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

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• Physik

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• Publié par: universitat_bayreuth
• Publié le: 01 janvier 2008
• Nombre de lectures: 11
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

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 ter