# Time Domain NMR in Polymer Science: From the Laboratory to the Industry
## 2 Sequences
### \({}^{1}\)H \(T_{2}\) Relaxation  
Numerically T2 is the time required to achieve 37% of dephasing of magnetization, flipped in the transverse (xy) plane respect to the static magnetic field, so it is called transverse relaxation time. Here it is useful to remind the reader that the orientation of the magnetic field defines the z-axis of both the laboratory frame and the rotating frame, thus x and y are transverse axes. The dephasing is caused by the interactions existing within the spins' ensemble, due to random variation in the local field experienced by each spin, and it is not associated with any energy exchange with the environment. Theoretically, T2 should be obtained by flipping the magnetization on the xy plane with a single \(\pi\)/2 pulse and then analyzing the free induction decay (FID)  
However, especially with low-field devices and with solid samples, field inhomogeneities cause an additional contribution to the decoherence of spins, producing a much shorter decay time, indicated as T2*, this phenomenon is substantially uncorrelated with the dynamic phenomena. To remove the field's effect, more complex sequences need to be applied, as shown in Figure 2, T\({}_{2}\) is prevalently influenced by slow motional dynamics. Hahn echo (HE) [13] (Figure 2a) and CPMG (Carr-Purcell-Meiboom-Gill) [14] (Figure 2b) experiments rely on the formation of spin echoes. The application of an initial \(\pi\)/2 pulse is followed by \(\pi\) pulses that refocus the dephasing of spins due to field inhomogeneities, with the creation of an echo. The echo amplitude reduction is determined only by the transversal relaxation processes. HE employs a two-pulse scheme, repeated for increasing separation times between the two pulses, while CPMG employs a single \(\pi\)/2 pulse followed by multiple \(\pi\) pulses, separated by the same echo time. CPMG is faster than HE, but it is subjected to imperfection in the pulses and off-resonance effects, especially for highly coupled systems, while HE cannot offset spatial dislocation of the nuclei, and becomes unsuitable for the study of fluids [15]. Both experiments can also be performed on NMR-MOUSE devices [6], with appropriate adaptation. First of all, the \(\pi\) pulse is obtained doubling the amplitude of the \(\pi\)/2 pulse, instead of doubling the length, to excite a specific slice, and secondly, the signal decays according to an effective relaxation time T\({}_{2,\text{eff}}\), which is a mixture of T\({}_{1}\) and actual T\({}_{2}\), because of the strong field gradient. T\({}_{2}\) decay is represented according to different models: for liquids or systems with high mobility, exponential decays describe the system rather well, but for strongly restricted mobility more complex representations, including Gaussian (\(I(0)e^{-\frac{1}{2}(\frac{t}{T_{2}})^{2}}(3)\)), Abragamian (\(I(0)e^{-\frac{1}{2}(\alpha t)^{2}}\frac{sin(\alpha t)}{bit}\) (4) where a\({}_{2}\)+b\({}_{2}\)/3=M\({}_{2}\) is the second Van Vleck moment) or stretched or compressed exponentials (\(I(0)e^{-\frac{(t}{T_{2}})^{n}}(5)\) with \(0<n<2\)) functions, are employed. The Abragamian function (Equation (4)) is often used to represent the shape of magnetization decays of crystalline lattices, which is influenced by the strong contribution of the second dipole moment, caused by the high number of interacting dipolar-coupled pairs. A compressed exponential, also called Weibullian function, so-defined when \(n\) in Equation (5) is comprised between one and two, gives a better description of more isotropic dynamics, for example in liquids or polymers above T\({}_{g}\), especially when the decay depends on dipolar couplings. Instead, for \(n\) below one the exponential is defined as stretched and indicates inhomogeneous distributions of exponential decays, that it is interpreted as inhomogeneity of the sample itself [16]. Multiphase systems usually generate multi-modal decays, described by combinations of the above functions (Figure 3 depicts the curve forms of the equations described above).