Neutron Scattering - JUWEL - Forschungszentrum Jülich

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J-NSE 9 3.2 Zimm dynamics Polymers in solution can be described by the Zimm model, where hydrodynamic interaction between the chain segments mediated by the solvent are dominant. Moving chain segments exert forces on other segments due to the flow of the surrounding solvent. Within some approximations the system can be described by a Langevin equation analogous to that of the Rouse model which includes the friction coefficient ξ =6πηaseg with η the viscosity of the solvent. The diffusion of a chain segment depends on its hydrodynamic radius aseg. More details can be found in literature [3]. In general the intermediate scattering function for polymers in solution is � I(Q, t) kB TQ = F I(Q, 0) 3 � t (16) 6 πη with a function F (x) which depends on the polymer conformation and the quality of the solvent. The relaxation rate ΓQ = kB TQ3 /(6 πη) is mainly determined by the viscosity of the solvent. Internal dynamics is dominant at higher scattering vectors Q, where also the typical Q3 dependence of the relaxation rate can be observed. At smaller scattering vectors the contribution from the center of mass diffusion is more prominent so that rather a Q2 dependence of the relaxation rate is expected (see below). For not too small Q values and long polymer chains, where the end-to-end distance of the chain segments follow Gaussian statistics (Gaussian chain), the function F (x) can be written as: F (x) = �∞ 0 � 2/3 2 exp −u − x π �∞ 0 cos(yux −2/3 ) y 2 � � × 1 − exp − y2/3 �� √ dy du 2 (17) This more complex function can be approximated by a stretched exponential function over a wide Q range: � � x �β � F (x) � exp − b (18) with the parameters b � 1.354 and β � 0.85. approximation of F (x) can be used. For the evaluation of this experiment this 3.3 Center of mass diffusion With NSE spectroscopy the movements on length scales in the order of nanometer and time scales in the order of nanoseconds can be observed. This matches e.g. the center of mass diffusion of macromolecules in solution or micelles. The mean square displacement of a particle is = 6D0t with the diffusion constant D0 = kBT/(6πηRG), where RG is the hydrodynamic particle radius and η the viscosity (Stokes-Einstein-relation). The dynamic structure factor which is measured by NSE is then I(Q, t)/I(Q, 0) = exp � −1/6 Q 2� =exp � −D0tQ 2� Asimple diffusion therefore has a quadratic dependence on the scattering vector Q. (19)
10 O. Holderer, M. Zamponi and M. Monkenbusch 4 Preparatory Exercises 1. How fast do neutrons with a wavelength of 8 ˚A fly? 2. What is the value of the earth’s magnetic field? 3. What is the magnetic field at the surface of a common permanent magnet? 4. How many mm fall neutrons on their way from the entrance of the spectrometer to the detector (about 7 m) due to gravity? 5. How many precessions does a neutron of λ =8 ˚A perform in the main coils if the Fourier time is set to 20 ns? (Angle Ψ=γ/v � Bdl with γ =2π · 2913.06598 × 10 4 (s · T ) −1 . 5 Experiment Procedure 5.1 The experiment itself First, the function of the key components of the neutron spin-echo spectrometer will be explained and demonstrated. The generation of the ”Spin Echo” will be demonstrated with an auxiliary phase coil, wound around one of the main precession coils with a simple wire. With a laboratory DC-powersupply connected to this coil, the magnetic field inside this main coil is slightly varied. A fully symmetrical setup with identical magnetic path integrals in both main coils results in a maximum count rate at the detector. Increasing the current in the auxiliary coil from this point results in an additional phase shift of the neutron spin and thus the intensity varies from the maximum to a minimum and further to the next maximum and so on. In this way, the echo group is scanned. The experimental sample under investigation is a polymer chain (PEP, polyethylenepropylene) with a molecular weight of 70 kg/mol in solution (deuterated decane). The PEP concentration is 3 wt %. The first experiment with the sample is to measure the elastic scattering by recording the spin-up and spin-down intensity at the detector. • The coherent and incoherent scattering of the sample shall be extracted from this reading and plotted versus the scattering vector Q. The dynamics of the sample is measured. For some selected scattering vectors Q, a series of Fourier times is measured for the sample, for a background sample containing everything but the objects under investigation, in this case the pure deuterated solvent (d-decane), and for an elastic scatterer as reference.
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Member of the Helmholtz Association
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Forschungszentrum Jülich GmbH Jül
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Page 78 and 79: SPHERES Backscattering spectrometer
Page 80 and 81: SPHERES 3 1 Introduction Neutron ba
Page 82 and 83: SPHERES 5 Fig. 2: The monochromator
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Page 86 and 87: SPHERES 9 neutron is scattered, it
Page 88 and 89: SPHERES 11 The stationary equilibri
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Page 92 and 93: ________________________ DNS Neutro
Page 94 and 95: DNS 3 1 Introduction Polarized neut
Page 96 and 97: DNS 5 Monochromator horizontal- and
Page 98 and 99: DNS 7 3 Preparatory Exercises The p
Page 100 and 101: DNS 9 important and always necessar
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Page 128 and 129: ________________________ KWS-3 Very
Page 130 and 131: KWS-3 3 1 Introduction Ultra small
Page 132 and 133: KWS-3 5 2. The standard Q-range of
Page 134 and 135: KWS-3 7 Table 2. Samples for practi
Page 136 and 137: KWS-3 9 References [1] Eskilden, M.
Page 138 and 139: ________________________ RESEDA Res
Page 140 and 141: RESEDA 3 1 Introduction Neutrons ar
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