Neutron Spin Echo Spectroscopy

Figure 3.17.: Setup of a neutron spin echo spectrometer.^{82, 83}

layer is a PNIPAM layer with an SLD of 0.8x10⁻⁶ Å⁻². The thickness of the third layer was fitted. The initial fit guess was derived from ellipsometry measurements.

For the low grafting density brush a four-layer model was assumed. In addition to the three-layer model described above, a fourth layer, consisting of the PNIPAM brush with a higher volume fraction of D₂O toward the solvent phase was added. The SLD of the initiator layer was adjusted to the ratio of initiator and dummy molecules on the surface.

More details can be found in Chapter 6 of this thesis.

3.2.7. Neutron Spin Echo Spectroscopy

Neutron spin echo spectroscopy (NSE) is well-suited to study the rather slow dynamics in polymer systems.^{79, 80} Due to its high energy resolution it grants access to the space and time evolution of segmental diffusion.

These days times up to a few 100 ns can be measured with NSE on large and intermedi-ate length scales. The technique measures energy changes in neutrons due to scattering events, with an energy resolution of up to 10⁻⁵ and a neutron velocity bandwidth of 20%.⁸¹

The neutron velocity of the incidenct and scattered neutron beam are compared, taking advantage of the Larmor precession of the neutron spin in an external homogeneous magnetic field.

More precisely the polarization of the neutron beam is measured. Figure 3.17 shows

the typical setup for a neutron spin echo experiment. The primary neutron beam passes a velocity selector, which typically selects a wavelength band of 20%. Afterwards, the neutron beam is polarized in a forward direction and all neutrons are polarized parallel to the beam direction. As the neutron beam passes a so-called π/2-flipper the neutron spin is flipped by 90^◦ and all neutrons are polarized perpendicular to the beam direction.

After entering the primary solenoid the neutron spin begins to precess around the lon-gitudinal field and undergoes ten to hundred thousands of precessions. As the neutrons have different velocities, after exiting the primary solenoid, they will have different pre-cession angles. In close proximity to the sample a π-flipper rotates the neutron spin by 180^◦. Behind the sample is the secondary solenoid, which is symmetric to the primary solenoid. Assuming that there is no energy exchange with the sample, the number of precessions in the secondary solenoid is equal to the primary. Upon reaching the second π/2-flipper all neutrons would have the same precession angle, pointing upwards, which is due to the π-flipper. This response is called the spin-echo. The second π/2-flipper rotates the neutron spin by 90^◦. An analyzer then transmits only neutrons with spin components parallel or antiparallel to the axis.⁸⁰

If however there is an energy exchange with the sample, a velocity changeΔνS from the scattering results. Therefore, the downward pointing spins will be rotated to the axially antiparallel direction by theπ/2-flipper and will be blocked by the analyzer, which will result in a reduced echo signal. The cosine of the final precession angle determines the transmission at the detector. The scattering functionS(Q, ω)describes the influence of the sample on the neutrons. Here,ωis the frequency proportional to the energy transfer between neutron and sample.

ω

2π = mn

2h[ν²−(ν+ ΔνS)²]. (3.29) The relation between the final polarization Pf and the initial polarization Pi can be expressed as:

The normalized dynamic structure factor describes the probability that a scattering event occurs at a certain wavelength change

δλ= (m/2π)λ³ω (3.31)

0 0.5

1.0 elastic scattering: Pdet= Pi

inelastic scattering: Pdet< Pi

Q₁

Figure 3.18.: Normalized intermediate scattering functions. A decay is only observed for inelastic scattering.

at a given momentum transfer Q. However, the result of a NSE measurement is the intermediate scattering function S(Q, τN SE). A NSE scan is done by variation of the guide field, which leads to a variation in Fourier time.⁸⁰

τN SE = 1.863x10⁻¹⁴HLλ³ (3.32) The intensity that reaches the detector is

IDet ∝ 1

path|B|dl the integral if the magnetic induction along the flight path of the neutron, andγ = 1.83033x10⁸ radian/sT. Therefore, the detected intensity strongly depends on the strength of the magnetic field and the neutron wavelength. However, at higher magnetic fields, inhomogeneities in the magnetic field will affect the measurement more. Furthermore, the maximum Fourier time depends on

τN SE =J λ³γ m²_n

2πh². (3.34)

Therefore, the maximum achievable Fourier time is limited by the neutron wavelength and the magnetic field.^{79, 80}

A schematic representation of different intermediate scattering functions (ISFs) is displayed in Figure 3.18. The dashed line represents a purely elastic scattering event, where no decay is observed and the polarization at the detector Pdet equals the initial

polarizationPi. The three solid lines represent inelastic scattering at differentQ-values, where Pdet is smaller than Pi. With increasing Q the decay becomes more rapid over τN SE.

Experimental Setup ISFs of PNIPAM microgels were recorded at the SNS-NSE at the spallation neutron source at the Oak Ridge National Laboratory in Tennessee, USA.

Combination of two neutron wavelengths (8 and 11 Å) allows coverage of a wide range of Fourier times between τN SE = 0.04 and 95 ns (s. Figure A.5). The Hellma quartz cells for NSE experiments had a neutron path way of 3 mm. To suppress center-of-mass diffusion the microgel concentration was 8 wt%. All measurements were conducted in D₂O at 20 ^◦C to ensure the swollen state of the microgel network.

Furthermore, a semidilute solution of linear PNIPAM (30 kDa) was measured. The concentration was chosen due to results from viscosimetry. The viscosity in D₂O as a function of the polymer concentration was measured with a Lauda iVisc glass capillary viscometer (LAUDA Scientific, Germany). The efflux timet was measured with micro-Ostwald-viscometers (type I and Ic), suited for different viscosity ranges. Samples sat in the temperature bath for 10 minutes prior to the measurement for thermal equilibration.

Results are the average over five measurements. As Ostwald-viscosimetry measures the kinematic viscosity ν =η/ρ, the polymer solution density had to be measured as well.

The polymer solution density was measured with an Anton Paar DMA 4500 density meter (Anton Paar, Austria).