Diffusion of Water Absorbed in Cellulose Fibers ... - ACS Publications

The diffusion of water and the cross relaxation between water and cellulose in a hydrated paper ... Natural Cellulose Fiber as Substrate for Supercapa...
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Diffusion of Water Absorbed in Cellulose Fibers Studied with 1H-NMR Daniel Topgaard* and Olle So¨derman Division of Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, S-221 00 Lund, Sweden Received July 12, 2000. In Final Form: December 22, 2000 The diffusion of water and the cross relaxation between water and cellulose in a hydrated paper system were followed with the Goldman-Shen and the pulsed-field-gradient stimulated-echo NMR pulse sequences as a function of moisture content. The diffusion coefficient measured along the plane of the paper sheet is very sensitive to the moisture content; more water leads to faster diffusion. The cross relaxation rate is independent of moisture content but high enough to severely influence the evaluation of the diffusion data. From the experiments, structural information about the porous network formed by cellulose microfibrils in the interior of the cellulose fibers is obtained.

Introduction Moisture affects the properties of paper. To predict and control these properties, understanding of the watercellulose interactions is necessary. There have been a number of earlier studies of paper with nuclear magnetic resonance (NMR), for example, cryoporometry,1 relaxation,2,3 and diffusion4,5 experiments. The main objective of all these studies was to determine pore sizes of the paper and the cellulose fibers. This article is written from the NMR point of view and describes how a special set of NMR techniques can be used to obtain information about the porous structure of cellulose fibers at low amounts of sorbed water. We have investigated how the self-diffusion coefficient (D) of water absorbed in paper depends on the moisture content (MC), defined as

MC )

mw mc

(1)

where mw and mc are the masses of water and cellulose (including other solid components, e.g., hemicellulose), respectively. The MC was determined from the time domain NMR signal according to the procedure described in Hartley et al.6 For the diffusion experiments, the pulsedfield-gradient stimulated-echo (PFG STE) NMR technique was used. Peschier et al.7 pointed out that the results obtained with this technique may be influenced by cross relaxation between the water and the macromolecular matrix. To our knowledge, the rate of cross relaxation has never been measured in paper, but there are some studies on related materials, for example, starch8 and wood.9 The cross relaxation was studied with the Goldman-Shen (GS) * Corresponding author. E-mail: [email protected]. Tel: INT +46 46 222 01 34. Fax: INT +46 46 222 44 13. (1) Furo, I.; Daicic, J. Nord. Pulp Pap. Res. J. 1999, 14, 221-225. (2) Ha¨ggkvist, M.; Li, T.-Q.; O ¨ dberg, L. Cellulose 1998, 5, 33-49. (3) Ha¨ggkvist, M.; Solberg, D.; Wågberg, L.; O ¨ dberg, L. Nord. Pulp Pap. Res. J. 1998, 13, 292-298. (4) Li, T.-Q.; Henriksson, U.; Klason, T.; O ¨ dberg, L. J. Colloid Interface Sci. 1992, 154, 305-315. (5) Li, T.-Q.; Ha¨ggkvist, M.; O ¨ dberg, L. Langmuir 1997, 13, 35703574. (6) Hartley, I. D.; Kamke, F. A.; Peemoeller, H. Holzforschung 1994, 48, 474-479. (7) Peschier, L. J. C.; Bouwstra, J. A.; de Bleyser, J.; Junginger, H. E.; Leyte, J. C. J. Magn. Reson., Ser. B 1996, 110, 150-157.

experiment.10 The measured D of water, which is a dynamic property, can be related to the connectivity or tortuosity, which are structural properties, of the porous network formed in the interior of the cellulose fibers. To gain further insight into the system, the water absorption isotherm was determined by means of a calorimetric method.11 We have chosen to study filter paper because it is a rather simple system which in contrast to other types of paper contains no additives (clay, fillers, etc.). Paper in general consists of a two-dimensional network of cellulose fibers (cf. Figure 1) in the form of flattened tubes having the approximate size 20 µm × 50 µm × 3 mm. The large internal cavity of the fibers, the lumen, has about half that size. Dry fibers have an external specific surface area (including the surface toward the lumen) of ∼1 m2/g according to nitrogen adsorption measurements.12 From electron microscopy13,14 on wood cell walls, it is known that the pores of the cellulose fiber walls are formed as voids between roughly cylindrical (o.d. ∼ 10 nm) cellulose microfibrils mainly oriented in the direction of the fiber. The pores have a complex shape, and the pore boundaries are ill defined. At exposure of dry paper to humid air, the fiber wall space is gradually filled with water.15,16 First, a monolayer is formed at all available surfaces, and subsequently (MC > 4-5%) capillary condensation takes place. At 100% relative humidity (RH), the fiber wall sites are saturated and the fiber saturation point (FSP) is reached. Further addition of water fills the lumen and the space between the fibers. In the fully wet state, a large amount of water (∼120-140%) is associated with the fibers according to the solute exclusion technique17 and NMR self-diffusion measurements.4 Water between (8) Tanner, S. F.; Hills, B. P.; Parker, R. J. Chem. Soc., Faraday Trans. 1991, 87, 2613-2621. (9) Oleskevich, D. A.; Gharamany, N.; Weglarz, W. P.; Peemoeller, H. J. Magn. Reson., Ser. B 1996, 113, 1-8. (10) Goldman, M.; Shen, L. Phys. Rev. 1966, 144, 321-331. (11) Wadso¨, I.; Wadso¨, L. Thermochim. Acta 1996, 271, 179-187. (12) Haselton, W. R. Tappi 1954, 37, 404-412. (13) Fengel, D.; Wegener, G. Wood: Chemistry, Ultrastructure and Reactions; Walter de Gruyter: Berlin, NY, 1984. (14) Hafre´n, J.; Fujino, T.; Itoh, T. Plant Cell Physiol. 1999, 40, 532541. (15) Weatherwax, R. C. J. Colloid Interface Sci. 1977, 62, 433-446. (16) Stone, J. E.; Scallan, A. M. Tappi 1967, 50, 496-501. (17) Stone, J. E.; Scallan, A. M. Cellul. Chem. Technol. 1968, 2, 343358.

10.1021/la000982l CCC: $20.00 © 2001 American Chemical Society Published on Web 04/05/2001

Diffusion of Water Absorbed in Cellulose Fibers

Figure 1. Schematic representation of a cellulose fiber. The fiber wall consists of closely packed cellulose microfibrils oriented mainly in the direction of the fiber.

the fibers and inside the lumen has bulk characteristics, whereas the water in the fiber wall is capillary water and thus strongly influenced by the surrounding microfibrils. The samples used in this study had MC values ranging from 10 to 20% corresponding to RH values from 70 to 90%. It is expected that the diffusion along the fiber axis is faster than in the perpendicular direction because of the extension of channels along the microfibrils. Because of this and the preferential orientation of the fibers in the plane of the paper, the results of the diffusion measurements depend on the orientation of the paper sheet. The PFG STE experiment measures molecular displacements in one dimension during a well-defined time. The displacements are on the scale of a few micrometers, which means that the features of the nanometer-scale microstructure are averaged out and few water molecules reach the border of the fibers. The permeability of the porous network is thus probed. In all experiments, we measured the diffusion parallel to the paper sheets, which gives a diffusion coefficient averaged over all fiber orientations in the plane of the paper sheet. A rough estimate of the fiber composition was obtained with an optical microscope. The results of the analysis were 70-80% cotton fiber and 20-30% bleached fiber from coniferous trees. The cotton fibers contain 95-97% cellulose and no lignin. The wood fibers contain 40-50% cellulose, 30-40% hemicellulose, and 20-30% lignin,18 the latter being removed during the bleaching process.16,19 The wood fiber walls thus gain some extra porosity compared to the cotton fiber walls. A more detailed knowledge of the fiber composition is not required because the relevant parameters needed to evaluate the diffusion experiment are actually measured with the described set of experiments. MacKay et al.20 studied hydrated cotton thread and filter paper with different relaxation and lineshape techniques and obtained similar results for the two samples. The diffusion experiments are however more sensitive to the porosity. (18) Hafre´n, J. Ultrastructure of the wood cell wall. Ph.D. Thesis, Royal Institute of Technology, Stockholm, Sweden, 1999. (19) Ha¨ggkvist, M. Porous Structures in Paper Studied by NMR. Licentiate Thesis, Royal Institute of Technology, Stockholm, Sweden, 1999. (20) MacKay, A. L.; Bloom, M.; Tepfer, M.; Taylor, I. E. P. Biopolymers 1982, 21, 1521-1534.

Langmuir, Vol. 17, No. 9, 2001 2695

Figure 2. The Goldman-Shen pulse sequence was used for the determination of the cross relaxation parameters. The first 90° pulse turns all magnetization into the transverse plane. The preparation time (τ1) is adjusted to let the cellulose magnetization disappear but leave large parts of the water magnetization. The second 90° pulse turns the magnetization back to the longitudinal direction, where it evolves during the mixing time (τ2) until the third 90° pulse detects it.

Experimental Section Sample. The sample used was a 1 mm thick filter paper from Munktells, Sweden. For the NMR experiments, the paper was cut into 2 cm squares, rolled into a cylinder, and put in 5 mm o.d. NMR tubes. After equilibration in 100% RH for 3 weeks, the paper was allowed to dry (hours to days) in 40% RH to reach the desired MC. Fifteen samples were prepared with MC in the range from 10 to 20%. Absorption Isotherm. The water absorption isotherm was determined with a twin double microcalorimeter according to the procedure described in Wadso¨ and Wadso¨.11 The apparatus consists of two connected vessels, one with the liquid and one with the substrate. The absorption process is followed continuously by monitoring the thermal power of vaporization and absorption in the two vessels as a function of time. Knowledge of the enthalpy of vaporization for the water and application of Fick’s law of diffusion make it possible to calculate the absorption isotherm. The method has previously been used to study for example phospholipid hydration.21 NMR Measurements. All experiments were performed at 25 °C on a Bruker DMX-200 spectrometer, with a 1H resonance frequency of 200 MHz, equipped with a Bruker diffusion probe capable of delivering 9 T/m at 40 A. In all experiments, 4 µs 90° pulses were used. The MC was evaluated from the free induction decay (FID) following a 90° pulse. The receiver dead time was set to 4.5 µs, and the dwell time was 1 µs using a fast analogue to digital converter to digitize 4096 points. Cross relaxation experiments were carried out with the Goldman-Shen pulse sequence10 (see Figure 2). τ1 was varied linearly between 50 and 500 µs in 10 steps, and τ2 was varied in a geometric sequence between 1 ms and 2.2 s in 20 steps. Diffusion measurements were performed with the PFG STE pulse sequence22 (see Figure 3). δ was 300 µs, τ1 was 900 µs, τ2 was varied in a geometric sequence between 66.67 and 337.5 ms in 5 steps, and G was varied linearly between zero and maximum strength in 19 steps. The time for the eddy currents to disappear was found to be shorter than 500 µs for a 500 µs gradient pulse of full strength. The gradient strength was calibrated using the same gradient pulses on a sample with known D. The gradient calibration experiment was also used to control the gradient linearity. A repetition time of 2.5 s was used throughout. The receiver dead time was for the diffusion and GS experiments set to 50 µs, and only the water signal was detected. (21) Markova, N.; Sparr, E.; Wadso¨, L.; Wennerstro¨m, H. K. J. Phys. Chem. B 2000, 104, 8053-8060. (22) Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy, 1st ed.; Oxford University Press: Oxford, 1991.

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Figure 5. Experimental FID for a sample with MC ) 20%. The water and cellulose components are easily separated. The line is the result of the fit of eq 5 to the experimental data.

Figure 3. The pulsed-field-gradient stimulated-echo pulse sequence used for the diffusion measurements. The three 90° pulses produce a stimulated spin-echo at time 2τ1 + τ2, and the gradients of amplitude G and duration δ tag and read the position of the spin-bearing molecules. ∆ is the time between the leading edges of the gradient pulses. The effective diffusion time td is ∆ - δ/3.

Figure 4. The water absorption isotherm determined with a twin double microcalorimeter. Fitting eq 2 to the initial part of the isotherm yields the monolayer capacity which can be related to the specific surface area of the cellulose and the radius of the cellulose microfibrils.

Results and Discussion Introduction. The outline of this section is as follows. First, we present and discuss the water absorption isotherm. After that, the free induction decay, cross relaxation, and diffusion experiments are treated. The final section is devoted to the conclusions drawn. Absorption Isotherm. In Figure 4, we show the water absorption isotherm determined with a twin double microcalorimeter. According to the figure, the MC ranges from 10 to 20% which corresponds to RH from 70 to 90%. To obtain a rough estimate of the specific surface area, the initial part of the isotherm (RH < 0.5) was analyzed with the Brunauer-Emmett-Teller (BET) expression23

cRH MC ) MCm (1 - RH)(1 + (c - 1)RH)

(2)

where MCm is the monolayer capacity (the MC where a monolayer of water is formed at all available surfaces) and c is a constant depending on the heat of sorption and the temperature. MCm is related to the specific surface area of the cellulose (Ac) through

NA Ac ) MCm aw Mw

(3)

where NA is Avogadro’s number, Mw is the molar mass of

water, and aw is the area of a water molecule projected to the cellulose surface (estimated to 14.8 × 10-20 m2 24). If we assume that the only contribution to the cellulose surface area is from the outer surface of the (monodisperse) microfibrils, the microfibril radius is given by

r)

2 FcAc

(4)

where Fc is the cellulose density (1.55 g/cm3 13). The fit of the BET isotherm yields a monolayer capacity of 4.6% which is equivalent to Ac ) 230 m2/g and r ) 5.7 nm. The values of MCm and Ac are in agreement with what is reported in the literature.25 Finally, we note that although the analysis performed above is somewhat crude as it does not take surface heterogeneity and specific interactions between water and hemicellulose into account, the obtained value of r is in good agreement with the results of other methods.14 Free Induction Decay. In Figure 5, we display an experimental FID consisting of two clearly discernible parts: a fast decaying component (with a T/2 ≈ 15 µs) from the cellulose and a slower decaying component (with a T/2 ≈ 700 µs) from the water. T/2 is the time for the signal to decay to 1/e of its initial value. The assignment of the two components is well established in the literature.6,9,20,26-29 Bulk (lumen) water would show up as a third component with T/2 in the range of tens to hundreds of milliseconds.29 The total time domain signal is the sum of the signals from the cellulose and the water,

S ) Sc + Sw

(5)

The cellulose part of the FID was described by a Gaussian-sinc equation30 (23) Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: London, 1994. (24) Li, T.-Q. Interactions between Water and Cellulose - Applications of NMR Techniques. Ph.D. Thesis, Royal Institute of Technology, Stockholm, Sweden, 1991. (25) The Physics and Chemistry of Wood Pulp Fibers; Page, D. H., Ed.; Technical Association of the Pulp and Paper Industry: New York, 1970; Vol. 8. (26) Flibotte, S.; Menon, R. S.; MacKay, A. L.; Hailey, J. R. T. Wood Fiber Sci. 1990, 22, 362-376. (27) Araujo, C. D.; MacKay, A. L.; Whittal, K. P.; Hailey, J. R. T. J. Magn. Reson., Ser. B 1993, 101, 248-261. (28) Xu, Y.; Araujo, C. D.; MacKay, A. L.; Whittal, K. P. J. Magn. Reson., Ser. B 1996, 110, 55-64. (29) Menon, R. S.; MacKay, A. L.; Hailey, J. R. T.; Bloom, M.; Burgess, A. E.; Swanson, J. S. J. Appl. Polym. Sci. 1987, 33, 1141-1155. (30) Abragam, A. The Principles of Nuclear Magnetism; Oxford University Press: London, 1961.

Diffusion of Water Absorbed in Cellulose Fibers 2 2

Sc ) S0c exp(-a t /2) sinc(bt)

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(6)

where S0c is the signal intensity of the cellulose at t ) 0 and a and b are constants. The second moment of the line width (M2) is given by30

M2 ) a2 +

b2 3

(7)

M2 was determined to (1.042 ( 0.008) × 1010 s-2 and did not show any dependence on MC in the investigated range. This indicates that the cellulose is unaffected by changes in MC in this range. MacKay et al.20 calculated the rigid lattice M2 for cellulose to 7.5 × 109 s-2 and measured a similar value on a fully hydrated (MC ≈ 160%) cotton thread sample at 24 °C and a 1H resonance frequency of 90 MHz. The reason for the discrepancy between our measured value and the calculated value of MacKay et al. is unclear. We do note that the magnitude of the difference is quite small. Moreover, Flibotte et al.26 observed a 20% increase in M2 when decreasing the MC from above FSP to almost dryness on a western red cedar sample. It is conceivable that M2 of our samples would also change if MC was changed beyond the range used in this study. Because the water part was neither purely exponential nor purely Gaussian, it was described by a Voigt line shape, which is produced by a Gaussian broadening of an intrinsically Lorentzian peak.31,32

Sw ) S0w exp[-πwLt - (πwGt/2xln 2)2]

(8)

where wL and wG are the Lorentzian and Gaussian full widths at half-maximum height of the peaks in the frequency domain, respectively, and S0w is the signal intensity of the water at t ) 0. The transverse relaxation time (T2) is given by32

T2 )

1 πwL

where

PFID )

S0w

(11)

S0c

and RSD is the relative spin density of cellulose compared to water. It is reasonable to assume that this is valid also for paper, but with a different constant of proportionality because of the difference in RSD between paper and wood. Hartley6 states that the RSD values for cellulose, hemicellulose, and lignin are 0.56, 0.60, and 0.73, respectively. In a gravimetric calibration experiment, we found that RSD ) 0.565 ( 0.021 for the paper used in this study. Cross Relaxation. In Figure 7, we show experimental curves from the GS experiment. The data were analyzed in terms of the two-site model given in Figure 8. The time dependent longitudinal magnetization of the water pool mw(t) with the equilibrium value meq w can be shown to be7,34 + + mw(t) ) meq w [1 + c exp(-R τ2) + c exp(-R τ2)] (12)

(9) 2R( ) kw + R1w + kc + R1c (

T2 was found to be proportional to MC, which is shown in Figure 6. See also the discussion about T2 in the diffusion section. The main purpose of analyzing the FID was to determine the initial intensities S0w and S0c . In principle, this could also be achieved by integration of the frequency domain peaks, but this procedure fails to account for the nonnegligible decay of the cellulose signal during the finite receiver dead time. The time origin was taken as the middle of the 90° pulse, because this point is the effective starting point for the evolution of the perturbed spin system.33 It should be noted that eqs 6 and 8 were chosen as a compromise between best fit to experimental data and least number of fitting parameters without regard to the mechanism for the line shape. Equation 6 could be replaced by a series expansion in even moments and eq 8 could be replaced by a sum of Gaussians, but we found that the chosen line shapes had superior numerical stability in the fitting procedure. For wood, the initial ratio between the water and cellulose signals (PFID) is proportional to the MC of the sample.6,29

MC ) RSD PFID

Figure 6. T2 of the absorbed water as a function of MC. The line is the result of the fit of eq 34 to the experimental data.

(10)

x(kw + R1w - kc - R1c)2 + 4kwkc kw + R1w - R-

c( ) (∆mw(τ1)

+

R -R

-

kw - ∆mc(τ1) + R - R-

where ∆mw/c is the normalized deviation from the equilibrium value of the water and cellulose magnetizations, respectively.

∆mw/c(t) )

eq mw/c(t) - mw/c eq mw/c

(13)

∆mw is given by

∆mw(τ1) ) exp[-πwLτ1 - (πwGτ1/2xln 2)2] - 1 (14) If the preparation time in the GS experiment (τ1 in Figure (31) de Beer, R.; van Ormondt, D. NMR: Basic Princ. Prog. 1992, 26, 201-248. (32) Bruce, S. D.; Higinbotham, J.; Marshall, I.; Beswick, P. H. J. Magn. Reson. 2000, 142, 57-63. (33) Barnaal, D.; Lowe, I. J. Phys. Rev. Lett. 1963, 11, 258-260. (34) Edzes, H. T.; Samulski, E. T. J. Magn. Reson. 1978, 31, 207229.

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PFID )

pw - psOH pc + psOH

(18)

where psOH is the fraction of cellulose protons (mainly -OH but possibly also other exchanging protons) in fast chemical exchange with the water

psOH ) pw -

Figure 7. Experimental curves from the Goldman-Shen experiment on a sample with MC ) 20%. The dips on the curves show the existence of cross relaxation. The lines are the result of the fit of eq 12 to the 2D experimental data. τ1 is increasing from top to bottom.

Figure 8. The two-site model used for evaluation of the cross relaxation and diffusion experiments. R1w and R1c are the intrinsic longitudinal relaxation rates, kw and kc are the cross relaxation rates, and pw and pc are the proton fractions (pw + pc ) 1) of the water and cellulose proton pools, respectively. Dw is the diffusion coefficient of the water pool, which also includes cellulose surface hydroxyl groups in fast exchange with the water. To get the true diffusion coefficient of water, Dw must be multiplied with the correction factor pw/(pw - psOH) where psOH is the fraction of surface hydroxyl groups.

2) is longer than 50 µs, Sc/S0c is less than 1‰ and

∆mc(τ1) ) -1

(15)

The equations have the same form as in Peschier et al.7 with the exception of eq 14 where the exponential of Peschier et al. has been replaced with the observed Voigt form (cf. eq 8). Fitting eq 12 to the experimental data yields estimates of kw, kc, R1w, and R1c. The fraction of protons in the two pools in Figure 8 is given by

pw )

kc kc + kw

pc )

kw kw + kc

(16)

As pointed out by Edzes and Samulski,34 the water pool may also contain a fraction of the cellulosic protons that are in fast (on the time scale of the GS experiment) chemical exchange with the water. For the sake of comparison with PFID, the ratio between the number of exchanging protons as determined in the GS experiment (PGS) can be calculated by

PGS )

pw pc

(17)

If the water pool contains some of the cellulose protons, it follows that PGS > PFID because

PFID PFID + 1

(19)

In the experiments reported on here, PGS was always larger than PFID, indicating that hydroxyl groups at the surface of the cellulose microfibrils are in fast exchange with the water on the time scale defined by the GS sequence but not on that of the FID experiment. Fast (∼1 ms) proton exchange in polysaccharide solution is believed to occur via a cyclic transition state with rings of water molecules hydrogen bonded to the hydroxyl groups.8 For proton exchange to be fast, some critical amount of water (more than a monolayer) must be associated with the surface. In water-poor systems, the rate of this mechanism is greatly reduced. PGS is in this interpretation related to the exposed area of the cellulose microfibrils, where exposed implies that the surface of the cellulose has more than the critical amount of water adsorbed to allow for fast proton exchange. PGS was found to be proportional to PFID with a proportionality constant of 1.401 ( 0.006. The exposed area is proportional to psOH/ (psOH + pc) which is plotted in Figure 9. This quantity is determined from a comparison between the FID and GS experiments without need for knowledge of the fiber composition. Assuming that the cellulose consists only of condensed sugar units (C6H10O5)n, the maximum value of psOH/(psOH + pc) is 30%. The corresponding value for hemicellulose can be calculated to ∼18% from the hemicellulose composition of aspen wood given in Hartley et al.6 The measured maximum value of psOH/(psOH + pc) is ∼10% which means that roughly one-third of the hydroxyl groups are exposed to water. The small amount of hemicellulose present in the sample is not enough to significantly alter this figure. To explain why the exposed area increases with MC, the simple model in Figure 10 is useful. At low MC, the microfibrils must be densely packed because of space filling constraints. The positions are not completely ordered, but to simplify calculations the cellulose microfibrils with radius r are assumed to form an array with hexagonal symmetry. In this model, the smallest distance d between the microfibril surfaces is

d)

(x (

) )

2π Fc MC + 1 - 2 r x3 Fw

(20)

A more complex model taking surface heterogeneity into account is not necessary to rationalize the experimental findings. Using Fc ) 1.55 g/cm3,13 r ) 5.7 nm (estimated from the isotherm), and MC in the range from 10 to 20% gives values of d in the range from 0.27 to 1.0 nm which corresponds to 1-4 layers of water. Earlier, it was argued that more than a critical amount of water must be associated with the cellulose surface for chemical exchange to be fast. A monolayer of water is clearly too little, but it is conceivable that four layers is enough. Although the microfibrils do not form this type of ordered structure, it is plausible that the distance between the microfibrils is increased and more surface is exposed when MC increases.

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Figure 9. The exposed cellulose area (defined in text) is proportional to psOH/(pc + psOH).

Figure 11. In a porous material, the observed diffusion coefficient depends on the diffusion time. At short times, the bulk behavior is observed, whereas at long times the connectivity of the porous network is probed.

measured with an inversion recovery experiment was ∼400 ms, which sets the upper limit of τ2. The PFG STE NMR technique measures the onedimensional root mean square distance of diffusion (x〈z2〉) over a well-defined time interval, td ) ∆ - δ/3. An apparent diffusion coefficient may be defined as

Dapp )

〈z2〉 2td

(22)

In general, in a porous material with pores of size a, three Figure 10. The model of the microfibril geometry. r is the microfibril radius, and d is the smallest distance between adjacent microfibrils.

The cross relaxation results are reported in the form of the cross relaxation time Tcr defined as

Tcr )

1 1 ) pwkw pckc

(21)

No pronounced MC dependence was found for Tcr. The average value was 43.8 ( 2.8 ms, which is in agreement with what is reported for other hydrated macromolecules/ biological materials using the same type of experiment (viz., aspen wood MC ) 10.8% and Tcr ) 16 ms,9 hydrated lysozyme MC ) 17.8% and Tcr ) 10 ms,35 mouse muscle tissue Tcr ) 33 ms,36 and rat tail tendon collagen Tcr ) 69 ms34). Diffusion. Here, we report what we believe are the first direct measurements of fiber wall water diffusion with NMR. Previous measurements4,5,37 have been on bulk (lumen or between fibers) water. The main experimental problem in obtaining the diffusion data is the fast transverse relaxation of the fiber wall water. T2 estimated from the FID was in the range from 0.8 to 1.5 ms (see Figure 6). Similar values were obtained with a simple spin-echo experiment. The magnetization suffers transverse relaxation during the τ1-periods in Figure 3. For technical reasons, 2τ1 could not be set shorter than 1.8 ms. During this time, a large part of the magnetization relaxes, which makes extensive signal averaging necessary for the samples with lowest MC and T2. The success of our experiments is due to our ability to apply short and strong magnetic field gradient pulses with minimum separation to echo acquisition. The longitudinal relaxation time T1 (35) Peemoeller, H. Bull. Magn. Reson. 1989, 11, 19-30. (36) Sobol, W. T.; Cameron, I. G.; Inch, W. R.; Pintar, M. M. Biophys. J. 1986, 50, 181-191. (37) MacGregor, R. P.; Peemoeller, H.; Schneider, M. H.; Sharp, A. R. J. Appl. Polym. Sci.: Appl. Polym. Symp. 1983, 37, 901-909.

regimes of td can be distinguished. When x〈z2〉 , a, very few molecules feel the influence of the pore boundaries and Dapp is roughly the same as the bulk value D0 (2.3 ×

10-9 m2/s for water at 25 °C38). At longer td, when x〈z2〉 ≈ a, Dapp is decreasing with increasing td. At very long td,

when x〈z2〉 . a, Dapp reaches a new constant value D∞ (shown in Figure 11). The tortuosity (T) describes the long range connectivity in a porous medium and is originally defined through electrical conductivity measurements. For the case of diffusion, T is given by39-41

1 D∞ ) T D0

(23)

In Figure 12, the echo attenuation curves at five different diffusion times on the same sample are displayed. For free diffusion in the absence of cross relaxation, the echo attenuation is given by the Stejskal-Tanner equation22

E ) exp[-(2πq)2D(∆ - δ/3)]

(24)

where q is the wave vector

q)

γGδ 2π

(25)

and γ is the gyromagnetic ratio of the spin-bearing nucleus. Plotting ln E versus (2πq)2D(∆ - δ/3) would according to eq 24 yield a straight line independent of the value of ∆. As can be seen in Figure 12, the curves are not straight and a ∆-dependence is observed. Evaluating the diffusion data using eq 24 leads to a D decreasing with increasing diffusion time. This could be taken as evidence for barriers (38) Mills, R. J. Phys. Chem. 1973, 77, 685. (39) Heil, S. R.; Holz, M. J. Magn. Reson. 1998, 135, 17-22. (40) Latour, L. L.; Kleinberg, R. L.; Mitra, P. P.; Sotak, C. H. J. Magn. Reson., Ser. A 1995, 112, 83-91. (41) Mair, R. W.; Wong, G. P.; Hoffmann, D.; Hu¨rlimann, M. D.; Patz, S.; Schwartz, L. M.; Walsworth, R. L. Phys. Rev. Lett. 1999, 83, 3324-3327.

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Figure 12. Experimental echo attenuation curves for five diffusion times on a sample with MC ) 20%. The nonequivalence of the echo decays could erroneously be interpreted in terms of restricted diffusion. The lines are the result of a global fit of eq 29 to the five experiments. The diffusion time is increasing from bottom to top.

to diffusion on the length scale defined by the diffusion experiment (≈5 µm). Cross relaxation could however also lead to the observed behavior. Because the cross relaxation times and diffusion times are of the same order of magnitude (Tcr ≈ 45 and td ≈ 70-350 ms), longer td leads to more complete cross relaxation. When cross relaxation effects are taken into account, the echo attenuation becomes7

E(q) )

E0 )

exp[-(A/2)τ2] {(A + B + C) exp[-(B/2)τ2] 2BE0 (A - B + C) exp[(B/2)τ2]} (26)

1 {(B0 + C) exp[-(B0/2)τ2] 2B0 (-B0 + C) exp[(B0/2)τ2]} A ) D(2πq)2 C ) kw + R1w - kc - R1c B ) x(A + C)2 + 4kwkc B0 ) xC2 + 4kwkc

The predictions of eq 26 for values of the pulse sequence parameters equal to the experimental ones, using D and values of the cross relaxation parameters from a sample with MC ) 20%, are plotted in Figure 13. As is the case for the experimental curves, the shape of the curves depends on the value of ∆. The Stejskal-Tanner expression, eq 24, yields a much higher attenuation for the same value of D. The explanation is that the stationary cellulose protons have much slower longitudinal relaxation rates (R1c < R1w). We note that eq 26 reduces to eq 24 when kw/c ) 0. In conclusion, the apparent dependence of the attenuation curves on the values of the diffusion times in Figure 12 is caused by the cross relaxation. Thus, we will use eq 26 in the analysis of the STE data. The second important feature of the decays in Figure 12 is that they are nonexponential. One conceivable reason for this effect is the presence of heterogeneity in the sample on a length scale longer than the typical diffusion distance. Regions with different chemical composition and orientation of the structural elements in the cellulose fiber can be recognized with the electron microscope.13 It seems

Topgaard and So¨ derman

Figure 13. The predicted echo attenuation curves using eq 26 (thin lines) and the same parameters as in Figure 12. The curves are nonequivalent, and a slight curvature can be observed. The thick line is calculated with the Stejskal-Tanner expression, eq 24.

reasonable that water in the different environments has different diffusion coefficients. The fact that we have both cotton and wood fibers in the sample probably also contributes to the observed polydispersity. Because of the preferential orientation of microfibrils in the direction of the fiber, the water diffusion in the fiber is anisotropic with faster diffusion along the fiber. Because we are measuring an average over all fiber orientations in the sheet plane, the anisotropic diffusion leads to a distribution of D. It is an often encountered problem in NMR relaxation and diffusion experiments42-44 that an infinite number of distributions, both discrete and continuous, satisfy the same set of experimental data. Here, we turn the problem into an advantage by ignoring the various causes that may lead to a distribution of complex shape, and instead we choose a simple distribution, with few variable parameters, that still gives a satisfactory fit. For dataanalysis purposes, we assume that the resulting distribution of diffusion coefficients could be described by the following formula:

P(D) ) Pmax exp{-(1/2)[(ln D - ln Dmax)/σ]2} (27) where Pmax is a normalization constant, Dmax is the diffusion coefficient with the highest probability, and σ gives the width of the distribution. The mean diffusion coefficient, which can be obtained from the initial slope of the echo attenuation curve,22 is given by

〈D〉 )

∫DP(D) dD

(28)

The echo attenuation curve is given by

E(q) )

∫P(D) E(q,D) dD

(29)

where E(q,D) is given by eq 26 and we have assumed that the cross relaxation rate is independent of the local value of D. Individual fitting of eq 29 with eqs 26 and 27 to data from the five diffusion times yielded the same values of the fitted parameters within experimental error. To improve the accuracy, a global fit was made for each sample, using the five different decays from the different diffusion times. The resulting distribution of diffusion coefficients is presented in Figure 14 for one sample with MC ) 20%. As can be inferred, the distribution is rather broad, covering roughly 2 orders of magnitude, and peaks (42) Whittal, K. P.; MacKay, A. L. J. Magn. Reson. 1989, 84, 134152. (43) Whittal, K. P. J. Magn. Reson. 1991, 94, 486-492. (44) Overloop, K.; Van Gerven, L. J. Magn. Reson. 1992, 100, 303315.

Diffusion of Water Absorbed in Cellulose Fibers

Langmuir, Vol. 17, No. 9, 2001 2701

because of a widening of the pores.17 To quantify these two effects, we first assume that the local diffusion of the water is given by the bulk diffusion coefficient D0 and that the (long-range) diffusion is given by D0 modified by an obstruction factor fo:

〈D〉 ) foD0

(31)

where fo is a function of the volume fraction of cellulose (Φc)

Figure 14. The distribution of diffusion coefficients obtained from the fit of eq 29 to the experimental diffusion data on a sample with MC ) 20%.

Figure 15. The mean diffusion coefficient of water absorbed in cellulose fibers as a function of moisture content.

around 10-11 m2 s-1. The average diffusion coefficients for the various MC values can now be obtained from eq 28. Because there is a fast exchange between cellulosic hydroxyl protons and water, this procedure yields the mean diffusion coefficient of the water pool in the two-site model 〈Dw〉 (cf. Figure 8). If the diffusion coefficient of the hydroxyl groups is small, the contribution from the hydroxyl groups can be corrected for with the following relation:

〈D〉 ) 〈Dw〉

pw pw - psOH

(30)

Φc )

(

Fc MC + 1 Fw

)

-1

(32)

The form of fo(Φc) for diffusion perpendicular to parallel cylinders on a hexagonal lattice was discussed by Jo´hanneson and Halle.48 fo reaches zero when the connection between the cylinders is closed at Φc ) 91%. If we assume that we have a 2D powder of hexagonally packed cylinders (corresponding to the microfibrils), we have to average over all orientations in the sheet plane and fo would have a lower limit of 1/2 because the water can move along the cylinders. The next step is to take the hydration effects into account. Close to a surface, one or two layers of water are affected and the diffusion is reduced by a factor of ∼5 (see e.g. ref 49 and references therein). From the isotherm (cf. Figure 4), we have an estimate of the amount needed to form one monolayer of water. If we for simplicity assume that the surface monolayer is stationary and all the other water diffuses with D0,

(

〈D〉 ) 1 -

)

MCm D MC 0

(33)

The assumption of fast exchange between a strongly affected monolayer with fast transverse relaxation (e.g., due to cross relaxation28) and a bulklike liquid with slow relaxation leads to a T2 proportional to MC (if MC > MCm),

MC T2 ) T2m MCm

(34)

where 〈D〉 is the diffusion coefficient of the water (not the water pool), pw is determined with the GS experiment, and psOH is from the GS and FID experiments using eq 19. Note that this procedure does not require knowledge of the sample composition. The main results of this study are presented in Figure 15, which shows that 〈D〉 depends exponentially on MC and is a factor of 20-200 slower than the bulk value of water. Araujo et al.27 obtained D ) 2.18 × 10-10 m2/s for cell-wall water in redwood sapwood at 26.5 °C and MC ) 291% by relating the surface relaxation parameter of the Brownstein-Tarr model45-47 to the cell-wall water D. Araujo et al. claimed to be the first to measure cell-wall water self-diffusion and also pointed out the experimental difficulties in performing PFG STE diffusion measurements on these kinds of samples. The general trend of the curve in Figure 15 can be readily explained: water molecules are slowed because of hydration at the cellulose surfaces and obstruction from the large volume fraction of microfibrils. Increasing the MC leads to a larger fraction of free water and less obstruction

where T2m is the transverse relaxation time of the surface monolayer, which is indeed observed in Figure 6. Fitting eq 34 to the experimental data using MCm ) 4.6% gives T2m ) 0.19. The appearance of Figure 6 gives some credibility to the assumptions underlying eq 33. Because our samples always have MC more than twice as high as MCm, the hydration can account only for a factor of 2 in reduced D. Taking the hydration and obstruction effects together, it is clear that they can only produce D values which are a factor of 4 lower than the bulk-water diffusion. The model with infinite, perfectly aligned, cylindrical microfibrils is obviously too simplistic to explain the slow diffusion. One alternative is to see the fiber wall as a porous material with elongated grains, corresponding to the microfibrils, preferentially aligned in the direction of the fiber, and a local geometry similar to the model in Figure 10, but where the larger scale structure is substantially more complex and where the channels between the microfibrils do not extend uninterrupted over micrometer-scale distances. During the diffusion time, the molecules move far compared to the local structure

(45) Brownstein, K. R.; Tarr, C. E. J. Magn. Reson. 1977, 26, 17-24. (46) Brownstein, K. R.; Tarr, C. E. Phys. Rev. A 1979, 19, 24462453. (47) Brownstein, K. R. J. Magn. Reson. 1980, 40, 505-510.

(48) Jo´hanneson, H.; Halle, B. J. Chem. Phys. 1996, 104, 68076817. (49) Balinov, B.; Olsson, U.; So¨derman, O. J. Phys. Chem. 1991, 95, 5931-5936.

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Langmuir, Vol. 17, No. 9, 2001

Topgaard and So¨ derman

Figure 16. The tortuosity of the cellulose fibers as a function of moisture content.

and the diffusion experiment is sensitive to the long-range connectivity of the porous network. The measured 〈D〉/D0 however contains contributions from both the connectivity and the hydration. Neglecting hydration, the tortuosity is given by eq 23. In our case, with hydration the expression is modified to

(

)

MCm 1 ) 1T MC

-1

〈D〉 D0

(35)

Strictly speaking, T is a bulk quantity, but we find it useful also for mesoscopic objects such as cellulose fibers. As shown in Figure 16, 1/T depends exponentially on MC in the investigated range. At this stage, we have no explanation as to why the dependence is exponential; we note also that recent experiments have shown that the exponential increase lasts no longer than to MC ≈ 25%. Decreasing the amount of pore water shrinks the distance between fibrils and decreases the diffusion in the channels between the fibrils. As stated above, it is also necessary to assume the existence of some sort of hindrance in the diffusion along the fibrils to rationalize the observed increase in the tortuosity. The exact mechanism of this is however still unclear but may possibly be revealed by experiments performed on well-defined fiber samples at various degrees of treatment known to affect the fiber structure. Such experiments are underway in our laboratory. Conclusions The effect of drying of paper hydrated below the fiber saturation point was studied using three NMR tech-

niques: the recording of free induction decays, GoldmanShen, and pulsed-field-gradient stimulated-echo experiments. Additional information about the system was obtained from the water absorption isotherm obtained using a twin double microcalorimeter. To obtain the selfdiffusion coefficients of water in the porous network formed by the microfibrils in the interior of the cellulose fibers, two additional steps in the standard evaluation of the NMR diffusion experiments had to be considered: magnetization exchange between the water and the cellulose protons on the time scale of 50 ms and chemical exchange between water and cellulosic surface hydroxyl groups on the time scale of 1 ms. From the analysis described above, the average diffusion coefficient of water as a function of MC (in the range from 10 to 20%) is obtained, the analysis of which conveys structural information about the porous network. This is achieved without the need for knowledge of the fiber chemistry and composition because all the relevant parameters are measured with the described set of experiments. The following picture of the diffusion process is in agreement with the experimental observations: At MC ) 20% (corresponding to Φc ) 76%), 23% of the absorbed water covers the cellulose microfibril surfaces and is considered as stationary and fast relaxing. There is a fast chemical exchange between water and 33% of the cellulosic hydroxyl protons. The self-diffusion coefficient of water is diminished by a factor of ∼0.8 because of hydration effects, and the tortuosity is 15 because of obstruction from the microfibrils. The smallest distance between microfibrils is 4 times the width of a water monolayer. Drying removes the nonhydrating water and deswells the fiber. At MC ) 10% (corresponding to Φc ) 87%), 46% of the water takes part in the hydration. The smallest distance between microfibrils is decreased to 1 layer of water, which is not enough to sustain fast chemical exchange. The hydration factor has decreased slightly to ∼0.5 whereas the tortuosity has increased 1 order of magnitude to 150 because of a constriction of the channels between and along the microfibrils. Acknowledgment. We are grateful to Natalia Markova for the isotherm and Bo Andre´asson and Boel Nilsson at SCA Research for the fiber analysis. This work was financially supported by the Swedish Foundation for Strategic Research (SSF). LA000982L