A detailed treatment of the measurement of transport coefficients in transient grating experiments [Elektronische Ressource] / von Marianne Hartung
146 Pages
English

A detailed treatment of the measurement of transport coefficients in transient grating experiments [Elektronische Ressource] / von Marianne Hartung

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A Detailed Treatment of the
Measurement of Transport Coefficients
in Transient Grating Experiments
Von der Universit¨at Bayreuth
zur Erlangung des Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat)
genehmigte Abhandlung
von
Marianne Hartung
geboren am 22.08.1978 in Wu¨rzburg
1. Gutachter: Prof. Dr. W. K¨ohler
2. Gutachter: Prof. Dr. A. Seilmeier
3. Gutachter: Prof. Dr. J. K. G. Dhont
Tag der Einreichung: 19. Juli 2007
Tag des Kolloquiums: 27. November 2007iiiii
Contents
Composition Variables and Partial Specific Quantities 1
1 Introduction 3
2 Thermodynamic–Phenomenological Theory 9
2.1 Entropy Production and Phenomenological Equations . . . . . . . . . . . . . 9
2.1.1 First Law and Definition of Heat . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Entropy Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.3 Phenomenological Equations and Onsager Coefficients . . . . . . . . . 23
2.2 Reference Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Reference Velocities and Diffusion Currents . . . . . . . . . . . . . . . 31
2.2.2 Prigogine’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.3 Definition of Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . 38
2.2.4 Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.5 Thermodynamic Driving Forces . . . . . . . . . . . . . . . . . . . . . . 53
2.2.6 Equations for the Analysis of Transient Grating Experiments . . . . . 61
3 Boundary Effects in Holographic Grating Experiments 62
3.1 Heat and Mass Diffusion Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.1.1 One-dimensional Model . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.2 Two-dimensional Model . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.1.3 Stationary Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.1.4 Time Dependent Solutions. . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Experimental Technique and Sample Preparation . . . . . . . . . . . . . . . 80
3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3.1 Measurements on Pure Toluene . . . . . . . . . . . . . . . . . . . . . . 81
3.3.2 Measurements on Binary Systems . . . . . . . . . . . . . . . . . . . . 84
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 Optical Diffusion Cell with Periodic Resistive Heating 89
4.1 Experimental Setup and Principles of Measurement . . . . . . . . . . . . . . . 89
4.1.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.1.2 Fabrication of Multilayer Structures . . . . . . . . . . . . . . . . . . . 91
4.1.3 Heterodyne Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 95iv Contents
4.1.4 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Heat and Mass Diffusion Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2.1 Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.2 Refractive Index Grating and Heterodyne Diffraction Efficiency . . . . 102
4.2.3 Stationary Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.4 Time Dependent Solutions . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2.5 Sample Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3 Validation of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.3.1 Measurement of Heat Diffusion . . . . . . . . . . . . . . . . . . . . . . 116
4.3.2 Thermal Stability of the Heterodyne Signal . . . . . . . . . . . . . . . 120
4.3.3 Measurement of Mass and Thermal Diffusion . . . . . . . . . . . . . . 123
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5 Summary 128
Deutsche Zusammenfassung 130
Bibliography I
Danksagung IX1
Composition Variables and Partial Specific
Quantities
Almost the same nomenclature as in the book by de Groot and Mazur [16] is used. Compo-
sition variables and partial specific quantities will be abbreviated as follows:
K number of components in the mixture
1m total mass of component kk
P
m= m total massk
N total number of particle of component kk
P
N = N total number of particlesk
V volume occupied by species kk
P
V = V volumek
P
ρ=m/V = ρ total mass densityk
2ρ =m /V mass density of component kk k
P
n =N/V = n total number densityk
n =N /V number density of component kk k
1M =m /N =ρ /n molecular mass of component kk k k k k
c =m /m=ρ /ρ weight fraction of component kk k k
x =N /N =n /n mole fraction of component kk k k
φ =V /V =υ ρ volume fraction of component kk k k k
U internal energy
u=U/m specific internal energy

∂Uu = partial specific internal energy of component kk ∂m p,T,m ,...,m ,m ,...mk 1 k−1 k+1 K
1Note that in the book by de Groot and Mazur [16] m and M are defined reversely.k k
2Only (K−1) mass densities ρ are independent, since, for given temperature T and pressure p an equationk
of state ρ =f(p,T,ρ ,...ρ ) holds in (local) thermodynamic equilibrium.K 1 K−12 Composition Variables and Partial Specific Quantities
S entropy
s=S/m specific entropy

∂Ss = partial specific entropy of component kk ∂m p,T,m ,...,m ,m ,...mk 1 k−1 k+1 K
H =U +pV enthalpy
h =H/m specific enthalpy

∂Hh = partial specific enthalpy of component kk ∂m p,T,m ,...,m ,m ,...mk 1 k−1 k+1 K
G =U−TS+pV Gibbs free energy
g =G/m specific Gibbs free energy

∂Gμ = chemical potential of component kk ∂m p,T,m ,...,m ,m ,...mk 1 k−1 k+1 K
′ ∂Gμ = =M μ chemical potential of component k per particlek kk ∂N p,T,N ,...,N ,N ,...Nk 1 k−1 k+1 K
υ =V/m=1/ρ specific volume

∂Vυ = partial specific volume of component kk ∂m p,T,m ,...,m ,m ,...mk 1 k−1 k+1 K3
Chapter 1
Introduction
The interest in transport coefficients of multicomponent liquid mixtures is rooted both in
their relevance for technical applications and in their fundamental importance for a better
theoretical understanding of fluids. During the last decade especially the number of publi-
cations on the Soret effect [60, 93], also known as the Ludwig-Soret effect, thermal diffusion,
or thermodiffusion, has constantly been growing. This off-diagonal effect accounts for the
occurrence of mass diffusion that is not driven by a concentration but rather by a temper-
ature gradient. Even though the phenomenon was discovered by Ludwig already in 1856,
it is still poorly understood at the microscopic level. There exists, however, a successful
~thermodynamic phenomenological theory [16], which relates the mass diffusion flux J in a
binary mixture to the gradients of temperature and concentration by
~ ~ ~J =−ρD∇c−ρD c(1−c)∇T. (1.1)T
c is the concentration of component 1 in weight fractions, ρ the density, and T the tempera-
ture. Of course, the magnitude of the mass diffusion coefficient D and the thermal diffusion
coefficient D can only be be determined from a microscopic theory. Nevertheless a deepT
understandingof the thermodynamic phenomenological theory is indispensable, since all mi-
croscopic theories have to be in agreement with thermodynamics. There are comprehensive
textbooks on irreversible thermodynamics by de Groot and Mazur [16] and by Haase [40],
which treat all classes of irreversible phenomena in a very general way. However, as the
underlying concepts are sometimes rather complex, it is difficult and time consuming for a
reader who is mainly interested in the Soret effect, to find the relevant information. Further-
more, since thermal diffusion is only one irreversible phenomenon among many others, these
books do not treat it to the last detail. We will therefore give a brief overview of the aspects
of the thermodynamic phenomenological theory being important for the description of diffu-
sion and thermal diffusion. Our considerations are based on the above mentioned books, but
go beyond them in some cases. To mention only two examples, the differences between irre-
versible and reversible mass transfer between the two homogenous phases of a heterogenous
system or the invariance of transport coefficients against shifts of entropy or enthalpy zero
and its consequences have not been considered in Refs. [16, 40] and will be treated in detail.
Moreover, we will briefly discuss recent literature work, where thermodynamic principles4 Chapter 1 Introduction
have not been correctly incorporated.
The rest of the thesis deals with the measurement of heat, mass, and thermal diffusion.
Eq. (1.1) is not suitable for the interpretation of time–resolved experiments. Usually, the
heat equation for the temperature T,
~ ~ ˙ρc ∂ T =∇·[κ∇T]+Q, (1.2)p t
and the extended diffusion equation for the concentration c,
~ ~ ~∂ c=∇·[D∇c+c(1−c)D ∇T], (1.3)t T
are used for the description of coupled heat and mass transport in binary liquids. Here cp
˙is the specific heat at constant pressure, κ the thermal conductivity, and Q a source term.
The derivation of Eqs. (1.2, 1.3) is not as trivial as it might appear at first glance. Strictly
speaking, they only hold if the center of mass velocity ~v vanishes. It will be shown, that
Eqs. (1.2, 1.3) can also be used in case of non-zero ~v, if all gradients are small and second
order terms may be neglected. For that purpose we will generalize the considerations of
Ref. [16] to non–isothermal systems.
AlthoughtheconceptexpressedbyEqs.(1.2,1.3)appearsrathersimple,experimentsthatare
not hampered by artifacts are not easily conducted. Especially unwanted convection caused
by the unavoidable temperature gradients is a major obstacle, and microgravity experiments
have been conducted [101] to overcome this problem. Over the years a certain body of
experimental data for the Soret coefficient had been accumulated but hardly any values had
been cross-checked by another group, and if so, agreement was not guaranteed [51]. In 2003
the results of a measurement campaign with five participating laboratories utilizing different
experimental techniques were published and reliable Soret coefficients could be established
for three equimolar reference systems [75].
Challenged bythe experimental difficultiesa numberofmethodshave beendeveloped, which
all have certain strengths and weaknesses. A popular classical method is based on the
determination of the degree of separation of the fluid components that can be obtained
in a thermogravitational experiment. This method has a long history and a large amount
of the thermal diffusion data accumulated in the literature has been obtained with this
technique [18, 58, 63, 26, 27, 28]. A comprehensive description can be found in the book
by Tyrell [98]. Nowadays mainly annular thermogravitational columns are used [9] and in
some experiments the space between the two cylinders is filled with a porous medium [14].
Another recent development in thisfield are thermogravitational columns with laser Doppler
velocimetry as optical detection [25, 76]. As all thermogravitational methods are based on
the interplay of thermodiffusion and convection, the interpretation of the measurements is
necessarily complex. The amount of material needed is substantial and can be a problem5
in case of expensive isotopic or biological samples. Additional complications arise in case of
negative separation ratios.
Another method where flow and thermal diffusion are combined is thermal field flow frac-
tionation (TFFF), which mainly aims at the separation of polymers and colloids in dilute
solution [87]. It is neither suitable for higher concentrations nor, because of their rapid
diffusion, for small molecules.
Methodsthatallowforminutesamplevolumesandsmalltemperaturegradientsaretypically
based on optical techniques, either for detection or both for detection and generation of the
temperature gradients.
In an optical beam deflection experiment a diffusion cell is heated from above and cooled
from below. The concentration gradient induced by the temperature gradient is detected by
deflection of a laser beam which passes through the cell in a direction parallel to the top
and bottom plates. The time dependence of the deflection angle contains a fast contribution
stemming from the temperature and a slow contribution from the concentration gradient.
Beam deflection is caused by changes of the refractive index of the liquid associated with the
nonuniform temperature and concentration. Optical beam deflection was already used by
Meyerhoff and Nachtigall, who employed a Schlieren technique [67, 66], and later by Giglio
and Vendramini [36, 37], by Kolodner et al. [54], Zhang et al. [116, 117], and Piazza et
al. [74]. Since the diffusion length, the distance between the two plates, is of the order of a
few millimeters to one centimeter, establishment of equilibrium is rather slow, especially for
systems with small diffusion coefficients such as polymers and systems close to the critical
point.
In holographic grating experiments (thermal diffusion forced Rayleigh scattering, TDFRS)
lightisusednotonlyfordetection butalsoforheatingofthefluid. Aholographicinterference
grating is written into the sample. An added dye absorbs and thermalizes the energy of the
lightfieldandatemperaturegratingbuildsup. Thetemperaturegradientsofthetemperature
grating give rise to thermal diffusion, and a secondary concentration grating is generated.
Both the temperature grating and the concentration grating are accompanied by a refractive
index grating, which can be read by Bragg diffraction of a readout laser beam. Thyagarajan
and Lallemand were the first who observed the Soret effect with forced Rayleigh scattering
in the binary liquid mixture CS /ethanol [97]. Later, Pohl studied a critical mixture of 2,6–2
lutidine/waterwiththesametechnique[77]. Byusingheterodynedetectionschemes[52]high
sensitivity is achieved and TDFRS has successfully been applied to a broad class of systems
ranging from small molecules [17, 111, 70] to polymer solutions of arbitrary concentration
[84, 45, 83, 108, 15, 50], colloidal suspensions [94, 69], and critical polymer blends [34, 33].
Anotherall-opticaltechniqueforheatinganddetectionisthethermal-lensmethod,wherethe
signalisderivedfromthedefocusingofalaser beamtransmittedthroughaslightlyabsorbing
sample. Thesuitabilityofthemethodhasbeendemonstratedinrecentexperiments[2,3,86],6 Chapter 1 Introduction
but convection problems are not easily avoided with this technique.
For some special systems, where fluorescent labels can be attached to large molecules such
as DNA, Duhr and Braun have demonstrated a further optical method for the study of
thermal diffusion [22, 24]. By local heating with an infrared laser and fluorescence detection
of the concentration distribution, Soret coefficients of aqueous colloidal solutions could be
determined within microfluidic devices.
Whencomparingallthesetechniques, TDFRShasanumberofindisputableadvantages. The
micrometer diffusion length within the grating reduces the diffusion time to the millisecond
range, which is easily accessible and about six orders of magnitude faster than in case of
macroscopic diffusion cells. In particular for systems with small diffusion coefficients of
−10 2the order of 10 cm /s, such as binary glass formers [84] or critical polymer blends [34],
diffusion times can still be kept within the range of seconds. In a thermal diffusion cell they
would already exceed a week, which is hardlyfeasible. Another major advantage is that only
a single Fourier component, the one of the grating vector q, contributes to the signal, which
makes itsinterpretation particularly simpleand even allows for adeconvolution into multiple
decay functions for e.g. determination of the molar mass distribution in polymer analysis
[53]. Due to the short diffusion lengths, the thin samples and the orientation of the fringes
of the grating parallel to the direction of gravity, convection problems can easily be avoided.
Being all-optical, the method is non-invasive and ideally suited for remote sensing without
direct contact to the sample. The sample volume can be below 1μL. Furthermore it should
be mentioned that this technique is not restricted to the determination of mass and thermal
diffusion but also allows for the measurement of heat conduction in liquids and solids. In
fact, the first holographic grating experiments by Eichler et al. [30] and Pohl et al. [78] in
1973 aimed at the measurement of heat conduction. Whereas Eichler et al. determined the
thermaldiffusivityofmethanol, glycerin andrubycrystals, Pohletal. studiedheattransport
in inorganic crystals at low temperatures. Later the method has been used to determine the
anisotropic thermal conductivity in liquid crystals [46, 95] and sheared polymer melts [102],
to mention only two examples.
There remain, however, unresolved questions and experimental problems of the holographic
grating technique. Two of them will be treated in this thesis.
The first one is related to the correct analysis of the detected diffraction efficiency. When
treating the problem in one dimension, an analytical expression for the time dependent
diffraction efficiency is easily found [8]. The thermal diffusivity D = κ/(ρc ), the massth p
diffusion coefficient D, and the thermal diffusion coefficient D are obtained from a fit ofT
the model to the measured diffraction efficiency. Most holographic grating experiments are
interpreted in terms of this one-dimensional description and heat flow into the cell walls and
a nonuniform temperature distribution along the optical axis are usually neglected. This is,
however,onlypermissible,ifthegratingperioddismuchsmallerthanthesamplethicknessl .s