Chain Heterogeneity in Simulated Polymer Melts: NMR Free Induction Decay and Absorption Line

Abstract

Retrieving information about the molecular motion of polymer chain in melt from the FID is based on the theoretical assumptions in which the heterogeneity of segmental dynamics and its anisotropy along chain are ignored. This article—the second and conclusive in the series—demonstrates, in detail, the FID calculation for various length chains considering these peculiarities revealed in the previous article. The experimentally observed FID components are assigned to the corresponding fragments of the entangled chain, and reasons for the deviation from theoretical values defined by the chain structure are proposed.

Description of the Interactive Toolkit

The supplement is the interactive analytical toolkit based on the MS Excel 2010 and contains the numerical Fourier transformation of the relaxation curves, which can be compared with the Pake doublet NMR-absorption line broadened by the Lorentz line. Some publications have described such specialized software suites that reduce time-consuming routine data processing [111, 125], as well as simulating data instead of their simple fitting [126–132], but these program are either not free or require non-standard computer software and hardware. The suggested toolkit is intended to work on typical modern personal computer.

The initial simple example is the stretch exponential from formula (6b) or (7b), where all parameters p_E, T_2E, and 0 ≤ β ≤ 2 can be changed. Inputting any parameter value instantly affects the spectral line shape by the discrete Fourier transform procedure (8). This example can be replaced by a digitized relaxation curve. Further, the spectral line curve can be manually copied or readdressed onto another Excel sheet where it is visualized together with the Pake doublet, whose shape, by-turn, is varied with the help of two parameters—splitting b ≡ b_P and broadening a ≡ b_К—which both have dimension of reciprocal second, s⁻¹. The physical interpretation of b is the characteristic RDC in a spin system with isotropic space orientation of the inter-spin vectors; a is the line half-width broadening the doublet, what is analytically described by the convolution equation which is equivalent to Eq. (8) [12, 49, 133–135]:

$${\text{g}}\left( {{\omega }} \right) = \mathop \smallint \limits_{ - \infty }^\infty P\left( {{{\omega '}}} \right)G\left( {{{\omega }} - {{\omega '}}} \right)d{{\omega '}}.$$

(A1)

Here, the Pake probability density P(ω') is a piecewise continuous function derived by converting the inter-spin orientation vector into the resonance frequency offset ω':

$$P({{\omega '}})\, = \,\left\{ \begin{gathered} \frac{1}{{\sqrt {48b} }}\,\frac{1}{{\sqrt {b - {{\omega '}}} }},\quad - {\kern 1pt} 2b < {{\omega '}} < - b \hfill \\ \frac{1}{{\sqrt {48b} }}\left[ {\frac{1}{{\sqrt {b\, - \,{{\omega '}}} }}\, + \,\frac{1}{{\sqrt {b\, + \,{{\omega '}}} }}} \right],~\quad - {\kern 1pt} b\, < \,{{\omega '}}\, < \,b \hfill \\ \frac{1}{{\sqrt {48b} }}\,\frac{1}{{\sqrt {b + {{\omega '}}} }},\quad b < {{\omega '}} < 2b. \hfill \\ \end{gathered} \right.$$

(A2)

Each infinitely narrow line in this doublet can be broadened by either Lorentzian (A3a) or Gaussian (A3b), absorption lines:

$$G\left( {{{\omega }} - {{\omega '}}} \right) = \frac{a}{\pi }\,\frac{1}{{{{a}^{2}} + {{{\left( {{{\omega }} - {{\omega '}}} \right)}}^{2}}}},\quad ~{{T}_{2}} = \frac{1}{a};$$

(A3a)

$$G\left( {{{\omega }} - {{\omega '}}} \right) = \frac{1}{{a\sqrt {2\pi } }}\exp \left[ {{{{\left( {\frac{{{{\omega }} - {{\omega '}}}}{{a\sqrt 2 }}} \right)}}^{2}}} \right].$$

(A3b)

For all melt temperatures, the broadening is caused by the DC of the fast fluctuating Kuhn segments, so, in this case, it is necessary to apply formula (A3a); the Gaussian broadening (A3b) is ordinarily appeared in solid-phase systems. If the transverse magnetic relaxation is dominated by a single contribution of any line shape but not two as in formula (7a), then, in both cases, the line half-width a determines the relaxation rate 1/T₂, which is given by the second term in Eq. (A3a).

The input of the mentioned parameters also instantly changes the Pake doublet shape observed on the background of the approximated spectrum. As a result, it is possible to obtain the maximal coincidence of two spectra or determine whether the explored relaxation function requires the Pake doublet superposition with some another spectrum, as mentioned. The coincidence visualization is added by common quantitative characteristic, the relative root-mean-square difference of the overlapped curves [104].

The time and frequency ranges on both sheets are varied arbitrarily, which is essential to obtain a relaxation curve with deep damping (ca. 0.01 of the amplitude value) and revise the spectrum wings. The limitations are only imposed by the prescribed discretized time and frequency ranges, 400 points. In addition, it must be remembered that the rectangular pulse in the time domain is converted by the Fourier transform into a zero-centered bell-shaped curve with slowly damping oscillations, a so-called, sinc-function [49, 104, 136, 137]. This peculiarity appears in the NMR Fourier spectroscopy if the acquisition time is insufficient to achieve the FID damping approximately 0.01 of its amplitude value. Practically, this degree of attenuation refers to the missing stepped signal in the FID tail. The necessary damping limit is obtained by gradually decreasing T_2EF; that is, the oscillations on spectrum wings vanish at some value [138, 139].

The depth of experimental data attenuation is defined by their previous digitization; therefore, it is more complicated to reduce the tail jump. In this case one might try to add the tail analytic extension until the necessary depth is reached [139] or to simply cut the wing oscillations by narrowing frequency range. Though, it is important to remember, the last technique changes the amplitude of normalized spectrum because its area is partly removed by cutting the wings.