) sin cosP q qR qR qR
qR
Minimum bei qR = 449
Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
Light Scattering Setup of the F-Practical Course PhysChem Mainz
Scattering volume defined by intersection of incident beam and optical aperture of the detection opticsvaries with scattering angle
Important scattered intensity has to be normalized
Scattering from dilute solutions of very small particles (ldquopoint scatterersrdquo)(eg nanoparticles or polymer chains smaller than l20)
222 2
040
4 ( )DD
L
nb n K
cN
in cm2g-2Mol
22 ( ) D
solution solvent
rR b c M I I
V
std abssolution solvent
std
IR I I
I
contrast factor
Absolute scattered intensity of ideal solutions Rayleigh ratio ([cm-1])
For calibration of the setup one uses a scattering standard Istd Toluene ( Iabs = 14 e-5 cm-1 )
Reason of ldquoSky Bluerdquo (scattering from gas molecules of atmosphere)
Scattering from dilute solutions of larger particles
- scattered intensity dependent on scattering angle (interference)
0q k k
0k
k
4 sin( )2Dnq
0k
The scattering vector q (in [cm-1]) length scale of the light scattering experiment
q
q = inverse observational length scale of the light scattering experiment
q-scale resolution information comment
qR ltlt 1 whole coil mass radius of gyration eg Zimm plot
qR lt 1 topology cylinder sphere hellip
qR asymp 1 topology quantitative size of cylinder
qR gt 1 chain conformation helical stretched
qR gtgt 1 chain segments chain segment density
0 2 4 6 8 10 1210
-5
10-4
10-3
10-2
10-1
100
P(q
)
qR
For large (ca 500 nm) homogeneous spheres
26
9( ) sin cosP q qR qR qR
qR
Minimum bei qR = 449
Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
Scattering volume defined by intersection of incident beam and optical aperture of the detection opticsvaries with scattering angle
Important scattered intensity has to be normalized
Scattering from dilute solutions of very small particles (ldquopoint scatterersrdquo)(eg nanoparticles or polymer chains smaller than l20)
222 2
040
4 ( )DD
L
nb n K
cN
in cm2g-2Mol
22 ( ) D
solution solvent
rR b c M I I
V
std abssolution solvent
std
IR I I
I
contrast factor
Absolute scattered intensity of ideal solutions Rayleigh ratio ([cm-1])
For calibration of the setup one uses a scattering standard Istd Toluene ( Iabs = 14 e-5 cm-1 )
Reason of ldquoSky Bluerdquo (scattering from gas molecules of atmosphere)
Scattering from dilute solutions of larger particles
- scattered intensity dependent on scattering angle (interference)
0q k k
0k
k
4 sin( )2Dnq
0k
The scattering vector q (in [cm-1]) length scale of the light scattering experiment
q
q = inverse observational length scale of the light scattering experiment
q-scale resolution information comment
qR ltlt 1 whole coil mass radius of gyration eg Zimm plot
qR lt 1 topology cylinder sphere hellip
qR asymp 1 topology quantitative size of cylinder
qR gt 1 chain conformation helical stretched
qR gtgt 1 chain segments chain segment density
0 2 4 6 8 10 1210
-5
10-4
10-3
10-2
10-1
100
P(q
)
qR
For large (ca 500 nm) homogeneous spheres
26
9( ) sin cosP q qR qR qR
qR
Minimum bei qR = 449
Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
Scattering from dilute solutions of very small particles (ldquopoint scatterersrdquo)(eg nanoparticles or polymer chains smaller than l20)
222 2
040
4 ( )DD
L
nb n K
cN
in cm2g-2Mol
22 ( ) D
solution solvent
rR b c M I I
V
std abssolution solvent
std
IR I I
I
contrast factor
Absolute scattered intensity of ideal solutions Rayleigh ratio ([cm-1])
For calibration of the setup one uses a scattering standard Istd Toluene ( Iabs = 14 e-5 cm-1 )
Reason of ldquoSky Bluerdquo (scattering from gas molecules of atmosphere)
Scattering from dilute solutions of larger particles
- scattered intensity dependent on scattering angle (interference)
0q k k
0k
k
4 sin( )2Dnq
0k
The scattering vector q (in [cm-1]) length scale of the light scattering experiment
q
q = inverse observational length scale of the light scattering experiment
q-scale resolution information comment
qR ltlt 1 whole coil mass radius of gyration eg Zimm plot
qR lt 1 topology cylinder sphere hellip
qR asymp 1 topology quantitative size of cylinder
qR gt 1 chain conformation helical stretched
qR gtgt 1 chain segments chain segment density
0 2 4 6 8 10 1210
-5
10-4
10-3
10-2
10-1
100
P(q
)
qR
For large (ca 500 nm) homogeneous spheres
26
9( ) sin cosP q qR qR qR
qR
Minimum bei qR = 449
Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
Scattering from dilute solutions of larger particles
- scattered intensity dependent on scattering angle (interference)
0q k k
0k
k
4 sin( )2Dnq
0k
The scattering vector q (in [cm-1]) length scale of the light scattering experiment
q
q = inverse observational length scale of the light scattering experiment
q-scale resolution information comment
qR ltlt 1 whole coil mass radius of gyration eg Zimm plot
qR lt 1 topology cylinder sphere hellip
qR asymp 1 topology quantitative size of cylinder
qR gt 1 chain conformation helical stretched
qR gtgt 1 chain segments chain segment density
0 2 4 6 8 10 1210
-5
10-4
10-3
10-2
10-1
100
P(q
)
qR
For large (ca 500 nm) homogeneous spheres
26
9( ) sin cosP q qR qR qR
qR
Minimum bei qR = 449
Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
q
q = inverse observational length scale of the light scattering experiment
q-scale resolution information comment
qR ltlt 1 whole coil mass radius of gyration eg Zimm plot
qR lt 1 topology cylinder sphere hellip
qR asymp 1 topology quantitative size of cylinder
qR gt 1 chain conformation helical stretched
qR gtgt 1 chain segments chain segment density
0 2 4 6 8 10 1210
-5
10-4
10-3
10-2
10-1
100
P(q
)
qR
For large (ca 500 nm) homogeneous spheres
26
9( ) sin cosP q qR qR qR
qR
Minimum bei qR = 449
Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
0 2 4 6 8 10 1210
-5
10-4
10-3
10-2
10-1
100
P(q
)
qR
For large (ca 500 nm) homogeneous spheres
26
9( ) sin cosP q qR qR qR
qR
Minimum bei qR = 449
Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
Dynamic Light Scattering
Brownian motion of the solute particles leads to fluctuations of the scattered intensity
( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr
mean-squared displacement of the scattering particle
2 6 sR D 6s
H
kT kTD
f R
change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein-Gl
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
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- Slide 7
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- Slide 13
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- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
2
2
( ) ( )( ) exp( ) ( ) ( ) 1
s s s s
I q t I q tF q D q E q t E q t
I q t
ltI(
t)I(
t+)gt
T
I(t)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+t)gt )
(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))
Siegert-Relation
2Basislinie I t
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
rdquoCumulant-Methodldquo for polydisperse samples Fs(qt) is a superposition of various exponentials
2 31 2 3
1 1ln
2 3sF q
1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient
2nd cumulant 22 42 s sD D q is a measure for sample polydispersity
ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)
22 2
2 1i i i iapp s gz z
i i i
n M P q DD q D K R q
n M P q
Data analysis for polydisperse (monomodal) samples
Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to
qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
Cumulant analysis ndash graphic explanation
Monodisperse sample Polydisperse sample
linear slope yields diffusion coefficient slope at t=0 yields apparent diffusioncoefficient which is an average weightedwith NiMi
2Pi(q)
log(
Fs(q
)
yx=-Dsq2
log(
Fs(q
)
yx=-Dsq2
larger slower particles
small fast particles
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )
0 1x1010
2x1010
3x1010
4x1010
00
50x10-15
10x10-14
15x10-14
20x10-14
Dm
2 s-1
q2cm
-2
s zD
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
Explanation for q-dependence of Dapp for larger particles due to Pi(q)
2
2
i i i iapp
i i i
n M P q DD q
n M P q
Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
100E-05
100E-04
100E-03
100E-02
100E-01
100E+00
P(q) 130 nm
P(q) 260 nm
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
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- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)
DLS of concentrated samples ndash influence of the static structure factor S(q)
From Gapinsky et al JChemPhys 126 104905 (2007)
S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
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- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
-
From Gapinsky et al JChemPhys 126 104905 (2007)
Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )
D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
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- Slide 10
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-