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Solid polymers: a challenge for NMR
Solid State Nuclear Magnetic Resonance 9 (1997) 21–27
Solid polymers: a challenge for NMR Vincent McBrierty* Physics Department, Trinity College, Dublin 2, Ireland
Abstract The sheer structural and motional complexity of polymers presents a formidable challenge to the power and versatility of NMR. This challenge is well met through exploitation of the plethora of experimental devices which have been developed over the past 50 years, among which the discovery of Magic Angle Spinning by Raymond Andrew features prominently. This paper presents a brief review of the subject in terms of a number of examples which illustrate the rich and detailed information which NMR provides. 1997 Elsevier Science B.V. Keywords: Solid polymers; NMR
1. Historical perspective A distinguished colleague’s 75th anniversary is both an occasion for celebration and for reflection. We celebrate the achievements of Raymond Andrew, a pioneer of nuclear magnetic resonance (NMR), while reflecting upon the manner in which the NMR technique has developed over more than half a century. Carolan [1] has quantified the ebb-and-flow which has characterized the overall growth of NMR; its epitaph has been written on more than one occasion only to be overtaken by the spontaneous development of ever more ingenious experiments and concomitant new areas of application. The historical perspective of NMR has an interesting Irish dimension. The Hamiltonian, *, which elegantly describes the different NMR interactions was invented by William Rowan Hamilton, an Irishman, who was Professor of Astronomy in Trinity College Dublin from 1827 to 1865. He was appointed the first
* Tel.: +353 1 6081676; fax: +353 1 6711759.
foreign Associate Member of the American Academy of Sciences in 1863, citing him at the time as ‘the greatest of living scientists’. Joseph Larmor, who graduated from Queen’s University Belfast in 1874, derived the Larmor Precession theorem which describes the motion of nuclear spins in a magnetic field; this theorem is at the heart of NMR. Among its many applications, NMR has a venerable history in unravelling the complexities of polymers, clarifying in particular the relationship between the intricate structure and dynamics of constituent macromolecules and their overall properties. Structural order is explored in a number of ways [2]: (i) through the inherent sensitivity of the shortrange, near-neighbour nature of dipole–dipole interactions; (ii) by virtue of the site-specific detail of high resolution spectra; and (iii) by monitoring the diffusive path lengths associated with the transport of spin energy through the spin system by means of spin diffusion. The fact that NMR is sensitive to motional correlation frequencies over some 10 decades (Fig. 1) renders the technique particularly useful in probing the wide
0926-2040/97/$17.00 1997 Elsevier Science B.V. All rights reserved PII S0926-2040 (97 )0 0039-8
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V. McBrierty / Solid State Nuclear Magnetic Resonance 9 (1997) 21–27
Fig. 1. Dynamic range of motional correlation frequencies probed by NMR.
range of molecular motions routinely encountered in polymers [3]. Collating NMR relaxation data with those from a range of diverse experiments, for example, dielectric spectroscopy, dynamic mechanical thermal analysis, differential scanning calorimetry and quasi-elastic neutron scattering, adds rich perspective to the overall relaxation profile of the polymer; conventionally, data are presented as plots of log vc(Hz) versus 1000/T(K), the so-called transition maps [4]. A number of examples will illustrate the way in which NMR probes the nature of molecular motion and structure and also their influence on macroscopic properties.
2. Molecular motion in polymers The partially crystalline polymer poly(vinylidene fluoride) (PVF2) is an interesting system from the perspective of spin dynamics, molecular motion and physical properties of practical importance. Low resolution T1, T1r and T2 relaxation data recorded as a function of temperature reveal three amorphous and one crystalline relaxation processes (Fig. 2a) [5]. Attention is focused on the crystalline a-relaxation
because of its significance in the development of piezoelectric properties in PVF2. The b-polymorph of the polymer is poled by suitably applying a strong electric field Ep (or corona discharge) for a period of time at an elevated temperature and then cooling the sample down while Ep is still switched on [6]. Poling proceeds with the alignment along Ep of the electric dipole moments which are collinear with the b-axes in the PVF2 crystals. The a-relaxation is identified with the small increase in the crystalline T2 component of ~1 ms at 120°C, initially attributed to rotation, or rotation of restricted amplitude, about the chain axis in the crystalline regions. The onset of rotational freedom is clearly linked to the poling process and, as such, the NMR data provide an indication of the temperature at which poling should be carried out. This preliminary motional assignment has been subsequently refined by the results of an elegant deuterium two-dimensional exchange analysis which reduced the 16 possible orientations of a pair of C-D bonds between the four polarization states in PVF2 to two, namely, reorientations through uniquely defined angles of 67° or 113° [7]. Guided by extant dielectric data [8], it was further deduced that poling involved electric dipole moment transitions along the molecu-
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lar directions only, accompanied by a conformational change, tgtg¯ → g¯tgt, which indicated a unique angle of 113° for the C-D bond rotations. The transient Overhauser effect has been used to examine cross relaxation between the 1H and 19F spin systems [9,10]. Any process leading to an Overhauser effect is a mixture of relaxation to the lattice and exchange of energy between the spin systems. Aside from the contribution of overlap of the tails of the two nuclear resonances, the mutual spin flip term Jo(qF − qH) will require an interaction with the lattice to conserve energy and is therefore allowed only in the presence of lattice motions at the difference frequency (qF − qH). For these two processes to contribute to cross relaxation j, the thermal motion must involve modulation of the direct interaction between the unlike spins. This process, observed in PVF2, is termed phonon assisted spin diffusion [11]. It contrasts with spin diffusion in a homonuclear system which proceeds via energy conserving pairwise near-neighbour dipolar interactions; most often it is the energy that diffuses through the spin system and not the spin itself. In PVF2, the Overhauser experiment provided (i)
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unusually good resolution of T1 components in the system because, in this case, it is the difference of two exponentials which is observed rather than the sum, (ii) quantification of cross relaxation effects and (iii) an additional sampling frequency for molecular motion at (qF − qH). The transition map in Fig. 2b collates the available relaxation data for PVF2.
3. Structural considerations The most useful commercial polymers are structurally inhomogenious. They include partially crystalline polymers, blends and composites, segregated block copolymers, filled and plasticized systems and polymers subjected to fabrication processes such as mechanical deformation. 3.1. Site-specific studies The characteristic features of composite low resolution spectra arising from different environments and molecular motions can label constituent structural
Fig. 2. (a) Proton resonance data for PVF2 recorded as a function of temperature. The long T1r component below the cross-over temperature (~100°C) and the short T2 above ambient temperature reflect crystalline behaviour. (b) Transition map for PVF2. Points labelled (X) are mechanical and dielectric relaxation data; the two data points (B) are from cross-relaxation experiment (see text). The remaining data points are from T1, T1r and T2 measurements. (Reprinted with permission from McBrierty et al. (1976). Copyright (1976) John Wiley and Sons.)
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Fig. 3. High resolution proton spectrum of polybutadiene terminated with phenyl groups. The numbers in parenthesis are theoretical estimates. (Reprinted from McBrierty and Packer (1993) with the permission of CUP.)
phases in the polymer. The delineation of T1r and T2 components from the crystalline and amorphous regions in Fig. 2a are an obvious example. This approach, dominated by dipole–dipole interactions, contrasts with the routine identification of individual chemical sites in high resolution spectra of solids, which is made possible by the development of techniques that selectively suppress the dominant dipolar contribution [2]. Fig. 3 typically portrays the high resolution proton spectrum of a moderately low molecular weight (Mn = 2600) polybutadiene terminated at both ends with phenyl groups. Phenyl, methylene and methine protons are clearly resolved and their spectral intensities agree closely with theoretical estimates.
tion, the anisotropy is spatially averaged to yield a single isotropic value (P = Px = Py = Pz). Consider uniaxial stretching which may be viewed as the progressive alignment of basic structural units (single crystals, chain segments or specific structural moieties on the polymer chain) in the bulk polymer along the draw direction (Fig. 4a). Macroscopic properties are altered quite dramatically, as in oriented fibres whose strength can be considerably enhanced by extensive uniaxial drawing [12]. The key elements of a model for the deformed polymer include (i) the intrinsic property of the molecular structural units (for example, their NMR, mechanical, electrical, optical response), (ii) the relative amounts of distinguishable macroscopic phases in the polymer (for example, crystalline/amorphous content) and (iii) the relative orientation of the constituent structural units relative to a chosen axis in the polymer sample (usually the draw direction) [2]. Information is required on the statistical distribution of the structural units in order to compute the aggregate response for the polymer as a whole. In NMR, one is interested in the orientation of a fundamental NMR parameter such as the direction of
3.2. Oriented polymers Subjection of polymers to specific fabrication procedures such as extrusion or mechanical deformation, or indeed in some cases by application of electric or magnetic fields (vide infra), can lead to preferred molecular orientation along a specific direction (or directions) within the polymer. In this way, the anisotropy normally displayed by material properties at the molecular level (Px Þ Py Þ Pz) can be at least partially recovered in the bulk where, prior to deforma-
Fig. 4. (a) Random orientation of structural units in an undrawn polymer. (b) Orientation of a typical structural unit relative to the draw direction in the sample and the orientation of the sample in the laboratory co-ordinate frame whose polar axis is in the direction of Bo. (Reprinted from McBrierty and Packer (1993) with the permission of CUP.)
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an internuclear vector, the principal axis of the chemical shift tensor, or some other specific direction in the structural unit relative to the direction of Bo, the steady laboratory magnetic field. P(Q) is the most general distribution function which describes the orientations of the z-axes of the molecular co-ordinate frame (xyz) relative to the sample frame (XoYoZo) where Zo is the draw direction in the polymer and Q denotes the Euler angles (abg) (Fig. 4b). The reasonable assumption of fibre symmetry, that is transverse isotropy about the fibre or draw direction, implies that the angle a is averaged over a circle. If the crystal symmetry of the structural unit itself can be treated as transversely isotropic, g is similarly averaged in which case the distribution function simplifies to P(b) which can be expanded in a series of Legendre polynomials as follows [13]. P(b) = ∑ (l + 1=2)〈Pl (cosb)〉Pl (cosb) l = 0, 2, 4… l
(1) where ,Pl(cosb). are the non-zero moments of the statistical distribution. If symmetry were not invoked, eqn (1) would involve a sum over rotation matrices, D(l) m,n(abg) rather than Legendre polynomials.
Fig. 5. Dependence of d31 on ,cos2d> in a number of samples of poled PVF2. X-ray data are denoted by (B). (Reprinted with permission from McBrierty et al. (1982). Copyright (1982) American Institute of Physics.)
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Fig. 6. Schematic of the way in which efficiently relaxing spins in region 1 of a polymer can relax spins in region 2 which are only weakly coupled to the lattice.
By way of illustration, consider the second, M2, and fourth, M4, moments of a dipolar broadened resonance line for a partially drawn fibre which can be expressed as 2N
MN (b1 ) = ∑ CN (b1 , l)〈PL (cosb)〉Pl (cosb1 ) N = 2, 4… l=0
(2) The coefficients CN(b1,l) are determined from known positions of nuclei (lattice sums) and fundamental constants and b1 is the angle between the draw or fibre direction of the polymer and Bo. M2(b1) or, equivalently, the free induction decay (FID), T2(b1), recorded as a function of fibre orientation b1 yields second, ,P2(cosb)., and fourth, ,P4(cosb)., moments of the distribution; M4(b1), in principle, provides moments to order 8. These moments have been used successfully to map out the statistical distribution of structural units (eqn (1)) and to follow the change in the distribution as a function of sample draw ratio. Because of the limited number of moments available, the predicted distributions are progressively less accurate at the higher draw ratios. In other applications, the moments of the distribution have been used to predict other properties such as birefringence and extensional modulus, for which only moments to order 2 and 4, respectively, are required [13]. Consider again the case of b-PVF2 where the analysis was used to examine the evolution of the electric dipole moment distribution relative to Ep. The aim of the experiment was to determine the contribution of dipole reorientation to the evolution of the piezoelectric d31 constant. d31 was found to correlate well with ,cos2d. where d is the angle between the poling direction and a typical b-axis in the PVF2 crystal, the direction of the electric dipole moment (Fig. 5) [6].
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Fig. 7. (a) Schematic representation of ester methyl and methylene protons relaxed by a-methyl protons via spin diffusion in MMA. (b) Energy transport via successive flip-flop transitions to the methyl groups (the sinks) which are tightly coupled to the lattice. The filled arrows denote the high energy spins and each row represents successive ‘snap-shots’ in time as the spin energy progresses to the sink. (c) Schematic representation of ‘spheres of influence’ surrounding the efficiently relaxing a-methyl groups on the MMA chain. All the protons in the MMA and the PVC protons in a compatible PVC/PMMA blend are within the spheres of influence and are therefore relaxed by spin diffusion to the sinks, yielding exponential decay. (d) Comparison of T1 relaxation for a physical mixture of PMMA and PVC (X, W) with the response for the compatible blend (B). (Reprinted from McBrierty and Packer (1993) with the permission of CUP.)
A number of interesting observations were drawn from the results, i.e. d31 evolved smoothly with poling time, np/3-fold rather than np-fold rotation was indicated, there was good agreement between NMR and X-ray predictions of electric dipole reorientation and dipole alignment was far from perfect along the poling direction even in the most strongly poled material.
3.3. Polymer blends Component polymers are often blended together to achieve desirable properties. In such cases it is important to ascertain the degree of mixing of the two components or, alternatively stated, the degree of structural heterogeneity in the blended system as mea-
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V. McBrierty / Solid State Nuclear Magnetic Resonance 9 (1997) 21–27
sured on an appropriate dimensional scale. The resolution of common experimental probes can span many orders of magnitude [2]. Even within a given technique, such as NMR, resolution can vary from microns as in NMR imaging to a few nanometers as in the case of T1r relaxation when spin diffusion is operative. Spin diffusion as revealed in T1 and T1r relaxation data permits efficiently relaxing protons in one component to fully or partially relax other protons in the spin system that are not efficiently coupled to the lattice (Fig. 6). The ‘sphere of influence’ of the relaxing sink can embrace protons in both polymer components in a well blended system (Fig. 7). The radius of the ‘sphere of influence’ is the root mean diffusive path length 〈r2 〉 where ,r2. = nDst. For threedimensional diffusion (n = 6) the diffusion coefficient Ds = 10 − 16m2s − 1 in the laboratory frame and half this magnitude in the rotating frame and t can be equated to T1 or T1r to obtain reasonable orders of magnitude of the spatial dimensions involved. Typically, in T1 and T1r experiments, 〈r2 〉 = 20 and 2 nm, respectively. The correlation length of the local field, which is a few near-neighbour distances, defines the lower limit of the spatial inhomogeneities that can be probed in this way. The analysis has been applied to a number of systems where methyl groups in one of the components act as the relaxation sink. Single exponential T1 decay in blends of poly(methyl methacrylate)/poly(styrene co-acrylonitrile) (PMMA/PSAN) and poly(vinyl methyl ether/polystyrene (PVME/PS) confirm an upper limit of 20 nm for the possible spatial inhomogeneity in the blends [2]. In cases where T1r for the blend is also exponential, miscibility is on the finer scale of 2 nm. The transient Overhauser effect has also been used
p
p
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to probe the degree of mixing of the two component polymers in a blend of PVF2/PMMA [14]. Crossrelaxation measurements, supported by T1, T2 and T1r data, confirm that a substantial number of amorphous PVF2 molecules (at least 20% in the 40/60 blend) see PMMA molecules at nearest neighbour distances.
References [1] J. Carolan, E. Raymond Andrew Conference, 1997. [2] V.J. McBrierty and K.J. Packer, Nuclear Magnetic Resonance in Solid Polymers, Cambridge University Press, Cambridge, 1993. [3] V.J. McBrierty, in D. Bloor, R.J. Brook, M.C. Flemings and S. Mahajan (Eds.), Encyclopedia of Advanced Materials, Vol. 3, pp. 1798–1803, Pergamon Press, New York, 1994. [4] D.W. McCall, in R.S. Carter and J.J. Rush (Eds.), Molecular Dynamics and Structure, National Bureau of Standards, pp. 475–537, Special Publication 301, Washington, DC. [5] V.J. McBrierty, D.C. Douglass and T.A. Weber, J. Polym. Sci.: Polym. Phys. Ed., 14 (1976) 1271. [6] V.J. McBrierty, D.C. Douglass and T.T. Wang, Appl. Phys. Lett., 41 (11) (1982) 1051. [7] J. Hirshinger, D. Schaefer, H.W. Spiess and A.J. Lovinger, Macromolecules, 24 (1991) 2428. [8] Y. Miyamoto, H. Miyaji and K. Asai, J. Polym. Sci.: Polym. Phys. Ed., 18 (1980) 597. [9] I. Solomon, Phys. Rev., 99 (1955) 559. [10] I. Solomon and N. Bloembergen, J. Chem. Phys., 25 (1955) 261. [11] V.J. McBrierty and D.C. Douglass, Macromolecules, 10 (1977) 855. [12] I.M. Ward and D.W. Hadley, An Introduction to the Mechanical Properties of Solid Polymers, Wiley, New York, 1993. [13] V.J. McBrierty and I.M. Ward, Br. J. Appl. Phys., Ser. 2, 1 (1968) 1529. [14] D.C. Douglass and V.J. McBrierty, Macromolecules, 11 (1978) 766.
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Solid polymers: a challenge for NMR Vincent McBrierty* Physics Department, Trinity College, Dublin 2, Ireland
Abstract The sheer structural and motional complexity of polymers presents a formidable challenge to the power and versatility of NMR. This challenge is well met through exploitation of the plethora of experimental devices which have been developed over the past 50 years, among which the discovery of Magic Angle Spinning by Raymond Andrew features prominently. This paper presents a brief review of the subject in terms of a number of examples which illustrate the rich and detailed information which NMR provides. 1997 Elsevier Science B.V. Keywords: Solid polymers; NMR
1. Historical perspective A distinguished colleague’s 75th anniversary is both an occasion for celebration and for reflection. We celebrate the achievements of Raymond Andrew, a pioneer of nuclear magnetic resonance (NMR), while reflecting upon the manner in which the NMR technique has developed over more than half a century. Carolan [1] has quantified the ebb-and-flow which has characterized the overall growth of NMR; its epitaph has been written on more than one occasion only to be overtaken by the spontaneous development of ever more ingenious experiments and concomitant new areas of application. The historical perspective of NMR has an interesting Irish dimension. The Hamiltonian, *, which elegantly describes the different NMR interactions was invented by William Rowan Hamilton, an Irishman, who was Professor of Astronomy in Trinity College Dublin from 1827 to 1865. He was appointed the first
* Tel.: +353 1 6081676; fax: +353 1 6711759.
foreign Associate Member of the American Academy of Sciences in 1863, citing him at the time as ‘the greatest of living scientists’. Joseph Larmor, who graduated from Queen’s University Belfast in 1874, derived the Larmor Precession theorem which describes the motion of nuclear spins in a magnetic field; this theorem is at the heart of NMR. Among its many applications, NMR has a venerable history in unravelling the complexities of polymers, clarifying in particular the relationship between the intricate structure and dynamics of constituent macromolecules and their overall properties. Structural order is explored in a number of ways [2]: (i) through the inherent sensitivity of the shortrange, near-neighbour nature of dipole–dipole interactions; (ii) by virtue of the site-specific detail of high resolution spectra; and (iii) by monitoring the diffusive path lengths associated with the transport of spin energy through the spin system by means of spin diffusion. The fact that NMR is sensitive to motional correlation frequencies over some 10 decades (Fig. 1) renders the technique particularly useful in probing the wide
0926-2040/97/$17.00 1997 Elsevier Science B.V. All rights reserved PII S0926-2040 (97 )0 0039-8
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V. McBrierty / Solid State Nuclear Magnetic Resonance 9 (1997) 21–27
Fig. 1. Dynamic range of motional correlation frequencies probed by NMR.
range of molecular motions routinely encountered in polymers [3]. Collating NMR relaxation data with those from a range of diverse experiments, for example, dielectric spectroscopy, dynamic mechanical thermal analysis, differential scanning calorimetry and quasi-elastic neutron scattering, adds rich perspective to the overall relaxation profile of the polymer; conventionally, data are presented as plots of log vc(Hz) versus 1000/T(K), the so-called transition maps [4]. A number of examples will illustrate the way in which NMR probes the nature of molecular motion and structure and also their influence on macroscopic properties.
2. Molecular motion in polymers The partially crystalline polymer poly(vinylidene fluoride) (PVF2) is an interesting system from the perspective of spin dynamics, molecular motion and physical properties of practical importance. Low resolution T1, T1r and T2 relaxation data recorded as a function of temperature reveal three amorphous and one crystalline relaxation processes (Fig. 2a) [5]. Attention is focused on the crystalline a-relaxation
because of its significance in the development of piezoelectric properties in PVF2. The b-polymorph of the polymer is poled by suitably applying a strong electric field Ep (or corona discharge) for a period of time at an elevated temperature and then cooling the sample down while Ep is still switched on [6]. Poling proceeds with the alignment along Ep of the electric dipole moments which are collinear with the b-axes in the PVF2 crystals. The a-relaxation is identified with the small increase in the crystalline T2 component of ~1 ms at 120°C, initially attributed to rotation, or rotation of restricted amplitude, about the chain axis in the crystalline regions. The onset of rotational freedom is clearly linked to the poling process and, as such, the NMR data provide an indication of the temperature at which poling should be carried out. This preliminary motional assignment has been subsequently refined by the results of an elegant deuterium two-dimensional exchange analysis which reduced the 16 possible orientations of a pair of C-D bonds between the four polarization states in PVF2 to two, namely, reorientations through uniquely defined angles of 67° or 113° [7]. Guided by extant dielectric data [8], it was further deduced that poling involved electric dipole moment transitions along the molecu-
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V. McBrierty / Solid State Nuclear Magnetic Resonance 9 (1997) 21–27
lar directions only, accompanied by a conformational change, tgtg¯ → g¯tgt, which indicated a unique angle of 113° for the C-D bond rotations. The transient Overhauser effect has been used to examine cross relaxation between the 1H and 19F spin systems [9,10]. Any process leading to an Overhauser effect is a mixture of relaxation to the lattice and exchange of energy between the spin systems. Aside from the contribution of overlap of the tails of the two nuclear resonances, the mutual spin flip term Jo(qF − qH) will require an interaction with the lattice to conserve energy and is therefore allowed only in the presence of lattice motions at the difference frequency (qF − qH). For these two processes to contribute to cross relaxation j, the thermal motion must involve modulation of the direct interaction between the unlike spins. This process, observed in PVF2, is termed phonon assisted spin diffusion [11]. It contrasts with spin diffusion in a homonuclear system which proceeds via energy conserving pairwise near-neighbour dipolar interactions; most often it is the energy that diffuses through the spin system and not the spin itself. In PVF2, the Overhauser experiment provided (i)
23
unusually good resolution of T1 components in the system because, in this case, it is the difference of two exponentials which is observed rather than the sum, (ii) quantification of cross relaxation effects and (iii) an additional sampling frequency for molecular motion at (qF − qH). The transition map in Fig. 2b collates the available relaxation data for PVF2.
3. Structural considerations The most useful commercial polymers are structurally inhomogenious. They include partially crystalline polymers, blends and composites, segregated block copolymers, filled and plasticized systems and polymers subjected to fabrication processes such as mechanical deformation. 3.1. Site-specific studies The characteristic features of composite low resolution spectra arising from different environments and molecular motions can label constituent structural
Fig. 2. (a) Proton resonance data for PVF2 recorded as a function of temperature. The long T1r component below the cross-over temperature (~100°C) and the short T2 above ambient temperature reflect crystalline behaviour. (b) Transition map for PVF2. Points labelled (X) are mechanical and dielectric relaxation data; the two data points (B) are from cross-relaxation experiment (see text). The remaining data points are from T1, T1r and T2 measurements. (Reprinted with permission from McBrierty et al. (1976). Copyright (1976) John Wiley and Sons.)
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Fig. 3. High resolution proton spectrum of polybutadiene terminated with phenyl groups. The numbers in parenthesis are theoretical estimates. (Reprinted from McBrierty and Packer (1993) with the permission of CUP.)
phases in the polymer. The delineation of T1r and T2 components from the crystalline and amorphous regions in Fig. 2a are an obvious example. This approach, dominated by dipole–dipole interactions, contrasts with the routine identification of individual chemical sites in high resolution spectra of solids, which is made possible by the development of techniques that selectively suppress the dominant dipolar contribution [2]. Fig. 3 typically portrays the high resolution proton spectrum of a moderately low molecular weight (Mn = 2600) polybutadiene terminated at both ends with phenyl groups. Phenyl, methylene and methine protons are clearly resolved and their spectral intensities agree closely with theoretical estimates.
tion, the anisotropy is spatially averaged to yield a single isotropic value (P = Px = Py = Pz). Consider uniaxial stretching which may be viewed as the progressive alignment of basic structural units (single crystals, chain segments or specific structural moieties on the polymer chain) in the bulk polymer along the draw direction (Fig. 4a). Macroscopic properties are altered quite dramatically, as in oriented fibres whose strength can be considerably enhanced by extensive uniaxial drawing [12]. The key elements of a model for the deformed polymer include (i) the intrinsic property of the molecular structural units (for example, their NMR, mechanical, electrical, optical response), (ii) the relative amounts of distinguishable macroscopic phases in the polymer (for example, crystalline/amorphous content) and (iii) the relative orientation of the constituent structural units relative to a chosen axis in the polymer sample (usually the draw direction) [2]. Information is required on the statistical distribution of the structural units in order to compute the aggregate response for the polymer as a whole. In NMR, one is interested in the orientation of a fundamental NMR parameter such as the direction of
3.2. Oriented polymers Subjection of polymers to specific fabrication procedures such as extrusion or mechanical deformation, or indeed in some cases by application of electric or magnetic fields (vide infra), can lead to preferred molecular orientation along a specific direction (or directions) within the polymer. In this way, the anisotropy normally displayed by material properties at the molecular level (Px Þ Py Þ Pz) can be at least partially recovered in the bulk where, prior to deforma-
Fig. 4. (a) Random orientation of structural units in an undrawn polymer. (b) Orientation of a typical structural unit relative to the draw direction in the sample and the orientation of the sample in the laboratory co-ordinate frame whose polar axis is in the direction of Bo. (Reprinted from McBrierty and Packer (1993) with the permission of CUP.)
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V. McBrierty / Solid State Nuclear Magnetic Resonance 9 (1997) 21–27
an internuclear vector, the principal axis of the chemical shift tensor, or some other specific direction in the structural unit relative to the direction of Bo, the steady laboratory magnetic field. P(Q) is the most general distribution function which describes the orientations of the z-axes of the molecular co-ordinate frame (xyz) relative to the sample frame (XoYoZo) where Zo is the draw direction in the polymer and Q denotes the Euler angles (abg) (Fig. 4b). The reasonable assumption of fibre symmetry, that is transverse isotropy about the fibre or draw direction, implies that the angle a is averaged over a circle. If the crystal symmetry of the structural unit itself can be treated as transversely isotropic, g is similarly averaged in which case the distribution function simplifies to P(b) which can be expanded in a series of Legendre polynomials as follows [13]. P(b) = ∑ (l + 1=2)〈Pl (cosb)〉Pl (cosb) l = 0, 2, 4… l
(1) where ,Pl(cosb). are the non-zero moments of the statistical distribution. If symmetry were not invoked, eqn (1) would involve a sum over rotation matrices, D(l) m,n(abg) rather than Legendre polynomials.
Fig. 5. Dependence of d31 on ,cos2d> in a number of samples of poled PVF2. X-ray data are denoted by (B). (Reprinted with permission from McBrierty et al. (1982). Copyright (1982) American Institute of Physics.)
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Fig. 6. Schematic of the way in which efficiently relaxing spins in region 1 of a polymer can relax spins in region 2 which are only weakly coupled to the lattice.
By way of illustration, consider the second, M2, and fourth, M4, moments of a dipolar broadened resonance line for a partially drawn fibre which can be expressed as 2N
MN (b1 ) = ∑ CN (b1 , l)〈PL (cosb)〉Pl (cosb1 ) N = 2, 4… l=0
(2) The coefficients CN(b1,l) are determined from known positions of nuclei (lattice sums) and fundamental constants and b1 is the angle between the draw or fibre direction of the polymer and Bo. M2(b1) or, equivalently, the free induction decay (FID), T2(b1), recorded as a function of fibre orientation b1 yields second, ,P2(cosb)., and fourth, ,P4(cosb)., moments of the distribution; M4(b1), in principle, provides moments to order 8. These moments have been used successfully to map out the statistical distribution of structural units (eqn (1)) and to follow the change in the distribution as a function of sample draw ratio. Because of the limited number of moments available, the predicted distributions are progressively less accurate at the higher draw ratios. In other applications, the moments of the distribution have been used to predict other properties such as birefringence and extensional modulus, for which only moments to order 2 and 4, respectively, are required [13]. Consider again the case of b-PVF2 where the analysis was used to examine the evolution of the electric dipole moment distribution relative to Ep. The aim of the experiment was to determine the contribution of dipole reorientation to the evolution of the piezoelectric d31 constant. d31 was found to correlate well with ,cos2d. where d is the angle between the poling direction and a typical b-axis in the PVF2 crystal, the direction of the electric dipole moment (Fig. 5) [6].
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Fig. 7. (a) Schematic representation of ester methyl and methylene protons relaxed by a-methyl protons via spin diffusion in MMA. (b) Energy transport via successive flip-flop transitions to the methyl groups (the sinks) which are tightly coupled to the lattice. The filled arrows denote the high energy spins and each row represents successive ‘snap-shots’ in time as the spin energy progresses to the sink. (c) Schematic representation of ‘spheres of influence’ surrounding the efficiently relaxing a-methyl groups on the MMA chain. All the protons in the MMA and the PVC protons in a compatible PVC/PMMA blend are within the spheres of influence and are therefore relaxed by spin diffusion to the sinks, yielding exponential decay. (d) Comparison of T1 relaxation for a physical mixture of PMMA and PVC (X, W) with the response for the compatible blend (B). (Reprinted from McBrierty and Packer (1993) with the permission of CUP.)
A number of interesting observations were drawn from the results, i.e. d31 evolved smoothly with poling time, np/3-fold rather than np-fold rotation was indicated, there was good agreement between NMR and X-ray predictions of electric dipole reorientation and dipole alignment was far from perfect along the poling direction even in the most strongly poled material.
3.3. Polymer blends Component polymers are often blended together to achieve desirable properties. In such cases it is important to ascertain the degree of mixing of the two components or, alternatively stated, the degree of structural heterogeneity in the blended system as mea-
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V. McBrierty / Solid State Nuclear Magnetic Resonance 9 (1997) 21–27
sured on an appropriate dimensional scale. The resolution of common experimental probes can span many orders of magnitude [2]. Even within a given technique, such as NMR, resolution can vary from microns as in NMR imaging to a few nanometers as in the case of T1r relaxation when spin diffusion is operative. Spin diffusion as revealed in T1 and T1r relaxation data permits efficiently relaxing protons in one component to fully or partially relax other protons in the spin system that are not efficiently coupled to the lattice (Fig. 6). The ‘sphere of influence’ of the relaxing sink can embrace protons in both polymer components in a well blended system (Fig. 7). The radius of the ‘sphere of influence’ is the root mean diffusive path length 〈r2 〉 where ,r2. = nDst. For threedimensional diffusion (n = 6) the diffusion coefficient Ds = 10 − 16m2s − 1 in the laboratory frame and half this magnitude in the rotating frame and t can be equated to T1 or T1r to obtain reasonable orders of magnitude of the spatial dimensions involved. Typically, in T1 and T1r experiments, 〈r2 〉 = 20 and 2 nm, respectively. The correlation length of the local field, which is a few near-neighbour distances, defines the lower limit of the spatial inhomogeneities that can be probed in this way. The analysis has been applied to a number of systems where methyl groups in one of the components act as the relaxation sink. Single exponential T1 decay in blends of poly(methyl methacrylate)/poly(styrene co-acrylonitrile) (PMMA/PSAN) and poly(vinyl methyl ether/polystyrene (PVME/PS) confirm an upper limit of 20 nm for the possible spatial inhomogeneity in the blends [2]. In cases where T1r for the blend is also exponential, miscibility is on the finer scale of 2 nm. The transient Overhauser effect has also been used
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to probe the degree of mixing of the two component polymers in a blend of PVF2/PMMA [14]. Crossrelaxation measurements, supported by T1, T2 and T1r data, confirm that a substantial number of amorphous PVF2 molecules (at least 20% in the 40/60 blend) see PMMA molecules at nearest neighbour distances.
References [1] J. Carolan, E. Raymond Andrew Conference, 1997. [2] V.J. McBrierty and K.J. Packer, Nuclear Magnetic Resonance in Solid Polymers, Cambridge University Press, Cambridge, 1993. [3] V.J. McBrierty, in D. Bloor, R.J. Brook, M.C. Flemings and S. Mahajan (Eds.), Encyclopedia of Advanced Materials, Vol. 3, pp. 1798–1803, Pergamon Press, New York, 1994. [4] D.W. McCall, in R.S. Carter and J.J. Rush (Eds.), Molecular Dynamics and Structure, National Bureau of Standards, pp. 475–537, Special Publication 301, Washington, DC. [5] V.J. McBrierty, D.C. Douglass and T.A. Weber, J. Polym. Sci.: Polym. Phys. Ed., 14 (1976) 1271. [6] V.J. McBrierty, D.C. Douglass and T.T. Wang, Appl. Phys. Lett., 41 (11) (1982) 1051. [7] J. Hirshinger, D. Schaefer, H.W. Spiess and A.J. Lovinger, Macromolecules, 24 (1991) 2428. [8] Y. Miyamoto, H. Miyaji and K. Asai, J. Polym. Sci.: Polym. Phys. Ed., 18 (1980) 597. [9] I. Solomon, Phys. Rev., 99 (1955) 559. [10] I. Solomon and N. Bloembergen, J. Chem. Phys., 25 (1955) 261. [11] V.J. McBrierty and D.C. Douglass, Macromolecules, 10 (1977) 855. [12] I.M. Ward and D.W. Hadley, An Introduction to the Mechanical Properties of Solid Polymers, Wiley, New York, 1993. [13] V.J. McBrierty and I.M. Ward, Br. J. Appl. Phys., Ser. 2, 1 (1968) 1529. [14] D.C. Douglass and V.J. McBrierty, Macromolecules, 11 (1978) 766.
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