3. Nuclear magnetic resonance 29
3.8. Simulation of NMR observables
To enhance the power of 2H NMR as a tool to investigate dynamics, Random-Walk(RW) simulations can be performed and compared with experiments. The simple orientation dependence of the QP interaction allows to calculate ob-servables, as for example spectra and stimulated echo signals, by this simple method. This way qualitative information on the kind, the geometry, and the timescale of the motion can be extracted.
In this work RW simulations of different geometric models are utilized. Detailed information on the simulation programs and the utilized models can be found in ref. [177]. This section will shortly illustrate the basic foundations of the sim-ulations. The simulations assume dynamics based on the Ivanov-model [178].
In this model the orientation of a particle is constant between two jumps and the jumps themselves happen instantaneously.
The NMR observables are calculated by averaging over multiple trajectories of molecular orientations ~Ω(t). The 2H NMR observables are calculated from the reorientation trajectory of the O-2H-bond. The important quantity is the
orientation of the interaction tensor towards the external magnetic field B0. The orientation-dependency of the frequency must be described by eq. (3.25):
ωQ= δ2
3 cos2(θ)−1−ηsin2(θ)cos(2φ) .
The constants δ and η are known from experiments or, in special cases, can be used as fit parameters, see below in sect. 3.8.2. With a given reorientation trajectory it is straight forward to calculate the signal for a desired pulse exper-iment. The calculation of NMR observables consists of two major parts: first, the generation of the trajectory Ω(~ t) and second, the calculation of the signal from this trajectory. The steps are discussed in the next sections.
3.8.1 Generation of trajectories
Generating an orientation trajectory Ω(t~ ) consist of three steps. It utilizes a random number generator (RNG) giving equally distributed numbersz∈[0; 1[. 1. In a first step, the geometrical model is implemented. Therefore, in models with a limited number of accessible sites, the vectors are pre-calculated.
To simulate a powder the resulting set of orientation vectors is rotated by Euler rotations in a random orientation. In supercooled liquids all ori-entations are isotropically distributed on a unit sphere, therefore the start orientation cannot simply be chosen by chance, but must be weighted pro-portional to its probability. This can be established by choosingΘ0∈[0;π[
in the form
Θ0=arccos(1−2z). (3.62) This is equal to weight it by its solid angle elementsinΘ0. From the avail-able orientations an initial orientation is diced, when there is no limitation of the number of accessible sites, the inital orientation is chosen by chance according to the probabilities of the angleΘ0.
2. In the second step a waiting time tw between two jumps must diced. Ac-cording to a Markov process the waiting times are independent from all other waiting times. For a proper description an exponential waiting time distribution characterized by a jump correlation timeτj is introduced:
P(tw) = 1 τj
exp
¨tw τj
«
(3.63)
From the distribution a waiting time can be calculated by
tw=−τjln(1−z) (3.64) when using the normalization relationP(z)dz=P(tw)dtwfor the random numbers z [157]. When considering the timescale the geometrical filter effect discussed for STE experiments must be taken into account as well.
For small angle jumps the jump correlationτj does not need to agree with the overall correlation timeτ. This can be expressed by the relation [179]
τj
τ = 3
2sin2γ (3.65)
in the case of isotropic jump with an angleγ.
3. In the third step a new orientation must be chosen from all available ones when a jump happens. This step strongly depends on the implied model of motion.
The complete trajectoryΩ(~ t)requires repetition of step 2. and 3. until the sum of all waiting times equals the experiment length. In order to achieve good statistics many particles must be simulated. The number of virtual molecules simulated may range from 104 to 108 depending on the simulated geometry and experiment.
3.8.2 Calculation of the NMR signal
The second part in calculating NMR observables is the extraction of the time signal from Ω(~ t). In this step the specific experiment plays an important role, since it is necessary to calculate the phase of the spin in every part of the ex-periment. For this work 1D SE and FID spectra as well as STE CFFxx in ze- and sa-order were simulated.
For all simulations the initial point in time (t = 0) is set after the first pulse.
This is valid due to the independence of the jump probability from the time bygone since the last reorientation [157]. In addition the time is also parted into constant, discrete increments of length ∆t, which define the resolution of the experiment.
1D Spectra
The time signal of FID and SE experiments is proportional to the phase of the virtual bond. This results from the density matrix eq. (3.36) and can be ex-pressed as [155, 157]
sFID(t, tp)∝cos
φ(0, t)
(3.66) in the simple case of an FID. For the SE spectrum it is:
sSE(t, tp)∝cos
φ(tp, t)−φ(0, tp)
(3.67) with the phasesφ(t1,t2)given by
φ(t1,t2) = Z t2
t1
ωQ(t0)dt0. (3.68) In the simulations the phases are given by the frequency ω0 which causes a phase shift of ω0 ·∆t in a time step ∆t. Hence, it is necessary to choose small time increments in order to account for all phase changes. Replacing the integrals in eq. (3.68) by discrete time steps gives
sSE(t)∝cos
Xt−1 ti=tp
ω0(ti)∆t−
tp−1
X
ti=0
ω0(ti)∆t
. (3.69)
Note that∆tmust be chosen such thattpcan be expressed as an integer multiple of∆t. The time signalStp(v)is the sum over the signalsstp(t)of all trajectories.
The spectrum can be obtained by Fourier transformation of the total signal. The transformation is performed starting at the time t=2tp in the SE case and at t
= 0 for the FID spectra.
Fitting experimental Pake spectra
The spectra obtained from RW simulation can be fitted to experimental spectra by varying δ and η. For reasons of speed a powder average is approximated by pre-calculated sets of orientations. For each orientation the corresponding quadrupolar frequency shift is calculated and used to generate a time signal by adding up the individual signals weighted by their solid angle contribution.
Additional Gaussian broadening was introduced to account for minor interac-tions and the spectrometer resolution. The influence of finite pulse length was attributed by convolution of a spectral resolution function A(ω) [153]:
Figure 3.15.:Measured (red) and fitted (black) SE spectrum of confined glycerol (sample P49G) at T = 163 K.
A(ω) =ω1∆π/2
sin
∆π/2Æ
ω21+14ω2 ∆π/2Æ
ω21+14ω2
3
. (3.70)
This function approximates the shape of the Fourier transformation of a rect-angle pulse. The frequency ω1 is given as ω1 = γB1. The value of the pulse length ∆p was fixed within experimentally set bounds, according to
∆p,fit = ∆p,exp±0.2 µs. By this procedure static spectra can be fitted using only the four free parameters δ, η, the pulse length ∆p,fit and the additional Gaussian broadening. An example of such a fit is shown in figure 3.15 for glycerol confined to MCM-41 with d = 4.9 nm. Here, the assumption of static dynamics fits the spectrum very well.
Zeeman order stimulated echoes
STE signals are calculated analogous to the SE signals. The STE pulse sequence can be expressed by
sze(t)∝
(cos
φ1(0, tp) cos
φ2(t0m+ ∆4, t)−φ2(t0m, t0m+ ∆4) sin
φ1(0, tp) sin
φ2(t0m+ ∆4, t)−φ2(t0m, t0m+ ∆4) (3.71) for ze and sa order, respectively. The signal depends implicitly on the evolution time tp, the mixing time t0m, and the time ∆4 in case of a four pulse experi-ment5. Analogous to the 1D case the phases can be discretized. The important contributions stem from the phases in the evolution time given by
5 In three pulse experiments∆4is set∆4= 0.
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.