Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models
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610 Alfred Leipertz and Andreas P. Fr¨oba
Fig. 15.22. Scattering geometry for the observation of the scattered light in trans-
mission direction.
the choice of the angle of incidence , resulting in a specific angle δ of the
refracted light, and the scattering angle Θ
S
the scattering vector q = k
I
−k
S
is determined and, by this, the wave vector of the observed surface vibration
mode, see Fig. 15.22. Here, k
I
and k
S
denote the projections of the wave
vectors of the refracted (k
I
) and scattered light (k
S
) in the surface plane, re-
spectively. For the observation of scattered light within the irradiation plane
and assuming elastic scattering (i.e., k
I
∼
=
k
S
), the modulus of the scattering
vector is obtained as
q =
k
I
− k
S
∼
=
2k
I
sin(Θ
S
/2) cos(δ − Θ
S
/2)
=
4πn
λ
0
sin (Θ
S
/2) cos(δ − Θ
S
/2),
(15.39)
where n is the fluid refractive index and λ
0
is the laser wavelength in
vacuo. The experimental set-up and the computation of the scattering vec-
tor are considerably simplified if the scattered light is observed perpendic-
ularly to the fluid interface, i.e. in a vertical direction with Θ
S
= δ,where
q =(2π/λ
0
)sinΘ
i
.
SLS experiments typically use small scattering angles of about 0.1
◦
up
to a few degrees. The according range of scattering vectors reaches from an
order of 10
4
m
−1
to about 10
6
m
−1
, where large q-vectors can be achieved
easier with a transmission geometry due to intensity considerations. An ad-
vantage of larger scattering vectors is that spectral broadening effects due to
an uncertainty ∆q in the scattering vector do not play a role in the larger
q-range.
15 Diffusion Measurements in Fluids by Dynamic Light Scattering 611
As in many applications of DLS also the variant surface light scattering
makes it possible to determine two properties simultaneously, in the case
of SLS this measurement is performed without any extra effort and yields,
with surface tension and viscosity, two quantities whose determination by
other means requires two completely different sets of experimental equip-
ment. While commonly the kinematic viscosity ν is regarded as the property
directly accessible in SLS experiments, it should be pointed out that this is
only true with the simplifying neglect of gas phase properties. With a proper
execution of the method and experimental care, however, no measurable dif-
ferences between the values obtained by SLS and conventional methods, in
part relying on bulk properties, could be found. An inaccuracy of 1% for
either property can be obtained with SLS [58]. This value is at least compa-
rable with the uncertainty of conventional methods, and, together with the
unanimous advantage of a simultaneous determination of surface tension and
Fig. 15.23. Kinematic viscosity of the liquid phase and surface tension of alterna-
tive refrigerants under saturation conditions from SLS [59].
612 Alfred Leipertz and Andreas P. Fr¨oba
Fig. 15.24. Dynamic viscosity of the liquid phase and surface tension of a silicone
oil at ambient pressure from SLS [54].
viscosity, makes the method interesting also for a routine measurement of
these properties of simple fluids [59] and mixtures [60, 61]. For both the os-
cillatory and the overdamped behavior of surface fluctuations, experimental
results are shown on various new refrigerants in Fig. 15.23 and on a silicone
oil in Fig. 15.24, respectively. For the silicone oil the quantity which has been
directly evaluated from the SLS experiment was the ratio of the dynamic vis-
cosity to the surface tension [54]. Assumed that the temperature dependence
of the dynamic viscosity and surface tension of the silicone oil can be reflected
by a simple Andrade-type and a linear equation, respectively, with the help
of a calibration point the dynamic viscosity and surface tension have been
derived over the whole temperature range investigated. This procedure is
shown in Fig. 15.24, where the dashed curves indicate the uncertainty result-
ing from the uncertainty of the calibration point. Also, apart from studying
specific surface properties, the contactless operation makes SLS ideally suited
15 Diffusion Measurements in Fluids by Dynamic Light Scattering 613
for the investigation of high-temperature melts [62], yet presently with clear
limitations in accuracy.
15.5.2 Forced Rayleigh Scattering
Forced Rayleigh scattering (FRS) differs from usual dynamic light scattering
in the respect that it does not probe statistical fluctuations in a sample but
employs a temperature gradient produced by laser irradiation [63,64]. For the
realization of a thermal grating the laser light, which is chopped for pulses
of a few hundreds of microseconds duration, is split and the two beams are
intersected in the sample to produce an interference pattern (see Fig. 15.25).
This causes the temperature to rise locally, where in some instances light
absorption is increased by doping the sample with an additional dye. The
grating is probed by an additional laser beam, where the intensity diffracted
into the first order is measured over time. Assuming one-dimensional heat
conduction, the grating decays with
δT(x, τ)=∆T cos(kx)exp(−τ/τ
C
), (15.40)
where x is the location in the sample and k is the modulus of the grating
vector. The grating relaxes with a characteristic time τ
C
=1/(aq
2
), which is
detected by measuring the intensity of the diffracted beam
I(τ) ∝ [∆T (τ )]
2
∝ exp(−2τ/τ
C
), (15.41)
Fig. 15.25. Typical setup for a FRS experiment [63].
614 Alfred Leipertz and Andreas P. Fr¨oba
Fig. 15.26. Distribution of thermal diffusivities within a PVC film, obtained by a
scanning FRS technique [64].
which is usually done by means of a photomultiplier and a transient recorder.
A major advantage of FRS as compared to conventional DLS is that a single
measurement may be performed very fast, i.e. in about 1 ms. On the other
hand, measurements are not performed in true thermodynamic equilibrium,
which makes experiments close to the saturation line or to a critical point
difficult, and the technique requires a dye for usual fluids. Additionally, there
are deviations from the ideal working model of one-dimensional heat conduc-
tion. However, the overall uncertainty of the method is found to be about
1.5% to 3% for usual liquids and is thus similar to that of conventional DLS.
Moreover, there are some fields of application, where FRS is especially
attractive. This includes some mixtures, where accurate DLS measurements
may be difficult because of a large signal due to concentration fluctuations,
water and aqueous mixtures (where DLS cannot be used due to the extreme
weakness or lack of the Rayleigh signal), and a number of high temperature
melts, e.g., molten salts. Another interesting feature of FRS is illustrated in
Fig. 15.26, where the technique has been used to obtain a “map” of ther-
mal diffusivities by scanning the laser beam across a sample. It should be
noted that besides the application of FRS to the measurement of thermal
diffusivities it is also possible to extend the technique to the determination
of some other properties. Similar to DLS, FRS may also be applied to the
measurement of the Soret coefficient, as it has been done for polymer solu-
tions [65]. Additionally, FRS has been used to measure the sound velocity
in hydrothermal solutions [66], where acoustic waves are generated by using
intense pump pulses.
15 Diffusion Measurements in Fluids by Dynamic Light Scattering 615
15.6 Conclusion
Light scattering techniques have been widely employed for the investigation
of diffusive processes and the determination of transport and other thermo-
physical properties. The reasons for the success of these methods may be
summarized as follows:
– Measurements may be performed in or close to thermodynamic equilib-
rium, almost without an input of energy.
– Experiments are conducted in a non-contact mode, allowing access to
regions of thermodynamic state which can hardly be probed by other
techniques.
– The methods allow the determination of a wide range of transport and
other thermophysical properties, in some instances even simultaneously.
– The techniques are based on simple and rigid working equations, where
the reliability of the measurements can easily be checked.
As the state of development is different for the various properties, major
tasks for future work will be both the routine application of light scattering
to data acquisition for interesting samples over a wide range of temperature
and pressure and the continuous improvement of the experimental techniques
themselves. Specific challenges are further applications to samples like high-
temperature melts and the simultaneous determination of several properties
within a single experiment.
Notation
A area
a thermal diffusivity, experimental constant
b experimental constant
c concentration, experimental constant
c
p
isobaric specific heat
c
S
speed of sound
c
v
isochoric specific heat
d particle diameter
D
P
particle diffusion coefficient
D
S
sound attenuation
D
12
mutual diffusion coefficient
E electric field strength
f stochastic complex quantity, strength, count rate
G correlation function
g normalized correlation function
H state variable
I intensity
616 Alfred Leipertz and Andreas P. Fr¨oba
k modulus of wave vector or grating vector
k
B
Boltzmann constant
n refractive index
p pressure
q modulus of scattering vector
R distance
S Landau-Placzek ratio
S
E
power spectral density of the electric field
t temperature, time
T temperature, time
Γ line width
δ angle of refraction
permittivity of the medium, angle of incidence
0
permittivity in vacuo
η dynamic (shear) viscosity
Θ
S
scattering angle
Θ
i
external angle of incidence
κ isentropic exponent
λ thermal conductivity
λ
0
laser wavelength in vacuo
ν kinematic viscosity
ρ density
σ surface tension, standard deviation
τ time
∆τ sample time
χ
e
electric susceptibility
χ
s
isentropic compressibility
Ψ random function
Ω solid angle
ω angular frequency
ACF autocorrelation function
DLS dynamic light scattering
FRS forced rayleigh scattering
PCS photon correlation spectroscopy
SLS surface light scattering
··· time average
Sub-/Superscripts
B Brillouin
C correlation
c concentration
ccritical
coh coherence
I incident
15 Diffusion Measurements in Fluids by Dynamic Light Scattering 617
i channel number
LO local oscillator
M modulator
q Fourier component
R Rayleigh, “ripplon”
S scattered, sound, sample
t temperature
projection in the surface plane
∗ complex conjugate variable
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16 Diffusion in Colloidal and Polymeric
Systems
Gerhard N¨agele, Jan K. G. Dhont, and Gerhard Meier
16.1 Introduction
In this chapter we shall discuss diffusion properties of spherical and rod-like
colloids, and of binary polymer melts. In addition, the principles of dynamic
light scattering (DLS), fluorescence recovery after photobleaching (FRAP)
and fluorescence correlation spectroscopy (FCS), which are methods com-
monly used to measure various types of diffusion coefficients, are introduced
on a basic level. In Sect. 16.2 we discuss light scattering on a heuristic level,
and we introduce various kinds of dynamic structure factors. These structure
factors describe different types of diffusion processes, like self-, collective-,
rotational- and interdiffusion. These diffusion processes, and their connection
to the corresponding types of structure factors, are discussed in Sect. 16.3,
first for very dilute systems (Sect. 16.3.1) and then for concentrated systems
(Sect. 16.3.2). Interdiffusion is especially relevant for the dynamics in binary
polymer melts. Since long-time self-diffusion is difficult to measure with light
scattering, especially for non-spherical particles, two very direct methods
to probe this type of diffusion process are introduced in Sect. 16.4: FRAP
(Sect. 16.4.1) and FCS (Sect. 16.4.2). These sections also contain experimen-
tal data obtained with these techniques for rod-like macromolecules. Sec-
tion 16.5 deals with diffusion of spherical colloids, where the various types
of diffusion processes that were treated in Sect. 16.3 on a heuristic level,
are quantified, and where theoretical short-time and long-time predictions,
and computer simulation results, are explored and compared to experimen-
tal findings. The systems considered comprise three-dimensional dispersions
of neutral and charged colloidal spheres, quasi two-dimensional suspensions
of charged particle monolayers between narrow plates, and magnetically in-
teracting particles confined to a liquid-gas interface. Section 16.5 addresses
further the diffusion in binary polymer blends, where earlier discussed general
relations for interdiffusion are explicitly quantified within the Flory-Huggins
approach and the dynamic random phase approximation (RPA).