Internal Structure of Isolated Cellulose I Fibril Aggregates in the Water

Chapter 5

Downloaded by CITY UNIV OF HONG KONG on October 25, 2017 | http://pubs.acs.org Publication Date (Web): October 23, 2017 | doi: 10.1021/bk-2017-1251.ch005

Internal Structure of Isolated Cellulose I Fibril Aggregates in the Water Swollen State Per Tomas Larsson,*,1,3 Rose-Marie Pernilla Karlsson,3 Per-Olof Westlund,4 and Lars Wågberg2,3 1Innventia

AB, P.O. Box 5604, SE-114 86 Stockholm, Sweden and Polymer Technology, Royal Institute of Technology, KTH Teknikringen 56, SE-100 44 Stockholm, Sweden 3The Wallenberg Wood Science Centre, KTH, Teknikringen 56, SE-100 44 Stockholm, Sweden 4Department of Chemistry, Biological Chemistry, Theoretical and Computational Chemistry, Umeå University, SE 901 87 Umeå, Sweden *E-mail: [email protected].

2Fibre

By combining 2H-NMRD and CP/MAS 13C-NMR measurements of water-based cellulose gels and of water swollen pulps it was possible to estimate the nature of the interior structure of cellulose fibril aggregates. A set of samples with high cellulose purity and low charge was used. The interpretation of data was based on a relaxation model describing the exchange dynamics for deuterium exchange between water molecules and cellulose hydroxyl groups. The theoretical model used made it possible to calculate cellulose surface-to-volume ratios (q-values) from both 2H-NMRD and CP/MAS 13C-NMR data. Good consistency between 2H-NMRD and CP/MAS 13C-NMR data was found. In all investigated samples the cellulose fibril aggregates showed a different degree of “openness” interpreted as the presence of interstitial water inside fibril aggregates. One result also showed that an increased degree of fibril aggregate openness results from the TEMPO-oxidation. Common to all samples was that in the water swollen state water molecules could access part of the fibril aggregate interior.

© 2017 American Chemical Society Agarwal et al.; Nanocelluloses: Their Preparation, Properties, and Applications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.


Background Isolation of cellulose I from plant materials such as wood can be achieved by large scale industrial processes such as chemical pulping. During chemical pulping, anatomical plant fibres are separated from their original composite environment and the composition of the fibres becomes enriched with respect to polysaccharides, especially cellulose. Depending on the details of the cooking process and subsequent bleaching processes it is possible to isolate cellulose rich fibres, i.e. separated fibres with the major anatomical features preserved but composed of mainly cellulose I. In the never-dried form, the fibre wall of a cellulose rich fibre can be viewed as a porous, water filled (swollen), network of cellulose I. The fibre wall cellulose network is composed of building blocks with high aspect ratios and with lateral dimension in the range of a few nanometres to several tenths of nanometres, separated by randomly shaped pores with typical dimensions in the tenths to hundreds nanometre range (1, 2). The building blocks of the cellulose rich fibre walls are a part of the supramolecular structure of cellulose I and are referred to as cellulose fibril aggregates in the present work. Internally the cellulose fibril aggregates are composed of largely co-axial cellulose fibrils. For cellulose isolated from wood the average lateral fibril dimensions are in the range of 3-5 nanometres and the average lateral fibril aggregate dimension is dependent on the isolation method used and the history of the sample but is typically found in the range from individual fibrils up to some 30 nanometres (3–5). Fibrils and fibril aggregates are the terms used by us to describe the supramolecular structure of isolated cellulose I. In this work, cellulose isolated from wood and cotton were studied. For fibrils in isolated cellulose we assume that the degree of crystallinity is limited by the lateral dimensions of the fibrils and that only polymers located at the fibril phase boundaries may come in direct contact with surrounding water molecules. The presence of separate surface signals in CP/MAS 13C-NMR spectra for cellulose I indicate that the polymers at the fibril phase boundary are not in a conformation identical to polymers in the crystalline fibril core, that’s why surface polymers are considered non-crystalline, limiting the maximum degree of fibril crystallinity (5). For example, in a 4 nanometres wide fibril some 50 % of the polymer content constitute surface, limiting the maximum degree of crystallinity to about 50 %. Cellulose isolated from wood and cotton have significantly different lateral fibril dimensions, manifested in their different degrees of crystallinity. Isolation of individual fibrils can be achieved by chemically modifying cellulose rich fibres, typically by adding charged groups, followed by some form of dispersing mechanical treatment (6–8). The yield of chemically modified individualized fibrils can be as high as 97 % (7) which suggests that, in a liquid swollen state, despite the high degree of aggregation of cellulose fibrils in the cellulose rich fibre wall, fibril surfaces inside the fibril aggregate can be targeted by the reagent molecules. This indicates that there may be two kinds of “porosities” of different origin in a liquid swollen cellulose rich fibre wall. The traditional concept of “pores” is defined to describe the liquid filled cavities between neighbouring cellulose fibril 92 Agarwal et al.; Nanocelluloses: Their Preparation, Properties, and Applications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.


aggregates, and the concept “channels” is used to describe liquid filled cavities present within a fibril aggregate, as defined in Figure 1. For samples consisting of water swollen cellulose rich fibres the average lateral fibril dimensions (LFD) and the average lateral fibril aggregate dimensions (LFAD) can be measured with good precisions using cross-polarization magic angle spinning solid state carbon-13 nuclear magnetic resonance spectroscopy (CP/MAS 13C-NMR) (9, 10). Due to presence of separate signals originating from fibril surfaces in the spectral C4-region the LFD and LFAD are calculated from a surface-to-volume ratio (q) of signal intensities estimated by spectral fitting of the C4-region. Both fibrils and fibril aggregates are assumed to have square cross-sections in the analysis of fitted spectral intensities and, although generally not the case in CP/MAS 13C-NMR, for cellulose I signal intensities are quantitative at sufficiently long contact times (10).

Figure 1. An idealized illustration showing the cross-sections of a dense or compact (left) and a less dense or open (right) fibril aggregate in the water swollen state. Annotations show the locations of pores and channels (see text) and identifies the cross-section of a fibril. Fibrils and fibril aggregates are shown with square cross-sections. For typical wood based cellulose fibrils the side lengths are in the 3 nm to 5 nm range and fibril aggregate side lengths are in the order of 15 nm to 30 nm. The depth dimension, perpendicular to the plane of the illustration, is not indicated. Nuclear magnetic relaxation dispersion (NMRD) spectroscopy is a form of solution state NMR where specially designed instruments are used to study only the magnetic field dependence in the spin-lattice relaxation rate (R1), i.e. the inverse of the relaxation time (T1). This is achieved by using switchable electromagnets operating at relatively low magnetic flux density, typically below 1 Tesla. Since the spin-lattice relaxation rate is intrinsically coupled to molecular motions, measuring the magnetic field dependence in the relaxation rate is a powerful method for studying the rate and kind of molecular motions causing the observable spin-lattice relaxation. The problem of interpreting the water-NMRD profiles for confined water in proteins has been discussed by several authors (11–14) but we take a somewhat different approach in this study. The purpose of this study was to investigate the presence of channels in water swollen fibril aggregates. The strategy was to augment the cellulose 93 Agarwal et al.; Nanocelluloses: Their Preparation, Properties, and Applications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.


structural information obtained CP/MAS 13C-NMR with dynamic information about the water obtained by 2H-NMRD measurements, performed for a set of samples differing in water content and structure. To span the sample space, water swollen cellulose rich samples of different origin and history were used. Further, to increase the structural differences in the sample set, one sample consisting of cellulose beads (15) was included. The cellulose beads are composed of a water-based gel of β-(1,4)-D-glucan polymers obtained by dissolving cellulose molecularly followed by precipitation in a non-solvent.

Experimental Section Materials and Samples Table 1 summarizes the materials used in this study. The cotton linters was supplied by Tumba Bruk (currently Crane AB), Sweden, the Eucalyptus dissolving pulps (ND96 and D96) were acetate grade (96α) pulps supplied by Sappi, Saiccor, South Africa, the softwood dissolving pulp (NDDIS) was a 60% Norwegian spruce and 40% Scots pine pulp from Domsjö Fabriker AB Aditya Birla, Örnsköldsvik, Sweden. The cellulose beads (CB29) were made from the softwood dissolving pulp; see below for a detailed procedure.

Table 1. Abbreviations, initial form and relative glucose content (rGlu) for the materials. The form state was the state in which the pulps were delivered. Abbreviation

Description

Initial form

rGlu (%)

DLI

Cotton linters

Dried

> 99

ND96

Eucalyptus 96a dissolving pulp

Never dried

> 98

D96

Eucalyptus 96a dissolving pulp

Dried

> 98

NDDIS

Softwood dissolving pulp

Never dried

> 96

CB29

Cellulose beads

Never drieda

> 96a

a For cellulose beads the values refers to the starting material (NDDIS) that was dried before

dissolution and preparation of cellulose beads.

Samples with different water content were prepared for 2H-NMRD measurements. An excess of deionized water was added to the material, followed by stirring for at least 3 h and centrifuge filtration (5000 G for 30 min at room temperature) using 5-mL disposable centrifuge filters (Millipore Ultrafree-CL Centrifugal Device, Millipore, Billerica, MA, USA) equipped with 5-micron PVDF membranes. Some of the samples were dried for a short time in a heating cabinet at 105 °C to decrease the water content further. Immediately after preparation the samples were packed into 10 mm NMR glass tubes and sealed with a plastic cap. 94 Agarwal et al.; Nanocelluloses: Their Preparation, Properties, and Applications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.


Cellulose beads were prepared from dried softwood dissolving pulp (starting from the NDDIS material) basically following the procedure by Berthold (16) and modified and refined by Carrick (15). The charge of the softwood dissolving pulp was 29 micromoles/gram determined by conductometric titration (17). The dry softwood dissolving pulp was disintegrated in deionized water (disintegrator model 95568, PTI, Austria) for 30 000 revolutions. The water was displaced by ethanol (96% VWR, Sweden) by filtration, repeated three times per day during two days, followed by immersion N,N-dimethylacetamide (DMAc, puriss p.a., >99.5% Sigma Aldrich ). The ethanol was displaced by DMAc exchange, three times per day for two days. 2.5 mg DMAc saturated pulp was dissolved by adding it to 258.5mL of 7 % (w/w) solution of lithium chloride (puriss p.a., anhydrous >99% Sigma Aldrich) in DMAc at 40 °C and left for slowly stirring at 5°C overnight. The solution was filtered using 45µm acrodisc PTFE membrane filters (VWR, Sweden) to remove undissolved material. The formation of the cellulose beads was performed by dropwise precipitation of the LiCl/DMAc cellulose solution into 0.03 M hydrochloric acid (HCl(aq)) using needles with a diameter of 1.25 mm. The beads were allowed to rest in the formed DMAc phase at the bottom of the beaker, to maintain their spherical shape and almost molecularly smooth surface (15), at 5°C to minimize acid hydrolysis. After 24 h the acidic water was gradually replaced by de-ionized water by decantation, three times per day for a week. The so prepared cellulose beads had a diameter of about 2 mm and a solids content of 2.5 % (w/w). One sample of softwood dissolving pulp, NDDIS-T, was TEMPO oxidized to a charge of 690 micromoles/gram dry mass. The chemicals used for the TEMPO oxidation were: NaClO2 (puriss p.a.), NaClO (14% solution), 2,2,6,6-tetramethyl-1-piperidineyloxy (TEMPO, free radical), all purchased from Sigma-Aldrich Sweden AB, Stockholm, Sweden. Sodium bromide (99+ %) was purchased from Alfa Aesar. All chemicals were used as received. The softwood dissolving pulp (NDDIS) was oxidized by TEMPO mediated treatment at neutral pH (18). The never-dried pulp was dispersed in phosphate buffer (90 mL of buffer/g of dry fibre) at pH 6.8 and 60°C, NaClO2 (10 millimole, 80%), TEMPO (0.1 millimole) and NaClO (1.0 millimole). The amount of chemicals was calculated per mass of dry fibre and the chemicals were added to the fibre suspension under stirring. The reaction time was adjusted in order to achieve a degree of charge in the range 600 to 800 micromoles/g. The TEMPO oxidized pulp was thoroughly washed with deionized water using filtration and subsequently washed to hydrogen counter-ion form (19), and stored refrigerated. The charge of the oxidized fibres was determined using conductometric titration (17). Methods Solids Content The solids content of the samples were determined gravimetrically after drying at 105 °C for a minimum of 2 h. Reported values are an average of at least three determinations. The solids content (mass solids per sample mass) was 95 Agarwal et al.; Nanocelluloses: Their Preparation, Properties, and Applications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.


recalculated to the reported mass solids per mass liquid as given in Table 2 below. The dry content of the cellulose beads was determined by thermogravimetric analysis (TGA/DSC 1 STAR Mettler Toledo). Prior to measurement the cellulose beads were drained for a few seconds on a moist filter paper to remove external excess water surrounding the cellulose beads. The temperature ramp was 35 °C to 100 °C at 10 °C/min followed by another temperature ramp from 100 °C to 220 °C at 5 °C/min both in performed under nitrogen atmosphere. The second ramp corresponds to the temperature regime in which cellulose loses adsorbed water (20). The temperature was kept constant at 220°C for 5 min to establish the solids mass. Subsequently the gas was switched to oxygen, 2 min at 220 °C, followed by a ramp from 220°C up to 650°C at 10 °C /min to completely degrade the cellulose. The dry content of the cellulose beads was calculated using the solids mass established at 220°C in nitrogen atmosphere and the total mass of the cellulose bead measured immediately after sample insertion: dry content (% w/w)= 100 % x (solids mass)/(total mass).

CP/MAS 13C-NMR Cross-Polarization Magic Angle Spinning Carbon-13 Nuclear Magnetic Resonance Spectra. All samples had a water content of at least 40 % and were packed uniformly in a zirconium oxide rotor. The CP/MAS 13C-NMR spectra were recorded in a Bruker Avance III AQS 400 SB instrument operating at 9.4 T. All measurements were carried out at 295 (±1) K with a magic angle spinning (MAS) rate of 10 kHz. A 4-mm double air-bearing probe was used. Data acquisition was performed using a cross-polarization (CP) pulse sequence, i.e., a 2.95 microseconds proton 90-degree pulse and an 800 microseconds ramped (100–50 %) falling contact pulse, with a 2.5 s delay between repetitions. A SPINAL64 pulse sequence was used for 1H decoupling. The Hartmann-Hahn matching procedure was based on glycine. The chemical shift scale was calibrated to the TMS-scale (tetramethylsilane, (CH3)4Si) by assigning the data point of maximum intensity in the alpha-glycine carbonyl signal to a shift of 176.03 ppm. 4096 or 16384 transients were recorded on each sample depending on solids content, leading to acquisition times between 3 h to 12 h. The software for spectral fitting was developed at Innventia AB and is based on a Levenberg-Marquardt algorithm (9). All computations were based on integrated signal intensities obtained from spectral fitting (10). The errors given for parameters obtained from the fitting procedure are the standard error of the mean with respect to the quality of the fit.

2H-NMRD

Deuterium Nuclear Magnetic Relaxation Dispersion measurements. The field dependence of the deuterium longitudinal relaxation rate, R1 = 1/T1, was measured on a 1 Tesla Stelar FFC2000 fast-field-cycling instrument (Stelar s.r.l. Mede, Pavia (PV), Italy) with polarization at 25 MHz and detection at 16.29 MHz (35.176 96 Agarwal et al.; Nanocelluloses: Their Preparation, Properties, and Applications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.

MHz for deuterium). The relaxation rate was measured at different magnetic field strengths corresponding to proton Larmor frequencies ranging from 0.01 to 40 MHz. The switching time was 3 milliseconds, and the 90-degree pulse length was 8.0 microseconds. The polarization and recovery time was set to 4 T1 and the number of accumulated transients was 4 for all samples. The sample temperature was controlled using the temperature unit of the Stelar spectrometer and was maintained within ±0.1 degree centigrade. The accuracy of the absolute value of the temperature was about ±1 degree centigrade.


Determining β-Values by Fitting The β-value is the scaling factor used to make the 2H-NMR profiles recorded for a pulp sample to superimpose onto the 2H-NMR profile recorded for the cellulose beads, both recorded at the same temperature. The β-values were obtained by a single parameter fit preformed using the Solver tool available in Microsoft Excel®. The fitting procedure was performed by determining the β-values that minimized relaxation rate weighted sum of squared residuals when fitting the 2H-NMRD profiles recorded on pulps onto the 2H-NMRD-profile recorded for the cellulose beads sample. The obtained β-values were recalculated to the reported surface-to-volume ratios (q(NMRD)-values, see the Theory section). Reported errors in the obtained q(NMRD)-values are one standard error based on the quality of the fit.

Results CP/MAS 13C-NMR spectra recorded for materials specified in Table 1 are shown in Figure 2. Table 2 summarizes structural and compositional data on the samples prepared for 2H-NMRD measurements. Spectral fitting of the C4-region of the CP/MAS 13C-NMR spectra was used to determine the stoichiometric surface-to-volume ratios for fibrils, q(A), and for cellulose fibril aggregates, q(F). From the surface-to-volume estimates the average lateral fibril dimensions (LFD) and the average lateral fibril aggregate dimensions (LFAD) was calculated. The spectral fitting and computational procedures are detailed in the work by Larsson (9), Wickholm (10) and Nocanda (5). The determination of cellulose surface-to-volume ratio based on 2H-NMRD measurements performed on the interstitial water in water swollen samples (q(NMRD)) are detailed in the Theory section. Figure 3 shows the 2H-NMRD profiles recorded for the samples listed in Table 2.

97 Agarwal et al.; Nanocelluloses: Their Preparation, Properties, and Applications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.


Figure 2. CP/MAS 13C-NMR spectra recorded on the samples, annotations per Table 1. All spectra were recorded for water saturated samples. Small spinning sidebands were the only distinguishable signals outside of the 40 ppm to 120 ppm range.


99


Table 2. Results from the CP/MAS 13C-NMR measurements and the solids content measurements performed for the samples. LFD: the average lateral fibril dimension, LFAD: the average lateral fibril aggregate dimension, Cr: the degree of crystallinity, SC: solids content, mS/mL: the ratio of solids mass to liquid mass, q(F): stoichiometric surface-to-volume ratio of fibrils, q(A): stoichiometric surface-to-volume ratio of fibril aggregates, q(NMRD): stoichiometric surface-to-volume ratio determined from the 2H-NMRD measurements (see the Theory section). The numeral in e.g. DLI and DLI-2 signifies that two samples were prepared, at different times, and with different resulting solids/water content. Sample

Cr (%)

SC

mS/mL (g/g)

q(F)

q(A)

q(NMRD)

32.2 (1.3)

68.3 (2.0)

0.64c

1.79c

0.317 (0.006)

0.070 (0.003)

0.176 (0.004)

6.6 (0.1)

32.2 (1.3)

68.3 (2.0)

0.67

2.00

0.317 (0.006)

0.070 (0.003)

0.227 (0.072)

D96

4.2 (0.1)

22.8 (0.7)

52.7 (0.5)

0.54

1.16

0.473 (0.005)

0.098 (0.003)

0.413 (0.010)

D96-2

4.2 (0.1)

22.8 (0.7)

52.7 (0.5)

0.58

1.38

0.473 (0.005)

0.098 (0.003)

0.457 (0.202)

ND96

4.1 (0.1)

18.4 (0.5)

52.0 (0.6)

0.41

0.69

0.480 (0.005)

0.120 (0.003)

0.336 (0.046)

ND96-2

4.1 (0.1)

18.4 (0.5)

52.0 (0.6)

0.44

0.78

0.480 (0.005)

0.120 (0.003)

0.505 (0.214)

NDDIS

4.2 (0.1)

16.9 (0.4)

53.5 (0.6)

0.43

0.75

0.465 (0.005)

0.130 (0.003)

0.160 (0.005)

NDDIS-T

4.2 (0.1)b

16.9 (0.4)b

53.5 (0.6)b

0.28

0.39

0.465 (0.005)b

0.130 (0.003)b

0.247 (0.007)

CB29

n/a

n/a

n/a

0.025

0.026

n/a

n/a

n/a

DLI

LFD (nm) (0.1)a

DLI-2

6.6

LFAD (nm)

a Values in parenthesis are one standard error. b For the NDDIS-T sample structural parameters are taken from the NDDIS sample. (SC) and in the solids-to-liquid mass ratio (mS/mL) are about ±4%.

Agarwal et al.; Nanocelluloses: Their Preparation, Properties, and Applications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.

c Errors in solids content


Figure 3. (a) 2H NMRD profiles recorded for samples in Table 2. (b) The 2H NMRD profiles from (a) after multiplication of the scaling factor β, superimposing all 2H NMRD profiles recorded for pulps onto the 2H NMRD profiles recorded for cellulose beads. The sample order is, from top to bottom: D96-2, D96, DLI-2, DLI, ND96-2, ND96, NDDIS, NDDIS-T, and CB29. Sample legends from Table 2. Two 2H-NMRD profiles recorded for a mixture of 50 % 2H2O in water are shown at the bottom of the charts. No dispersion was observed in the 2H2O/water samples.



Assuming the dynamic process responsible for the dispersions observed in Figure 3 was a chemical exchange process and utilizing the 2H-NMRD profile recorded for the cellulose bead sample it was possible to estimate cellulose surface-to-volume ratios for the pulp samples based on the 2H-NMRD measurements (q(NMRD) given in Table 2, see Theory section). The q(NMRD) values are compiled with the q-values for fibrils (q(F) and q(A) for fibril aggregates) obtained from CP/MAS 13C-NMR measurements for water swollen cellulose in Figure 4.

Figure 4. A compilation of surface-to-volume ratios (q-values) measured by CP/MAS 13C-NMR and 2H-NMRD spectroscopy. The abbreviations are: q(F): average surface-to-volume ratio of cellulose fibrils, q(A): average surface-to-volume ratio of cellulose fibril aggregates, q(F) and q(A) were calculated from CP/MAS 13C-NMR spectra recorded for water swollen cellulose, q(NMRD): cellulose surface-to-volume ratio based on 2H-NMRD measurements performed for the interstitial water in water swollen cellulose. Error bars indicate one standard error and represents the quality by which the experimental data could be fitted to the respective models used. Sample legends from Table 2.

Theory The dynamic processes causing relaxation in the measured magnetic field range were comparatively slow, with correlation times in the 0.1 microseconds to 100 microseconds range. Due to this observation, an attempt was made to model the dynamic process as an exchange process during which deuterium (2H) nuclei are being exchanged between the water molecules and the hydroxyl groups in cellulose. The exchange process was modelled by a two-site model, which for many reasons is a coarse model for the system, but judged sufficient as a first approximation. In the model used it is the exchange process itself that constitute 101 Agarwal et al.; Nanocelluloses: Their Preparation, Properties, and Applications ACS Symposium Series; American Chemical Society: Washington, DC, 2017.


the dynamic process causing the observable spin-lattice relaxation, hence no well-defined spin-lattice relaxation times are introduced for the bound and free sites. The dynamic process modelled is described by Equation 1. In what follows concentrations are expressed as dimensionless fractions, consequently the unit of rate constants k1 and k2 are the inverse of time (1/s), below 2HB and 2HF indicate deuterium atoms in bound (B) or free (F) sites.

The spin Hamiltonian used for describing the system was

HZ is the Zeeman term describing the interaction energy between the magnetic nuclei and the external magnetic field, HQ(t) is the nuclear quadrupole term describing the time dependent interaction energy between the electric nuclear quadrupole moment and the electrical field gradient at the nucleus. All other interactions are neglected. This is an approximation assuming that HQ(t) is a single and dominating perturbation term in the system studied. Linking the dynamic process to the observable spin-lattice relaxation rates is made in a series of steps. A suitable mathematical model is formulated describing the dynamic process believed responsible for the observable relaxation rate. Some form of averaging and/or temporal coarse graining (separation of time scales) of the mathematical model is used to define the stochastic model counterpart of the dynamic model; the correlation function which describes the average temporal behaviour of the dynamic process. The correlation function is used to calculate the spectral density, which gives a measure of how efficient the underlying dynamic process is at causing relaxation. In NMR relaxation theory, the spectral density is theoretically linked to the observable relaxation rate (21). A stationary random telegraph process (22), between bound (B, hydroxyl group) and free (F, water molecule) sites, was used to describe the exchange process given in Equation 1. During this process nuclei jumps between the bound and free sites that are characterized by two distinct values of the interaction energy, HQ(t) that can assume either of the values f or b. Symbols k1 and k2 are the rate constants introduced in Equation 1.The symbol Gs(t) in Equation 3 is the correlation function, sub-index “s” indicates that it is a stationary process, i.e. describing the dynamics of the system in a situation where any macroscopic gradients has vanished. In this context, it serves to emphasise that the notions “free” and “bound” stems from the model and only refers to whether or not deuterons are attached to hydroxyl groups on the β-(1,4)-D-glucan polymers. All deuterons attached to water molecules are considered free, independent of the distance between the deuteron containing water molecule and any cellulose surface.



From the exponential function in Equation 3 the correlation time (τ) is identified as

In the stationary limit, i.e. equilibrium populations of deuterium in bound and free sites, the rate constants k1 and k2 can be related to the population fractions; nB the fraction of 2H nuclei in site B and nF the fraction of 2H nuclei in site F by

Using Equation 5 and that nB+nF = 1, Equation 3 can be rewritten as

Where the correlation time τ can now be written as

The relaxation rate for spin = 1 nuclei (R1) is related to the spectral density (j(ω)) by (23)

The spectral density was calculated as the real part of the Fourier transform of the stationary correlation function, Equation [6]


Substituting Equations 9 and 10 in Equation 8 yields


In the limit nB

Internal Structure of Isolated Cellulose I Fibril Aggregates in the Water

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