3. Nuclear magnetic resonance 29
3.9. Other experimental methods
φ1(0, tp) =
tp−1
X
ti=0
ωQ(ti)∆t (3.72)
and the phase in the detection time given by
φ2(t0m+ ∆4, t) =
t0m+∆4−1
X
ti=t0m
ωQ(ti)∆t (3.73)
The abbreviation t0m =tm+tp is used. The calculations assume infinte short rf pulses.
Figure 3.16.: Typical dielectric spectra at two temperatures. Figure adapted from [182].
The dielectric susceptibilityχˆDSand equivalent the dielectric permittivityεˆDS= 1−χˆDScontain information about the reorientational correlation time of dipole moments connected to molecules in the sample. The relaxation strength of such a process is connected to the number of contributing molecules and their dipole moment. The relaxation peaks can often be described by the CC, CD, or HN functions introduced in section 2.1.3. Additionally, information about the electric conductivity and electronic mobility may be extracted.
A typical dielectric spectrum is sketched in figure 3.16, showing the peaks of the structural α relaxation, a possibleβ-process and the excess wing. In addition, higher frequency phenomena such as the Boson peak and electronic relaxation are indicated, but are out of scope of this work. Due to the dependence of dipole reorientation on the frequency, DS measures the rotational auto-correlation of the first Legendre polynomial F1(t):
F1(t1, t2)∝
cos(Θ(t1))cos(Θ(t2)
(3.75) Here, Θ represents the orientation of the dipole and cross-correlations are ne-glected.
Quasi elastic neutron scattering
In QENS thermal neutrons are used to investigate dynamics and structure of systems at the (sub-)nanometer scale [183]. The De Broglie wavelength of neutrons at 300 K has a mean value of 〈λ〉 = 1.8 ˚A. The measured quantity is the double differential scattering cross section
Figure 3.17.: Sketch of a neutron scattering experiment. The monochromator selects a velocity. Energy changes are detected by time of flight methods. Figure modified from [183].
d2σ
dΩdω = |k|
|k0|S(q,ω) (3.76)
In this equation, k0, k are the wave vectors of the incident and the scattered neutron wave, q is the scattering vector and ω is the energy transfer in units of ħh, see figure 3.17. The information is contained in the dynamics structure factor S(q,ω). Its information content is best seen in the Fourier transform I(q, t) =F
S(q,ω)
called theintermediate scattering function(ISF). It can be split into a coherent and an incoherent contributionIcoh(q, t)and Iinc(q, t). The coherent contribution depends on the phase relation of the waves scattered by different atoms, and thus, results in a measure of collective motion. In contrast, the incoherent part depends only on single atom contributions and results in a measure for self-correlation.
Whether coherent or incoherent scattering is measured depends on the atoms in the sample. The quantities Ii(q, t) ∝b2i, i = {coh, inc}, are proportional to the square scattering length b2i, which is an element and isotope specific con-stant. For example hydrogen atoms have a large incoherent scattering length and hence, the resulting ISF is dominated by their incoherent scattering. In contrast, deuterons only show a weak incoherent scattering and larger coher-ent scattering.
By spatial Fourier transformation the van-Hove self-correlation function can be calculated, i.e. G(r, t) = Fr
I(q, t)
. In the case of incoherent scattering, G(r, t)can be interpreted as the probability density of a particle displacement, and thus gives a measure for single particle dynamics [183]. The long time
limit of the incoherent ISF, i.e. Iinc(q, t→ ∞), is of particular interest to obtain information about the type of motion. It is called theelastic incoherent structure factor(EISF) and gives the inverse of the average volume explored in the con-figuration space. The EISF is analogous to the residual correlation F∞measured in2H NMR STE experiments [169].
The usual time window of dynamic that can be observed is from some picosec-onds to a hundred nanosecpicosec-onds, by a combination of different NS techniques such as Time of flight, backscattering, and neutron spin echo. By observing the spectra at different detector positions, i.e. at different values of q, the characteristic length scale at which dynamical processes are observed can be varied.
Molecular dynamics simulations
The dynamical and structural properties of supercooled liquids can be modeled in classical MD simulations. A detailed introduction into MD can be found in [28] and references therin. The term classical refers to the fact that the interaction of the atoms are approximated by effective potentials. Those are usually calculated using quantum chemical methods and also include effective and instantaneous electronic contributions.
In MD, the time evolution of the model system is calculated by iteratively solv-ing Newton’s equation of motion. The resultsolv-ing trajectories contain all infor-mation about the system, from which all experimental observables can be cal-culated. MD simulations can be employed to unravel microscopic properties of a large variety of systems, that are limited only by computational costs and the availability of proper potentials. On modern computers, systems with roughly 105 atoms can be simulated on time scales up to the microseconds regime. In contrast to experiments, MD simulations can be used to investigate nonphysi-cal systems were certain properties can be tuned to demonstrate the influence of certain properties, e.g. the charge distribution in a molecule. This was for example used to simulate water behavior in neutral pores [28,84]. Those possi-bilities render MD simulations a rather powerful tool that, in combination with experiments, can help to resolve the problems encountered in current research.
MD simulation also exhibits some drawbacks. The time scales of dynamics upon cooling towards the dynamic glass transition increases by orders of magnitude, becoming too slow for MD time scales. In addition, the system size is rather limited: Problems involving biological materials, e.g. proteins, may require too large systems or simulation times. The classical approach also neglects electronic contributions and quantum mechanical effects. Therefore, processes based on those effects, like chemical reactions and or tunneling effects, can-not be addressed with classical MD simulations. Parts of those problems can be tackled by using ab initio MD simulation. In this technique, the density
functional theory is applied to solve the Schr¨odinger equation. The problem of system size is currently addressed by coarse graining.
Comparison of QENS, DS, and MD with2H NMR
Figure 3.18 compares the accessible time and frequency ranges of the different methods. While very fast dynamics are accessible by MD and QENS, DS allows investigations in a very broad frequency range. The 2H NMR time scale spans from some hundreds of picoseconds up to the seconds regime using different techniques.
τ / s
ps ns μs ms s
ν / Hz
THz GHz MHz kHz Hz
QENS MD
2
H NMR DS
Figure 3.18.:Time scales of different methods used to characterizes the dynam-ics of supercooled liquids.
QENS experiments are successfully used at higher temperatures, where dynam-ics are still fast. There, they can provide valuable information about time and length scales of dynamical process and additionally on the mechanism of mo-tion. Even more detailed information can be obtained by MD simulations, which rely on to the availability of appropriate potentials.
When the sample is cooled and dynamics become slow, both methods reach their limits. In this range2H NMR provides a useful tool to acquire information about the dynamics. Especially, on time scales between microseconds and sec-onds, this method can provide many insights into the dynamical process. The overlap of time scales with QENS and MD can be used to test the conformity of both methods.
DS can be used in an even broader frequency range. It is a very versatile method to obtain information about the time scales of the dynamical process. Neverthe-less, DS usually can not provide specific information on the geometry of motion.
Therefore, additional 2H NMR and QENS experiments are necessary to gain a full understanding of the present dynamics.
Heat Flow / mW
Temperature / K
Exo down
0 2 4
-2
-4
223.4 K
100 150 200 250 300
Figure 3.19.:Exemplary DSC thermogram of water confined to MCM-41 (sample P25). Cooling and heating rates were 10 K
min (inner circle) and 40 K
min (outer circle). Exothermic heat flow is negative. Measured by C. Lederle.
Differential scanning calorimetry
The thermal behavior of a sample upon temperature change can be quantified using DSC [180]. It is mostly used to gain information about the specific heat capacity cp, which in turn provides insights into thermodynamic and chemical properties. When studying the behavior of supercooled liquids, cp can be used to identify the glass transition or crystallization in the sample.
Usually, the heat flux from or to the sample is measured [180] and is compared to a reference. The desired thermodynamic quantities can be extracted from the temperature difference of the sample and the reference. In section 2.1.1, the temperature behavior of the entropy and the specific heat was introduced.
A crystallization or melting event releases or requires large amounts of heat flux from or into the sample. This causes sharp peaks in a DSC thermogram. In contrast, a glass transition results in a step in the heat flow.
As an example, a thermogram obtained for water confined to MCM-41 with a pore diameter of d = 2.5 nm (P25) as obtained in this work is show in fig-ure 3.19. The different cycles in the figfig-ure correspond to measfig-urements upon heating and cooling with two different cooling rates. The measurement was performed by C. Lederle and shows a broad crystallization peak around 218 K and a melting peak at ca. 223 K . The DSC measurements in this work have been performed in the lab of AG St¨uhn, Technische Universit¨at Darmstadt, on a TA instrumentsQ1000.