Furthermore, a new method for the determination of the stability regions of mesomorphic phases and similar liquid-liquid equilibria needs to be established, as the exact knowledge of the phase diagrams is essential. An automated instrument23, much more precise than just the visual observation and characterized by a high sample throughput, is desired.
1.3 Systems investigated
An outline of all investigated binary systems is given in Fig. 1.1. 1,4-Dioxane, characterized by its ring structure, is the most rigid non-ionic compound studied here. It has a negligible overall dipole moment and no dielectric mode was found for 1,4-dioxane up to several 100 GHz.
1,4-Dioxane (OC H )
"Oligo(ethylene glycol) dimethyl ethers CH -(OC H ) -OCH
! " n !Oligo(ethylene glycol) monomethyl ethers
CH -(OC H ) -OH
! " nOligo(ethylene glycol) monoalkyl ethers C H -(OC H ) -OH
n 2n+1 " m- flexible molecules - H-bond acceptor only - no mesostructure - rigid
- H-bond acceptor only - no mesostructure
- flexible molecules - H-bond donor/acceptor - no mesostructure
- flexible molecules - H-bond donor/acceptor - various mesostructures
Figure 1.1: Investigated aqueous binary systems.
Oligo(ethylene glycol) dimethyl ethers are open-chain molecules with much higher degrees of conformational freedom, but, as there are only H-bond acceptor places present, no intermolecular H-bonds can be formed within these liquids. They are miscible with water and many organic solvents in any ratio and can dissolve large quantities of polar gases, what
4 CHAPTER 1. INTRODUCTION
makes them important chemicals for industrial applications. From the dimethyl ethers, it is only a short step to the monomethyl ethers. These are equipped with a free OH group, enabling them to form intermolecular H-bonds.
The CnEm, maybe the most important class of non-ionic surfactants, represent oligo(ethy-lene glycol) monoalkyl ethers. Although they differ from the oligo(ethyoligo(ethy-lene glycol) mono-methyl ethers only by the length of the hydrophobic alkyl side chain, a completely different physico-chemical behavior is observed, arising from the various mesostructures formed in aqueous solutions.
Ternary surfactant/oil/water systems can show even more complicated structures. As an example, C12E5 /water/n-octane microemulsions were investigated in the oil-rich region, because these mixtures are well characterized by other methods24.
In some cases, D2O was substituted for water in order to use the isotope effect for the interpretation of the relaxation behavior and for comparison with data from other methods that require isotope exchange (e.g. NMR spectroscopy or neutron scattering).
Chapter 2
Theoretical background
2.1 Basics of electrodynamics
2.1.1 Maxwell and constitutive equations
The theory of electromagnetic fields is based on the four Maxwell equations25,26. These four equations
rot H =j+ ∂
∂t
D (2.1)
rot E =−∂
∂tB (2.2)
divD =ρel (2.3)
divB = 0. (2.4)
express how electric charges (electric charge density, ρel) produce electric fields (electric field strength, E; Gauss’s law, Eq. 2.3), the absence of magnetic charges (Eq. 2.4), the generation of magnetic fields,H (magnetic field strength), by currents (extended Amp`ere’s law, Eq. 2.1), and how changing magnetic fields produce electric fields (Faraday’s law of induction, Eq. 2.2). B and D account for the magnetic and electric induction, also called magnetic flux density or electric displacement field, respectively.
Together with the Newton equation
m∂2
∂t2r=q(E +v×B), (2.5)
a full set of linear partial differential equations is obtained which enables us, at least in principle, to calculate all kinds of electromagnetic phenomena.
Now we want to constrain ourselves to homogenous, non-dispersive, isotropic materials at 5
6 CHAPTER 2. THEORETICAL BACKGROUND
low fields (linear regime) and introduce the constitutive equations
D =εε0E (2.6)
j=κ E (2.7)
B =µµ0H, (2.8)
where the D and H fields are related toE and B by time- and field strength-independent scalars (material properties): the relative electrical permittivity,ε, specific conductivity,κ, and relative magnetic permeability, µ. ε0 and µ0 are the absolute permittivity of vacuum and the permeability of vacuum, respectively.
2.1.2 The electric displacement field
The constitutive equations (2.6-2.8) are valid only for the special case of a time-independent field response.
Considering the dynamic case, with an electric field E that harmonically oscillates with the amplitude E0 and angular frequency ω = 2πν,
E(t) = E0cos(ωt), (2.9)
most condensed systems show above a certain frequency, typically of the order of 1 MHz to 1 GHz, a significant phase delay, δ(ω), between the electric field and the electric displace-ment field so that
D(t) = D0cos(ωt−δ(ω)). (2.10) Eq. 2.10 can be written as
D(t) = D0cos(δ(ω)) cos(ωt) +D0sin(δ(ω)) sin(ωt), (2.11) and by introducing
D0cos(δ(ω)) = ε(ω)ε0E0 (2.12) D0sin(δ(ω)) = ε(ω)ε0E0 (2.13) the electric displacement field can be expressed as
D(t) = ε(ω)ε0E0cos(ωt) +ε(ω)ε0E0sin(ωt), (2.14) with a phase shift
tan(δ(ω)) = ε(ω)
ε(ω). (2.15)
2.1. BASICS OF ELECTRODYNAMICS 7
Hence, the relation betweenD(t) and E(t) is not longer characterized by an amplitude D0 and a phase shift δ(ω), but by the real and imaginary part of the complex permittivity,
ˆ
ε(ω) = ε(ω)−iε(ω). (2.16) The dispersive part of the electric displacement field,ε(ω)ε0E0cos(ωt), is described by the real part and the imaginary part is the dissipative contribution, ε(ω)ε0E0sin(ωt). Note that the latter is phase shifted by π/2 with respect to the driving electric field.
To simplify the mathematical treatment, complex field vectors E(t) andˆ D(t) can be in-ˆ troduced:
ˆ
E(t) = E0cos(ωt) + iE0sin(ωt) =E0exp(iωt) (2.17) ˆ
D(t) = D0cos(ωt−δ) + iD0sin(ωt−δ) =D0exp[i(ωt−δ)]. (2.18) Thus, for the non-static case, the constitutive equations (2.6) to (2.8) have to be rewritten as27
ˆ
D(t) = ˆε(ω)ε0E(t)ˆ (2.19) j(t) = ˆˆ κ(ω)E(t)ˆ (2.20) ˆ
B(t) = ˆµ(ω)µ0H(t),ˆ (2.21) with the complex conductivity ˆκ(ω), and the complex relative magnetic permeability, ˆµ.
Thus, Eqs. (2.19)-(2.21) are suitable for the description of the frequency-dependent linear dielectric response of dissipative systems.
2.1.3 Wave equations
The Maxwell equation (2.1) for harmonic fields ˆ
E(t) = E0cos(iωt) (2.22)
ˆ
H(t) = H0cos(iωt) (2.23)
can be transformed with the help of the complex constitutive equations (2.19) - (2.21) into rot H0 = (ˆκ(ω) + iωε(ω)εˆ 0)E0. (2.24) In a similar way, the equation
rot E0 =−iωµ(ω)µˆ 0H0 (2.25) is obtained from the Maxwell equation (2.2).
By application of the rotation operator on Eq.(2.24) and by using the Legendre vectorial identity,
rot rot H0 =grad div H0− H0 =grad (0) − H0 =− H0, (2.26)
8 CHAPTER 2. THEORETICAL BACKGROUND
and Eq.(2.25), the reduced form of the wave equation of the magnetic field can be obtained:
H0+ ˆk2H0 = 0 (2.27)
The propagation constant ˆk is given by kˆ2 =k20
The propagation constant of the vacuum,k0, is defined by
k0 =ω√
ε0µ0 = 2π
λ0 (2.29)
c0 = 1
√ε0µ0, (2.30)
where c0 is the speed of light andλ0 the wavelength of a monochromatic wave in vacuum.
For a source-free medium (divE = 0) a reduced wave equation forE can be obtained, Eq.
2.31.
Eˆ0+ ˆk2Eˆ0 = 0 (2.31)
In the case of non-magnetizable substances (ˆµ = 1), the complex propagation constant, Eq.(2.28), can be written as
ˆk2 =k02
Its real part is given by
η(ω) =ε(ω)− κ(ω)
ωε0 (2.33)
and the imaginary part by
η(ω) = ε(ω) + κ(ω)
ωε0 . (2.34)
Equations (2.32) to (2.34) show that the dielectric properties and the conductivity of the system cannot be measured separately. In electrolyte systems, the theory28 suggests some dispersion of the complex conductivity, ˆκ. However, at microwave frequencies this effect can be neglected29, especially at the low electrolyte concentrations of the materials covered by this study.
Therefore we assume
κ(ω) = κ (2.35)
and
κ(ω) = 0. (2.36)