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Structural analysis of nanocrystalline cellulose using solid-state NMR Lemke, Clark H. 2011

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Structural Analysis of Nanocrystalline Cellulose Using Solid-State NMR by Clark H. Lemke B.Sc., The University of British Columbia, 2009 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Physics) The University of British Columbia (Vancouver) April 2011 © Clark H. Lemke 2011 ii Abstract Nanocrystalline cellulose (NCC) shows very unique properties – in suspension, it sponta- neously forms a chiral nematic phase and, in high purity, exhibits iridescence. While native cellulose has historically been extensively studied in solution and solid-state NMR with success, the physical structure between NCC nanocrystallites is not fully known. Due to the complex structure of the nanocrystallites, conventional diffraction techniques cannot fully determine the structure. In this work 1H-2H exchange coupled with the following NMR techniques were used to investigate the crystallite structure of NCC: 13C CP/MAS, 13C T1, T2 and T1ρ measurements and 13C-2H and 13C-31P REDOR. Results suggest a broad distribution of regions possessing varied dynamical and structural properties. Based upon previously assigned peaks arising from crystalline and amorphous regions, approximately 40% of the NCC particles are characteristic of amorphous and/or surface regions. Even though NCC preparation is designed to remove amorphous regions, this result is remarkably similar to native, untreated cellulose. Proton-deuterium exchange experiments suggest an unequal proton exchangeability between the different possible ex- change sites, and suggest that the samples consist not of sharply defined exchangeable and unexchangeable regions, rather they are more uniformly partially exchanged. We also describe a method to determine the number of surface phosphate groups remaining after hy- drolysis treatment. 13C-31P REDOR experiments conclude that 2.6±0.2 phosphate groups are attached to C2 or C3 per 100 monomers. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Classical Model of NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Quantum Mechanics and the Density Matrix Representation . . . . . . . . 6 2.3 Summary of Relevant Hamiltonians . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Dipolar Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.2 Chemical Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.3 Quadrupolar Interaction . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Average Hamiltonian Theory . . . . . . . . . . . . . . . . . . . . . . . . . 16 iv 3 NMR Techniques for Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 Cross Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Magic Angle Spinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Heteronuclear Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4 Cellulose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1 Native Cellulose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Cellulosic Nanocomposites and Nanocrystalline Cellulose . . . . . . . . . 32 4.3 Solid-state NMR of Native Cellulose and NCC . . . . . . . . . . . . . . . 36 5 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2 NMR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.2.1 13C T1, T2 and T1ρ Relaxation Measurements . . . . . . . . . . . . 40 5.2.2 REDOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2.3 2H REDOR Simulation . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2.4 2H-1H Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.1 13C NCC CP/MAS Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.2 Relaxation Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.3 2H REDOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.3.2 2H-1H Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.4 31P REDOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 v7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 A Example of SIMPSON Simulation of REDOR Dephasing Curve . . . . . . . 86 B SIMPSON Input Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 vi List of Tables 4.1 Typical properties of cellulose nanofibrils, softwood kraft pulp and stain- less steel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.1 T 01 (slow relaxation component) values . . . . . . . . . . . . . . . . . . . . 51 6.2 T 11 (fast relaxation component) values . . . . . . . . . . . . . . . . . . . . 52 6.3 T1 values from stretched exponential . . . . . . . . . . . . . . . . . . . . . 52 6.4 T2 values of carbon resonance peaks in milliseconds. . . . . . . . . . . . . 54 6.5 T1ρ measurements on air and freeze-dried NCC at room temperature mea- sured in milliseconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.6 T1 values of deuterated NCC . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.7 T1 values from stretched exponential . . . . . . . . . . . . . . . . . . . . . 57 6.8 Summary of 2H REDOR fit values . . . . . . . . . . . . . . . . . . . . . . 60 6.9 2H REDOR SIMPSON fit values . . . . . . . . . . . . . . . . . . . . . . . 64 6.10 2H REDOR SIMPSON deuterium exchange fractions . . . . . . . . . . . . 70 B.1 13C-2H Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 vii List of Figures 2.1 FID and its Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Connection between rectangular coordinates x,y,z and spherical coordi- nates r,θ ,φ between two spins I1 and I2. . . . . . . . . . . . . . . . . . . . 11 2.3 Powder spectra for various chemical shift asymmetry parameters η with isotropic shift set to ω = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Schematic diagram of the quadrupolar interaction . . . . . . . . . . . . . . 15 3.1 Diagram of cross polarization pulse sequence transferring polarization from abundant 1H to dilute 13C. . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Magic-angle spinning of cylindrical rotor at rapid speeds about a spinning axis where θ = 54.74° with respect to the static magnetic field. . . . . . . . 23 4.1 Diagram of cellulose monomer repeat unit . . . . . . . . . . . . . . . . . . 29 4.2 Hydrogen-bonding patterns for cellulose Iα and Iβ . . . . . . . . . . . . . 31 4.3 Schematic diagram of cellulose fibril structure. . . . . . . . . . . . . . . . 32 4.4 Scanning electron micrograph of NCC strands. . . . . . . . . . . . . . . . 33 4.5 Chiral nematic structure diagram . . . . . . . . . . . . . . . . . . . . . . . 35 4.6 CP/MAS 13C NMR spectra of native cellulose . . . . . . . . . . . . . . . . 37 5.1 Pulse sequence used for carbon T1 measurements. . . . . . . . . . . . . . 41 5.2 Hahn echo pulse sequence used for T2 measurements. . . . . . . . . . . . . 42 5.3 31P-13C REDOR pulse sequence . . . . . . . . . . . . . . . . . . . . . . . 44 viii 5.4 13C-2D REDOR pulse sequence . . . . . . . . . . . . . . . . . . . . . . . 46 5.5 2H quadrupole powder Pake pattern. The pattern is composed of two over- lapping signals which give the pattern its characteristic shape. . . . . . . . 48 6.1 13C NCC Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.2 Comparison of possible fitting functions for T1 relaxation data of the C2,3,5 peak for freeze-dried NCC at room temperature. . . . . . . . . . . . . . . 51 6.3 Normalized T1 relaxation curves for C2,3,5 for 3 samples. The slow com- ponent decay times are included. . . . . . . . . . . . . . . . . . . . . . . . 52 6.4 13C NCC spectrum after a delay of 150 seconds where only the slow com- ponent remains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.5 Semi-logarithm plot of the T1 decay of C6 for D2O soaked and unsoaked samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.6 2H-13C REDOR Difference Spectrum . . . . . . . . . . . . . . . . . . . . 58 6.7 2H-13C REDOR Dephasing Curve . . . . . . . . . . . . . . . . . . . . . . 59 6.8 Proportion of population that has exchanged 1H for 2H as a function of soaking time according to the C2,3,5 peak. . . . . . . . . . . . . . . . . . . 61 6.9 Diagram of some of the C-D bond lengths used for fitting. . . . . . . . . . 62 6.10 Examples of simulated REDOR dephasing curves for C1 in cellulose Iβ with 1 (closest deuteron), 2 (second closest deuteron) and both possible sites occupied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.11 Probability, Fraction and both Superposition model fits for cellulose Iα in A conformation for 2 week D2O soaked NCC. . . . . . . . . . . . . . . . 65 6.12 Probability, Fraction and both Superposition model fits for cellulose Iα in B conformation for 2 week D2O soaked NCC. . . . . . . . . . . . . . . . 66 6.13 Probability, Fraction and both Superposition model fits for cellulose Iβ in A conformation for 2 week D2O soaked NCC. . . . . . . . . . . . . . . . 67 6.14 Probability, Fraction and both Superposition model fits for cellulose Iβ in B conformation for 2 week D2O soaked NCC. . . . . . . . . . . . . . . . 68 6.15 2H FID with signal of first rotor echo highlighted red. . . . . . . . . . . . 71 ix 6.16 Build up curve for vapour soaked samples. The dashed line shows the signal for a liquid soaked sample. . . . . . . . . . . . . . . . . . . . . . . 72 6.17 REDOR difference spectrum multiplied by x50. . . . . . . . . . . . . . . 74 6.18 Phosphate group geometry at the end of a cellulose chain. . . . . . . . . . . 75 6.19 With the dipolar coupling fixed, only one data point was used for the fit. . . 75 xAcknowledgments First and foremost I would like to thank my advisers Carl Michal and Ronald Dong. I would also like to thank my lab-mates Andy Reddin, David Katz and Tom Depew. Lisa Thomson, your love has helped me in ways I may never fully understand – thank you. I want to give a huge thanks for my family: Mom and Dad for providing inspiration, Jane and Jeremy for motivation and comfort. I want to thank my friends for being there when I needed entertainment, a laugh, a hard night of partying or generous advice: Sam, Connor, Jeff, Adam, John, and Daniel. Finally I would also like to thank Wadood Hamad for providing ample NCC samples whenever, and however, I needed them. 1Chapter 1 Introduction Novel cellulose-based nanomaterials potentially have significant benefits in a wide variety of applications due to their ability to impart improved physical strength and their unique optical and biological properties [1]. A particular cellulose-based nanomaterial, nanocrys- talline cellulose (NCC), shows very unique properties - in suspension, it spontaneously forms a chiral nematic phase and in high purity, exhibits iridescence. The tunable nature of the chiral nematic phase has recently been used to produce remarkable tunable reflec- tive filters and sensors [2]. NCC is produced by careful treatment of wood pulp with acid hydrolysis. While much work has been done on the hydrolysis of cellulose from various sources, there are only scant studies on the characterization of the structure, morphology, and rheology of the resulting nanomaterials. Further characterization is needed for under- standing the full potential of the functionality and processability of the extracted nanocrys- talline cellulose and the extent of its applications. Since being discovered in 1951 by Ranby [3], NCC has mainly remained a scientific curios- ity. But over the last few years, research into applications of NCC has multiplied. Plastic packaging made from high and low-density polyethylene contribute substantially to land- fills worldwide as plastic packaging does not biodegrade. To compound the environmental hazard, the production and fabrication of plastic bags and packaging from non-renewable petrochemical starting materials contributes significantly to global CO2 levels. A renew- able solution to these environmental problems is adding NCC to plastic packaging or sim- ply replacing plastic in packaging with NCC. As NCC possesses very high tensile strength this would also strengthen packaging. Other potential applications include strengthening textiles and other composite materials or exploiting the iridescence for security papers and cosmetics. Characterization of the structure of cellulose by conventional X-ray diffraction and imaging techniques have proved difficult due to its complex morphology. One technique, nuclear 2magnetic resonance (NMR) spectroscopy, has historically been an effective method of elu- cidating cellulose’s structure and has been used with success in many cellulose variants [4]. Applying NMR to cellulose and cellulose derivatives is far from trivial as the origin of var- ious components of solid-state cellulose 13C NMR spectra are complicated by overlapping regions. In particular, native celluloses exist as a mixture of two (or more) crystalline com- ponents, crystalline and noncrystalline (or amorphous) regions simultaneously contribute to the NMR signal, and the NMR signals from the chains on the surface of the cellulose crystallites overlap with those on the inside. By using different NMR pulse sequences, one can circumvent these problems and probe various structural regions of NCC to avoid am- biguity in analysis. This thesis characterizes the structure of this unique nanomaterial with the hope that these characterization techniques will help optimize preparation, treatment and chemical yield. This thesis begins with an outline of the necessary background and basics of NMR in Chapter 2. Specific solid-state NMR techniques and recent structural studies on native cellulose and acid-hydrolysis cellulose are included in Chapter 3. An overview of the chemistry and properties of native, untreated cellulose and acid-hydrolyzed NCC follows in Chapter 4. Chapter 5 explores the various NMR pulse sequences used in this thesis as well as a brief outline of NCC preparation. Chapter 6 includes the results of the experiments and finally Chapter 7 summarizes the thesis. 3Chapter 2 Nuclear Magnetic Resonance Solid-state 13C NMR has historically provided very significant insights regarding the struc- tures of cellulose due to its ability to distinguish chemically equivalent carbons that occur in magnetically non-equivalent sites. As such, the primary tool used in this work to study NCC is solid-state NMR though references will be made to other characterization tech- niques. The following sections include a brief summary of a typical NMR experiment and the relevant background NMR concepts required to investigate NCC structure. For a full treatment of magnetic resonance concepts see the standard texts of Abragam [5] and Slichter [6]. For specific solid-state NMR references see Duer [7] or Schmidt-Rohr and Spiess [8]. 2.1 Classical Model of NMR In order for a nucleus to be NMR active, it must possess an intrinsic nuclear magnetic moment ~M (created by an odd number of protons and/or neutrons). The basis of all NMR experiments arise due to this quantum phenomena: the intrinsic nuclear magnetic moment ~M and the angular momentum~J are related through: ~M= γ~J, (2.1) where γ is the gyromagnetic ratio [5]. If a nucleus is placed in a homogeneous magnetic field ~B (such as in an NMR spectrometer) a torque T is exerted on the magnetic moment ~M: T= ~M×~B. (2.2) 4The above equations can be combined with the basic equation of rotational motion into the equation of motion for the magnetization: d dt ~M= γ~M×−→B , (2.3) and by taking ~B to exist only in the z direction (~B0 = (0,0,B0)) we can solve the Bloch equations without relaxation [9]: Mx(t) = Mx(0)cos(γB0t)−My(0)sin(γB0t) My(t) = My(0)cos(γB0t)+Mx(0)sin(γB0t) Mz(t) = Mz(0). (2.4) We define the Larmor frequency ω0 [5]: ω0 =−γB0. (2.5) If we include exponential relaxation of Mx, My and Mz towards an equilibrium value we find the time dependence of the magnetic moment ~M(t) as: Mx(t) = (Mx(0)cos(γB0t)−My(0)sin(γB0t))e− t T2 My(t) = (My(0)cos(γB0t)+Mx(0)sin(γB0t))e − tT2 Mz(t) = Mz,eq− [Mz,eq−Mz(0)]e− t T1 , (2.6) where Mz,eqis the magnetization at thermal equilibrium, T1 is the longitudinal relaxation time and T2 is the transverse relaxation time [5]. These relaxation times are easily observed characteristic timescales describing the rate of magnetization recovery in the sample. NMR experiments generally consist of a series of electromagnetic pulses~B1 which contain an oscillatory time dependence in the laboratory frame. The oscillatory time dependence can be removed by changing to a rotating frame of reference that precesses around ~B0. If one chooses a frame frequency equal to the applied magnetic field ω1, ~B1 now contains no explicit, oscillatory time dependence and the net magnetization vector will nutate around the ~B1 direction. In this frame we also see an addition of an effective field that mostly (or entirely) cancels the static field, so that in this frame, Eq. 2.3 still holds, but with an 5effective field given by: ~Be f f =~B0+ ω1 γ . (2.7) Typical frequencies are located in the radio-frequency (rf) band when used with ~B0 fields on the order of several Teslas and so are often referred to as “rf-irradiation”. The duration of rf pulses are generally discussed in terms of their flip angle θr f . The flip angle refers to the angle that the pulse rotates the magnetization during a time tr f : θr f = ω1tr f = γB1tr f . (2.8) A flip angle of 90° is often used in simple NMR experiments as it excites the maximum NMR signal in the x− y plane. A typical NMR experiment begins by placing a system of nuclear spins (the sample of interest) into a large magnetic field (typically order of a few Teslas). The sample is then subjected to radio-frequency pulses generated by a coil which tightly surrounds the sample. If the frequency of the pulses is close enough to the resonance frequency ω0 of a particular nucleus, the system becomes excited. The precession of the nuclear spin magnetization induces an emf in the coil which is detected, over time t, in the form of a free-induction- decay (FID). The signal generated in the coil then undergoes Fourier transformation (FT) to frequency space S(Ω): S(Ω) = ∞̂ 0 s(t)e−iΩtdt, (2.9) where s(t) is the signal of the FID in the time domain. The transformed signal constitutes the NMR spectrum and is composed of a superposition of individual spin contributions as illustrated in Fig. 2.1. 6Time Frequency FT Figure 2.1: Diagram of an FID for a single species and, after Fourier transformation, its peak in the frequency domain. 2.2 Quantum Mechanics and the Density Matrix Representation So far we have only concerned ourselves with the semi-classical model of NMR. We will now dive into a full quantum discussion of the individual spins. We begin with examin- ing individual spins and, from this, we can generate an expression for the macroscopic magnetization of a sample. The most basic spin system consists of a single isolated particle in a uniform magnetic field. The Hamiltonian Ĥ for the single spin is called the Zeeman Hamiltonian [8]: Ĥ =−µ̂B0, (2.10) where µ̂ is the nuclear magnetic moment operator and B0 is the static applied magnetic field. µ̂ can be written in terms of the nuclear spin operator Î: µ̂ = γ h̄Î. (2.11) Conventionally the direction of the static B0 is taken to be the z-axis so Ĥ becomes: Ĥ =−γ h̄B0Îz. (2.12) The eigenfunctions of the Hamiltonian are the wavefunctions of the possible states |I,m〉 where the quantum number m can take values differing by integers between I and −I, 7i.e., −I, −I + 1, −I + 2, · · · , I. The eigenvalues are found by applying Ĥ to these spin wavefunctions: Ĥ |I,m〉 = EI,m |I,m〉 , (2.13) where EI,m is the energy of the eigenstate wavefunction. By putting Eq. 2.12 into 2.13 we find that: Ĥ |I,m〉=−γ h̄B0Îz |I,m〉=−γ h̄B0m |I,m〉 . (2.14) Thus the energies of the eigenstates are easily read off by comparing the above equations: EI,m =−γ h̄B0m. (2.15) For a spin 1/2 particle, there are are two possible eigenstates (m =−12 or m = 12 ) separated by a transition energy of γ h̄B0. In frequency units, this is equal to the Larmor frequency ω0 of the semi-classical model in Eq. 2.5. We can extend this behaviour to an ensemble of spin 1/2 nuclei through the Boltzmann distribution at thermal equilibrium [7]: pi = Ni Nt = e−Ei/kBT ∑i e−Ei/kBT , (2.16) where pi is the probability of being in a particular state i, Ni is the number of spins in state i, Nt is the total number of spins in the sample, Ei is the energy of state i, kB is the Boltzmann constant and T is the temperature of the system. The relative populations and energy levels of the spin system uniquely define the behaviour of the system and ultimately the results of NMR experiments. In an NMR experiment, one observes the ensemble average of the system. One can extend this treatment to the more general density matrix or density operator ρ̂ [7]. Here we examine a sample of identical spins in state Ψ, though this time the spins can be in any one of N possible energy states. The general expression for the possible states in the Zeeman eigenfunction basis is: Ψ=∑ i cψiφi, (2.17) where φi’s must be all the possible eigenstates of the ensemble and cψi is the amplitude for a particular eigenstate φi. If one takes the expectation value 〈  〉 = 〈Ψ|  |Ψ〉 of an 8observable operator  in this basis, one finds 〈  〉 =∑ i′ pi′∑ i,, j c?ΨicΨ j 〈φi|  ∣∣φ j〉 , (2.18) and that the matrix elements of 〈φi|  ∣∣φ j〉 are independent of the sample state Ψ. The first parts of Eq. 2.18 can be rewritten as another matrix ρ so that the equation becomes: 〈  〉 =∑ i ∑ j Ai jρ ji, (2.19) where Ai j = 〈φi|  ∣∣φ j〉 and ρ ji is called the density matrix with corresponding operator ρ̂ . Hence the density operator can be used to elegantly describe an ensemble of many identical systems. The density operator ρ̂ provides a natural bridge between quantum mechanics and statisti- cal mechanics. The time dependence of the density operator ρ̂(t) is expressed by the von Neumann equation which is derived from the time-dependent Schrödinger equation [10]: d dt ρ̂ =−i[Ĥ, ρ̂], (2.20) which has the following solution for a time-independent Hamiltonian: ρ̂(t) = Û(t)ρ̂(0)Û−1(t), (2.21) where Û(t) is commonly referred to as the propagator Û(t) = e−iĤt and evolves the spin system from time 0 to time t. 2.3 Summary of Relevant Hamiltonians The behaviour of nuclear spins during an NMR experiment is dominated by the Zeeman interaction as illustrated in Section 2.2 above: Ĥ0 =−γB0Îz =−∑ j γ jB0Î jz , (2.22) 9where the classical components are substituted with quantized operator counterparts. Here, and from now on, we will be using frequency units for the Hamiltonian as is common and convenient when discussing NMR. While the Zeeman interaction does not contain any molecular or structural information, one can probe the local fields of the nuclear spins and these contain a wealth of information. For solid polymers these fields include: the dipolar couplings between spins, the electron shielding cloud which alters the net effect of ~B0, and the effects of the nuclear electric quadruple moments. In static fields of several Tesla, the local fields are very small compared with~B0 and so can be treated as a first-order perturbation of Equation 2.22. If we take a basis of Îz eigenfunctions, the energy shifts are given by the diagonal elements of the perturbing Hamiltonians and so only the diagonal (secular) parts of the local-field operators contribute. The secular parts of the perturbing Hamiltonians are easily identified as those that commute with Îz. As described in the classical representation above, a rotating frame of reference is com- monly used when discussing behaviour of magnetization during an NMR experiment. The direction of ~B1 in the rotating frame is conventionally given by the phase of the pulse φr f which is the angle ~B1 makes with the x-axis. The rotation operator takes the form: R̂ = e−iφr f Îz, (2.23) which can be used to generate a rotating frame Hamiltonian: ĤR = R̂−1ĤR̂−ωr f Îz. (2.24) In this frame, the effect of the Zeeman interaction is greatly reduced (as ωr f is usually near the Larmor frequency). In solid materials, NMR lines can be much broader than typically found in solution. Mech- anisms responsible for this broadening can be divided into two categories, homogeneous and inhomogeneous, depending upon whether each spin contributes to all parts of the line (homogeneous) or to a narrow portion (inhomogeneous). The following sections will ex- plore a number of causes that contribute to the broadening of a resonance line. 10 2.3.1 Dipolar Coupling Homogeneous broadening in NMR is most often due to networks of strongly coupled neighbouring spins. Here we examine the magnetic dipolar coupling between various nu- clei. Magnetic dipoles generate their own magnetic field. This local field interacts with other dipole fields through space and can significantly broaden the resonance line. As it is orientation dependent, dipolar coupling can be used to extract spatial information of molecular structure [7]. The classical interaction energy U between two magnetic moments µ1 and µ2 is: U = µ0 4pi ( µ1 ·µ2 r3 −3(µ1 ·~r)(µ2 ·~r) r5 ) , (2.25) where ~r is the radius vector connecting the spins µ1 and µ2. Making the quantum me- chanical operator substitution µ1 = γ1h̄Î1 and µ2 = γ2h̄Î2 we find the dipolar quantum Hamiltonian to be: ĤD = µ0 4pi γ1γ2h̄2 ( Î1 · Î2 r3 −3(Î1 ·~r)(Î2 ·~r) r5 ) . (2.26) Expressing Eq. 2.26 in terms of raising and lowering operators Î+1 , Î − 1 , Î + 2 , Î − 2 and express- ing the rectangular coordinates x,y,z in terms of spherical coordinates r,θ ,φ we find the conventional form for the Hamiltonian: ĤD =− µ04pi γ1γ2h̄ 2(A+B+C+D+E +F), (2.27) 11 where A = −Îz1Îz2(3cos2θ −1) B = 1 4 (Î+1 Î − 2 + Î − 1 Î + 2 )(3cos 2θ −1) C = 3 2 (Îz1Î+2 + Î + 1 Îz2)sinθ cosθe −iφ D = 3 2 (Îz1Î−2 + Î − 1 Îz2)sinθ cosθe iφ E = 3 4 (Î+1 Î + 2 )sin 2θe−2iφ F = 3 4 (Î−1 Î − 2 )sin 2θe2iφ . (2.28) Only the A and B terms commute with Ĥ0 and according to first-order perturbation theory, these terms make up the secular or adiabatic part of ĤD [8]. B0 x y z θ φ I I 1 2 r Figure 2.2: Connection between rectangular coordinates x,y,z and spherical coordinates r,θ ,φ between two spins I1 and I2. Dipolar coupling interactions occur as one of two cases: homonuclear dipolar coupling between two like spin species or heteronuclear dipolar coupling between two different spin species. In organic materials, one often considers a rare nucleus like 13C or 15N in the presence of abundant protons. In these cases, the abundant spins (protons) are usually des- ignated as the I spins while the rare spins (13C, 15N, etc.) are the S spins. Rearranging the secular parts of the Hamiltonian and summing over all spins, one can rewrite the truncated 12 Hamiltonian for homonuclear dipolar coupling [8]: ĤI,ID =− µ0 4pi h̄∑ j ∑ k< j γ2 r3jk 1 2 (3cos2(θ jk)−1)(3Î jz Îkz − ˆ̃I j · ˆ̃Ik), (2.29) and for heteronuclear dipolar coupling: ĤS,ID =− µ0 4pi h̄∑ j ∑ k< j γ IγS r3jk 1 2 (3cos2(θ jk)−1)2Î jz Ŝkz , (2.30) where θ jk is the angle between the field and the axis which connects nuclei j and k, and r jk is the magnitude of the vector from nucleus j to k. Thus two variables that NMR pulse sequences can determine and exploit are the 1/r3 internuclear distance between the spins and the relative orientations of the spins with respect to B0. 2.3.2 Chemical Shift Under the influence of the ~B0 field, the electron cloud surrounding a spin generates an additional local field which shields the nuclear spin from the external applied~B0. In a large molecule, the chemical environment may be slightly different for each nucleus and so each spin will precess at a slightly different frequency. This gives rise to small, but measurable, resonance shifts in the NMR spectrum [7]. These shifts contain an orientation dependence whose exploitation provides one of the most important foundations of the spectroscopic power of NMR. From a semi-classical perspective, a diamagnetic shielding effect is generated by the in- duced current in the electron cloud when the sample is inserted into the static magnetic field ~B0. Due to Lenz’ law, the induced current opposes the direction of ~B0 and has mag- nitude proportional to the static field. There is also an induced paramagnetic effect which is due to the partial alignment of orbital angular momenta of the electron cloud. This chemical shielding is represented by a second rank tensor σcs describing the directional dependence of the atom or molecule with respect to the static field. It is possible to choose an axis frame for σcs (or other second-order interaction tensors) such that in this frame, the tensor is diagonal [8]. This frame is called the principal axis frame (PAF) and the diagonal components (σPAFxx ,σPAFyy ,σPAFyy ) are called the principal values. The overall Hamiltonian 13 for the chemical shielding and Zeeman interaction is: Ĥ = Ĥ0+ Ĥcs =−γ h̄Î ·B0+ γ h̄Î ·σcs ·B0. (2.31) As the chemical shielding is very small compared with the Zeeman interaction, we can treat the chemical shielding with perturbation theory. Making the secular approximation (only operators that commute with Îz contribute) to first order and transforming into a rotating frame we find: Ĥ0cs = γ ÎzσzzB0. (2.32) The absolute resonance frequency of the observed NMR signal is: ω = γ(Ĥ0+ Ĥ0cs) ω = γ(Ĥ0+ γ ÎzσzzB0). (2.33) Absolute resonance frequencies are B0-field dependent so comparing spectra of identical compounds between different magnets may be very different. To standardize NMR spectra, one measures the relative frequency with respect to a reference compound to determine the chemical shift δ . This is measured in units of ppm (parts per million) of the Larmor frequency and is defined by the IUPAC δ -scale: δ = νobs−νre f νre f ×106, (2.34) where νobs is the observed frequency of the resonance peak for the spin of interest and νre f is the spectral frequency in the reference compound. Conventionally resonance frequencies increase from right to left and large ppm values are called “down-field” and small ppm values are designated “up-field”. δ11, δ22, δ33 (δ11 ≥ δ22 ≥ δ33) are the labeled principal value components and are conventionally used to express the isotropic chemical shift δiso, 14 the chemical shift anisotropy ∆cs and chemical shift asymmetry ηcs parameters: δiso = 1 3 (δ11+δ22+δ33) (2.35) ∆cs = δ33−δiso if |δiso−δ33|> |δiso−δ11|δiso−δ33 otherwise (2.36) ηcs =  δ11−δ22 ∆ if |δiso−δ33|> |δiso−δ11| δ33−δ22 ∆ otherwise. (2.37) If one expresses the magnetic field in polar coordinates with respect to the PAF (θ ,φ as done for dipolar couplings in Eq. 2.27), then one finds the frequency contribution of the chemical shift in terms of the δiso, ∆cs and ηcs parameters: ωcs =−ω0δiso+ω0 12∆cs(3cos 2θ −1−ηcs sin2θ cos2φ), (2.38) where the quantity ω0δiso is the isotropic chemical shift frequency of a bare nucleus. For a powder sample, all molecular orientations are present and each different orientation has a different chemical shielding hence the spectrum takes the form of a characteristic “powder pattern” depending on the symmetry of the shielding tensor as shown in Fig. 2.3. -0.51.0 -0.51.0 -0.51.0 η=0 η=0.2 η=1 Chemical Shift Figure 2.3: Powder spectra for various chemical shift asymmetry parameters η with isotropic shift set to ω = 0. 2.3.3 Quadrupolar Interaction Nuclei with spin I > 12 also possess an electric quadrupole moment Q that depends on the electric field gradient tensor V (second spatial derivatives of the electric potential). 15 Qualitatively this arises due to electric effects that affect the energies of the nuclear spin states of the nucleus. Fig. 2.4 illustrates this. +q +q -q -q +q +q -q -q Figure 2.4: Quadrupolar interaction on a uniformly charged, non-spherical nucleus in a field of four external charges. The left arrangement is preferential as the positive charge of the nucleus is closest to the negative external charges. The quadrupolar interaction is described by the following Hamiltonian: ĤQ = eQ 2I(2I−1)h̄ Î ·V · Î, (2.39) where e is the elementary charge. Using the Wigner-Eckart theorem, one can expand the Hamiltonian for an arbitrary orientation: ĤQ = eQ 6I(2I−1)h̄ ∑α,β Vαβ [ 3 2 (Îα Îβ + Îβ Îα)−δαβ Î2], (2.40) where α = 1,2,3, β = 1,2,3 correspond to respective x, y, and z coordinates of the par- ticular reference frame for the external potential sources and δαβ is the Kronecker delta. The quadrupolar interaction can be simplified by choosing a set of principal axes relative to which Vαβ = 0 for α 6= β . In tensor form (similar to the chemical shift tensor), the quadrupolar coupling is: ĤQ = eQ 2I(2I−1)h̄ Î  V PAF xx 0 0 0 V PAFyy 0 0 0 V PAFzz  Î, (2.41) 16 or written explicitly: ĤQ = eQ 6I(2I−1)h̄ [V PAF xx (3Î 2 x − I2)+V PAFyy (3Î2y − I2)+V PAFzz (3Î2z − I2)], (2.42) V must also satisfy Laplace’s equation ∑αVαα = 0 so we can simplify this expression: ĤQ = eQ 4I(2I−1)h̄ [V PAF zz (3Î 2 z − I2)+(V PAFxx −V PAFyy )(Î2x − Î2y )]. (2.43) Conventionally the quadrupolar anisotropy parameter, eq and the quadrupolar asymmetry parameter, ηQ are defined as: eq =V PAFzz (2.44) ηQ = V PAFxx −V PAFyy V PAFzz . (2.45) Note that there is no isotropic parameter like the chemical shift due to the fact that the trace of the electric field gradient tensor V is zero. Substituting eq and ηQ into Eq. 2.43, the high-field quadrupole Hamiltonian (secular part) in the PAF becomes: ĤQ = e2qQ 4I(2I−1)h̄ [3Î 2 z − Î2+ 1 2 ηQ(Î2x − Î2y )] = CQ 4I(2I−1)[3Î 2 z − Î2+ 1 2 ηQ(Î2x − Î2y )], (2.46) where the so-called quadrupolar coupling constant CQ is given by (units Hz): CQ = e2qQ h̄ . (2.47) Quadrupole coupling values range from a few kHz up to several MHz and typically dom- inate the NMR spectrum at the Larmor frequency even for a deuteron which possesses a very small nuclear quadrupole moment [11]. 2.4 Average Hamiltonian Theory The time dependence of the density operator as expressed in Eq. 2.21 assumes that the Hamiltonian is constant during the time period t - though this is not always the case. Dur- ing a multi-pulse sequence, Û(t) collects a number of exponential operators, each with a 17 potentially different Hamiltonian. Thus, one must perform another density matrix calcula- tion of the equations of motion for the spin system for each time the Hamiltonian changes during t. Average Hamiltonian theory is a simple method to save much effort to follow the dynamics of a spin system. Average Hamiltonian theory can be applied to any time varying Hamiltonian but we will limit ourselves to piecewise constant Hamiltonians of the form: H(t) =  H1 for t1 H2 for t2 ... ... Hk for tk, (2.48) where t = t1+ t2+ · · ·+ tk. The resulting propagator is:: Û(t) = e−iĤktk · · ·e−iĤ2t2e−iĤ1t1, (2.49) note that each Hamiltonian appears in reverse chronological order. To deal with this series of exponential operators we can calculate the average Hamiltonian H̄ such that Û(t) = e−iH̄t . This should come as no surprise due to the intrinsic oscillatory framework of NMR, approximating Hamiltonians by averaging over time periods provides a method of examining or manipulating local-field interactions after or during pulse sequences. To tackle the problem of averaging Hamiltonians, Haeberlen and Waugh have developed a simple method [12] using the Magnus expansion to evaluate the series of exponential op- erators procured during a pulse sequence: eÂeB̂ = eÂ+B̂+ 1 2! [Â,B̂]+ 1 3! ([Â,[Â,B̂]]+[[Â,B̂],B̂]+... (2.50) If we apply the Magnus expansion to a sequence of pulses, the average Hamiltonian is given in terms of a series of orders with decreasing importance: H̄(t) = H̄(0)+ H̄(1)+ · · · , (2.51) 18 where H̄(0) = 1 t (Ĥ1t1+ Ĥ2t2+ . . .+ Ĥ6t6) H̄(1) = − i 2t ([Ĥ2t2, Ĥ1t1]+ [Ĥ3t3, Ĥ1t1]+ [Ĥ3t3, Ĥ2t2]+ . . .), where higher orders are given by increasingly higher order sums of commutators. In cases where all the terms commute with each other, only the first order term H̄(0) is required for a good description of the average Hamiltonian and this is simply the average of the individual piecewise Hamiltonians. When the Hamiltonians which operate over a periodic time interval do not commute with each other, one can transform to a new frame, called the toggling frame, where the non-commuting terms disappear. In this frame, the first order term H̄(0) may be an accurate approximation for the total average Hamiltonian [7]. 19 Chapter 3 NMR Techniques for Solids 3.1 Cross Polarization Cross polarization (CP) experiments are commonly used in solid-state NMR as they enable magnetization transfer, typically from abundant ( 1H) to dilute ( 13C) spins, because it produces a large sensitivity enhancement. A characteristic of 13C NMR in rigid solids is a long spin-lattice relaxation time T1, and because T1 values determine how rapidly a pulsed NMR experiment can be repeated, long T1 values can quickly become a problem with respect to experiment waiting times. The prohibitive waiting times for signal averaging can be overcome by using CP. CP was developed by Hartmann and Hahn [13] and usually involves transferring polariza- tion from protons to rare spins (in this case 13C). CP is obtained by continuous simultane- ous irradiation at both resonance frequencies, protons (I spins) and the desired X-nuclei (S spins), such that the B1 field strengths satisfy the Hartmann-Hahn condition: γ IBI1 = γ SBS1⇔ ω I1 = ωS1 . (3.1) To understand properly why the Hartmann-Hahn condition allows polarization transfer, one must determine the average Hamiltonian as described in Sec. 2.4. The CP pulse sequence is shown in Fig. 3.1. The sequence begins with a 90° pulse applied to the abundant spins (in most cases protons are used) to tip their magnetization along the -y-axis in the rotating frame. This is followed by a CP contact pulse during which the abundant spins are locked (also called spin-lock) in place due to constant applied resonant power. During this CP time, the dilute spins are also subjected to constant resonant rf. In the rotating frame during spin-lock, the spins are aligned along the effective field and they are likely to remain there in a state of quasi-equilibrium. After the CP time the dilute spin signal is acquired. 20 C 13 H 1 90° CP CP Decoupling Figure 3.1: Diagram of cross polarization pulse sequence transferring polarization from abundant 1H to dilute 13C. We have the Hamiltonian for the system during the CP period: Ĥ = Ĥz+ ĤI−I + ĤI−S+ Ĥxpulse, (3.2) where Ĥz = ω I0∑ i Îiz+ωS0 Ŝz (3.3) ĤI−I = −∑ i> j CI−Ii j (3ÎizÎ jz− ˆ̃Ii · ˆ̃I j) (3.4) ĤI−S = −2∑ i CI−Si (ÎizŜz) (3.5) Ĥxpulse = ω I 1∑ i Îix cos(ω I0t)+ω S 1 Ŝx cos(ω S 0 t), (3.6) with the pre-factors CI−Ii j and C I−S i in frequency units: CI−Ii j = µ0 4pi h̄ γ2I r3i j 1 2 (3cos2(θi j)−1) (3.7) CI−Si = µ0 4pi h̄ γIγS r3i 1 2 (3cos2(θi)−1), (3.8) where Ĥz is the Zeeman interaction for both abundant and dilute spins, ĤI−I and ĤI−S rep- resent the homonuclear and the heteronuclear dipolar interactions respectively and Ĥxpulse 21 represents the contact pulse (assumed to be perfectly on resonance). We now transform into a “double-rotating” frame - one that precesses about B0 at ω I0 for the abundant spins and at ω I0 for the dilute spins. This is represented by the transformation: Ĥrot = R̂−10 ĤR̂0−ω I0∑ i Îz−ωS0 Ŝz, (3.9) where R̂0 is the familiar rotation operator but now it has two pieces in it - one for each spin species: R̂0 = e−iω I 0∑i Îxte−iωS0 Ŝxt . In this frame the Zeeman effects are removed and the pulse Hamiltonian is time independent: Ĥxpulse,rot = ω I 1∑ i Îx+ωS1 Ŝx. (3.10) We now transform further into the toggling frame (one that follows the effects of the con- tact pulse so that in this frame, the pulse Hamiltonian is zero). Mathematically this is represented by: Ĥ∗ = R̂−11 Ĥrot R̂1−ω I1Îz−ωS1 Ŝz, (3.11) where R̂1 = e−iω I 1∑i Îxte−iωS1 Ŝxt . In this frame the dipolar part of the Hamiltonian becomes: Ĥ∗I−I = − 1 2∑i> j CI−Ii j (Î i · Î j−3ÎixÎ jx ) (3.12) Ĥ∗I−S = − 3 2∑k ∑i CI−Sik [(Î i zŜ k z + Î i yŜ k y)cos(ω I 1−ωS1 )t +(ÎizŜ k y + Î i yŜ k z)sin(ω I 1−ωS1 )t]. (3.13) Ĥ∗I−I is constant with time and manifests itself as proton spin diffusion. When ω I1 6= ωS1 , Ĥ∗I−S is very small when averaged over the entire contact pulse. But if the Hartmann-Hahn condition is satisfied (ω I1 ∼= ωS1 ), the time dependence disappears and Ĥ∗I−S produces a double resonance effect between the two spin species and contains terms of the form Îii+Ŝ k k− and Îii−Ŝ k k+. As both species have the same energy level separation (due to Hartmann-Hahn condition), the carbon and protons equilibrate to equal magnetizations during the contact time. As γH is around 4 times larger than γC, the probability of polarization for protons at room temperature is much greater than for 13C. This translates into a much larger 13C signal after CP. Another source of signal enhancement due to CP follows from the initial magnetization of the protons (as opposed to the direct polarization straight to carbon). 22 Proton T1 times are generally much shorter than 13C T1 times in solids thus the experiment repeat delay can be shortened accordingly, drastically improving signal averaging. 3.2 Magic Angle Spinning Solid-state NMR usually concerns itself with powder samples (many crystallites at random orientations). Due to anisotropic interactions, powder sample spectra exhibit overlapped, broad resonance shapes. These anisotropies are averaged away naturally in liquids due to molecular motion but can also be averaged away in solid-state samples by macroscopic spinning of the sample at the “magic angle” [14, 15]. Magic angle spinning (MAS) requires spinning the sample macroscopically at θm = arccos( √ 1/3) = 54.74° in a rotor with re- spect to the −→ B 0 field and is used to obtain structural information contained in isotropic chemical shifts. MAS is not without complications and the spinning speed may need to be unfeasibly high to average out the undesired anisotropies. Proton-proton and proton- carbon couplings may be stronger than 20 kHz and are not averaged away by MAS unless the spinning speed is greater then 20 kHz. When spinning in this intermediate region (not fast enough to average completely), MAS will break the anisotropic chemical-shift broad- ening into spinning sidebands in which intensity is split into frequency components spaced by multiples of the rotation frequency ωr from the isotropic chemical shift resonance. It may be difficult to distinguish spinning sidebands from true isotropic peaks - especially if sidebands from multiple peaks overlap - though one can always change the spinning speed and only the sidebands will move or use pulse sequences which can eliminate the spinning sidebands (eg. TOSS [16]). 23 θ B 0 Figure 3.2: Magic-angle spinning of cylindrical rotor at rapid speeds about a spinning axis where θ = 54.74° with respect to the static magnetic field. As illustrated in sections 2.3.1 and 2.3.2, the molecular orientation dependence of the vari- ous spin interactions is of the form 3cos2θ −1 for symmetric interactions η = 0 where θ is the angle of the z-axis of a particular spin interaction tensor (dipolar coupling, chemical shielding, etc. with respect to −→ B 0). In a powder sample, θ takes on all values. If one spins the sample at an angle θr with respect to −→ B 0, then θ for each spin changes as a function of time. The average orientation dependence would then take the form: 〈 3cos2θ −1〉= 1 2 (3cos2θr−1)(3cos2β −1), (3.14) where β is the angle between the z-axis of a particular spin interaction and the spinning axis. Even if η 6= 0, the anisotropic factor is still multiplied by (3cos2θr− 1)/2. While θ and β are fixed for a particular system, θr can be varied freely. By setting θr = 54.74°, 3cos2θr−1 is set to zero setting the average also to zero. Thus if the spinning rate is rapid enough so that θ is averaged well when compared with the anisotropy of a particular spin interaction, the anisotropy will average to zero. Solid-state NMR literature routinely refers to rotor or rotational echoes. Rotor echoes arise during the acquisition of the FID in an NMR experiment under MAS. After a 90° pulse, the magnetization lies in the x-y plane of the rotating frame. As the sample spins, the frequency recorded for a particular isochromat in the FID varies due to rotation of crystallite orientation. When the rotor finishes one complete rotation, the isochromat has returned to its original point and the evolution frequency returns to its original value. This process repeats itself for each rotor rotation and gives rise to observed echoes in the FID. 24 It is common in solid-state NMR to use CP and MAS simultaneously to enjoy the benefits of each technique while very little interference is introduced. The dipolar Hamiltonians in the toggling frame mentioned in section 3.1 must be modified to contain a MAS time dependence. Now the pre-factors CHCik of Ĥ ∗ HC in Eq. 3.13 are no longer constant with time but oscillate ±ωr [17]. This modifies the Hartmann-Hahn condition to the so-called “side- band match condition” ωH1 −ωX1 = ωr though intensity will still be found at the traditional ωH1 −ωX1 = 0 as long as the spinning frequency is not fast compared to the H-C coupling strength. The amount of polarization transfer at each condition depends on spinning speed with the sideband match condition increasing as spinning speed increases and the static condition decreasing as spinning speed increases. 3.3 Heteronuclear Decoupling Heteronuclear dipolar coupling between dilute spins (13C is around 1% abundant) and abundant spins (1H) can severely broaden dilute spectral lines even after using CP/MAS. A simple technique to reduce this effect involves high power rf-irradiation to the abundant spins during acquisition of the dilute spin signal. This excitation causes the abundant spins to undergo energy transitions at a fast rate dictated by the rf amplitude. If these transitions are very fast relative to the strength of the heteronuclear dipolar coupling, only the time- averaged dipolar coupling will affect the dilute spin signal. For a 13C - 1H spin pair under high rf power, the z-component of the 1H spin oscillates between ±12 which averages to zero. 3.4 Relaxation If a sample is left undisturbed for a long period of time in a magnetic field, it reaches a state of thermal equilibrium. Radio frequency (RF) pulses disturb the spin system from its equilibrium state and may generate population coherences [7]. Nuclear spin relaxation occurs as the system attempts to regain the equilibrium state, through interactions between the spin system and the thermal molecular environment. Relaxation studies have long been used to characterize molecular motion and reorientation in solids [18] as relaxation 25 times are dependent on the spectral density of molecular dynamics (Fourier transforms of the auto-correlation functions of the fluctuating magnetic interactions) with rates within the range of the characteristic frequency ωr = γBr [5, 6] where Br is the strength of the relevant ~B field. For longitudinal relaxation in the laboratory frame (T1) or rotating frame (T1ρ ) this is the ~B0 or ~B1 respectively. T1 probes the spectral density at the Larmor frequency at tens to hundreds of MHz and T1ρ probes at tens of kHz [8]. Relaxation may be incorporated into the Bloch equations (via T1 and T2) but little information regarding the underlying physics and mechanisms of relaxation may be extracted from the equations. To describe relaxation in more detail, one begins by expressing the Hamiltonian of the relaxing system: Ĥ = Ĥ0+ Ĥ1, (3.15) where Ĥ0 is the static Hamiltonian and Ĥ1 is the spin-lattice coupling responsible for all the relaxation interactions and is a random function of time with zero average value. Due to the random nature of relaxation, one can express the spin-lattice Hamiltonian as a sum of terms each of which is the product of a spin operator by a random function of time: Ĥ1 =∑ α VαFα(t), (3.16) where Vα is a spin operator, and Fα is a random function of time with zero average. By using the Liouville von Neumann equation, expressing the Hamiltonians in the interaction representation and applying iterative formal integration, one can isolate the effect of the spin-lattice Hamiltonian on the time evolution of the density matrix to first non-zero order [5]: d dt ρ̂ =−∑ α [Vα , [V †α ,(ρ̂− ρ̂eq)]]Jα(ωα). (3.17) This is the so-called master equation of relaxation where ρ̂eq is the density matrix at thermal equilibrium and Jα is the spectral density. This equation can also be applied to the time dependence of physical variables, for the longitudinal relaxation of identical spins subject to the Zeeman interaction one finds: d dt 〈Iz〉=− 1 T1 (〈Iz〉−〈Iz〉eq), (3.18) 26 where 1 T1 =∑ m m2 Tr(VmV †m) Tr(I2z ) Jm(mω0). (3.19) The most common and important mechanism for relaxation is the magnetic dipole-dipole interaction between the magnetic moment of a nucleus and the magnetic moment of another nucleus or spin. As described in Sec. 2.3.1, this depends on the distance between the spins and on their orientation relative to B0. Other relaxation mechanisms also exist: chemical shift anisotropy (CSA) relaxation arises whenever there is a non-spherical electronic envi- ronment around a nucleus with the magnitude of the shielding dependent on the molecular orientation relative to B0, spin rotation (SR) relaxation arises from an interaction between the spin and a coupling to the overall molecular rotational angular momentum (like the rotation of methyl group). While the CSA and chemical exchange contribute to relaxation, dipolar relaxation is often the most important mechanism and illustrates the features of CSA relaxation, hence will be the only one examined further. The dipolar spin operator between two identical spins (I1 and I2) separated by distance r for use in Eq. 3.16 is: Vm =−γ h̄r3 Tm, (3.20) where: T0 = 1√ 6 (3Î1z Î 2 z − I1 · I2) (3.21) T±1 = ∓12(I 1 ±I 2 z + I 1 zI 2 ±) (3.22) T±2 = 1 2 I1±I 2 ±. (3.23) If we plug these operators Vm in Eq. 3.19 and assume an exponential correlation function e−t/τc (due to rotational Brownian motion), we find a meaningful expression for T1: 1 T1 = 3 10 γ4h̄2 r6 τc[ 1 1+ω20τ2c + 4 1+4ω20τ2c ], (3.24) where τc is called the correlation time and is the average time constant of the exponen- tial correlation functions and represents random molecular tumbling. This result assumed isotropic rotational diffusion as is found in a liquid. If the dynamics contributing to relax- ation are restricted as they often are in solids, a more involved calculation depending on 27 the details of the dynamics is required. Using similar calculations as for T1 above, one can derive an expression for the transverse relaxation: 1 T2 = 3 20 γ4h̄2 r6 τc[3+ 5 1+ω20τ2c + 2 1+4ω20τ2c ]. (3.25) Eqs. 3.24 and 3.25 are the so-called BPP equations (Bloembergen, Purcell and Pound) [19]. In the limit ω0τc 1 (fast molecular motion limit), we find: 1 T1 = 1 T2 = 3 2 γ4h̄2 r6 τc. (3.26) Hence, in this regime relaxation times are independent of magnetic field strength and in- verse to the correlation time. In the slow motion limit ω0τc 1, we find: 1 T1 = 3 5 γ4h̄2 r6 1 ω20τc (3.27) 1 T2 = 3 10 γ4h̄2 r6 τc. (3.28) Hence, in this regime T1 is proportional the magnetic field strength squared (as ω0 =−γB0) and directly proportional to the correlation time. One way to study slow motion is to observe the relaxation in the rotating frame of one of the spin species. In this frame, only part of the original dipolar Hamiltonian will be secular. T1ρ is used to classify the relaxation time of the secular part of the rotating frame Hamiltonian. In this frame rotating at frequency ω , the effective magnetic field is given by: Be f f = B1i+(B0−ω/γ)k, (3.29) which means that the equilibrium magnetization will be parallel to B1 rather than to B0. BPP predicts the minimum value for T1ρ will occur when: 1 T1ρ = K τc 1+4ω21τ2c , (3.30) where ω1 = γB1 [20]. As this minimum occurs when τc≈ 12ω1 , this may be several orders of magnitude longer than τc for a T1 minimum. As B1 is typically orders of magnitude smaller 28 than B0, this minimum occurs at much lower frequency motions than the laboratory frame T1. The formalism presented here is conventionally used for liquids as it assumes an ideal case where there is a single correlation time and isotropic motion. Solid molecules such as NCC may possess a distribution of correlation times and one must also account for solid-state motional restrictions. Lipari and Szabo [21] have used a model free approach with great success for a single correlation time undergoing anisotropic motion. They have shown that by slight modification of the spectral density J(ω) (Fourier transform of the correla- tion function), one can more accurately determine the correlation times and dynamics of macromolecules. 29 Chapter 4 Cellulose 4.1 Native Cellulose Cellulose (C6H10O5)n is the primary component of cell walls in plants and is the most common organic molecule on earth. From a raw material production perspective, cellu- lose represents an almost infinite source of raw biomass (around 1.5×1012 tons annually) [22]. Most cellulose that is used commercially is processed in the form of wood pulp for use as paper or packaging or in the form of cotton or linen for textiles – though a small percentage of chemically modified cellulose is used in a wide variety of industries such as optical films, cosmetics and pharmaceuticals. Native plant cellulose usually exists alongside other substances - hemicellulose, lignin and pectin - though these are normally removed during industrial processing. The considerable interest in cellulose bio-polymers arises to due its intriguing chemical structure - as illustrated in Figure 4.1. Due to their Figure 4.1: Diagram of cellulose monomer repeat unit spatial configuration and chemical composition, cellulose molecules collect into highly ordered chains of linearly linked β (1→4) D-glucose units whose length depends on the origin and treatment/preparation of the cellulose. In contrast with lower molecular weight carbohydrates, the biochemistry of cellulose depends on a wealth of molecular attributes: 30 intermolecular interactions, cross-linking reactions, chain lengths, chain-length distribu- tion, and functional group distribution along the chains [23]. These properties determine a uniquely catenulate structure which imparts cellulose with characteristic micro and macro- scopic properties such as hydrophilicity, chirality, and biodegradability. Cellulose from wood pulp has typical chain lengths between 300 and 1700 units; where other sources (other plants or bacteria) have chain lengths ranging from 800 to 10,000 units. Molecules with very small chain length resulting from the breakdown of cellulose are known as cel- lodextrins and are typically soluble in water and organic solvents whereas longer chain molecules are usually not water soluble. Relative to other polysaccharides (such as starch), cellulose is not easily broken down chemically and is much more crystalline. For instance, starch undergoes a crystalline-to-amorphous transition when heated beyond 70°C in wa- ter, whereas cellulose requires a temperature of 320°C and pressure of 25 MPa to become amorphous in water. While the molecular structure in Fig. 4.1 appears relatively unremarkable, it is the supra- molecular fiber structure of cellulose that provides much of the structural complexity. Cel- lulose has historically been thought to consist of areas of both high order (crystalline) and low order (amorphous) [24]. In 1984, VanderHart and Atalla first showed with 13C cross-polarization magic angle spinning (CP/MAS) NMR that native cellulose crystalline regions contain a mixture of different structures differing in the location of hydrogen bonds connecting strands. They found that native cellulose consists of a mixture of crystalline polymorphs Iα and Iβ . Fig. 4.2 shows the different hydrogen bonding networks in the two different allomorphs [25]. Bacteria produced cellulose contains greater amounts of the Iα allomorph while cellulose derived from plants possesses mainly Iβ . Yamamoto et. al. [26] demonstrated the use of X-ray diffraction and 13C NMR spectroscopy to determine rela- tive amounts in native cellulose. Another form of crystalline cellulose is called cellulose II and it is the most thermodynamically stable polymorph and is derived from natural re- generated fibers. One difference between cellulose I and II is that the O6-HO3 inter-chain hydrogen bonding is dominant in cellulose I, whereas in cellulose II, O6-HO2 is the main inter-chain hydrogen bonding [27]. There are also other polymorphs of cellulose obtained artificially by chemical or heat treatments and easily identified with X-ray diffraction [28]. After being subjected to ammonia solution, cellulose does not revert to its initial crystal form but adopts another, designated cellulose IIII or IIIII, depending on the initial crystal form (cellulose I or II) [29]. Cellulose IIII or IIIII can be further modified into Cellulose IVI, IVII by heating in glycerol at 260°C for 20 minutes [30]. While the crystal structure 31 of cellulose has been studied with success, the hydrogen bonding network structures found in the amorphous regions remain unknown [31, 32, 33]. Figure 4.2: Hydrogen-bonding patterns for cellulose Iα and Iβ based on the work of Nishiyama and co-workers [25, 34]. Top left and right: two alternative networks of cel- lulose Iα . Bottom left: principle network in cellulose Iβ at the origin of the unit cell. Bottom right: cellulose Iβ chains at the center of the unit cell. Adapted with permission from [35] with permission. The characteristic fiber morphology of cellulose gives it the required strength and proper- ties for both the natural environment and for industrial applications. The hierarchy of the fibers in the conventional model of cellulose morphology is defined by elementary fibrils, microfibrils and microfibrillar bands [36]. These different fibrils range in size from 1.5 nm to lengths of several hundred nm. The microfibrils are thought to organize into different models depending on preparation and treatment. The fringed fibrillar model (as illustrated in Fig. 4.3) with crystallites and noncrystalline regions has proved successful for the de- scription of the structure of microfibrils and the partial crystalline structure of cellulose in connection with its surface chemistry and reactivity. Another successful model includes 32 Figure 4.3: Schematic diagram of cellulose fibril structure. Adapted from [37]. additional “skin and core regions” for each crystallite though this has mainly been used to study Cellulose II. The crystallinity of the skin and core regions are almost the same and the main difference between the regions is seen with an electron microscope: a presence of voids in the skin (on the microscopic scale) and a much higher orientation of the crys- talline regions (on the nanoscale) [38]. The crystallinity (percentage of the volume of the material that is crystalline) of cellulose has historically been the source of debate. Deter- mination methods and techniques have historically provided conflicting results prompting more precise definitions of crystallinity with respect to the fibril structure [39, 40]. 4.2 Cellulosic Nanocomposites and Nanocrystalline Cellulose When one examines cellulose fibers, in a piece of paper for example, it is difficult to fully appreciate the fibrillar nanostructures composing the material. The properties of paper arise due to nano-scale domains of cellulose, hemicellulose and lignin present in natural wood [41]. By manipulating the composition and structure of these nano-domains, one can de- velop nano-scale components that achieve properties not found in nature. One remarkable 33 and relatively easy way of cellulose manipulation is acid treatment. The production of sus- pensions of cellulose by acid hydrolysis of cellulose fibers was first proposed by Ranby in 1951 [3]. Ranby proposed that cellulose was comprised of micellar strings of uniform width, which could be freed into suspension by mechanical or chemical means. Hydrolysis was viewed as cutting the micellar strings into short fragments, or micelles, while retaining their width. Frey-Wyssling’s experimental observations in 1954 added to Ranby’s theory by providing evidence that elementary microfibrils, separated (and joined) by paracrys- talline cellulose, represent the basic building units of cellulose [42]. Paracrystalline cellu- lose lacks the high degree of hydrogen bonding found in crystalline cellulose, thus giving it a structure that is less ordered. Also during the early 1950s, Battista used hydrochloric acid hydrolysis of wood pulp cellulose fibers, followed by sonification treatment, to produce microcrystalline cellulose (MCC) [43, 44]. MCC possesses attractive binding properties and has been used in a variety of applications including the pharmaceutical industry as a tablet binder, as a texturizing agent and fat replacer in food, and as an additive in paper and composites applications. In 1992 Revol et. al. developed what would become nanocrys- talline cellulose (NCC) [45] by demonstrating that, above a critical concentration, a chiral nematic-ordered phase spontaneously formed during acid hydrolysis. While MCC consists of colloidal particles on the order of a micron, NCC is comprised of nanoscale strands. NCC has also been called “nanorods” [46], “nanowires” [47] or “whiskers” due to their appearance in electron micrographs as shown in Fig. 4.4 [1, 48, 49]. Figure 4.4: Scanning electron micrograph of NCC strands. 34 NCC is typically extracted from kraft softwood pulp by cleaving cellulose fibrils at struc- tural defects. This is usually done with sulfuric or phosphoric acid hydrolysis which results in short nanocrystals. These whisker-like nanocrystallites possess diameters of 2-20 nm and lengths up to 500 nm. The acid hydrolysis frees the NCC crystallites from within the lignocellulosic fiber structure of the pulp [50] into suspension. The crystallites are then prepared as either air-dried, iridescent films or freeze-dried powders. These crystallites have impressive mechanical properties with a tensile strength much greater than, and an elastic modulus similar to stainless steel as illustrated in Table 4.1. In addition to these impressive mechanical properties, NCC crystallites display interesting and valuable optical and magnetic properties. Property Cellulose Nanofibil Softwood Kraft Pulp Stainless Steel Length (nm) 500 1500000 N/A Diameter (nm) 5 30000 N/A Aspect Ratio 100 50 N/A Tensile Strength (MPa) 10000 700 1280 Elastic Modulus (GPa) 150 20 210 Table 4.1: Typical properties of cellulose nanofibrils (derived from native cellulose), soft- wood kraft pulp and stainless steel. Adapted from [1]. When acid-hydrolyzed cellulose is suspended above a critical concentration in water, the crystallites spontaneously organize into a chiral nematic (cholesteric) liquid crystalline phase [51]. The degree of phase-forming ability of the cellulose crystallite suspension depends on the mineral acid chosen for hydrolysis [52]. While both sulfuric and phos- phoric acid yield the desired chiral nematic phase, sulfuric acid is the most studied. Upon drying (air or freeze), the chiral nematic order persists in the resulting films. In the chiral nematic phase, molecules within a plane are all aligned along the director, but the align- ment axis winds in a helix as illustrated in Figure 4.5. The cholesteric phase affects the optical properties of NCC and gives rise to iridescence - as seen in bubbles or certain insect wings. This is due to multiple reflections from multi-layered, semi-transparent surfaces of NCC in which phase shift and interference of the reflections modulates the incident light (by amplifying or attenuating some frequencies more than others) [53]. Shortly after the discovery of NCC, Sugiyama et. al. showed that under high magnetic field, cellulose crystallites in dilute aqueous suspension orient themselves with their long 35 Figure 4.5: Representation of the chiral nematic phase [1]. Nematic liquid crystals have a high degree of long-range orientational order but no long-range translational order. In chiral nematics, each nematic plane consists of aligned crystallites that differ slightly in angle (about the cholesteric axis) from adjacent planes. axes perpendicular to the applied magnetic field [54]. This arises due to the negative dia- magnetic anisotropy of cellulose chains. Hence, application of high magnetic field to chiral nematic suspensions aligns the cholesteric axis parallel with the magnetic field. This has prompted research into nanocomposite production under high field. Recently, Kvien et. al. produced a very strong uni-directional reinforced nanocomposite by aligning NCC in polyvinyl alcohol under high magnetic field though oddly the nanocomposite possesses a higher dynamic modulus in the direction of the magnetic field compared with the transverse direction [55]. While NCC can provide composite materials with a variety of desirable characteristics, they currently possess a few potential weaknesses in their use as a widespread industrial composite [56]: the hygroscopic nature of cellulose, thermal degradation and high cost of nano treatment. The water-absorbent behaviour of native cellulose unfortunately is also present in cellulosic nanocomposite structures – severely limiting desirable water-proofing abilities. Matsumura et. al. have found a solution, albeit incomplete, by making fibers suf- ficiently hydrophobic - though this comes at a substantial cost of physical properties [57]. Cellulosic nanocomposite structures have been shown to degrade or completely breakdown at sufficiently elevated temperatures. Chemical modification of the surface of cellulose nanocrystals has been shown to affect the temperature at which decomposition begins - though researchers have shown conflicting results with sulphate groups remaining after 36 acid hydrolysis playing an important role [58, 59]. The high cost associated with produc- tion, formulation and surface treatment of cellulosic nanomaterials is a major hurdle before NCC materials can be competitive on a large scale commercial market. As pointed out by Zafeiropoulos et. al, traditional cellulose chemical treatments may be rendered economi- cally unfeasible due to the inherent high surface area per unit mass of nanomaterials [60]. By studying the crystalline structure of NCC, new light may be shed on how to best tackle these obstacles. 4.3 Solid-state NMR of Native Cellulose and NCC Solid-state 13C NMR spectra have provided many insights into the molecular and supra- molecular structure of native cellulose [61]. Examples of 13C CP/MAS spectra of native cellulose are illustrated in Fig. 4.6. Using Fig. 4.1 as a legend, we begin in the up-field part of the spectrum; the region around 65 ppm (with slight shoulder extending to 60 ppm) is assigned to C6 of the primary alcohol group. The next, relatively large, resonance peaks between 70 and 78 ppm are attributed to C2, C3 and C5 - the ring carbons not part of the glycosidic linkage. The region between 85 to 95 ppm corresponds with C4 and the sharp resonance around 105 ppm corresponds with C1. One of the most studied properties of cellulose is its crystallinity. Many studies using infrared (IR), X-ray diffraction (XRD) and NMR have shown that the crystallinity index of celluloses increases during kraft pulping due to the removal of less ordered structures (hemicellulose, lignin) [63, 64]. However, it has been shown that the crystallinity index (% of sample possessing crystalline structure) varies substantially depending on the choice of instrument and data analysis technique implemented [39]. The two most commonly used techniques for determining the crystallinity index are NMR and XRD. Cellulose samples have been shown with 13C CP/MAS NMR to possess between 36% to 70% crystallinity depending on the native source and treatment processes [65] - while XRD measurements consistently show higher results. With NMR, crystallinity is conventionally determined by deconvolution of the peaks using Lorentzian line shapes and then taking a ratio between the area under the sharp peaks and neighbouring broad shoulders. Though a novel crystallinity determination technique developed by Park et al. may be accurate [39]. This method in- volves subtracting a 100% amorphous cellulose 13C NMR spectrum from the spectrum of 37 Figure 4.6: CP/MAS 13C NMR spectra of cellulose I from different sources. From bottom to top the order is: Valonia cellulose, Cladophora cellulose, Halocynthia cellulose, cotton linters and bleached birch pulp [62]. Reused with permission from Elsevier. the sample of interest to create a “crystalline” spectrum. The relative size of the peaks between the “crystalline” and reference spectra can be used to determine the crystallinity. This simple technique is claimed to have the advantage of simultaneously providing infor- mation on the interior-to-surface (I/S) ratio of the cellulose crystallites. Zhao et al. have studied the structural changes and crystallinity percentages of native cel- lulose derived from cotton after various levels of hydrolysis [66]. They found that although dramatic morphology changes of cellulose microfibrils are observed, the relative ratio be- tween the amorphous and crystalline regions is not significantly changed by hydrolysis - though there is a large difference in reactivity between the regions. This leads to fast re- moval of surface amorphous followed by surface crystalline while the core sections leech much slower. Other studies on the crystallinity of cellulose after acid hydrolysis using the Ruland peak deconvolution method of XRD have been determined to be greater than 90% suggesting that amorphous regions have been removed [1]. Other solid-state NMR studies of native and chemically treated cellulose include proton and carbon relaxation studies [67]. Proton spin-lattice relaxation measurements have been used to examine the dynamics of various cellulose/synthetic polymer blends. These studies have provided significant insight into the dynamics associated with cellulosic nanocomposite materials [68]. Carbon spin-lattice relaxation times of native and treated cellulose have 38 shown two components which have been attributed to the crystalline (relaxing on the order of 200 seconds) and amorphous (relaxing in ∼ 1−20 seconds) regions [69]. With regards to nanocomposite production, thermostability of cellulose crystals may be vital. Roman and Winter have shown that even at low levels, sulfate groups on the surface cause a significant decrease in degradation temperatures [59]. Though no studies have been performed exclusively on the effect of phosphate groups, one would predict similar behaviour. 39 Chapter 5 Experimental This chapter describes the NMR pulse sequences and experiments performed and how the NCC samples were prepared. Based on previous structural studies of native or acid hydrolyzed cellulose, the objective of these experiments is to examine the surface, the “near-surface” and the core of the crystallites with crystalline/amorphous ratios and surface chemistry and phase structure in mind. 5.1 Sample Preparation NCC was produced by FPInnovations Vancouver [70]. This began by obtaining fully bleached western red cedar kraft pulp from Canfor Pulp Limited Partnership. The pulp was milled to pass through a 0.5-mm screen in a Wiley mill (so as to ensure particle size uniformity). Sulfuric acid solutions of 64 wt.% concentration were formed by adding con- centrated acid under magnetic stirring to water cooled in an ice bath. The milled pulp was then hydrolyzed with the sulfuric acid at 85°C under stirring with an impellar for 25 minutes. Extra water was added to the suspension to stop the hydrolysis and the resulting cellulose was separated from the extra liquid by a combination of centrifugation and de- canting. The resulting white-film suspension was placed inside dialysis membrane tubes for 1-4 days and finally dispersed by ultrasonic treatment. The dispersed suspension was then air-dried and ground up into a crystalline powder. Samples of phosphoric acid hy- drolyzed NCC were prepared in similar fashion as the sulfuric acid hydrolyzed samples described in the previous paragraph with sulfuric acid replaced with phosphoric acid. For 2H exchange experiments, NCC samples were either soaked in D2O or exposed to D2O vapour for times ranging from a few seconds up to 5 months. Liquid soaked samples were centrifuged and decanted then dried under vacuum. Vapour exposed samples were 40 dried under vacuum. Deuterium exchange of hydroxyl protons was performed by exposing an NCC sample to a saturated D2O atmosphere or immersed in 99% D2O liquid at room temperature for varying lengths of time ranging from a few seconds up to 5 months. The samples were then vacuum dried so as to remove non-exchanged D2O from the sample. 5.2 NMR Spectroscopy All MAS experiments were performed on a Varian Inova 400 MHz (9.4T) NMR spectrom- eter using a Varian Chemagnetics T3-HFX 3-channel 4 mm magic angle spinning probe, spinning at 4 or 5 kHz in airtight rotors. Radio frequency (rf) fields on the various irradia- tion channels had nutation frequencies of 50 kHz for the relaxation experiments and 83 kHz for the REDOR experiments. TPPM [71] proton decoupling was used during the data ac- quisition. 13C chemical shifts were referenced relative to the up-field peak at 29.50±0.10 ppm in an external adamantane reference [72]. A Highland Technology Model L950 Tem- perature Controller was used to control the temperature of the sample. Wideline 2H NMR spectra were collected on a home-built NMR spectrometer based upon a 4.7T Oxford In- struments magnet, using a modified Bruker probe and ENI power amplifier producing an rf field strength of 100 kHz. 5.2.1 13C T1, T2 and T1ρ Relaxation Measurements As discussed in Sec. 3.4, relaxation studies can provide insight into the chemical environ- ment of spins. Three sets of carbon relaxation experiments were conducted to characterize the structural environment of NCC: T1, T2 and T1ρ . As described in Sec. 3.4, T1 and T2 represent the longitudinal and transverse relaxation in the laboratory frame respectively and T1ρ refers to the longitudinal relaxation time in the rotating frame. T 1 values were de- termined through the use of the modified CP pulse sequence as published by Torchia [73] and as illustrated in Fig. 5.1. T1 measurements were performed at both room temperature (21°C) and 70°C for air-dried and freeze-dried NCC samples. The sequence begins with an initial CP between 1H and 13C. After the desired 13C polarization has been established due to the Hartmann-Hahn condition, the proton resonant field is turned off and a 90° pulse is applied on the carbons which rotates the magnetization to the z-axis. The magnetization is 41 allowed to evolve and will relax exponentially to its equilibrium value with a time constant T1. After a delay, the polarization is rotated into the xy plane by a final carbon 90° pulse before signals are acquired (proton decoupling is turned on for signal detection). CP Decoupling 90° H 1 C 13 CP 90° 90° d 1 Figure 5.1: Pulse sequence used for carbon T1 measurements. To determine T1 values from this experiment, one performs the experiment with varying evolution delay times d1 (from 0 to 300 seconds here) and plots the signal intensity of the specific carbon peak as a function of evolution delay time d1. For a homogeneous sample, the intensity is expected to follow an exponential decay: S(d1) = S0e−d1/T1, (5.1) where S0 is the signal intensity with no d1 delay. Transverse carbon relaxation values T2 were determined by the Hahn spin-echo pulse se- quence [74] as illustrated in Fig. 5.2. The pulse sequence is similar to a normal CP experi- ment except for an extra 180° carbon pulse surrounded by two delay times d1 and d2 placed just after the contact time and just before signal acquisition. 42 CP Decoupling 90° H 1 C 13 CP 90° d 1 d 2 Figure 5.2: Hahn echo pulse sequence used for T2 measurements. The magnetization following the CP time decays due to both spin-spin relaxation and any inhomogeneous effects which cause different spins to precess at different rates (such as a distribution of chemical shifts). The 180° carbon pulse serves to isolate the relaxation effects as relaxation leads to an irreversible loss of magnetization (decoherence), but the inhomogeneous effects can be re-phased. The inversion pulse is applied after a period d1 and the inhomogeneous evolution will re-phase to form an echo at time 2d1. This experi- ment is performed incrementing the delay d1 (d2 is set just right to be equal to d1 minus an additional delay for filter response) and the resulting peaks are fit to an exponential similar to T1: S(d1) = S0e−d1/T2. (5.2) Spin-lattice relaxation in the rotating frame T1ρ values were determined by modifying the standard CP pulse sequence with an extra time during which only carbon is spin-locked parallel to B1 just after the contact time and before acquisition. The length of the new spin-lock time d1 was varied from 0 to 8 milliseconds and an exponential function of time similar to those used for T1 and T2 was used to determine T1ρ : S(d1) = S0e−d1/T1ρ . (5.3) If populations of a sample exist in different dynamical environments, it is possible for the T1, T2 and T1ρ values of the populations to be significantly different. In this case, the time constants for each population can be determined by fitting with a modified function: S(d1) =∑ α S0αe−d1/Tiα , (5.4) 43 where S0α is the initial signal from each population with relaxation time Tiα (i= 1, 2, or 1ρ ). Relative values of S0α also represent the proportion of the sample existing in the various environments. All the relaxation fits were performed using the Marquardt-Levenberg-algorithm for fitting and the gnuplot function evaluation mechanism for calculation. 5.2.2 REDOR Another pulse sequence that can be used to characterize local structure is the rotational- echo double resonance (REDOR) experiment [75] designed to measure heteronuclear dipo- lar couplings [76]. Under magic angle spinning, the transverse magnetization of I spins will dephase during the first half of the rotor period due to I-S heteronuclear dipolar coupling, and, due to the same heteronuclear coupling, will be refocused during the second half of the rotor period [7]. The goal of REDOR is to measure the effect of applied perturbing rf pulses on the refocusing, as this can be used to determine heteronuclear dipolar couplings. Depending on the spin species of interest, different versions of REDOR are employed. In the chosen version for the case of 31P-13C couplings (two spin 1/2 nuclei), one applies rotor-synchronized 180° pulses to the 31P spins and acquires a signal on the carbon channel. These pulses decrease the height of the 13C resonance peaks with the more pronounced the decrease, the stronger the heteronuclear dipolar interaction. To perform a REDOR experiment one collects two spectra: one with and one without 180° dephasing pulses. Typically proton dipolar decoupling is used while transverse magnetization of the observed nucleus evolves, thus one needs a 3-channel probe. The pulse sequence, as illustrated in Fig. 5.3, begins with a standard CP contact time to en- hance 13C magnetization followed by 1H channel decoupling (which remains on throughout the rest of the sequence). During the reference experiment when the 31P pulses are omitted, heteronuclear dipolar couplings are refocused every rotor period as expected under MAS. In the other experiment, a train of xy-4 phased (xyxy)n rotor-synchronized 31P 180° pulses are applied to interrupt the averaging of the heteronuclear dipolar Hamiltonian due to MAS. The effect of the 180° pulses is to change the sign of the heteronuclear dipolar coupling 44 CP Rotor P C H CP 1 13 31 Decoupling 0 6 Tr 180° xy-4 180° xy-4 180° Figure 5.3: 31P-13C REDOR pulse sequence for a dipolar evolution time of 6 rotor periods. during the second half of each rotor cycle: IzPIzC →−IzPIzC. A single 180° pulse is ap- plied to the 13C channel at the midpoint of the dipolar evolution time to refocus isotropic chemical shifts and produce an echo at the start of acquisition [75]. The first-order Hamiltonian for the interaction between an isolated 31P-13C spin pair is: Ĥhetero(t) =−2C(t)Î31Pz Î 13C z , (5.5) where C(t) = µ04pi h̄ γ31Pγ13C r3 1 2(3cos 2(θ)−1) in frequency units. Applying average Hamilto- nian theory (as described in Sec. 2.4) in the case of an isolated 31P-13C spin pair to a single rotor cycle with 31P 180° pulses, one finds the average Hamiltonian over a rotor period: H = 1 τR τRˆ 0 Ĥhetero(t)dt (5.6) where τR is the rotor period [7]. Without the 180° pulses one finds H = 0, hence the only difference arises due to the heteronuclear dipolar coupling. Applying the time evolution operator to the density matrix ρ(0) = S0Ix and averaging over all possible orientations of the spin pair in the rotor shows that the ratio of the acquired signals between the two experiments depends only on the dipolar coupling frequency ωD of a spin pair (assuming 45 perfect 180° pulses): S S0 = √ 2pi 4 J1/4( √ 2NωD/ωR)J−1/4( √ 2NωD/ωR), (5.7) where S0 is the signal from the reference experiment, S is the signal from the dephasing experiment, J±1/4 are Bessel functions of the first kind of order±14 , N is the number of rotor cycles and ωR is the rotor spin frequency. ωD can be used to determine the heteronuclear dipolar interaction distance (in SI units) [77]: ωD = µ0γSγI h̄ 8pi2r3 . (5.8) Eq. 5.7 is called the universal dephasing curve and the expression ∆S= 1− SS0 is often used for REDOR analysis. If only a fraction of 13C spins have a 31P neighbour, ∆Sideal must be scaled by a factor C: (1− S S0 )measured =C · (1− SS0 )ideal, (5.9) where C represents the proportion of population dephased by nearby 31P nuclei. Hence the dipolar distance and proportion of population with 13C spins having a 31P neigh- bour can be determined by measuring SS0 for several values of N and fitting the data to the scaled universal dephasing curve. REDOR experiments were also performed on D2O soaked NCC samples to determine the proportion of carbons which were nearby 2H nuclei and to probe the hydrogen bonding geometry of the exchanged regions. The treatment above is a simple and effective way to determine heteronuclear dipolar couplings between isolated pairs of spin-1/2 nuclei. However, measurement of 13C-2H dipolar interactions may be hindered by the deuterium quadrupolar interaction. It has been shown that a modified REDOR scheme as shown in Figure 5.4 [78, 77] is preferable in this case. This sequence similarly uses a train of rotor-synchronized xy-4 phased 180° pulses, which last for N rotor cycles, applied to the 13C channel during the dipolar evolution period and the deuterium channel employed to apply just a single 90° pulse. The time interval between the 180° pulses is again half a rotor period. Thus a modified REDOR scheme swaps the pulse train from the X-nuclei to the observe nuclei and the solo 180° observe pulse switches to a 90° pulse applied to the X-nuclei. 46 CP CP Decoupling 90° 90° H 1 C 13 H 2 xy-4 xy-4 φ Rotor 0 10 Tr φ Figure 5.4: The 13C-2D REDOR pulse sequence for a ten rotor cycle dipolar evolution period. This is modified from the spin-1/2 case as in Figure 5.3. All carbon 180° pulses follow an xy-4 phase cycle. This pulse sequence modifies the universal dephasing curve for a powder sample [77]: S S0 = 1 6 (1+4 √ 2pi 4 J1/4( √ 2NωD/ωR)J−1/4( √ 2NωD/ωR)+ √ 2pi 4 J1/4(2 √ 2NωD/ωR)J−1/4(2 √ 2NωD/ωR)). (5.10) While determining information from REDOR is straightforward for isolated S–I spin pairs, for an S spin coupled simultaneously to many I spins (such as in NCC) the analysis becomes more difficult. As the NCC samples contain a mixture of deuterated and unexchanged molecules, the 13C resonances will contain contributions from 13C spins coupled to 2H spins and from 13C spins that have no deuterium neighbor. Consequently, the signal S is not what is obtained experimentally but instead Sm is directly measured. There are three reasons why S and Sm are different. First, as with the 31P case, only a fraction of 13C spins have 2H neighbours. The second issue is some 13C may have more than one 2H neighbour. The final reason is that the CP dynamics for 13C spins on deuterated molecules may differ from those on unexchanged molecules. Calculating the difference due to differing CP dynamics is not trivial [78, 77]. Fortunately each carbon in a cellulose monomer will be near a proton regardless of whether or not the monomer has undergone complete 2H 47 exchange. Thus the difference in CP dynamics is not expected to significantly effect results so S' Sm. 5.2.3 2H REDOR Simulation To examine the spin system in more depth, modeling of the system has been performed using the SIMPSON NMR simulation software [79] which functions by numerical integra- tion of the von Neumann equation (Eq. 2.20). This requires a complete description of the Hamiltonian of the system and a starting density matrix. The crystal and molecular struc- ture together with the hydrogen-bonding systems of cellulose Iα and Iβ have been deter- mined to great precision by Nishiyama et. al. [25, 34] and were used to determine dipolar coupling frequencies and the geometry between carbon and nearby deuterium spins. For each carbon, the two nearest deuterium spins were used to approximate the dipolar interac- tion. The quadrupolar interaction was measured by examining the width of a 2H spectrum (details are in the following section). The chemical shift anisotropies (CSAs) of cellulose Iα and Iβ obtained by Hesse and Jager were used [80]. A typical SIMPSON input file is included in the appendix. 5.2.4 2H-1H Exchange As 2H is a spin-1 nucleus with a small quadrupole moment, one finds quadrupolar coupling constants in the range of 140-220 kHz [7]. The powder spectra take on the characteristic Pake pattern due to the two possible spin transitions as illustrated in Fig. 5.5. The relative width of the pattern is dependent on the quadrupolar coupling constant and for 2H, is sen- sitive to molecular motions with correlation times of 10−4− 10−6Hz [7]. This makes 2H spectroscopy well suited for analysis of partially-ordered materials (such as NCC) through relaxation or exchange experiments. 48 -0.51.0 Chemical Shift Figure 5.5: 2H quadrupole powder Pake pattern. The pattern is composed of two overlap- ping signals which give the pattern its characteristic shape. To further investigate the supra-molecular structure of NCC, a series of simple 2H exchange experiments were conducted. These experiments are designed to examine the exchange rate of protons to deuterium when NCC to subjected to liquid or gas D2O in order to examine the crystallite structure. This technique has been used to examine a variety of biological compounds including protein structure [81] and bone matrix water content [82]. Quantification of the absolute number of exchanged nuclei in an NCC sample was per- formed by comparison of 2H spectra of an NCC sample with a PMMA sample polymer- ized from per-deuterated monomers. Assuming 3 potential exchange sites per cellulose monomer, one can compare the resulting 2H signal intensities of the two samples to deter- mine the percentage of the NCC sample that has exchanged. 49 Chapter 6 Results and Discussion 6.1 13C NCC CP/MAS Spectra The first experiment conducted was a standard 13C CP/MAS NMR experiment with spin- ning at 5 kHz as illustrated in Fig. 3.1. Fig. 6.1 shows the 13C CP/MAS NCC spectrum for 33 mg of NCC. The narrow resonance peaks have been assigned previously [83, 84]. The broad shoulder C4′ has been assigned by VanderHart to the amorphous and surface regions while the sharper C4 peak is believed to come from the crystalline regions [85]. Based on our spectrum, the C4′ peak indicates that there is a substantial amorphous content. C6 often displays a similar behaviour, though, as seen here, C6′ is not always separately resolved. The remaining carbon signals do not display separate peaks for the amorphous and crys- talline regions. The spectra we see are similar to what has seen seen for native cellulose, though the resolution is poorer, suggesting imperfect crystalline structure. 50 Figure 6.1: NCC spectrum spinning at 5 kHz. The highlighted surfaces represent the areas used for integration in all subsequent analysis. 6.2 Relaxation Measurements Fig. 6.1 shows the areas under each peak which were used for relaxation measurements. Most relaxation curves did not fit a single exponential function (S(t) = S(0)e−t/TA) but, as has been used previously to analyze 13C T1 relaxation data in cellulose [69], both 21°C and 70°C fit a double exponential function very well: S(t) = S0e−t/T 0 1 +S1e−t/T 1 1 , (6.1) where t is the delay time and S0 and S1 are the relative weights of populations with relax- ation times T 01 and T 1 1 and T 0 1 > T 1 1 . A stretched exponential function (S(t) = S(0)e −(t/TA)β ) was also fit to the data. Stretched exponential functions are frequently used to fit relaxation data in solid polymer systems when broad distributions of relaxation times are present [8]. When β = 1 we recover the single exponential and 0 < β < 1 corresponds to broader dis- tributions of decay times. The stretched exponential fits were generally indistinguishable from the double exponential fits as illustrated in Fig. 6.2 though the uncertainty of the parameters was lower. Parameters from two-component fits to relaxation measurements at room temperature and 70°C for air-dried and freeze-dried NCC samples are displayed in 51 Tables 6.1 and 6.2 and the relaxation times determined from stretched exponential fits are displayed in Table 6.3. An example of the fits’ quality showing the differences observed for the C2,3,5 peak is shown in Fig. 6.3. 0 50 100 150 200 250 300 Delay Time (seconds) I n t e n s i t y  ( a r b .  u n i t s ) Double Exponential Single Exponential Stretched Exponential Figure 6.2: Comparison of possible fitting functions for T1 relaxation data of the C2,3,5 peak for freeze-dried NCC at room temperature. Air-dried (21°C) Freeze-dried (21°C) Air-dried (70°C) Freeze-dried (70°C) C1 275±80 (46%) 248±96 (47%) 95±13 (44%) 126±40 (74%) C2,3,5 266±38 (51%) 218±27 (43%) 140±19 (57%) 174±23 (38%) C4 468±137 (57%) 324±101 (69%) 158±87 (58%) 177±100 (63%) C4′ 289±143 (42%) 305±110 (52%) 59±12 (58%) 176±151 (41%) C6 191±43 (51%) 156±18 (57%) 123±21 (42%) 111±17 (49%) Table 6.1: T 01 (slow relaxation component) values of air-dried and freeze-dried NCC sam- ples measured in seconds. The percentage refers to the population of the sample which follows the specified relaxation time. 52 0 100 200 300 400 500 Delay Time (seconds) I n t e n s i t y  ( a r b .  u n i t s ) T1 = 266 sec. Air-dried (21°) Freeze-dried (70°) Air-dried (70°) T1 = 140 sec. T1 = 218 sec. Figure 6.3: Normalized T1 relaxation curves for C2,3,5 for 3 samples. The slow component decay times are included. Air-dried (21°C) Freeze-dried (21°C) Air-dried (70°C) Freeze-dried (70°C) C1 41±12 (54%) 49±18 (53%) 23±31 (56%) 10±10 (26%) C2,3,5 37±4 (49%) 30±6 (57%) 20±4 (43%) 26±5 (62%) C4 46±21 (43%) 45±18 (31%) 49±67 (42%) 15±13 (37%) C4′ 30±10 (58%) 57±13 (48%) 2±2 (42%) 20±9 (59%) C6 11±4 (49%) 9±2 (43%) 6±1 (58%) 4±3 (51%) Table 6.2: T 11 (fast relaxation component) values of air-dried and freeze-dried NCC samples measured in seconds. Air-dried 21°C (β ) Freeze-dried 21°C (β ) Air-dried 70°C (β ) Freeze-dried 70°C (β ) C1 110±10 (0.7) 108±8 (0.7) 83±6 (0.9) 77±5 (0.7) C2,3,5 83±3 (0.6) 98±4 (0.6) 55±2 (0.6) 54±3 (0.6) C4 175±26 (0.5) 150±14 (0.7) 115±9 (0.9) 78±7 (0.8) C4′ 53±12 (0.5) 82±14 (0.6) 43±10 (0.7) 29±13 (0.7) C6 42±15 (0.4) 44±7 (0.4) 10±6 (0.6) 6±6 (0.6) Table 6.3: T1 values of air-dried and freeze-dried NCC samples measured in seconds deter- mined from stretched exponential fits. A few important trends are seen in these measurements. The fact that the T1’s are generally quite long is consistent with a rigid structure without much motion. In most cases, C6 has 53 the shortest T1’s which can be explained due to molecular structure – it is off the ring, has 2 protons directly bonded to it (each of the other carbons has only one) and is likely more mobile. Native cellulose has historically been characterized as a mixture of crystalline and amorphous regions [4, 85] with the C4′ and C6′ peaks arising from the amorphous regions. However, the fact that C4′ shows a large component with a very long T1 suggests that at least some of the C4′ peak arises from very rigid regions. This in turn suggests that the division between C4 and C4′ into crystalline and surface/amorphous may not be as abso- lute as has been suggested in the literature. Further evidence supporting this interpretation will be discussed in the REDOR and 2H exchange results below. In Table 6.3, β always gets closer to 1 (or stays the same) as temperature increases – meaning more nearly single exponential behaviour. C4′ always has smaller (or same) β and shorter mean T1 as C4, indicating broader, more mobile distributions. The C4′ T1 values show the most variation between the samples. In the air-dried sample, the C4′ T1’s are shorter, and much more sen- sitive to temperature than C4. This suggests that C4′ is in regions whose structure depends on drying conditions – this is consistent with the C4′ as surface/amorphous. As the double and stretched exponential functions fit all the peaks well, either the NCC crystallites must contain two drastically different areas or they contain a broad distribution of dynamic en- vironments. This result applies to both the C4 (previously assigned to be crystalline) and C4′ (previously assigned to be amorphous) peaks separately. From a dynamics point of view, the C4′ shoulder (conventionally attributed to C4 carbons possessing an amorphous chemical environment) shows components with both fast and slow relaxation timescales. From the two-component fits, the relative percentages in either relaxation time can be used to give rough estimates as to the percentage of the NCC crystallites that exist in the “fast” and “slow” components. These fits suggest between around 40% and 60% exist in the slow region, while the remaining molecules exist in the fast region. Another point to note is that by changing the temperature from 21°C to 70°C, one sees a decrease in relaxation time among the slow and fast relaxation populations though trends in the fast relaxation population are difficult to discern due to high relative uncertainties. The fact that all the relaxation times decrease when temperature increases is probably due to an increase in the amplitude of vibrational motions as temperature increases. When we compare our results with cotton cellulose (fast T1’s between 0.5 s and 12.4 s [69]), we see that, for all peaks, our “fast” T 11 relaxation times are at least 4 times as long – suggesting our samples contain less highly mobile material, either due to the different species, or because mobile regions were removed in processing. On the other hand, our slow relaxation times are much more similar 54 to those of native cellulose (∼ 200 s) suggesting that the crystalline regions have not been altered significantly during treatment. Our relaxation measurements have been analyzed with two component fits to compare with previous measurements, but we find that trends between samples under different circumstances (freeze vs. air-dried and 21°C vs. 70°C) are much more easily observed with the stretched exponential fits. Because the stretched exponential fits include one fewer parameter but provide indistinguishable quality of fit, these results tend to support the view of broad distributions of dynamical and structural environments. T2 measurements were performed on the same samples as above with the exception of freeze-dried at 21°C. Delay times were incremented from 0 to 8 ms. We found T2 values ∼ 8 ms but were unable to increase delay times in order restrict heating in the probe. In this case, a single exponential function fit the data well. Table 6.4 displays the measured values. Air-dried (21°C) Air-dried (70°C) Freeze-dried (70°C) C1 7.1±0.3 7.5±0.5 6.4±0.2 C2,3,5 5.5±0.2 5.9±0.2 5.0±0.2 C4 7.1±0.3 8.4±0.7 6.0±0.3 C4′ 5.0±0.2 5.6±0.4 5.1±0.2 C6 2.3±0.1 2.6±0.2 2.4±0.1 Table 6.4: T2 values of carbon resonance peaks in milliseconds. The T2 values show a slight increase upon raising temperature. The fact that the spin- spin relaxation data accurately fits a single exponential but not a double as the spin-lattice relaxation above suggests that T2’s are more uniform. This is because the T2’s are probably set by residual couplings to protons, this is supported by the fact that the C6 T2 times are much shorter that all the others, and C6 has two attached protons while each of the other carbon sites only has one. 13C spin-lattice relaxation in the rotating frame (T1ρ ) values were determined for room temperature air and freeze-dried NCC using the same samples as above. The T1ρ measure- ments fit a single exponential similar to that of T2. The tabulated results are in Table 6.5. Surprisingly, the C6 T1ρ times are amongst the longest and, as the T1ρ ’s are more similar to each other than the T1 measurements, this suggests more uniformity in the low frequency motions. 55 Air-dried (21°C) Freeze-dried (21°C) C1 137±13 112±10 C2,3,5 79±8 56±11 C4 101±7 157±18 C4′ 35±20 69±7 C6 133±10 118±7 Table 6.5: T1ρ measurements on air and freeze-dried NCC at room temperature measured in milliseconds. In general, relaxation measurements show that the broad C4′ peak relaxes faster then that of the sharp C4 peak. This is generally consistent with previous assignments [69] of the shoulders as arising from the amorphous and surface regions. The freeze-dried C4/C4′ two- component fits to the T1 measurements appear to show the opposite behaviour, but even in this case, the “average T1” of the C4′ is shorter than C4 due to the difference in populations corresponding to the slow and fast components. This is much more easily seen in the stretched exponential fits. The principle goal of the remaining experiments was to further analyze the phase structure of the NCC crystallites in hopes of concluding the locations and relative ratios of crystalline, paracrystalline and amorphous regions. One can look preferentially at the slow component by collecting spectra after a d1 delay (refer to T1 pulse sequence in Fig. 5.1) with a duration greater than the short T1 times and less than the long T1 times. This is useful to see if it produces a higher resolution spectrum as might be expected if the slowest parts were the most highly crystalline. Fig. 6.4 shows the spectrum collected after a delay d1 of 150 seconds. The delayed spectrum looks very similar except that C4′ has largely disappeared. The relative intensities are different because they are weighted according to their T1’s. The fact that the resolution does not get much better might again tend to support a broad distribution of relaxation times and structural environments. 56 60708090100110120 Chemical Shift (ppm.) I n t e n s i t y  ( a r b .  u n i t s ) Reference Slow Component Figure 6.4: 13C NCC spectrum after a delay of 150 seconds where only the slow component remains. T1 relaxation measurements were also performed on unsoaked and 3 month D2O liquid soaked NCC 35 mg samples in order to see if removal of protons from water accessible regions had any significant impact on the relaxation. The experiments were conducted on freeze-dried samples at room temperature. The samples used for this experiment were different than that used above (hydrolysis and treatment conditions were changed). The resulting two component and stretched exponential relaxation values are shown in Tables 6.6 and 6.7. In general, there is no significant change in relaxation times induced by soaking – though the C6 peak shows a slight change in the “slow” component times as illustrated in Fig. 6.5. C4 and C4′ are very similar and the plot is virtually indistinguishable due to the S/N. 57 Slow Component Fast Component D2O Freeze-dried Freeze-dried D2O Freeze-dried Freeze-dried C1 171±17 (67%) 180±36 (71%) 33±30 (33%) 19±11 (29%) C2,3,5 185±16 (61%) 179±20 (60%) 22±4 (39%) 25±5 (40%) C4 150±16 (72%) 161±22 (80%) 2±1 (28%) 3±4 (20%) C4′ 136±43 (50%) 84±26 (68%) 11±6 (50%) 16±12 (32%) C6 133±33 (52%) 98±16 (60%) 8±3 (48%) 6±3 (40%) Table 6.6: T1 values of deuterated and normal freeze-dried NCC samples at room tempera- ture measured in seconds from two component fits. D2O Freeze-dried (β ) Freeze-dried (β ) C1 107±5 (0.8) 108±10 (0.7) C2,3,5 86±3 (0.6) 85±4 (0.6) C4 131±15 (0.8) 140±16 (0.8) C4′ 32±18 (0.4) 44±9 (0.8) C6 32±14 (0.4) 37±10 (0.5) Table 6.7: T1 values of deuterated and normal freeze-dried NCC samples at room tempera- ture measured in seconds from stretched exponential fits. 0 50 100 150 Time (ms) I n t e n s i t y  ( a r b .  u n i t s ) Non-deuterated Deuterated Figure 6.5: Semi-logarithm plot of the T1 decay of C6 for D2O soaked and unsoaked sam- ples. 58 6.3 2H REDOR The goal of the next set of experiments was to examine the phase structure and characterize the different regions of NCC via the accessibility of deuterium-proton exchange. The NCC samples were prepared by soaking in liquid D2O for various times before drying, and 2H- 13C REDOR experiments were then performed. Fig. 6.6 illustrates the reference, dephased and difference carbon spectra obtained for a 2 day soaked NCC sample. The integrals of 406080100120140 Chemical Shift (ppm.) I n t e n s i t y  ( a r b .  u n i t s ) Not-dephased Dephased Difference Figure 6.6: Example of REDOR spectra for a 2 day liquid soaked sample after a dipolar evolution time of 3.2 milliseconds. each peak were determined in the same manner as those for the relaxation measurements except an additional baseline correction was performed prior to integrating. The baseline was determined graphically by estimation and, accordingly, the error-bars were taken to include extreme baseline estimates. As stated above, the REDOR dephasing curve will be scaled depending on the fraction of the 13C population with 2H attached. Hence, we examine the fraction attached as a function of soaking time to probe the phase structure of the NCC particles, expecting that 2H will exchange rapidly in the amorphous regions, very slowly in the crystalline regions, and potentially at an intermediate rate in the paracrys- talline. The amount of relative signal arising from 2H and neighbouring 13C nuclei depends 59 on the number of exchanged protons. Only those 3 protons per cellulose monomer that are part of the OH groups are expected to exchange – those that are attached to C2, C3 and C6. The carbon signal from these three peaks is expected to be modulated by dipolar couplings to the nearby deuterium, though the signal from the other three peaks (namely C1, C4 and C5) will also be modulated as the distances are relatively short. Fig. 6.7 shows an example of a dephasing curve, along with a curve showing the best fit to the ideal universal REDOR dephasing curve from Eq. 5.10. This fit makes the (unrealistic) assumption of isolated 2H-13C spin pairs, and requires two parameters, a distance and fraction of 13C with 2H neighbours. Fit parameters for each of the resolved carbon sites are shown in Table 6.8. Figure 6.7: Example of a REDOR dephasing curve fit. The dipolar coupling constant derived from the fits of the dephasing curve for C2,3,5 corresponds to a length of 2.5±0.2 Å which is in agreement with the literature for interaction distances of native cellulose [86, 87] of 2.4±0.1 Å [34, 25]. 60 % Dephased Distance (Å) C1 48±3 2.42±0.05 C2,3,5 55±4 2.38±0.01 C4 58±7 2.56±0.08 C4′ 75±10 2.64±0.05 C6 62±12 2.43±0.15 Table 6.8: Table of average percentage of signal that has dephased and average interaction distance as determined by fitting to the dephasing curve of Eq. 5.10 and using Eq. 5.8 to find distances. C6 in this case refers to both the sharp and broad shoulder. The numbers are averaged over all measured soaking times. The experimental data were virtually identical for all soaking times so the values in Ta- ble 6.8 are averages for all soaking times. Because C4′ has been previously assigned to surface and amorphous regions, one might expect that the C4′ peak would correspond to regions in the NCC that exchange fully, while the C4 peak would be expected to show only minimal exchange. While we do see a greater fraction of the C4′ exchanged than C4, the difference between them, 75 vs. 58%, is not nearly as great as would be expected based on straightforward assignments of surface/amorphous and crystalline. The fraction of the C2,3,5 population that has dephased as a function of soaking time is shown in Figure 6.8. With less than one minute soaking time, 55± 4% of the signal has dephased and does not increase for soaking times up to at least 30 days. This trend was mirrored with the other carbon peaks. This suggests that one region exchanges extremely quickly while another is very resistant. Presumably the proportion of population would continue to rise until all the expected protons had exchanged, but the timescale for complete exchange appears to be on the order of years. One cannot soak and subsequently dry the sample any faster than a few minutes so in order to probe the region of fast exchange (≤minute soaking time), vapour soaked samples were used instead (as described in the experimental section). As the experimental time for collecting a single REDOR dephasing curve was on the order of several days, a simple alternative 2H experiment was performed to measure the time constant for the fast exchange as described in Sec. 6.3.2. 61 Figure 6.8: Proportion of population that has exchanged 1H for 2H as a function of soaking time according to the C2,3,5 peak. 6.3.1 Simulation Results The fits to the REDOR dephasing curves yielding the parameters shown in Table 6.8 are based upon the unrealistic model of isolated 13C-2H spin pairs. In our NCC samples how- ever, it is likely that a given 13C will see multiple 2H neighbours nearby. To model the system with more precision, the data were then simulated with several models of increasing sophistication. The first two different models were both based upon SIMPSON simulations of REDOR curves taking into account two neighbouring deuterium sites. Restricting to the two closest sites should be satisfactory as they dominate the net dipolar coupling as the next nearest possible site was, on average, over 2 Å farther. In the first case, it was assumed that the two sites had an equal probability, p, of being occupied, such that p2 is the probability that both sites are occupied, p(1− p) is the probability that each of the sites are occupied individually. SIMPSON simulations of REDOR curves for all possible combinations of site occupancies were performed using the bond lengths illustrated in Fig. 6.9. The distances and angles were based on structures determined from neutron diffraction and occupancies of 1H/2H were allowed to vary as fit parameters. Figure 6.10 shows a few simulated curves. The data were fit with the following combined curve: 62 (a) Iα geometry. (b) Iβ geometry. Figure 6.9: Diagram of some of the C-D bond lengths used for fitting. For full geometrical information, see Nishiyama et. al. [25, 34] or appendices. 63 0 5 10 Time (ms) 0 0.2 0.4 0.6 0.8 1 ∆S /S 0 1 2 1 and 2 Figure 6.10: Examples of simulated REDOR dephasing curves for C1 in cellulose Iβ with 1 (closest deuteron), 2 (second closest deuteron) and both possible sites occupied. ∆S S0 = p(1− p)S1+ p(1− p)S2+ p2S12, (6.2) where Si are the simulated dephasing curves with sites 1 and/or 2 occupied as denoted by the subscripts. The second model divided the carbons into two groups. The first group assumed p = 1, the second group assumed p = 0, resulting in a REDOR dephasing curve described by: ∆S S0 = f ·S12, (6.3) where f represents the fraction of the population having both sites occupied. These models have the advantage of only having a single free parameter from which to fit the data. We refer to the first model as the “Probability” model, and the second as the “Fraction” model. The Probability model assumes that there exists a single deuterium occupancy that is uni- form throughout the sample, while the Fraction model assumes two regions, one in which protons have completely exchanged with deuterium, and another where no exchange has occurred. Prior to fitting, some data manipulation was necessary. First, the REDOR curves generated for C2, C3 and C5 were averaged to produce one simulated C2,3,5 curve. Secondly, cellu- lose Iα alternates ring orientations along a chain and cellulose Iβ consists of two parallel cellulose chains having slightly different orientations and organized in sheets packed in a 64 “parallel-up” fashion [25]. To deal with this, in each case the dephasing curves from the two orientations were averaged after simulation but prior to fitting. Finally, Nishiyama et. al. have shown that the location of the D3 (the deuterium attached to the oxygen attached to C3) is well defined whereas D2 and D6 appear in neutron diffraction results to each partially occupy two different sites, denoted A and B [25, 34]. Therefore, fits were performed for both A and B with the results summarized in Table 6.9 and illustrated in Figs. 6.11, 6.12, 6.13 and 6.14. χ2/d values were determined by the standard formula [88]: χ2 = N ∑ 1 ( yi− f (xi) σi )2, (6.4) and dividing by the degrees of freedom d. Here yi is the modeled value, f (xi) is the exper- imental measurement and σi is the uncertainty on the experimental measurement. Iα Probabilityp (χ2/d) Fraction f (χ2/d) A B A B C1 0.28 (1.6) 0.35 (0.6) 0.49 (4.2) 0.53 (1.9) C2,3,5 0.38 (7.5) 0.29 (5.7) 0.44 (9.6) 0.38 (10.7) C4 0.38 (3.0) 0.29 (3.2) 0.47 (5.1) 0.42 (8.3) C4′ 0.51 (2.0) 0.39 (2.6) 0.61 (3.2) 0.56 (4.5) C6 0.41 (5.2) 0.43 (4.4) 0.49 (7.2) 0.51 (6.2) Iβ Probabilityp (χ2/d) Fraction f (χ2/d) A B A B C1 0.31 (0.9) 0.40 (0.7) 0.45 (8.6) 0.55 (2.7) C2,3,5 0.42 (5.6) 0.37 (3.2) 0.49 (6.8) 0.45 (6.0) C4 0.45 (1.9) 0.39 (1.5) 0.56 (2.8) 0.52 (4.2) C4′ 0.60 (2.1) 0.52 (2.4) 0.69 (2.9) 0.67 (3.2) C6 0.45 (3.5) 0.39 (2.8) 0.54 (4.9) 0.54 (6.0) Table 6.9: Best fit values based on cellulose Iα geometry for both Probability and Fraction models for 2 week D2O soaked NCC. In some cases (e.g. C1) the fits are quite good, but in others (e.g. C2,3,5) neither of these models adequately describes the data. For all the carbon peaks, the REDOR dephasing curve is better described by the Probability model than the Fraction model suggesting that most of the sample is partially exchanged rather than containing fully exchanged and fully unexchanged regions. The Fraction model says 38-69% occupied, with the highest occu- pancy in the C4′ . The Probability model shows similar results ranging between 28% and 65  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C1 Probability Fraction Superposition Superposition Prime  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C2,3,5 Probability Fraction Superposition Superposition Prime  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C4 Probability Fraction Superposition  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C4’ Probability Fraction Superposition Prime  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C6 Probability Fraction Superposition Superposition Prime Figure 6.11: Probability, Fraction and both Superposition model fits for cellulose Iα in A conformation for 2 week D2O soaked NCC. 66  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C1 Probability Fraction Superposition Superposition Prime  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C2,3,5 Probability Fraction Superposition Superposition Prime  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C4 Probability Fraction Superposition  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C4’ Probability Fraction Superposition Prime  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C6 Probability Fraction Superposition Superposition Prime Figure 6.12: Probability, Fraction and both Superposition model fits for cellulose Iα in B conformation for 2 week D2O soaked NCC. 67  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C1 Probability Fraction Superposition Superposition Prime  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C2,3,5 Probability Fraction Superposition Superposition Prime  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C4 Probability Fraction Superposition  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C4’ Probability Fraction Superposition Prime  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C6 Probability Fraction Superposition Superposition Prime Figure 6.13: Probability, Fraction and both Superposition model fits for cellulose Iβ in A conformation for 2 week D2O soaked NCC. 68  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C1 Probability Fraction Superposition Superposition Prime  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C2,3,5 Probability Fraction Superposition Superposition Prime  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C4 Probability Fraction Superposition  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C4’ Probability Fraction Superposition Prime  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  2  4  6  8  10 ∆S /S 0 Time (ms) C6 Probability Fraction Superposition Superposition Prime Figure 6.14: Probability, Fraction and both Superposition model fits for cellulose Iβ in B conformation for 2 week D2O soaked NCC. 69 60% occupancy. The C4′ peak consistently shows 10-15% greater 2H exchange than the C4 peak, but none of these fits shows exchange greater than 70%. In general, the Iβ model fits better than the Iα model though there are a few exceptions. As cellulose is a complex hierarchically structured fiber, neither model is a complete description of the distribution of deuterium – which is likely very complicated. In order to address this, another model referred to as the “Superposition” model was con- structed. We divide the sample into two regions, in the model a fraction (1-fe) is completely unexchanged, while the remaining fraction fe is partially exchanged. In the partially ex- changed regions, f2, f3 and f6 represent the occupancy of the three exchangeable sites. The model function was: ∆Sk Sk0 = fe(fi(1− f j)Sk1+ f j(1− fi)Sk2+ fif jSk12), (6.5) in which fi and f j take on the values of f2, f3 and f6 depending upon which two exchangeable sites are nearest to Ck. Modeled curves for C2, C3 and C5 were again averaged prior to fitting. A single fit was then performed to a combined data set including dephasing for C1, C2,3,5, C4 and C6 with a single set of parameters (fe, f2, f3 and f6). This process was repeated with Ski j’s generated for each of the Iα (A), Iα (B), Iβ (A) and Iβ (B) structures. Because C4′ is separately resolved, the entire process was repeated again substituting the C4′ data in place of C4. A more realistic model might combine the C4′ data into a single multi-compartment incorporating f2, f3, f6, f2′ , f3′ and f6′ in a single fit. This was not deemed feasible here due to the limited signal to noise of the data. Each of the fits reported here includes either the C4 or the C4′ data (along with dephasing data for each of the other peaks) but never both simultaneously. Fit parameters were constrained to lie between 0 and 1. These fits, along with those from the Fraction and Probability models described earlier are shown in Figs. 6.11, 6.12, 6.13 and 6.14 and the best fit parameters are collected in Table 6.10. 70 Iα A B A’ B’ fe 1.0 1.0 1.0 1.0 f2 0.19 0.25 0.0 0.15 f3 0.44 0.08 0.59 0.27 f6 0.50 0.54 0.54 0.49 ft 0.38 0.29 0.38 0.31 χ2/d 4.6 3.3 4.5 3.0 Iβ A B A’ B’ fe 1.0 0.81 1.0 0.88 f2 0.15 0.34 0.0 0.10 f3 0.47 0.16 0.61 0.37 f6 1.0 1.0 1.0 1.0 ft 0.54 0.41 0.54 0.43 χ2/d 7.4 2.6 8.4 2.5 Table 6.10: Best fit values for Superposition model and total fraction ft of total deuterium exchanged for 2 week D2O soaked NCC. Also included in the tables is the “total deuterium exchanged” ft – this is the overall fraction of deuterium that has exchanged in the sample based on the model fits and is calculated by taking an average of f2, f3 and f6 and scaling the result by fe. ft will be compared with the 2H-1H exchange results in the following section. The best fits arise from the Iβ (B) model, yielding χ2/d of ∼ 2.5. These fits appear to capture most of the features of the data with a remarkably small number of parameters. Again all the fits suggest that most (or all) of the sample is partially exchanged. The best fits (with the Iβ (B) structure) suggest that C6 is fully exchanged within the exchanged region while C2 and C3 have much lower deuterium occupancy. Based on the results of the Superposition model, the results from the simpler Probability and Fraction models are misleading as every model predicts an uneven distribution of exchangeable protons. Comparing the 3 models across all the geometry conformations and carbon peaks a few trends emerge. The fact that we find the best fit with an Iβ structure agrees with results in the literature suggesting that cellulose derived from treated wood is comprised primarily of cellulose Iβ [84, 27]. We expect to have the prime models report more 2H since the REDOR signal for the C4 carbon used is from the C4′ peak, which shows greater dephasing, and is expected to arise from the surface/amorphous regions – though this is only the case for the B models. This is not terribly surprising as the data from all of the other carbon sites was identical, highlighting a limitation of the model. For the purpose of comparing the traditionally defined crystalline versus surface/amorphous regions, the probability and fractions models discussed earlier provide greater insight. The remaining discrepancies between the superposition model best fits and the data are likely due to the fact that the samples are almost certainly composed of a mixture of well defined structures (probably including both Iβ (A) and Iβ (B)) along with less well structured regions. 71 One limitation of the SIMPSON simulations presented here is the inclusion of only the two closest deuterium sites for each carbon peak. We settled with the two site option to simplify analysis and for most carbons only two exchangeable sites were within 4 Å. The impact to the fits of this approximation is expected to be relatively minor as curves including farther couplings were very similar to those used, over the range of dephasing times measured. One consequence of this restriction however is that the deuterium contents found are likely slight overestimates because of the experiment, carbons with no deuterium in either of the two nearest possible sites will still exhibit some minor dephasing from more distant sites, while in the simulation, such sites do not dephase at all. 6.3.2 2H-1H Exchange Because the liquid D2O soaked NCC samples showed no dependence on soaking time, and our initial goal in the exchange experiments was to quantify exchange dynamics, exchange experiments in D2O vapour were carried out. Due to the long acquisition times associated with a full REDOR experiment, quantification of the 2H content was determined using the magnitude of the first 10 rotor echoes of the 2H MAS FID (normalized for number of scans). From these echoes an exponential fit was used to determine the amount of signal at t = 0. An example of an FID from the resulting experiment is shown in Fig. 6.15. Figure 6.15: 2H FID with signal of first rotor echo highlighted red. 72 To quantify the absolute number of 2H nuclei that have exchanged into the sample as a function of time, a static 2H wide-line spectrum was also collected from a 2H labeled PMMA sample (the sample was not spun because its solid-chunks are not rotor friendly). Only the broad (static, non-methyl) part of the PMMA wide-line spectrum was used for the calibration and was compared with a static 2H wide-line spectrum of a 3 hour liquid soaked static NCC sample. Due to ring-down artifacts, the signal at zero echo time was determined by collecting spectra for a range of echo times (between 30 and 40 µs), and extrapolated back to zero echo time using an exponential fit of the integrals. The resulting build-up curve calibrated with respect to PMMA is shown in Fig 6.16. To compare the REDOR measurements of the liquid soaked samples, the experiment was additionally performed on a 3 month liquid soaked sample. 0 20 40 60 80 100 120 140 Exchange Time (hours) 0 5 10 15 20 25 30 35 40 P e r c e n t a g e  o f  2 H  E x c h a n g e d Figure 6.16: Build up curve for vapour soaked samples. The dashed line shows the signal for a liquid soaked sample. The deuterium spectra reveal that, with a time constant of 30±2 hours, the vapour soaked samples rise to 37.8±3.0% of expected exchangeable protons. The 3 month liquid soaked sample showed 37.2± 3.0% exchanged therefore we conclude that the same regions are exchanging in both vapour and liquid soaked. This result is consistent with the simulated ft results of Table 6.10 where the best fits to the Iβ (B) model found 41-43% exchanged. 73 With the caveat above that the REDOR results likely slightly overstate the total 2H content, this is remarkably good agreement. From these experiments we see only one exchange time scale. These data do not appear to be consistent with a model for NCC consisting of well-defined crystalline, para-crystalline, and amorphous regions, unless exchange into the para-crystalline is either too fast or too slow to be distinguished from one of the other regions. The relaxation measurements find either sharply defined crystal/amorphous regions (from double exponential) with fractions in each varying with temperature and drying conditions or a broad distribution of structural regions (from stretched exponential). As the quality of the fits predicting a broad distribution of regions is higher than the two-component fit and uses fewer fitting parameters and the REDOR experiments on the proton-deuterium exchanged samples find that almost all the sample is partially exchanged, we conclude that NCC consists not of well defined highly crystalline and poorly structured amorphous regions, but rather as a mixture of regions having a broad variety of structural order and crystalline perfection. This conclusion is consistent with the broad 13C CP/MAS NMR spectra seen here and may be consistent with the very broad X-ray diffraction peaks il- lustrated by Wadood and Hubbe [70]. This is not what was predicted given that acid hydrolysis treatment is expected to remove the amorphous regions and greatly increase crystallinity and uniformity. Based on the relative abundance of crystalline and amorphous peaks in the diffraction pattern, the X-ray diffraction measurements report a much higher crystallinity index (80-90%) compared with our results. The fact that the C4′ peak shows an unexchanged component might be explained by the presence of some amorphous re- gions completely enclosed within impenetrable crystalline regions, though this explanation seems unlikely. 6.4 31P REDOR To acquire a 13C NMR spectrum from only the surface layer of the NCC crystallites, sam- ples produced with phosphoric acid hydrolysis rather than the more usual sulfuric acid hydrolysis were prepared. Hydrolysis with sulfuric or phosphoric acid can result in the introduction of sulfate or phosphate esters at the surface of the cellulose crystallites. Sulfur is invisible to NMR (as the abundant isotope 32S, 95%, possesses zero magnetic moment) 74 but by preparing NCC with phosphoric acid hydrolysis instead, the sulfate groups are re- placed with phosphate, and the surface of the NCC crystallites is now observable through 31P NMR. As 31P is 100% abundant and NMR active, producing a carbon spectrum of the surface could be done with a simple 31P-13C CP experiment. While the signal to noise of this experiment is relatively low, the spectrum appears to support the hypothesis that the phosphate groups are bound to C2 and/or C3. The difference spectrum from a 31P-13C REDOR experiment is shown in Fig. 6.17. This represents a very small subset of the entire structure and with the 1% natural abundance of 13C, the signal to noise in this experiment is very low. This is after a week of experimental time and only the C2,3,5 peak is truly visible above the noise. Figure 6.17: REDOR difference spectrum multiplied by x50. Due to low S/N and long experiment times (on the order of weeks), only one REDOR experiment was conducted; that of 3.2 ms dephasing time. To fit the 13C-31P REDOR curve with only one data point, the bond length and geometry were modeled and the molecular distance set to 2.60 Å. This is based on bond length distances of 1.59 Å for P-O and 1.44 Å for O-C and an angle of 118° [86] as illustrated in Fig. 6.18. Thus the only fitted parameter was the fraction of 13C population dephased by nearby 31P nuclei. We used the C2,3,5 peak as it had the most signal (we cannot use C6 as the S/N is too low) and found the fraction dephased to be 0.85± 0.12%. As only C2, C3 and C6 are candidates to have 75 31P attached, we multiply this result by 1.5 to find 1.3± 0.2% of C2 and C3 dephased or 2.6±0.2 phosphate groups attached to C2 or C3 per 100 monomers. This result is consistent with previous studies of sulfuric acid hydrolyzed NCC using elemental analysis and/or conductivity titration which find between 1.4 and 6.7 sulfate groups per 100 glucose units [89, 70]. Other studies have been attempted with Fourier transform infrared spectroscopy (FTIR) though no peaks have been assigned to the sulfate groups when in low relative concentration [70]. P OH HO O O C 1.79Å 1.44Å 2.60Å 118° Figure 6.18: Phosphate group geometry at the end of a cellulose chain. The dashed lines in the top right represent the links to the rest of the glucose molecule. Based on distances from [90, 34, 25]. Figure 6.19: With the dipolar coupling fixed, only one data point was used for the fit. 76 Chapter 7 Concluding Remarks The results of this work provide substantial new insight into the structure of NCC. De- spite the prediction of 80− 90% crystalline structure from X-ray diffraction, CP/MAS NMR spectra show that, based on the amplitude of the C4′ peak, NCC retains a substantial (>∼ 40%) amorphous component. The T1 relaxation experiments find that both the crys- talline and amorphous regions (defined by the resolved C4 and C4′ peaks) consist of broad, overlapping distributions of dynamical regions. The breadths of the peaks in the NMR spectra further suggest a broad distribution of structures. The 2H-13C REDOR experiments indicate the NCC samples consist not of sharply defined exchanged and unexchanged re- gions, but rather are more uniformly partially exchanged. Within the exchanged regions however, the three exchangeable sites appear to be unevenly exchanged, with the OH group of C6 being much more more likely to exchange than those of C2 or C3. While the results do seem to be broadly consistent with the notion that the C4′ peak arises from more dynamically active, loosely structured and surface accessible regions, in total our results suggest that the separation of C4 into crystalline and C4′ into amorphous is not as clear cut as has been suggested. These results rather suggest a more finely varied distri- bution of environments. The breadths of the NMR spectra, the distributions of relaxation times, and the results of the deuterium-proton exchange experiments all appear to sug- gest that these samples contain very little well-structured rigorously crystalline material, but rather contain regions of varying degrees of crystalline order. There is no sign in the exchange dynamics of a paracrystalline region characterized by an intermediate exchange rate. 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Chem., 87:5121–5129, 1983. 86 Appendix A Example of SIMPSON Simulation of REDOR Dephasing Curve As an example of the REDOR simulations we performed, the SIMPSON program for sim- ulating REDOR dephasing curve for a single 13C-2H pair is shown here. spinsys { channels 13C 2H nuclei 13C 2H dipole 1 2 -319.246 0 0 0 quadrupole 2 1 199993 0.192 0 0 0 shift 1 65.6p -29.1p 0.415 0 0 0 } par { proton_frequency 400e6 spin_rate 5000 sw spin_rate/2.0 np 32 crystal_file rep100 gamma_angles 18 start_operator I1x detect_operator I1p verbose 1101 variable rf1 52083 variable rf2 83333 } proc pulseq {} { 87 global par maxdt 1.0 set t90 [expr 0.25e6/$par(rf2)] set tr4 [expr 0.125e6/$par(rf2)] set t180 [expr 0.5e6/$par(rf1)] set tr5 [expr 0.25e6/$par(rf1)] set tr2 [expr 0.5e6/$par(spin_rate)-$t180] set tr3 [expr 0.5e6/$par(spin_rate)-$tr4-$tr5] set tr6 [expr $tr3-$tr2] reset $tr5 delay $tr2 pulse $t180 $par(rf1) x 0 x delay $tr2 pulse $t180 $par(rf1) y 0 x store 1 reset $tr5 delay $tr2 pulse $t180 $par(rf1) x 0 x delay $tr3 pulse $t90 0 x $par(rf2) x delay $tr6 store 2 reset $tr5 delay $tr2 pulse $t180 $par(rf1) x 0 x delay $tr2 store 3 reset acq 88 for {set j 0} {$j < $par(np)-1} {incr j} { reset delay $tr5 prop 1 $j prop 2 prop 1 $j prop 3 acq } } proc main {} { global par set f [fsimpson] fsave $f $par(name).fid } 89 Appendix B SIMPSON Input Distances Included here are the distances used for the nearest two deuterium sites of the “up” con- formation for the SIMPSON REDOR simulations. Complete geometry can be found in the supporting materials of [25, 34]. Iα A B 1 (Å) 2 (Å) 1 (Å) 2 (Å) C1 2.51 (D2) 2.71 (D3) 2.71 (D3) 2.75 (D6) C2 1.87 (D2) 3.11 (D3) 1.88 (D2) 2.73 (D6) C3 1.86 (D3) 2.97 (D6) 1.86 (D3) 2.16 (D2) C4 2.26 (D3) 2.90 (D2) 2.26 (D3) 2.49 (D6) C5 2.91 (D6) 3.19 (D3) 2.31 (D6) 3.19 (D3) C6 1.85 (D6) 2.55 (D5) 1.88 (D6) 2.72 (D6) Iβ A B 1 (Å) 2 (Å) 1 (Å) 2 (Å) C1 2.44 (D2) 2.52 (D3) 2.52 (D2) 2.87 (D6) C2 1.94 (D2) 3.06 (D6) 1.95 (D2) 2.73 (D6) C3 1.94 (D3) 2.77 (D6) 1.94 (D3) 2.66 (D2) C4 2.53 (D3) 3.26 (D2) 2.53 (D3) 2.88 (D6) C5 3.25 (D6) 3.44 (D2) 2.66 (D6) 3.26 (D6) C6 1.94 (D6) 2.80 (D2) 1.96 (D6) 2.34 (D2) Table B.1: 13C-2H Distances

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