ARTICLES Qualitative and Quantitative Analysis of Solid State Free Induction Decay ( 1 H NMR) Curves Using a Combination of the Methods of Gardner and Prony: Isotactic Polypropylene as a Case Study S. Schreurs and J.-P. Franc ¸ ois* Limburgs UniVersitair Centrum, Institute of Materials Science, Department SBG, Research Group Analytical Chemistry, UniVersitaire Campus, B-3590 Diepenbeek, Belgium P. Adriaensens and J. Gelan Limburgs UniVersitair Centrum, Institute of Materials Science, Department SBG, Research Group Organic and Polymer Chemistry, UniVersitaire Campus, B-3590 Diepenbeek, Belgium ReceiVed: July 10, 1998; In Final Form: NoVember 3, 1998 A combination of the methods of Gardner and Prony is used for the qualitative and quantitative analysis of solid-state free induction decay ( 1 H NMR) curves of isotactic polypropylene, recorded at 40, 60, 80, and 100 °C at constant time intervals. These multicomponent decay curves contain information about the physico- chemical structure of the polymer sample: each component is characterized by a specific model function containing a spin-spin relaxation time T 2 and an intensity as parameters depending on the properties and amount of the corresponding phase. The number of phases is still an unanswered question. Each FID curve is analyzed by using an improved method of Gardner et al. in order to reveal the number of phases, to predict the best fitted model function, and to estimate the T 2 value of each component. The quantification of the parameters (T 2 and fraction) with Prony’s improved method is discussed. The presence of an intermediate phase is unambiguously demonstrated. The influence of temperature on the physicochemical structure of the polymer is examined. The precision of the computed parameters is discussed for the analyses of three consecutively measured FID curves of isotactic polypropylene at 60 °C. Introduction Solid state proton wide-line (broad-band) NMR provides a sensitive probe for the molecular state of a nuclear environment through the short range nature of magnetic dipolar interactions. For heterogeneous polymer systems (physical or chemical), the molecular mobility of each component and the fraction of the component can be directly estimated from the measurement of the spin-spin relaxation time T 2 . 1 A solid echo technique provides a way to collect the complete response of the system to a 90° pulse near the resonance frequency without influence of the system recovery time. The recorded free induction decay (FID) curve is a multicomponent decay curve having for each component a characteristic line shape, a spin-spin relaxation time T 2 , and an intensity depending on the characteristics of the corresponding phase. Unfortunately, the analysis of the recorded FID curve is a difficult task. Up till now, there is still uncertainty about the number of components and on their mathematical model functions in the decay curve. A lot of techniques have been proposed yet to analyze multicomponent decay curves based on, e.g., graphical extrac- tion, 2 linear 3,4 and nonlinear 5 (NLLS) least-squares methods, singular value decomposition, 6-8 Gardner’s method, 9-14 or Prony’s method. 15-18 The success of these techniques is often hampered by the ill conditioning property of the exponential- sum fitting problem. Furthermore, the final results seem to depend strongly upon the input of the correct number of exponential components and upon the initial estimates of the preexponential factors N° i and decay constants λ i . The original method of Gardner et al. 9 has been developed to decompose a multicomponent decay curve, which is a sum of independent exponential functions. The technique is based on nonlinear transformations of the variables and on Fourier transformations. From the theoretical point of view, the method results in a spectrum of delta functions whose number is equal to the number of components and their positions are equal to the decay constants. From the practical point of view, this method did not gain much success because at that time the numerical evaluation of the Fourier transformations was very cumbersome and tedious. After development of the fast Fourier transform (FFT) technique 19 the method was revised by Schlesinger 10 and, independently, by Smith and Cohn-Sfetcu. 11-14 Additionally, they combined the technique with an adequate low-pass filter in order to decrease the effects of computational noise and of the high frequency noise normally present in experimental data. They also showed that the improved method of Gardner et al. was able to analyze all kinds of multicomponent decay curves as long as the model functions were known. The analysis * To whom correspondence should be sent E-mail: francois@luc.ac.be. 1393 J. Phys. Chem. B 1999, 103, 1393-1401 10.1021/jp9829694 CCC: $18.00 © 1999 American Chemical Society Published on Web 02/17/1999