Self-diffusion in polymer systems studied by magnetic field-gradient spin-echo NMR methods

https://doi.org/10.1016/j.pnmrs.2010.04.002Get rights and content

Introduction

Polymers are macromolecules that consist of repeating structural units, called monomers, connected by covalent chemical bonds. Homopolymers consist of only one type of repeating unit, while copolymers consist of two or more different monomers. To the first group of polymers belong polyethylene, polystyrene and polyisoprene (rubber). To the second group belong examples such as nylon, and natural polymers like proteins and polysaccharides. Copolymers may have the different monomers spread randomly along the chains, forming random copolymers, or grouped together in block structures, forming so-called block copolymers. The molecular weight of polymer molecules may be in the range a few hundreds to millions of Daltons (Da). The polymer chains may be linear, branched, or even cross-linked. In addition, side-groups that differ in chemical composition from the monomers in the main chain may be chemically linked to the polymer chain, forming what is known as grafted polymers. Naturally produced polymers, like proteins, are usually monodisperse, i.e. are characterized by one single molecular weight. However, for synthetically made polymers there will always be a certain distribution of molecular weights in a given sample. The degree of polydispersity is dependent on the synthetic procedure, and certain polymers may be made in the laboratory with only a minor degree of molecular weight polydispersity. Some polymers may crystallize at a temperature below a characteristic polymer dependent crystallization, or melting, temperature, Tm. Above this temperature, they exist as a melt, i.e. a liquid of pure polymer. At a certain temperature below Tm, there exists a characteristic temperature called the glass-transition temperature, Tg, with, usually, 0.5 < Tg/Tm < 0.8 [1]. When polymers that crystallize are below their Tg, they behave as a glass. Between their Tg and Tm, this kind of polymer may have the right mechanical properties for use as a material.

Polymers may be dissolved in solvents of low molecular weight, forming polymer solutions. For the solution to be thermodynamically stable, the molecular interactions between the solvent molecules and the polymer units must be favorable, so that the units are solvated. This stability is also manifested through the second virial coefficient for the solution, in that the coefficient must be sufficiently large and positive for the solution to be stable. In that case, the solvent is termed a good solvent for the actual polymer. In the other extreme, when the second virial coefficient is negative for a certain combination of polymer and solvent, the solvent is termed a bad solvent. In this case, the solution is not thermodynamically stable, and the polymer will separate from the solvent. There is also a border-case, when the combination of polymer and solvent makes the second virial coefficient equal to zero. The solvent is then termed a theta solvent.

For block copolymers one usually has the problem that whatever the solvent, it may only be a good solvent for one of the blocks. This circumstance may, however, be turned into an advantage, in that the blocks for which the solvent is a poor one may form intermolecular physical associations. This may lead to polymer network formation, which is important in e.g. gel systems, that belong to the type of systems generally known as soft materials. The thermodynamics in such systems, as measured by the second virial coefficient, is generally temperature dependent.

General texts on polymer systems and polymer solutions may be found in Refs. [1], [2], [3], [4], [5]. Doi and Edwards have written a text on the theory of polymer dynamics [6]. The dynamics in polymer systems may be studied using many different techniques. NMR techniques may be used to determine the self-diffusion of the various components in such systems, giving detailed information on dynamics and even on structure at the molecular level; this review is focused on the latter approach.

The pulsed field-gradient spin-echo (PGSE) NMR method dates back to the year 1965 [7]. In the 45 years that have elapsed since this pioneering work, a large number of publications have dealt with the study of molecular self-diffusion in systems of various complexity. In 1968 Vold and coworkers [8] suggested Fourier transformation of the (second half) of the spin-echo to extract the individual diffusion coefficients in multicomponent liquid systems from partly relaxed spectra. In the most basic form, the PGSE experiment is based on the 90°–τ–180° spin-echo radio-frequency (RF) sequence used in, for example, the Carr–Purcell method for spin–spin (T2) relaxation measurements [9] to which magnetic field-gradients have been added. At a time τ after the 180° pulse (usually of the order of 10–100 ms) a spin-echo is formed. If the Fourier transformation starts at this point, the resulting spectrum will be essentially identical to the one obtained using an ordinary 90° RF pulse sequence, except for the attenuation of the various signals, brought about by the spin–spin relaxation over the time span of 2τ. This attenuation is in principle given by exp(−2τ/T2). The values of T2 are different from one component to the other, and also in principle from one NMR signal to another in the same component. Therefore, there will always be a restriction on the range of τ that may be used in such experiments.

Magnetic field-gradients must be absent for the Fourier transformation of the second half of the echo to yield a high-resolution spectrum. In practice, the magnetic field (B) will not be completely homogenous over the sample, but during acquisition of the echo time signal the conditions in the probe have to correspond to those needed for producing high-resolution spectra. This means that one can not observe spin-echoes intended for Fourier transformation if a steady gradient of the magnitude needed for the study of molecular self-diffusion in liquids, is present during the acquisition. However, it is possible to measure self-diffusion in systems with one or a few diffusing components in a steady magnetic field-gradient that is present during acquisition of the spin-echo, without Fourier transformation, provided that the components have sufficiently different self-diffusion coefficients. Steady gradient self-diffusion studies of polymer systems have been reviewed by von Meerwall [10].

The pulsed gradient experiment in its most basic form is also based on the simple Carr–Purcell spin-echo experiment. A magnetic field-gradient of magnitude G (units: T m−1) is generated by special coils present in a probe designed for self-diffusion measurements. It is turned on for a short time δ between the 90° and the 180° RF pulses. After the 180° RF pulse and a total time elapse counting from the beginning of the first gradient pulse of duration Δ (usually of the order 100 ms for liquid systems using protons (see below)), a new pulsed field-gradient identical in magnitude and duration to the first one, is applied to the system. The key feature is that it is turned off before the echo is formed, and the acquisition of the second half of the echo is performed without any magnetic field-gradient present. This basic experiment is illustrated in Fig. 1. Since the magnetic field during the influence of the magnetic field-gradient varies linearly along the sample (typically in the z-direction) according to B = B0 + G × z, where B0 is the magnetic field without any gradient, every spin in the sample acquires a phase angle in the rotating frame of reference of θ = G × z × δ. The 180° RF pulse inverts the precession direction of the individual spins, and if these do not move in space during the time interval Δ, the spins will refocus at the time and loose intensity only according to unavoidable spin–spin (T2) relaxation effects (see above). However, if the spins move to another point in space because of self-diffusion during Δ, this will produce a loss of echo intensity because the spins will not refocus completely at . This loss may be calculated theoretically by taking into account that molecules will be displaced a root-mean-square distance Z (in z-direction) in a time interval t due to self-diffusion according to Z2 = 2 × D × t, where D is the self-diffusion coefficient of the molecule containing the nucleus. The theoretical result for the echo intensity I is given by Eq. (8). See also Section 3 and Ref. [7]. In Eqs. (7), (8), γ is the magnetogyric ratio of the nucleus under study, which in most cases is the proton, 1H. This nucleus has the next highest γ value for all nuclei (only the radioactive tritium nucleus, 3H, has a higher γ value), and hence has a high sensitivity for diffusion (cf. Eqs. (7), (8)). It is also present in most polymers, with usually relatively long spin relaxation times. See more on this point, below. It deserves to be mentioned that the corresponding result for a steady gradient is contained in Eqs. (7), (8) by letting δ = Δ = 2τ.

Pulsed field-gradients may in modern commercial NMR instruments be produced with strengths of the order of 10 T m−1 (i.e. 1000 Gauss cm−1), or even larger. This makes it possible to determine diffusion coefficients in the range 10−9–10−14 m2 s−1, covering the range of self-diffusion coefficients that may ordinarily be expected in liquid-like systems.

By keeping the time interval τ constant in the experiments the effect of relaxation on reducing the spin-echo amplitude is constant (see Section 3). Therefore, the attenuation of this intensity as a function of changing either G or δ, comes about only as a result of self-diffusion of the molecules that the spins belong to. In the case of restricted self-diffusion, it may be shown that the equivalence between changing either G or δ no longer holds, and G is the proper experimental parameter to vary [11]. More on the topic of restricted self-diffusion is provided below. It is important that temperature gradients are absent in the sample, since these may cause convection in liquid samples. Yoshida and coworkers have recently constructed a probe for high-temperature pulsed field-gradient NMR work that eliminates problems with convection even in such low-viscosity samples as pure water [12].

One problem using ordinary Hahn-echoes in pulsed field-gradient work, i.e. echoes formed using the conventional 90°–τ–180° RF pulse sequence shown above, is that protons in polymers often have short spin–spin relaxation times, T2, but considerably longer spin–lattice, T1, relaxation times. This fact warrants the use of so-called stimulated echoes instead [13]. Here, a series of three 90° RF pulses are used, and the main point is that the spins relax along the z-direction, that is, relax according to the spin–lattice relaxation time T1 during the diffusion observation time, Δ. There is a loss, however, in intensity of the stimulated echo as compared to the ordinary Hahn echo formed after the same time interval 2τ. Therefore, T1 must in general be of the order of three times longer than T2 to obtain any echo signal intensity recovery using this RF pulse sequence. However, this criterion is often met with polymers in the liquid state, as found in polymer solutions, melts or gels. The first magnetic field-gradient pulse must occur in the time interval between the first and the second 90° RF pulses, as illustrated in Fig. 1b. This means that the duration of the field-gradient pulses, δ, has to be very short, in practice of the order of 1–3 ms, since the spins relax according to their T2 during this period. Therefore, very intense magnetic field-gradient pulses are generally needed for such studies of the self-diffusion of polymer chains.

Eddy-currents are formed in the probe chassis due to the very quick changes of magnetic field during start and stop of (nearly) rectangular pulses, and the eddy-currents produce distortions in the echoes unless this effect is counteracted. Detailed discussions on how to improve the situation with regard to eddy-currents can be found in the monographs of Callaghan [11] and Price [24].

As noted above, during the time interval Δ the spins (molecules) diffuse a mean-squared displacement (MSD) Z2 in the z-direction (i.e. the direction of the magnetic field-gradient). As an example, for a molecule with self-diffusion coefficient of 10−11 m2 s−1 observed for Δ = 0.1 s (100 ms), (Z2)1/2  10−6 m. The molecule will thus diffuse a micrometer in this direction during the observation time in a PGSE experiment. This distance is much larger than the dimension of most molecules and usually larger than most supramolecular structures met in chemical systems. The “nano-world”, or the domain of classical colloidal systems, is 10−9–10−6 m approximately. This means that measured self-diffusion coefficients of molecules in organized systems may contain information on structures on those length scales in such systems [14].

Section snippets

Scope of the present review

In this review we present a summary of field-gradient NMR work on polymer systems, with some emphasis on systems that can be grouped under the topic “soft polymer systems” and on work that has been published in the last two decades. We also include studies on polymer solutions, since a lot of theoretical models for polymer and solvent dynamics have been tested on such systems, that are in many respects simpler than more organized systems. We do not claim to provide a complete up-to-date review

Background

The PGSE experiment is commonly based on either a two-pulse spin-echo or a three-pulse stimulated echo RF pulse sequence as shown in Fig. 1. The NMR signal is encoded for molecular displacements using magnetic field-gradient pulses with duration δ and amplitude G inserted into the defocusing and refocusing part of the RF pulse sequence. The time between the leading edges of the gradient pulses is denoted Δ. For an isotropic and homogeneous system, the evolution of the transverse magnetization

Polymer solutions

With “polymer solutions” we refer to two-component systems with polymeric molecules dissolved in a (usually) low molecular weight solvent. Depending on the nature of the polymer, i.e. the chemical composition of its repeating units or monomers, whether it is a homopolymer with only one type of repeating unit, or it is a copolymer with more than one (usually two) different units with or without block structure etc., the choice of solvent is critical. Most PGSE NMR investigations in such systems

First page preview

First page preview
Click to open first page preview

References (174)

  • O. Söderman et al.

    Prog. Nucl. Magn. Reson. Spectrosc.

    (1994)
  • P. Stilbs

    Prog. Nucl. Magn. Reson. Spectrosc.

    (1987)
  • T. Nose

    Annu. Rep. NMR Spectrosc.

    (1993)
  • G. Lindblom et al.

    Prog. Nucl. Magn. Reson. Spectrosc.

    (1994)
  • W.S. Price

    Annu. Rep. NMR Spectrosc.

    (1996)
  • H. Yasunaga

    Annu. Rep. NMR Spectrosc.

    (1997)
  • L. Masaro et al.

    Prog. Polym. Sci.

    (1999)
  • Y. Yamane et al.

    Annu. Rep. NMR Spectrosc.

    (2006)
  • F. Stallmach et al.

    Annu. Rep. NMR Spectrosc.

    (2007)
  • P.T. Callaghan et al.

    Anal. Chim. Acta

    (1986)
  • C. Leal et al.

    Biochim. Biophys. Acta

    (2008)
  • S.W. Provencher

    Comput. Phys. Commun.

    (1982)
  • K.P. Whittall et al.

    J. Magn. Reson.

    (1989)
  • P.T. Callaghan et al.

    Carbohydr. Res.

    (1987)
  • T. Cosgrove et al.

    Polymer

    (1995)
  • T. Cosgrove et al.

    Polymer

    (1995)
  • K. Shimada et al.

    J. Chem. Phys.

    (2005)
  • S. Rathgeber et al.

    J. Chem. Phys.

    (1999)
  • P. Munk

    Introduction to Macromolecular Science

    (1989)
  • P.C. Painter et al.

    Fundamentals of Polymer Science: An Introductory Text

    (1997)
  • L.H. Sperling

    Introduction to Physical Polymer Science

    (2006)
  • S.F. Sun

    Physical Chemistry of Macromolecules. Basic Principles and Issues

    (1994)
  • I. Teraoka

    Polymer Solutions. An Introduction to Physical Properties

    (2002)
  • M. Doi et al.

    The Theory of Polymer Dynamics

    (1986)
  • E.O. Stejskal et al.

    J. Chem. Phys.

    (1965)
  • R.L. Vold et al.

    J. Chem. Phys.

    (1968)
  • H.Y. Carr et al.

    Phys. Rev.

    (1954)
  • E.D. Von Meerwall

    Adv. Polym. Sci.

    (1983)
  • P.T. Callaghan

    Principles of Nuclear Magnetic Resonance Microscopy

    (1991)
  • K. Yoshida et al.

    J. Chem. Phys.

    (2005)
  • J.E. Tanner

    J. Chem. Phys.

    (1970)
  • L. Masaro et al.

    Can. J. Anal. Sci. Spectrosc.

    (1998)
  • W.S. Price

    NMR Studies of Translational Motion: Principles and Applications

    (2009)
  • P.T. Callaghan et al.

    Biopolymers

    (1985)
  • A. Svensson et al.

    J. Phys. Chem. B

    (2003)
  • D. Topgaard et al.

    J. Phys. Chem. B

    (2002)
  • P.G. De Gennes

    J. Chem. Phys.

    (1971)
  • P.T. Callaghan et al.

    Phys. Rev. Lett.

    (1992)
  • P. Stilbs

    Diffusion and electrophoretic studies using nuclear magnetic resonance

  • F. Perrin

    J. Phys. Radium

    (1934)
  • T.M. Garver et al.

    Macromolecules

    (1991)
  • H. Wassenius et al.

    Starch/Staerke

    (2006)
  • B. Jönsson et al.

    Colloid Polym. Sci.

    (1986)
  • H. Walderhaug et al.

    Langmuir

    (2008)
  • H. Johannesson et al.

    J. Chem. Phys.

    (1996)
  • P.T. Callaghan et al.

    Macromolecules

    (1983)
  • K.F. Morris et al.

    J. Am. Chem. Soc.

    (1992)
  • K.F. Morris et al.

    J. Am. Chem. Soc.

    (1993)
  • M. Nyden et al.

    Macromolecules

    (1998)
  • B. Håkanson et al.

    Colloid Polym. Sci.

    (2000)
  • Cited by (77)

    • Recent MRI and Diffusion Studies of Food Structures

      2017, Annual Reports on NMR Spectroscopy
    View all citing articles on Scopus
    1

    Tel.: +46 46 22 28 603; fax: +46 46 22 24 413.

    2

    Tel.: +46 46 22 28 204; fax: +46 46 22 24 413.

    View full text