Through-Plane Diffusion of Moisture in Paper Detected by Magnetic

Ind. Eng. Chem. Res. 2002, 41, 6555-6565

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Through-Plane Diffusion of Moisture in Paper Detected by Magnetic Resonance Imaging Johannes Leisen,*,† Barry Hojjatie,‡ Douglas W. Coffin,‡ Sergiy A. Lavrykov,§ Bandaru V. Ramarao,§ and Haskell W. Beckham† School of Textile and Fiber Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0295, Institute of Paper Science and Technology, Atlanta, Georgia 30318-5794, and Empire State Paper Research Institute, SUNY-ESF, Syracuse, New York 13210

Magnetic resonance imaging (MRI) was used to measure the through-plane moisture distribution in paper samples. This information provides insight into the mechanism of moisture transport through the plane of paper sheets. The moisture distribution was measured as a function of time for a thick two-ply paper using two different experimental arrangements: (1) g95% relative humidity (RH) on one side and 0% RH on the other side and (2) g95% RH on both sides of the paper sample. The data clearly support a model in which moisture is transported in two stages: as vapor through the void spaces of the paper and subsequent sorption by the fibers on a longer time scale. Introduction Many of the mechanical properties of paper strongly depend on the moisture content.1,2 Because of the chemical structure of the cellulose used as one of the main components in all types of paper, paper has a strong tendency to absorb water which alters such properties as stiffness or strength.3 Thus, mechanical properties are a function of the overall moisture content. Additionally, if the moisture is not distributed uniformly, mechanical properties may vary across the product. For applications in which the mechanical performance is important, for instance, in packing materials, understanding how moisture is sorbed, transported, and spatially distributed should enable product performance prediction, product design, and property improvement. The relationship between the relative humidity (RH) in the surrounding ambient air and the overall moisture uptake of various types of paper has been reported.4,5 To achieve a theoretical understanding of these data, the sorption process, that is, the kinetics of moisture uptake, must be quantified. In addition, the interrelationship between sorption and diffusion or transport through a paper sample must be described. Paper is a porous material consisting of layers of cellulose fibers. Moisture transport through such a construction may occur by two mechanisms:6 (1) diffusion of vapor through the void spaces (interfiber diffusion) and (2) diffusion of sorbed moisture through the cellulosic fibers (intrafiber diffusion). Indeed, moisture transport can occur by either or both mechanisms simultaneously. For instance, a given water molecule could be transported for some time via diffusion through the void spaces and then absorb into a fiber and be transported via diffusion through the fiber. Ramarao has given a mathematical description of moisture * Corresponding author. E-mail: johannes.leisen@ tfe.gatech.edu. Tel: 404-894-9241. Fax: 404-894-9766. † Georgia Institute of Technology. ‡ Institute of Paper Science and Technology. § Empire State Paper Research Institute, SUNY-ESF.

transport in paper samples based on the two mechanisms.6 However, the relative importance of each of the two mechanisms has not been fully established through experiments. This is largely due to the lack of information on the moisture distribution within a paper sample. Moisture transport studies are primarily conducted using gravimetric methods, which provide very accurate information on the total moisture content within a paper sample but do not yield information on internal moisture distributions. Because theoretical models can be used to predict moisture distributions as well as the total moisture content, experimental verification requires information on both. Thus, discrimination between theoretical models should be largely assisted by experimental measurements of internal spatial distributions of moisture. Noninvasive measurements of spatially resolved moisture concentrations in opaque substrates can be accomplished with scanning neutron radiography7 and with magnetic resonance imaging (MRI).8 Based on the absorption of neutrons, scanning neutron radiography is costly and requires access to large-scale research facilities containing neutron sources. In addition, achievable resolutions (100 µm) are inferior to those commonly possible with MRI (10 µm). MRI is well established for visualizing spatial distributions of fluids in all kinds of materials.9 In fact, MRI has been employed to study paper drying.10-14 These studies, however, focused on pulp and paper samples with rather high moisture contents of over 100%. Investigating the sorption and diffusion of moisture from humid air requires the detection of much lower moisture contents, near 0% to about 20%, which is much more challenging. At these low concentrations, low signal-to-noise ratios typically require lengthy signal averaging sequences, during which time moisture transport may occur. For this reason, the approach described by Harding et al.14 is not well suited for real-time moisture sorption and transport measurements. In the study reported here, MRI was used to investigate moisture transport through paper with moisture concentrations from near 0% to about 20%. It will be demonstrated that MRI provides semiquantitative data

10.1021/ie0204686 CCC: $22.00 © 2002 American Chemical Society Published on Web 11/13/2002

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Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002

Figure 1. Experimental timing diagrams for the following: (a) the basic SE sequence. A magnetization echo is created using two rf pulses spaced by an adjustable TE/2. Recording of magnetic resonance images requires repetition of the basic sequence (as shown) separated by a repetition time, TR, using different settings of magnetic field gradients for subsequent sequences. (b) Variation of the basic sequence used for the calibration of image intensities with gravimetrically determined moisture contents. By subsequently recording the magnetization for a single-pulse excitation (no field gradients) and SE (field gradients set for an imaging experiment), the direct comparison of intensities may be used for construction of a calibration curve.

to complement gravimetric data, so that discrimination between different moisture transport mechanisms is greatly facilitated. Principles of MRI. The application of MRI to paper materials has been described.11,12 Therefore, only a brief outline of the technique is given here. We will focus on certain aspects necessary to understand the experimental challenges encountered in this study. More detailed information can be found in a variety of excellent references.8,9,15,16 To perform a MRI experiment, a sample is placed in a strong magnetic field. For certain nuclei (e.g., the hydrogen atoms of water molecules), this has the effect of creating discrete states separated by an energy increment that depends on the strength of the magnetic field. When energy is supplied to the sample, transitions between these discrete states occur, followed by a return to the pre-excited states. This energy is supplied in the form of electromagnetic radiation in the radio-frequency (rf) range. Where only one chemical species is present (for instance, the hydrogen atoms of water), the frequency of the rf radiation is given as

of such magnetic field gradients, Gr ) ∂B/∂r, in addition to the static field introduces a spatial dependence (r) to the magnetic resonance frequency:

ω(r) ) -γ[B0 + rGr]

(2)

Various experimental sequences exist in which the magnetic field gradients are applied with different orientations in conjunction with rf excitation to provide frequency data sets that are subsequently processed into magnetic resonance images.8 One of the most commonly used techniques is the spin-echo (SE) sequence consisting of two rf pulses (see Figure 1). This sequence is primarily characterized by two time delays, the echo time (TE) and the recovery time (TR). The lengths of these delays are variable experimental parameters that significantly influence the signal intensity for each pixel of the image. Contrast within SE images is governed by the relative intensities of the pixels, I(r), and generally given by the following expression:

(1)

I(r) ∝ F(r) F1[TR,T1(r)] F2[TE,T2(r)] F3[TE,G,D(r)] (3)

where γ is simply a proportionality constant that relates the frequency of the rf radiation (i.e., the energy between the nuclear states) to the strength of the magnetic field (B0, typically fixed for a given MRI scanner). A magnetic field gradient can be superimposed on the static field by additional coils, causing a linear spatial variation of the magnetic field. The application

The spin density, F(r), of MRI-active nuclei is the desired quantity for characterization of moisture distributions and diffusion. This value is attenuated by an amount depending on the local physical characteristics of the sample and the settings of the experimental parameters, as described by the last three terms of eq 3. These terms

ω ) -γB0

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002 6557

(F1, F2, F3) are called contrast functions because they describe the signal attenuation for each pixel resulting from variations in local physical characteristics across a sample. The first contrast function is that due to T1 relaxation, caused by interactions between the nuclei and the overall sample, the so-called lattice:17

F1[TR,T1(r)] ) 1 - exp[-TR/T1(r)]

(4)

Equation 4 is only valid for excitation pulses that tip the net magnetization by 90° (π/2 pulse). Signal attenuation due to F1 will disappear for values of the repetition delay TR . T1. Thus, it is always possible to completely eliminate the effect of F1 as long as sufficient time exists to perform a single MRI experiment. If the phenomenon under investigation occurs too rapidly, a compromise must be accepted between the temporal resolution of the measured image(s) and the magnitude of this contrast factor. The second contrast function is that due to T2 relaxation,17 caused by internuclear interactions:

F2[TE,T2(r)] ) exp[-TE/T2(r)]

(5)

Signal attenuation due to F2 can be minimized by setting TE as short as experimentally possible. However, when using the standard SE sequence in conjunction with pulsed magnetic field gradients, TE cannot be set short enough to completely eliminate the influence of F2 because of scanner hardware limitations. The third contrast function is that due to molecular diffusion or self-diffusion, the displacement of molecules via Brownian motion, or other types of incoherent mass transport, represented in eq 3 by D(r). Similar to F2, it is not possible to completely eliminate the effect of this contrast function. However, for samples containing low concentrations of bound fluid (with low self-diffusion coefficients), F3 should not contribute significantly to the signal attenuation. This is expected to be the case for paper samples containing low moisture contents in which diffusion of water bound to cellulose is restricted. Note that self-diffusion does not result in mass transport and is therefore not the type of diffusion that is monitored through the paper samples described below. Equations 4 and 5 are valid only for the simplest cases in which moisture exists in a single well-defined environment. Often, moisture in porous materials such as paper is distributed among environments defined by different binding strengths, which leads to more complicated expressions than those shown above. Generally, only liquid molecules are detectable with MRI. Gaseous molecules occur in concentrations too low for detection and often exhibit unfavorably long T1 relaxation times. Molecules in the solid state exhibit short T2 relaxation times. Consequently, for most solid materials, the F2 contrast factor in eq 5 approaches zero so that no signal is detected using standard SE imaging sequences. Specialized techniques, however, do exist for imaging gases and solids9 and also for imaging liquids while minimizing the effects of the contrast functions.18,19 In fact, a sequence referred to as single-point imaging (SPI) has been used to quantitatively image moisture concentrations in porous materials.20 However, these techniques did not provide a sufficient signal intensity to be used successfully for the work reported below.

Materials and Methods Paper Specimens. All experiments were performed using Formette Dynamique sheets formed from a southern pine bleached kraft (SWBK) pulp, refined to a Canadian Standard Freeness of 519 CSF in a Valley beater. Two 468 g/m2 sheets were formed and pressed between blotter papers at a pressure of 50 psi for 5 min to form a single two-ply sheet with a basis weight of 936 g/m2. The two-ply sheet was restrain-dried and then stored in a conditioned room at a constant RH of 50% and a temperature of 22 °C. The high-basis-weight sheets were needed so that moisture profiles in the through-sheet direction could be resolved with MRI. The average sheet thickness was 1.2 mm as measured using a caliper. Prior to the MRI experiments, circular samples with diameters of 1.7 cm were cut from the sheets and then dried at 70 °C for several hours to moisture regains of close to 0%. MRI Experiments. All experiments were performed in a magnetic field of 9.4 T using a Bruker DSX-400 NMR spectrometer with a microimaging accessory (Bruker Biospin GmbH, Rheinstetten, Germany). Standard SE imaging sequences8 were employed; TE and TR are given in the respective figure captions. Using a 25 mm rf coil, 90° pulse lengths of 35 µs were achieved. Experiments were performed using “hard” rf pulses with gradients switched off; thus, two-dimensional (2D) images correspond to 2D projections of the moisture concentration across the entire sample width. A matrix of 128 × 128 complex data points was recorded using 128 gradient steps and a spectral width of 100 kHz. The read gradient with a duration of 0.5 ms was aligned along the through-plane axis of the paper sheets. Data were multiplied with a squared sinusoidal function, a relatively strong apodization filter. The magnetic resonance images of Figures 4 and 5 were then obtained by Fourier transformation without further zero-filling. Because the through-plane moisture distribution is of interest, the magnetic field gradients were set to provide higher resolution in this direction. The field of view in the direction parallel to the paper surface (in-plane) was 34 mm, while the field of view in the direction perpendicular to the paper surface (through-plane) was 5.9 mm. MRI data were analyzed using the XWIN NMR and ParaVision software packages supplied with the spectrometer. Additional data processing was completed using a personal computer with IGOR Pro (Wave Metrics, Inc., Lake Oswego, OR). The simulation in Figure 8 was coded in FORTRAN 90 and performed on a personal computer. Experimental Setup for MRI Experiments. Moisture sorption and diffusion through paper were investigated using the experimental assembly shown in Figure 2. The edges of the dried, circular paper samples (1.2 mm thickness × 1.7 cm diameter) were sealed with Teflon tape. The samples were tightly affixed in the middle of the cylindrical sample cell. The lower part of the sample cell contained distilled water to obtain a RH near 100% on one side of the paper sample. The upper part of the sample cell was kept dry by placing a granular desiccant, anhydrous MgSO4, on the surface of the paper. Moisture uptake from both surfaces was observed by using an identical edge-sealed dry paper sample in a larger-diameter sample cell. Because this sample could not be tightly affixed against the walls of the chamber, a polymeric grid supported the paper sample in the

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Figure 2. Experimental assembly to observe moisture diffusion through a piece of paper. A circular paper sample was sealed on its edges with Teflon tape and fixed tightly in the sample chamber to act as a barrier between two regions with different RHs. Magnetic resonance images were measured as a function of time.

Figure 3. Signal intensities from the tandem sequence of Figure 1b measured as a function of gravimetrically determined moisture content. Both data sets were normalized at the maximum moisture content. The SE data (TE ) 1.4 ms and TR ) 3 s) were used as a calibration curve to relate signal intensities to actual moisture contents.

middle of the cell, which allowed moisture to pass to the upper part of the chamber from the distilled water reservoir in the lower part. Desiccant was not placed on the top surface of the paper. Instead, a piece of fabric was saturated with distilled water and fixed above the paper sample at the top of the sample chamber. For quantitative imaging, the contrast functions must be reduced to unity so that the measured signal intensity is directly proportional to the actual moisture concentration, that is, I(r) ∝ F(r). In practice, however, this is often difficult to achieve because the imaging sequences that allow such measurements typically return low signal intensities. The next best option is to use the standard SE sequence, which provides contrasted signal intensities, but establish the relationship between the measured signal intensity and the gravimetrically determined moisture concentration by measuring a calibration curve. For low moisture contents in paper below 20%, in which the water exists in a variety of environments, this semiquantitative approach is the most suitable. Such an approach has been demonstrated previously for MRI studies of moisture in porous catalysts.21 Note that this calibration method becomes ambiguous if the volume of the porous solid matrix changes upon absorption or desorption of moisture. This effect was not observed to be significant for the paper samples examined here. The calibration curve was created by imaging a wet paper sample (conditioned at 90% RH for 4 days) while it air-dried inside the MRI scanner. The sample was removed frequently to determine its weight. The calibration curve is shown in Figure 3 as the SE data fitted to a fifth-degree polynomial. The equation describing

this fit was used to convert experimental image intensities to actual moisture contents. Also shown in Figure 3 are data collected using a single-pulse excitation sequence. These data were collected in tandem with the SE calibration data using the pulse sequence shown in Figure 1b. The phase gradient was omitted from the SE portion of this sequence, which simply allows rapid collection of the calibration data; seconds are needed as opposed to measuring times that often approach 1 h for magnetic resonance images with acceptable signal-to-noise ratios. A 20° excitation pulse (θ in Figure 1b) was employed for the single-pulse portion to minimize the signal attenuation due to T1 effects and to enable the use of the same receiver gain setting for both signals (i.e., single-pulse and SE) from the tandem sequence. The single-pulse excitation data of Figure 3 exhibit a linear relationship with the gravimetrically determined moisture contents. Because the TE duration does not appear in the single-pulse excitation sequence, contrast functions such as F2 and F3 in eq 3 are irrelevant, and the detected signal intensity should be linearly proportional to the actual moisture content when the repetition delay is sufficiently long (i.e., TR . T1). Calibration data obtained with repetition delays of 2 and 3 s were identical within experimental error, evidence that TR ) 2 is indeed long enough to reduce the F1 attenuation function to unity. Comparison of the single-pulse excitation with the SE data in Figure 3 allows direct observation of the effects of the contrast functions. With decreasing moisture content, the SE signal attenuation becomes stronger so that virtually no signal is detected for moisture contents of less than 6%. This effect has been previously observed in paper,11,12 where it was attributed to shrinkage of the sample upon drying. However, the deviation from linearity in the calibration curve can be described simply as a T2 effect; during a drying process, the last moisture to leave exists in the smallest pores, interacts more strongly with the cellulose matrix, and consequently exhibits the shortest T2 relaxation times. This was confirmed by directly measuring the T2 relaxation times; signal intensities were recorded as a function of the TE duration using a SE sequence. Nonexponential relaxation behavior was clearly observed for moisture in paper conditioned at high levels of RH. The average relaxation time, T2, was found to decrease significantly with reduced moisture contents. Hence, the F2 contrast function in eq 3 is mainly responsible for the nonlinearity observed for SE signal intensities plotted as a function of the moisture content. Gravimetric Experiments. The moisture uptake was determined gravimetrically with an IGASorp gravimetric analyzer (Hiden Analytical Ltd., Warrington, U.K.) using a paper sample with exactly the same dimensions as the sample used for the MRI experiments. The edges of the circular sample were sealed with Teflon tape to ensure that moisture entered through the surfaces and not through the edges. The sample was suspended inside a chamber with welldefined temperature and RH. The initial RH and temperature were set to 0% and 22 °C, respectively, corresponding to the same initial conditions as those of the MRI experiments. After equilibrium was reached (determined when the sample weight reached a minimum), the RH level was set to 95% and the moisture uptake (i.e., sample weight) was followed as a function of time. A complete sorption isotherm was also recorded


Figure 4. Series of SE images depicting moisture ingress into a paper sample through one surface (TE ) 1.4 ms and TR ) 2 s). The physical dimensions of the paper sample are outlined on each image with dashed lines. The area below the sample was maintained near 100% RH, while the area above the sample was maintained near 0% RH by placing a granular desiccant directly on the paper sample.

Figure 5. Series of SE images depicting moisture ingress into a paper sample through both surfaces (TE ) 1.4 ms and TR ) 2 s).

at 22 °C by determining the sample weight from 5% to 95% RH in increments of 10% RH. Results and Discussion Qualitative Analysis. The series of magnetic resonance images shown in Figure 4 display moisture diffusion through a paper sample for which one side was maintained near 100% RH while the other side was maintained near 0% RH. These images exhibit the contrasted signal intensities; that is, they have not been corrected for the actual moisture content using the calibration curve. Hence, they only allow qualitative observation of water ingress into the paper sample. Because of a relatively poor signal-to-noise ratio, the time required to accumulate sufficient scans for each image was 68 min. Therefore, each image does not represent the moisture distribution at a specific discrete time but a time average. During the first 60 min, almost no moisture can be detected within the paper. Moisture with contents detectable by this MRI technique (>6%) appear in the paper sample between 68 and 136 min but only near the surface facing the humid environment. The moisture content, as reflected by image brightness, increases with increasing exposure time. No experimental evidence exists for a moisture front moving into the paper specimen. Moisture diffusion into a paper sample exposed to 100% RH on both surfaces is revealed by the series of MR images displayed in Figure 5. Compared to exposure

on one surface, higher overall moisture contents were achieved. Consequently, it was possible to measure MRI data at a rate of 17 min/image. The reduced signal intensities at the left and right edges of some images are simply a consequence of projecting the signal from a cylindrical sample onto a 2D plane. The reduced signal intensity at the center of some images (most notably in the 85-102 min image) is attributed to the nonuniform excitation profile of the rf coil. Signal intensities in the through-plane direction, the direction of interest, are not affected by either effect. Note that the MRI signal is due entirely to liquid water and not water vapor. The images of Figure 5 reveal that water is sorbed into all areas of the paper samples from the beginning of the experiment; there is no clear indication of a moisture front moving from the surfaces toward the inner regions. The absence of a clearly defined moving front into the samples suggests that moisture transport into this paper occurs initially via interfiber diffusion of vapor. Apparently, moisture vapor diffuses through the paper rapidly to establish a concentration profile governed by the moisture concentrations (i.e., the RHs) at the two different surfaces. As a secondary, slower process, moisture is adsorbed by the cellulose fibers from the vapor phase. While interfiber diffusion of moisture occurs by molecules in the vapor phase, intrafiber diffusion occurs by water molecules adsorbed onto cellulose fibers. Intrafiber diffusion is expected to occur with much slower diffusion constants

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Figure 6. Through-plane moisture profiles for ingress through both surfaces of the paper (left) and through one surface of the paper (right). The paper sample was initially dry, and profiles were collected as a function of time. (a) Simulated profiles by integrating eq 30 over the respective measuring times indicated in Figures 4 and 5. Ingress through both surfaces (left): c1 ) c2 ) 21%; ki ) 1.7 × 10-4 s-1. Ingress through one surface (right): c1 ) 14%; c2 ) 0; ki ) 1.3 × 10-4 s-1. (b) Calculated MRI profiles obtained from the simulated profiles of part a using the calibration curve (cf. Figure 3) relating the moisture content and MRI signal intensity and a resolution function describing the broadened MRI signals of bound water. (c) Experimental MRI profiles extracted from the center of the images shown in Figures 4 and 5.

than interfiber diffusion; therefore, it does not contribute significantly to moisture transport through the sample. Thus, the actual content of liquid water sorbed by the cellulose fibers is determined predominantly by the kinetics of moisture sorption by cellulose fibers. If moisture sorption occurs on a comparable or faster time scale than vapor diffusion, consequent swelling of fibers could reduce the void space and thereby lower the overall moisture transport through the paper. Another consequence of lowered vapor diffusion through the sample would be preferential moisture adsorption at the paper surfaces (where the moisture concentration is highest). This effect is somewhat visible in the 2D images of Figure 5; note the increased brightness near the surfaces of images labeled 68-85, 85-102, and 102119 min. Using the experimental arrangement described for the images measured in Figure 5, a sample was exposed to >95% RH for over 1000 min. Following this treatment, the water reservoir was replaced with a reservoir of heavy water. While heavy water is chemically very similar to water, it contains deuterons instead of hydrogen nuclei and therefore does not provide a detectable signal using a standard MRI scanner tuned

for hydrogens. Thus, it can be used to monitor the exchange of adsorbed water by simply following the reduction in image signal intensities. This experiment therefore provides a measure of moisture transport through a paper sample equilibrated at 100% RH. Beginning with the uniform moisture distribution depicted in Figure 5, exposure to heavy water led to a uniform signal reduction across the entire image until no signal was detectable. Because the sorption/desorption process was uniform, it must occur more slowly than diffusion of vapor through the sample; otherwise, the regions near the sample surfaces would exhibit signal reduction at a faster rate than interior regions. Thus, even for a sample equilibrated at >95% RH, moisture transport occurs predominately by interfiber vapor diffusion. Mathematical Modeling and Approximate Analysis. One-dimensional moisture profiles were extracted from the center of each MR image shown in Figures 4 and 5. These profiles are displayed in Figure 6c. Great care was taken to ensure that the extracted profiles coincide with the axis perpendicular to the plane of the paper. The edges of the profiles are not perfectly sharp because bound water gives rise to broad MRI signals


(i.e., short T2 values) and limits the spatial resolution. Despite these images, it is certain that a sharp transition exists between the air and the wet paper. Enhanced moisture concentrations near the paper surfaces are especially apparent in the extracted profiles shown in Figure 6c, left. Such enhanced moisture concentrations are not simply an artifact caused by truncation of the MRI time signal (in fact, apodization with a squared sinusoidal function ensures that this cannot occur); furthermore, images of the completely saturated paper samples (not shown) did not exhibit enhanced moisture concentrations near the paper surface. As discussed above, the MRI signal intensities do not correspond linearly to the actual moisture content. In fact, for moisture contents of 6% and less, the signal intensity is always zero (see Figure 3). In other words, it is impossible to distinguish moisture levels below 6% in these paper samples using SE MRI. Thus, calculation of actual moisture profiles from the experimental MRI profiles is wrought with ambiguity. A better approach is to calculate the MRI profiles for a variety of defined moisture profiles obtained by assuming different models of moisture sorption/diffusion into the paper. This can be easily accomplished using the calibration curve of Figure 3, followed by convolution with an appropriate line-broadening function (e.g., a Gaussian or Lorentzian line shape). Once a calculated set of MRI profiles matches the experimental MRI profiles, the operative moisture sorption/diffusion model is identified. Such comparisons of simulated to experimental data are commonly and often exclusively used for data analysis in the field of molecular science (e.g., scattering and NMR experiments).22 To demonstrate this approach, we develop a model for moisture transmission and consider a limiting case based on the observations discussed above. The concentrations of water vapor outside the surfaces, c1 and c2, are constant. Consider moisture inside the paper to reside as vapor of concentration c(t,z), where t is time and z is the spatial coordinate. Moisture is also present in its condensed phase inside the fiber matrix [represented by a locally volume-averaged field, q(t,z)]. Note that q(t,z) is the quantity measured by the MRI experiment. The transport of moisture by diffusion through both of the pathways is described by the following model:6,23-25

∂2 c ∂c + Fpki[qsat(c) - q] ) Dp 2 ∂t ∂z

[

]

∂q ∂q ∂ D (q) + ki[qsat(c) - q] ) ∂t ∂z q ∂z

(6) (7)

In this model, Dp represents the diffusivity of water vapor through the void space of the paper, Dq(q) represents the diffusivity of condensed water through the fibers and is allowed to be a function of the local moisture content, ki represents a rate constant for the local sorption of moisture, and qsat is the equilibrium value of the moisture content corresponding to the local concentration of water vapor, c. is the porosity of the sheet, and Fp is the sheet density. Thus, in this model, moisture transport is considered to occur by diffusion in the z dimension along two parallel pathways in the c and q forms with local interchange represented by the interchange coefficient or rate constant, ki. A definition of the local interchange coefficient, ki, has been derived

Table 1. . Parameters for Moisture Equilibrium (GAB Isotherm) and Diffusion in a Sample of Bleached Kraft Paper Board6 parameter

value for samples in the present experiments

value by Bandyopadhyay et al.6

MGAB KGAB CGAB Dp, cm2 s-1 Dq0, cm2 s-1 m ki, s-1

0.04035 0.82881 17.1241 5.0 × 10-3 8 × 10-9 465 1.6 × 10-4

0.0485 0.814 75.924 5.069 × 10-3 6.010 × 10-9 465.64 0.0035

by averaging the concentration fields within the paper structure by Ramarao.23 The paper structure is modeled as periodic cells with the representative elemental volume (REV) consisting of a single cylindrical pore surrounded by an annular fiber matrix. Using this model, Lavrykov and Ramarao derived a comprehensive definition of the interchange coefficient, ki, and showed its variation with the geometry of the REV as well as system parameters such as the diffusivities.26 Earlier work by Bandyopadhyay et al.6,24 showed that, by comparison of model predictions with experiments, ki is 0.0035 s-1 for a sample of bleached kraft paper board of a basis weight of 240 g/m2. The diffusivity of condensed moisture inside the fibers can be estimated from steady-state experiments as shown by Radhakrishnan et al.27 An exponential variation with the moisture content, q, represented by the equation shown below was found.

Dq ) Dq0 exp(mq)

(8)

The equilibrium moisture content, qsat, is a function of the humidity and thus the water vapor concentration, c. It was found by Bandyopadhyay et al. and Ramarao et al. that the Guggenheim-Anderson-de Boer (GAB) model for the isotherm can be effectively used for representing this relationship for transient moisture studies.24 The GAB equation is given as

qsat(c) ) GAB(c) )

CGABMGABKGABc (1 - KGABc)[1 - KGABc(1 - CGAB)] (9)

Table 1 shows the parameters determined for a bleached kraft paper board by Bandyopadhyay et al.6 To study the diffusion model in detail, we first scale the dependent and independent variables. We choose a diffusion time scale and length scale as the thickness of the sample. The concentrations are scaled based on the initial and expected final values. After the nondimensionalization, eqs 6 and 7 are

∂2C ∂C + Rβ[Qsat(C) - Q] ) 2 ∂τ ∂ζ

[

]

∂Q ∂Q ∂ F (Q) + β[Qsat(C) - Q] ) ∂τ ∂ζ Q ∂ζ

(10) (11)

where C is the dimensionless vapor concentration in the void space and Q is the dimensionless moisture content in the fibers. Note the appearance of the dimensionless groups R, β, and FQ, which is a scaled form of the diffusion coefficient for moisture through the fibers. The definitions of these groups are provided in the Symbols section. This nondimensional model has been provided

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by Bandyopadhyay et al., who analyzed its limiting cases in considerable detail.28 Because the paper specimens are initially dry, the initial condition is

C(τ)0,ζ) ) Q(τ)0,ζ) ) 0

Φn(ζ) ) x2sin(nπζ)

(12)

At the bottom of the sheet, the environment is assumed to be saturated such that the RH is 100%. In this case, the boundary condition is

C(τ,ζ)0) ) Q(τ,ζ)0) ) 1

where

Θn(τ) )

(14)

When symmetrical conditions are imposed, i.e., both surfaces of the sheet are exposed to saturated conditions, we have

C(τ,ζ)1) ) Q(τ,ζ)1) ) 1

(15)

Let us consider a simplification when the sorption isotherm can be linearized. Under this assumption, the equation for the sorption isotherm in dimensionless form is simply

Qsat(C) ) C

Ψn(τ) )

∂2C ∂C + Rβ[C - Q] ) 2 ∂τ ∂ζ ∂Q ) β[C - Q] ∂τ

( [

(17)

)] ]

m2em1τ - m1em2τ β 1+ m1m2 m1 - m2 m1τ

These two equations have been used by Ramarao and Chatterjee.25 Notice that, because they are linear, a solution can be found subject to the initial and appropriate boundary conditions (eqs 12-15). We used the method of finite Fourier transforms to convert the space domain (ζ) and subsequently the time domain (τ) into the Laplace domain (s) and then solved the resulting equations algebraically and inverted the Laplace and Fourier transformations. For more details of this approach, see work by Deen.29 The final solution of the equations for the nonsymmetrical case is given as

(23)

m1 ) -(β + Rβ + n2π2) + x(β + Rβ + n2π2)2 - 4β(nπ)2 2 (24a) m2 ) -(β + Rβ + n2π2) - x(β + Rβ + n2π2)2 - 4β(nπ)2 2 (24b) The lead terms in eqs 19 and 20 are equal to the Fourier series expansion of 1 - ζ. Thus, we obtain

Q(τ,ζ) ) 1 - ζ +

2πβ

C(τ,ζ) ) 1 - ζ +

2π

∞

(

n

∑

n)1m1 ∞

∑ n)1m

1

n

e1 τ

- m2 m1

-

)

em2τ m2

[ ( ) ( )]

- m2

em1τ 1 +

em2τ 1 +

β

m2

β

sin(nπζ) (25) -

m1

sin(nπζ) (26)

The steady-state solution for C can be approximated as

Css(ζ) ) 1 - ζ (18)

(22)

m2τ

m2e - m1e x2nπ β 1+ m1m2 m1 - m2

(16)

Furthermore, we note that the fiber diffusivity Dq is generally significant only at very high moisture contents. Therefore, we can neglect it for the region between 0 and about 80% RH.27 To obtain a simplification, we neglect the moisture diffusion term proportional to Dq, that is, FQ in dimensionless form in eq 11. Under these approximations, the system of two differential equations (10) and (11) simplifies into

)

x2nπ em2τ - em1τ + m1 - m2

(13)

For the nonsymmetrical case in which one surface of the sheet is dry,

C(τ,ζ)1) ) Q(τ,ζ)1) ) 0

[(

(21)

(27)

Note that this solution is obtained if the sink term in eq 10 (i.e., Rβ) is negligible. When steady state is attained for the vapor concentration profile in the void space, the fibers can still continue to absorb. The approach to steady state of the fiber moisture content is then exponential as required by eq 11. A simple solution of eq 11 is

Q(τ,ζ) ) (1 - ζ)(1 - e-βτ)

(28)

Reintroducing parameter dimensions results in

(

)

qsat,2 - qsat,1 [1 - exp(-kit)] L

q(t,z) ) qsat,1 + z

(29)

∞

C)

∑ Θn(τ) Φn(ζ) n)1

Q)

∑ Ψn(τ) Φn(ζ) n)1

(19)

∞

(20)

qsat,1 and qsat,2 are the equilibrium moisture contents at the surfaces of a sample with thickness L. Equations 27 and 28 are based on the simplifying assumption of a linear relationship between qsat and c (cf. eq 16). However, the more accurate relationship between the steady-state vapor concentration (eq 27) and the equi-


Figure 7. Moisture content in the paper sample determined gravimetrically as a function of time. The dashed line represents the RH of the sample chamber, which quickly stabilized at 95%. The solid line is an exponential fit to the data using a time constant, ki ) 1.3 × 10-4 s-1. Only the range for which RH ) 95% was fitted.

librium moisture content is given by eq 9, which can be easily incorporated into eq 29:

(

)

c2 - c1 [1 - exp(-kit)] (30) L

q(t,z) ) GAB c1 + z

Note that eq 30 is only valid under the assumption that the moisture uptake constant ki is independent of the actual moisture vapor concentration and the concentration of moisture sorbed into the paper. Moisture profiles calculated using eq 30 are shown in Figure 6a for two cases: (1) the symmetrical case, in which moisture concentrations at both sample surfaces are equal (c1 ) c2), and (2) the nonsymmetrical case, in which the moisture concentration at one surface is zero (c2 ) 0). The corresponding calculated MRI profiles are shown in Figure 6b. The time constant ki and to a small extent the concentration c1 (corresponding to the humid side of the sample) were adjusted to achieve the best fit to the experimental moisture profiles displayed in Figure 6c. The calculated profiles of Figure 6b agree reasonably well with the experimental profiles of Figure 6c. It is especially satisfying that both simulations returned moisture uptake constants ki that are experimentally indistinguishable: ki ) 1.7 × 10-4 s-1 when c1 ) c2 and ki ) 1.3 × 10-4 s-1 for the c2 ) 0 case. In fact, these values are very close to the one measured gravimetrically from the data displayed in Figure 7. In this experiment, a paper sample was exposed to 95% RH at both surfaces and the moisture uptake was measured as a function of time. Beginning at the point when 95% RH was reached in the sample chamber, the moisture uptake data were fit to the following equation:

h sat - Q h 0) exp(-kit)] Q h (t) ) Q h sat - [(Q

(31)

Q h is the total moisture content measured gravimetrically as a function of time, Q h sat is the total equilibrium moisture content, and Q h 0 is the total initial moisture content (i.e., when RH ) 95%). Equation 31 assumes the same exponential rate law as that used in eq 18 for the moisture uptake. Using eq 31, the best fit to the data shown in Figure 7 was achieved using a moisture uptake constant of ki ) 1.3 × 10-4 s-1, which is in excellent agreement with the moisture uptake constants used to fit the MRI profiles. The theoretical moisture uptake trajectory, according to eq 31, is shown as a solid line in Figure 7. An apparent deviation exists between experiment and theory, suggesting that the moisture uptake constant is a function of the amount of sorbed moisture. This was examined by measuring the gravi-

Figure 8. Through-plane moisture profiles for the symmetrical case of moisture ingress through both surfaces of the paper (c1 ) c2) as calculated by numerically solving eqs 10 and 11, subjected to the conditions defined by eqs 12, 13, and 15. The parameters for this calculation are given in Table 1. (a) Simulated profiles representing the average moisture concentration during the measuring times indicated in Figure 5. (b) Calculated MRI profiles obtained from the simulated profiles of part a using the calibration curve (cf. Figure 3) relating the moisture content and MRI signal intensity and a resolution function describing the broadened MRI signals of bound water.

metric moisture uptake kinetics while increasing the RH in small steps. The average sorption constant, ki, increases with increasing RH up to about 60%. For higher RHs, ki decreases significantly. This observation may be explained by a combination of sorption mechanisms including direct sorption of moisture by fibers at low RHs, combined with capillary condensation at high RHs, as was recently suggested from NMR studies of paper samples by Topgaard and So¨dermann.30 Even though eq 30 provided a reasonable fit to the experimental data, some of the features of the MRI profiles are not reproduced by the simulations shown in Figure 6. For example, the MRI moisture profile for the nonsymmetrical case (Figure 6c, right) is shorter along the spatial dimension than the simulated profile. Inability to observe moisture contents below about 6% is the most likely explanation for these shortened profiles. Another feature of the MRI moisture profiles that is not reproduced by the simulations using eq 30 is the dip in the center of the experimental profiles for c1 ) c2 (Figure 6c, left). This dip reflects reduced moisture concentrations in the center of the paper sheet and may be taken into account if moisture sorption and intrafiber diffusion are considered along with interfiber diffusion through voids. To obtain a more accurate estimate, we solved the complete model represented by eqs 10 and 11 numerically, subjected to the conditions (12)-(15). We obtained best-fitting values of the four parameters Dp, Dq0, m, and ki (cf. Table 1). Figure 8 shows the best results achieved with the above model to fit the symmetrical case. The dip in the center of the experimental moisture profiles has been successfully reproduced. The model reproduces the overall moisture content reasonably well, and the value of ki is in good agreement with the moisture uptake constant used in eq 30. The simulated profile of Figure 8 provides a good

6564


description for the experimental data of the symmetrical case. Any deviations between experiment and simulations are within the experimental error of the MRI experiment, which are primarily embodied in the calibration curve of Figure 3. Conclusions It has been demonstrated that MRI can be used to measure through-plane moisture distributions in paper and thereby provide useful insight into the mechanism of moisture uptake. At low concentrations of water bound to cellulose, MRI SE data are best analyzed by comparing experimental MRI profiles to simulated profiles based on a given moisture sorption/diffusion model. For the two-ply paper investigated here, moisture diffuses rapidly into the sample as a vapor; steady-state vapor concentrations across the sample thickness are established quickly and are governed by the RHs at the two paper surfaces. Sorption of the vapor by the cellulose fibers occurs as a slower process. Intrafiber moisture diffusion plays a noticeable role in the transport of moisture through the paper sample at long times. Some features of the experimental moisture content profiles are only reproduced if this intrafiber moisture diffusion is included in the model.

q ) moisture content of fibers (mass of water/mass of dry fibers) q0 ) initial moisture content of fibers qsat ) equilibrium moisture content of fibers at constant vapor concentration c q1, q2 ) moisture contents in fibers on surfaces (1 and 2) of the paper sheet Q ) dimensionless moisture content of fibers ) (q - q0)/ (qsat - q0) Q h ) total moisture content measured gravimetrically r ) spatial coordinates in MRI experiments t ) time T1 ) spin-lattice relaxation time in MRI experiments T2 ) spin-spin relaxation time in MRI experiments TE ) echo delay in MRI experiments TR ) repetition delay in MRI experiments z ) thickness coordinate R ) Fp(q1 - q0)/(c1 - c0) β ) kiL2/Dp γ ) gyromagnetic ratio in MRI experiments ) porosity of the paper sample Fp ) density of the paper sample F ) spin density as measured by MRI ζ ) dimensionless distance ) z/L τ ) dimensionless time ) Dpt/L2 ω ) frequency in MRI experiments

Literature Cited Acknowledgment The research described here was supported through a seed grant program jointly administered by Georgia Tech and the Institute of Paper Science and Technology. Access to NMR instrumentation has been made possible by a NSF DMR instrumentation grant (DMR-9503936). Symbols B0 ) magnetic field c ) water vapor concentration inside the sheet c0 ) initial concentration of water vapor in the sheet c1,2 ) concentration of water vapor on surfaces (1 and 2) of the paper sheet C ) dimensionless vapor concentration ) (c - c0)/(c1 - c0) CGAB ) constant in the GAB equation, describing the relationship between qsat and c Dp ) diffusivity of water vapor in the void spaces of the sheet Dq ) diffusivity of condensed moisture through a fiber matrix Dq0 ) diffusivity of condensed moisture through a fiber matrix, limiting value for a dry sheet D(r) ) molecular self-diffusion coefficient Fi ) contrast function i (i ) 1-3) in MRI experiments FQ ) dimensionless diffusion term for moisture through fibers Gr ) magnetic field gradient in MRI experiments I ) signal intensity as measured by MRI ki ) coefficient describing the rate of moisture uptake to the equilibrium value qsat(c), also referred to as the fiber moisture interchange or exchange coefficient KGAB ) constant in the GAB equation, describing the relationship between qsat and c (measured in units of %RH) L ) thickness of the paper sheet m ) exponent in the equation for condensed phase diffusivity MGAB ) constant in the GAB equation, describing the relationship between qsat and c

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Received for review June 21, 2002 Revised manuscript received September 24, 2002 Accepted September 26, 2002 IE0204686

Through-Plane Diffusion of Moisture in Paper Detected by Magnetic

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