PROTON MAGNETIC RESONANCE OF WOOD AND WATER IN WOOD. Cynthia D. Araujo B. Sc. (Physics) University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1993 © Cynthia D. Araujo In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia 1956 Main Mall Vancouver, Canada Date: Abstract Proton nuclear magnetic resonance ( 1 1-1 NMR ) was used to investigate protons in solid wood and compartmentalized water in the wood cell walls and lumens. A lineshape second moment study found the second moment of protons in ovendry wood to be about 23% lower than the rigid lattice calculation, indicating a rigid structure with some anisotropic molecular motion of the polymeric constituents. Above 5% moisture content, the second moment decreased by a further 13 to 16% implying a "loosening" of the molecules in the solid with the increased moisture content. The T2 of the cell wall water was found to be single exponential and increased with moisture content. The 1 1-1 NMR measured fibre saturation point of the cell wall water agreed with the value calculated from the moisture isotherm. Two T2 techniques for characterization of water in wood are demonstrated. First a technique for analysing multi-exponential relaxation in terms of a continuous distribution of relaxation times was applied to T2 analysis of lumen water in wood. The lumen water T2 times vary as a function of the wood cell radius and are therefore expected to reflect the cell size distribution, which is continuous. A technique of selectively imaging water environments on the basis of T2 was applied for a range of moisture contents. The moisture density profile of the hound water was found to be independent of moisture content above the fibre saturation point. Spin- spin relaxation measurements of lumen water in wood were interpreted using a diffusion theory which models the lumen water T2 relaxation in terms of the cell radius distribution, the bulk water diffusion coefficient and a surface relaxation parameter. Agreement between theory and experiment was excellent. Evidence was found for the existence of higher order T2 relaxation modes ii predicted in the slow diffusion regime, using a sample with rather large cell lumens and at low temperatures. Using this diffusion model, T2 relaxation decay data were fitted to give a cell size distribution, comparable to scanning electron microscope results, when the bulk water diffusion coefficient and the surface relaxation parameter were known. A two region diffusion model was considered with free water in the cell lumens and water in the cell walls. The surface relaxation parameter was found to depend on the spin-spin relaxation time and diffusion coefficient of the cell wall water. Consequently, the cell wall water diffusion coefficient may be estimated from spin—spin relaxation times and the relative populations of lumen and cell wall water. The cell wall diffusion coefficient of maximum hydrated redwood sapwood was found to be 0.2 x 10' m 2 /s at room temperature, and from the temperature dependence the activation energy was found to be 6700 cal/mol, about 40% higher than the free water value. Numerical simulations of the two region diffusion model were developed. The lumen water T2 was found to be independent of the simulated cell wall thickness, simplifying to a surface relaxation as modeled with the surface relaxation parameter in the one region model. The simulated effect of exchange on the fibre saturation point measurement was found to be an over estimate compared to experimental results. Three techniques were used to investigate the spin-lattice relaxation of the solid wood and the water in wood. Separate T 1 measurements of the solid and water, separate T1 measurements of water in the earlywood and latewood regions, and separate T1 measurements of the cell wall water and lumen water were acquired. The results indicated that, on the T 1 time scale of 100 ms, all proton environments are mixed by diffusion of the water. The T1 of the water in the lumen and the cell wall and the protons of the solid were found to have the same T1 , which is an average of the T 1 of the three environments. The T1 was found to be dependent on the proportion of cell wall to lumen volume. Thick walled latewood cells had a lower T 1 than thin walled earlywood cells. Lastly, the cross relaxation of the protons in solid wood and the cell wall water was found to be the dominant mechanism for 7'2 relaxation of the cell wall water. iv PAGINATION ERROR.^ ERREUR DE PAGINATION. TEXT COMPLETE.^ LE TEXTE EST COMPLET. NATIONAL LIBRARY OF CANADA.^BIBLIOTIIEQUE NATIONALE DU CANADA. CANADIAN THESES SERVICE. ^ SERVICE DES THESES CANADIENNES. Table of Contents ii Abstract^ List of Tables^ vi List of Figures^ ix Acknowledgement 1 1 Introduction 1.1 Motivation ^ 1 1.2 Wood and Water in Wood ^ 2 1.2.1^Structure of Wood ^ 2 1.2.2^Cell Wall Composition and Structure ^ 8 1.2.3^Water in Wood ^ 1.3 1.4 2 1 1-I NMR and Wood ^ 12 14 1.3.1^1 11 NMR Relaxation Theory ^ 14 1.3.2^Review NMR work of water in wood ^ 17 Overview of Thesis ^ 19 General Materials and Methods 21 2.1 Samples ^ 21 2.2 SEM Images 21 2.3 SEM Moisture Content Measurement ^ 22 2.4 NMR Equipment ^ 22 ii ^ 3 4 2.5^NMR M2 and Moisture Content Measurements ^ 23 2.6^T2 Relaxation ^ 24 Second Moment and Cell Wall Water T2 26 1 3.1 Summary ^ 26 3.2 Introduction ^ 26 3.3 Materials and Methods 27 3.4 Results and Discussion ^ 28 3.5 Concluding Remarks ^ 36 T2 Techniques for Characterization of Water in Wood 37 4.1 Summary ^ 37 4.2 Introduction ^ 38 4.3 Materials and Methods ^ 38 4.3.1^Samples 38 4.3.2^One-dimensional Imaging ^ 39 Results and Discussion ^ 40 4.4 4.4.1^Fits of T2 Relaxation Data for White Spruce 4.5 5 2 ^ 40 4.4.2^One-dimensional Imaging ^ 44 Concluding Remarks ^ 51 Diffusion Model of Compartmentalized Water 3 53 5.1 Summary ^ 53 5.2 772 Relaxation of Compartmentalized Water: Lumen Water in Wood 54 5.2.1^Introduction 54 ^ chapter closely follows a previously submitted paper. [Araujo et a1.1993b]. chapter closely follows a previously published paper. [Araujo et a1.1992] 3 Part of this chapter closely follows, with some modifications, part of a previously published paper. [Araujo et a1.1993] 1 This 2 This iii 5.2.2^Theory ^ 5.2.3^Materials and Methods 5.3 6 ^ 6.2 62 5.2.5^Concluding Remarks ^ 74 Recovering Compartment Sizes from NMR Relaxation Data ^ 77 81 81 Determination of the Cell Wall Water Diffusion Coefficient in Wood 6.2.1^Introduction 7 4 Summary ^ from 12 Relaxation Measurements ^ 6.3 62 5.2.4^Results and Discussion ^ Diffusion Model of Two Regions of Compartmentalized Water 6.1 54 ^ 82 82 6.2.2^Results and Discussion ^ 87 6.2.3^Concluding Remarks 92 Numerical T2 Simulations ^ 94 6.3.1^Numerical Method ^ 94 6.3.2^Numerical Applications ^ 99 Spin-lattice Relaxation and Cross Relaxation 102 7.1 Summary ^ 102 7.2 Introduction ^ 103 7.3 Materials and Methods 103 7.3.1^Samples ^ 103 7.3.2^Spin-lattice Relaxation of Wood and Water in Wood ^ 104 7.3.3^Spin-lattice Relaxation of the One Dimensional Water Image ^. . 104 7.3.4^Two Dimensional 12 - T1 Dependence of Water in Wood^. . . . 105 4 Part of this chapter closely follows, with some modifications, part of a previously published paper. [Araujo et a1.1993] iv 7.3.5 Cross Relaxation of Protons in Solid Wood and Water ^ 105 7.4 Results ^ 106 7.4.1 Spin-lattice Relaxation of Wood and Water in Wood ^ 106 7.4.2 Spin-lattice Relaxation of the One Dimensional Water Image ^109 7.4.3 Two Dimensional T2 - T, Dependence of Water in Wood^110 7.4.4 Cross Relaxation of Protons in Solid Wood and Water ^ 110 7.5 Discussion ^ 116 7.6 Numerical T1 Simulations ^ 121 7.7 Cross Relaxation and T2 ^ 7.8 Concluding Remarks ^ 123 125 Bibliography^ 126 A Field Gradient NMR^ 131 B Diffusion Model for Rectangular Geometry^ 135 C Two Regions with Cylindrical Geometry^ 138 D Two Regions with Spherical Geometry^ 141 E Numerical T1 Simulations^ 144 List of Tables 1.1 Cell Wall Composition in Wood ^ 3.1 Transition moisture content and average M 2 's with standard deviations. 5.1 Geometric dependence of the fundamental T2 mode, T2(o) (for a fixed 9 31 surface sink parameter, M), and of M (for a fixed T2 ( o )), for cells with a cross-sectional area of 2826,am 2 (T2free 5.2 = 1.4s and D = 2.2 x 10 -9 m 2 /s). 61 Fit of the diffusion model to the CPMG decay curves for the redwood sample. ^ 72 5.3 White Spruce Results 73 6.1 Cell wall water T2, fibre saturation point, moisture content, surface sink parameter and cell wall water diffusion coefficient for redwood sapwood at temperatures 4 to 55°C. ^ 6.2 Numerical T2 Simulations ^ 7.1 Fast Exchange Model of T1 for typical cedar cells. ^ vi 89 99 119 List of Figures 1.1 Earlywood (large cells) and latewood tracheid cells (small, thick walled cells) in a softwood. ^ 3 1.2 Ring-porous hardwood. 1.3 Diffuse-porous hardwood. ^ 6 1.4 Logslice of a softwood ^ 7 1.5 Diagram of two cellulose repeat units. 9 1.6 Cominom forms of lignin. ^ 1.7 SEM showing the direction of the fibrils in the Si, S2, and S3 layers in ^ ^ 5 10 a earlywood tracheid (left side) and a latewood tracheid (right side) cell walls. ^ 11 12 1.8 Pit pairs. 1.9 Moisture Isotherm^ 13 3.1 Lodgepole Pine Isotherm. ^ 29 3.2 Free Induction Decay. 31 3.3 Second Moments of Protons in Solid Wood versus Moisture Content. . . 3.4 Moisture Content (ovendry method) versus NMR Moisture Content of ^ ^ lodgepole pine heartwood ^ 3.5 3.6 32 33 T2 Distributions of Water in Lodgepole Pine at a Moisture Content of 11.3% ^ 34 Lodgepole Pine cell wall T2 versus Moisture Content ^ 35 vii ^ 4.1 Relative numbers of cells and protons as a function of cell lumen radius for the white spruce sapwood and heartwood compression wood samples. 41 4.2 CPMG data and the fit from a continuous T2 spectrum. 42 4.3 T2 results as a continuous curve and as a discrete fit of 2 exponential components. ^ 42 4.4 Continuous T2 spectra of the water in white spruce juvenile wood and heartwood. ^ 44 4.5 Continuous T2 spectra of the water in rehydrated samples of white spruce heartwood compression wood, heartwood, and heartwood with incipient decay. ^ 45 4.6 Radial Moisture Density Profile of the total water distribution. ^ 46 4.7 Radial Moisture Density Profile of the cell wall water distribution. ^48 4.8 Radial Moisture Density Profile of the lumen water distribution.^49 5.1 The first 3 A n ,10 (71„r1R) modes 57 5.2 In amplitudes as a function of MR/D ^ 5.3 The diffusion model gives T2( „ ) versus cell radius ^ 58 59 5.4 The diffusion model gives T2( „ ) versus diffusion coefficient ^ 60 5.5 The diffusion model gives T2( „ ) versus surface sink parameter ^ 60 5.6 Radius distributions of redwood sapwood, white spruce sapwood, and white spruce heartwood compression wood samples. ^ 63 5.7 The CPMG decay curve of redwood at 18°C. ^ 64 5.8 T2 relaxation plots for the redwood sample. ^ 65 5.9 Average T2 of lumen water for the redwood sample.^ 66 5.10 T2 plots for white spruce sapwood, and rehydrated white spruce heartwood compression wood samples^ viii 67 ^ 5.11 The first 3 modes and the sum of the first 3 modes predicted for the redwood radius distribution at 4°C. ^ 71 5.12 Redwood Radius Distribution from NMR ^ 78 5.13 White Spruce Radius Distributions from NMR ^ 79 5.14 Alder and Douglas Fir Radius Distributions from NMR ^ 80 6.1 Two region diffusion problem ^ 83 6.2 Cell wall water decay curves of redwood sapwood. ^ 88 6.3 T2 relaxation plots of cell wall water decay curves from redwood sapwood. 89 6.4 Calculated cell wall water diffusion coefficient for redwood sapwood. ^. 90 6.5 Discrete steps in position and time. ^ 95 6.6 Lumen and Cell Wall Water Exchange. ^ 100 6.7 NMR Measurement of FSP 101 7.1 r dependence of FID in inversion recovery experiment ^ 107 7.2 T1 plots for liquid and solid signals. ^ 108 7.3 r dependence of amplitude images in modified inversion recovery experiment. ^ 109 7.4 Amplitude and T1 images at 4 moisture contents ^ 111 7.5 T1 -T2 plots for high moisture contents ^ 112 7.6 T1 -T2 plots for low moisture contents. ^ 113 7.7 The FID following the cross relaxation sequence. ^ 114 7.8 Reappearance of the solid signal following the cross relaxation sequence ^ 115 7.9 Cross section of a cylindrical c ell^ 117 7.10 Simulations of spin-lattice relaxation in cylindrical cells. ^ 122 7.11 Spin-lattice decay of the protons in solid wood. ^ ix 124 Acknowledgement I thank my supervisor Alex MacKay for his enthusiasm and guidance, Myer Bloom for being my professor for many years and sharing his lab, Elliott Burnell for his involvement in the teaching of NMR theory, Stavros Avramidis for the valuable collaboration, and all the people, prior and present from Lab 100, whose work and friendship have made this thesis possible and enjoyable. I also thank my first family, Mom, Dad and Shelly for their encouragement. I will always be thankful to my husband Dan for his understanding, caring and patience. Chapter 1 Introduction 1.1 Motivation The interpretation of 1 H NMR relaxation of water in biological systems is complicated by a number of factors including heterogeneity, chemical exchange, paramagnetic solutes, compartmentalization, magnetic susceptibility, and diffusion [Mathur—De Vre 1979, Belton and Ratcliffe 1985]. Although it has much of the complexity of general biological systems, wood is a better candidate for quantitative 1 11 NMR studies since its regular and robust structure makes it easy to work with, and simplifies the interpretation. Proton magnetic resonance is a useful tool for wood science since signals from the solid, cell wall water and lumen water can be distinguished to give the moisture content and allow investigation of the separate proton environments and the interaction between these environments. In this thesis, I have been able to i) show that the fibre saturation point of the cell wall water can be measured accurately, ii) image the cell wall moisture profile for a full range of moisture contents, iii) recover cell lumen size distributions iv) measure the cell wall water diffusion coefficient and activation energy, and v) measure the cross relaxation of water and solid wood in the cell wall. These techniques could be used to study wood drying and be applied to species identification. Diffusion of water in wood has played an important role in mixing water environments and the effect of diffusion on the relaxation times has been modeled in this thesis. Understanding diffusion as a mechanism of mixing water environments on 1 Chapter 1. Introduction^ 2 the 1 H NMR time scale is necessary for interpretation of relaxation data of biological systems [Belton and Ratcliffe 1985], and the qualities of wood make it ideal for diffusion studies. Diffusion of water in wood is similar to that in human tissue, however the wood geometry is simple and measurable by scanning electron microscopy and the cell walls act as a diffusion barrier, whereas tissue cells are in general permeable to water. Proton magnetic resonance has been successful in three dimensional imaging (MRI) applied to human tissue, but relaxation measurements applied to more medically relevant biological systems have rarely proven to be diagnostic [Bottomley et a/.1987]. The methods and approach developed for water in wood can be applied in tissue to study tissue pathology, and the state of water. 1.2 Wood and Water in Wood 1.2.1 Structure of Wood Trees are divided into two categories: Hardwoods and softwoods, based on the characteristics of their seeds. Softwoods have needle like leaves and are commonly called evergreens. The seeds of softwoods are basically unprotected, usually inside scaly cones, and are therefore often referred to as conifers. Hardwoods have broad leaves, which generally change color and drop in the autumn in temperate zones: Hardwoods produce seeds within acorns, pods or other fruiting bodies. The woods formed by softwoods and hardwoods contain different types of cells. Hardwoods are not necessarily more dense or harder than softwoods, as their names imply [Haygreen and Bowyer 1982]. In softwoods, there are three main reservoirs for lumen water: earlywood and latewood tracheids, which are oriented longitudinally along the tree trunk, and ray cells which are oriented radially. The earlywood tracheids, which grow during the spring and early summer, are larger in radius than the latewood tracheids, which grow in Chapter 1. Introduction^ Figure 1.1: Earlywood and latewood tracheids in redwood, a softwood.x85.[Haygreen and Bowyer 1982] 3 Chapter 1. Introduction^ 4 late summer and autumn. The latewood tracheids have thick cell walls and give the latewood region a higher solid wood density. The alternation of these cell types gives the appearance of annual growth rings, as shown in Fig. 1.1. Tracheids vary in size from species to species ranging from 5 to 50 pm in radius and are 4 to 5 mm long. Ray cells typically contain only about 5-7% of the total lumen volume and have radii of about 8 pm. The tracheid cell walls are one to several microns thick and provide a diffusion barrier for water, since the translational diffusion coefficient of cell wall water is about one order of magnitude less than for free water in the cell lumens [Siau 1984]. In hardwoods, there are four main reservoirs for lumen water: vessels, fibers and longitudinal parenchyma cells which are oriented longitudinally along the tree trunk, and ray cells which are oriented radially. Short, large diameter cells joined end to end with perforated unrestricted holes are called vessels or pores and are unique to hardwoods. Vessels give rise to two types of hardwoods, as shown in Figs. 1.2, and 1.3. Ring-porous hardwoods have smaller vessels in the latewood and diffuse-porous hardwoods have uniform vessel sizes. Fiber cells are simular to softwood tracheids, but are shorter, only 1 mm in length, and are rounder in cross section. Longitudinal parenchyma cells are thin walled storage units, divided by cross walls. Ray cells are found either alone or in groups up to 30 cells wide. Ray cells make up about 17% of the total lumen volume and sometimes more than 30%. Ray cells are distorted from radial orientation in the vicinity of large vessels. Figure 1.4 shows a cross-section log slice of a softwood. A log slice of a hardwood is similar, except that often the growth rings are less distinct. Cell division occurs in an outer layer called the cambium. The cells become part of the bark or part of the wood. The newer cells are referred to as sapwood, and in a freshly cut or green sample these cells are filled with water. In the heartwood region all cells have died, losing their nuclei and protoplasm. Cells in the heartwood have lower moisture content, especially Chapter 1. Introduction^ Figure 1.2: Ring-porous hardwood, northern red oak. x 35. [Haygreen and Bowyer 1982] 5 Chapter 1. Introduction^ Figure 1.3: Diffuse-porous hardwood, yellow poplar. x 80.[Haygreen and Bowyer 1982] 6 Chapter 1. Introduction^ LATEWOOD PITH EARLY WOOD ANNUAL INCREMENT ( GROWTH RING) Figure 1.4: Cross section of tree showing various wood zones. [Bramhall and Wellwood 1976] 7 Chapter 1. Introduction^ 8 in softwoods, and are usually high in extractives, which are the decomposition products of starches and sugars giving fats, waxes, oils, resins, gums, tannin, and aromatic and colouring materials [Haygreen and Bowyer 1982]. The sapwood is thought of as the cells storing and conducting water and minerals, and the heartwood gives the tree mechanical support. A juvenile region, the first 5-20 central growth rings, contains cells from early stages of the tree's existence. These cells are shorter and have fewer latewood cells than mature wood cells, although there is no abrupt division to mature wood. The high proportion of thin walled cells having low wood density in juvenile wood gives it lower strength than mature wood. Reaction wood forms in both softwoods and hardwoods. In softwoods, reaction wood forms on the bottom side of leaning stems and branches and is called compression wood. Compression wood cells are typically 30% shorter, have 10% less cellulose and 89% more lignin and hemicellulose. Compression wood has a high proportion of latewood and often forms wide growth rings so that growth is eccentric about the pith. In hardwoods, reaction wood forms on the top side of leaning stems and branches and is called tension wood. Tension wood has fewer and smaller vessels, fewer ray cells, and thick walled, small lumen fibers. The secondary cell wall layer is almost pure cellulose and loosely connected to the primary wall. Tension wood also has wide growth rings so that growth is eccentric about the pith. One cause of reaction wood formation has been shown to be gravity, not stress [Haygreen and Bowyer 1982], and it is the effect of gravity on the distribution of growth stimulators (auxins) which is thought to induce reaction wood formation. 1.2.2 Cell Wall Composition and Structure Wood is mainly composed of cellulose, hemicellulose and lignin [Haygreen and Bowyer 1982]. Cellulose is a long straight chain of glucose molecules. Figure 1.5 shows two Chapter .1. Introduction^ 9 repeat units of cellulose, and 5000 to 10000 units form cellulose making it at least 5/cm long and 8A in diameter [Aspinall 1983]. In wood 60-70% of the cellulose is hydrogen bonded to form crystalline cellulose. A single cellulose chain may extend into several crystalline and amorphous regions and in this way restrict the swelling of the structure. Hemicelluloses are low molecular weight branch chain polymers of various C1 H20 H^ C F71 H C‘ NI\ OH \ 1 C 1 H CH2OH I 0\^ C^0\ H \r--,,.^ 1,/^H^\i'-• C C^0_^C H^-------] OH H^i H 1 H C C C 1 1 i OH H OH /I Figure 1.5: Diagram of two cellulose repeat units. types of sugars, which form a matrix in the wood and are associated with the cellulose and the lignin. Lignin is made up of high molecular weight polymers of phenylpropane units, as shown in Fig. 1.6, which are the binding agent and make the cell walls rigid. For example, cotton is about 99% cellulose and yet it is non-rigid and so lignin acts as a stiffening ingredient. Lignin stiffens the structure, but it is the cellulose that gives it its strength. These composites arrange into long bundles called microfibrils of a few nanometers Table 1.1: Cell Wall Composition in Wood Cellulose Hemicellulose Lignin Softwood Hardwood 40-44% 40-44% 20-32% 15-35% 25-35% 18-25% Chapter I. Introduction^ 10 Hardwoods Softwoods 1 1 C HN , ^ C— v 1-1 NC 7H \ \ c7 N H OCH, CH 30 7 c^OCH 3 O Figure 1.6: Commom forms of lignin. in diameter. Hemicellulose and lignin bind microfibrils together to form the cell wall, which is found to be composed of distinct layers, as shown in Fig. 1.7. The S1 and S3, outer and inner layers, respectively, are formed with their fibrils wound perpendicular to the long axis of a cell. The centre layer, S2, is the thickest and its fibrils are aligned along the axis of the cell. The S2 layer is much thicker in the latewood cells compared to the earlywood cells. Cells in both softwoods and hardwoods are joined by pits, either simple, bordered or half-bordered, as described in Fig. 1.8. Pit regions are thin spots in the cell wall and may have an overhanging border. Softwood bordered pits, found between tracheids, usually have an additional thickening of the central pit membrane, called a torus. A net of radially arranged microfibrils may form around the torus, called the margo. Western redcedar is one known softwood that lacks the tori in bordered pit membranes. In drying, softwood bordered pits with tori become aspirated so that the opening is blocked, preventing normal moisture conduction through cells. Chapter 1. Introduction ^ Figure 1.7: SEM showing the direction of the fibrils in the Si, S2, and S3 layers of the cell wall [Siau 19S4]. 11 Chapter 1. Introduction^ 12 Figure 1.8: Profile of various types of pit pairs.[Haygreen and Bowyer 1982] 1.2.3 Water in Wood In green wood, water inside the cell walls, is typically 25 to 30% of the mass of the solid wood. It is well known [Siau 1984] that as wood dries, lumen water comes out first followed by the cell wall water. As cell wall water is removed the wood shrinks, typically by about 7 to 17% in volume. A number of physical properties of wood [Haygreen and Bowyer 1982] are independent of moisture content at higher moisture contents but are found to change as the bound water is removed from wood. For example: i) The heat required to evaporate a unit weight of lumen water from wood is the heat of vaporization of free water while there is an extra amount of heat, the heat of wetting, that is needed to remove all the bound water [Stamm 1964, Bodig and .Jayne 1982, Siau 1984], ii) the mechanical strength of wood increases as the bound water is removed, but is not affected by the amount of lumen water, iii) the thermal Chapter I. Introduction^ 13 and electrical conductivities of wood are also altered by the removal of bound water but are independent of the amount of lumen water. The moisture content at 100% relative humidity is defined as the fibre saturation point (FSP), and it is assumed that the cell walls are saturated and that there is no free water in the lumens. But, more practically, the FSP is defined as the moisture content where abrupt changes in the physical properties of wood occur [Stamm 1964]. 25 ^ 20^40^60^80^100 I-1 (%) Figure 1.9: Moisture isotherm showing equilibrium moisture content (M) as a function of relative humidity (H). The isotherms of the water of hydration (Mh ) and the water of solution (Ms ) components are shown. Cell wall water is closely associated with the cell—wall substance through hydrogen bonding. The major part of cell wall water adsorption in wood involves replacing solid—solid interfaces by solid—liquid—solid interfaces, and as a result adsorption of water swells wood by forming a solid solution. In the hygroscopic range (between 0% and FSP), the relationship between equilibritun moisture content of wood and relative humidity at a given temperature is called the sorption isotherm, as shown in Fig. 1.9. Chapter 1. Introduction^ 14 It has been shown to be of a sigmoid shape and has been classified as a type II isotherm [Stamm 1964, Siau 1984, Skaar 1988], and the cell wall water is generally considered to consist of two components, one strongly and the other weakly attracted by the sorption sites [Stamm 1964, Skaar 1988]. When the sorption process is considered to be a surface phenomenon, the strongly bound part is called monomolecular water whereas the weakly bound one is called polymolecular water. When the sorption process is considered a solution phenomenon, then they are called water of hydration (Mh ) and water of solution (Ms ), respectively. Most of the current sorption theories [Simpson 1973, Skaar 1988] divide the sigmoid isotherm into two curves; the type I and III, corresponding to the strongly and weakly cell—wall bound water parts, respectively. Simpson [1973] has examined several of the most popular models of this kind and concluded that the Hailwood—Horrobin (HH) model [Hailwood and Horrobin 1946], belonging to the group that treats the sorption process as a solution phenomenon, gave the most satisfactory results. Numerically, the total moisture content at a particular relative humidity and temperature will be equal to MI?, plus Ms . The water of hydration, Mh , at maximum humidity, has been found to range between 4 and 6% for most temperate zone wood species [Skaar 1988]. 1.3 1 H NMR and Wood 1.3.1 1 11 NMR Relaxation Theory Protons have spin angular momentum I = 1/2, and possess a magnetic moment /7 = yhr. In a large static magnetic field, H o = H0 . the spins see Zeeman energy levels , with the Hamiltonian 7-1 = = The equilibrium state of a spin system at a temperature T is characterized by a Boltzmann population of spins in these energy Chapter I. Introduction^ levels such that 15 — ticda N_^o f = e kT = e kT N+ for spin 1/2, where E = ±1/2-yhHo , and w o = -yHo . Therefore the magnetization, defined as M = [7, is non zero and is given by Curies Law as follows; I(I + 1)7 2 ti, 2 Ho^ 3kT^ < M >— (1.2) A vector < /17/ > is defined in Cartesian coordinates with < >, < M, >, and < Mr, > as the x, y and z components, respectively. This expectation value of the magnetization follows classical equations of motion. d < > dt = 7 < >x (1.3) resulting in precession of < M > about z when only the main static field H o is applied, but also predicts the motion of < M > when a resonant RF magnetic field, H 1 , is ti applied. Power from an applied RF magnetic field , perpendicular to 170, with frequency w is absorbed when the resonant condition w = w o holds. In the rotating frame of frequency co o about 17 0, < M > precesses through an angle co i t p = 71/ 1 t p , where t p is the length of time the RF field is applied. The lineshape of absorption at w o is broadened by dipolar interactions of neighbouring spins in the system. For two spins, the dipolar Hamiltonian is H12^ 71732 h2 7' 7172h2 7' 3 [I; ^. f)(1--2 n)] (A-1-B+C-FD+E+F)^(1.4) where A = /1 ,/2 ,(1 — 3 cos t 0) B = -- (/-0-- + /T4)(1 — 3 cos t 0) 4 Chapter 1. Introduction^ 16 3 f T+ T 1 2' + Ilz in Sill 9 cos Be -2 d' 2 3 D =^ 1- 1 2Z + z In Sill 9 cos Oe i `k 2 I E = __ (/+/-+) sin 2 0e -2i cs 4 I 2 ^ F = - 4 (/-/2-- ) sin 2 0e 2i g9 (1.5) where r is the separation distance between the two spins, and h is the unit vector in the direction joining them and making angles 0, and 0 with respect to the main magnetic field H0 , in spherical coordinates. The A and B terms give the secular or truncated Hamiltonian 7-(d) = 1^ 172h 2 r3 ^ (1^fi 3 cos 2 8)(3/1J2z - • 12) (1.6) which only gives transitions between states having the same Zeeman energy, and are the only terms used in the first order perturbation theory. C and D terms give energy transitions of tiw 0 , and E and F terms give energy transitions of 2t1w 0 . Motion averages the dipolar interaction in the limit where M 2 7,2 << 1. M2 is the second moment of the dipolar broadened lineshape in the absence of motion, and 7-, is a characteristic correlation time of the motions. Relaxation is described by Redfield's theory [Abragam 1961,Slichter 1980]. For spins on the same molecule, so that r is constant, and assiuning that the relative orientation varies isotropically, one has 1^41;2 =^14r6 it [J(wo) 4 J( 2 w0)] T1 1 y 4 h2 /2 16v6 rn [6J(0) 10J(co 0 ) 4J(2w0)] (1.7) (1.8) The spectral density, J(w), describes the fluctuating part of the Hamiltonian, and is the the Fourier transform of the correlation function. The correlation function is commonly assumed to be exponential, as follows ^ ^r .T(t).F*(t + ) = (1.9) Chapter I.^Introduction 17 The relaxation rates are now written as 1^3 74h2^1 Tl^10^r 6^COdTc2^1 Tc[1 3 74h2^ 1^ 5 20^r 6^1 Tc[3 + + W ciTc2 + 4 444 1 (1.10) 1 + 42(.4) Wi (1.11) ( For short correlation time, such that co o l-, << 1, one has 1^1 Tr^T2 = 114-2 7- C (1.12) In bulk water, Tc is of the order 10 -12 s, and T1 and T2 are equal and are 1 to 3 s. Water in biological systems contains hydration water, which is hydrogen bonded to macromolecules. The relaxation rates of water are affected by the strength of the local magnetic interaction, i.e. the dipole-dipole coupling, and by the motions. The motion of the hydration water molecules is restricted and anisotropic due to the bonding to the macromolecules. Also the motion of the macromolecule influences the motion of the hydration water. Typically, the motions must be described by more than one Tc or even a distribution of 7-, times. These correlation times are generally longer than that of bulk water. Enhanced proton transfer, and the cross relaxation of proton magnetization, between the hydration water and the macromolecules influence relaxation rates. Also, water in a nonspecific region surrounding the macromolecules may be affected, and therefore show slightly lower rates of rotation than bulk water. Consequently, the measured relaxation times, T1 and T2, are faster than for bulk water, and in general T2 < T1 [Mathur-De Vre 1979]. 1.3.2 Review NMR work of water in wood The wide-line 1 11 NMR absorption spectrum [Nanassy 1973,Nanassy 1974] of wood exhibits a narrow water line superimposed on a broad line from hydrogens of the solid Chapter 1. Introduction^ 18 wood fibre. The narrow line amplitude scaled to the amount of water, but was broader than that of pure water, implying that water in wood was less mobile than free water. It was demonstrated that the solid wood and water 1 I1 NMR FID signals are separable [Sharp et a1.1978,Riggin et a1.1979,Menon et al.1987]. The solid component can be fit to a second moment expansion to give an absolute moisture content measurement by 1 H NMR [Menon et a/.1987]. Furthermore, the water signal could be separated on the basis of T2 into cell wall water and lumen water [Hsi et a/.1977, Riggin et a1.1979, Menon et al.1987]. Drying studies confirmed the assignment of the T2 components, and comparison with anatomical data from SEM images confirmed the relative amplitude of lumen water components, and the ratio of T2 times scaled as the radius of cell type as predicted by a fast exchange model of free lumen water in exchange with a fraction of water on the cell wall surface [Menon et al.1987]. Further investigation of western red cedar [Flibotte et a/.1990] showed that the solid wood signal has a second moment of about 5 x 10 9 s -2 which increased by about 20 % at low moisture contents below the FSP. Heartwood and juvenile wood were found to have substantially less water and shorter cell wall water T2 relaxation times than the sapwood, which would enable the sapwood-heartwood boundary to be distinguished in a cross sectional image of a cedar log, but the heartwood- juvenile wood boundary would be more difficult to discern. Decayed wood was found to have high moisture contents and so would be easily identifiable, especially if in heartwood or juvenile wood. Pulsed field gradient methods on water in wood [MacGregor et al.1983, Peemoeller et a/.1985] showed that water in wood lumens undergoes restricted diffusion in all directions, even in the long direction along the cell lumens. Water squeezed from the wood was measured to have the same self diffusion coefficient as free water [MacGregor et al.1983]. Water environments have been selectively imaged on the basis of T2 to produce onedimensional images of cell wall and lumen water separately. One-dimensional images of Chapter 1. Introduction^ 19 earlywood, latewood and rays, and cell wall water for western red cedar were acquired [Menon et a/.1989]. One—dimensional radial moisture profiles of western red cedar sapwood [Quick et al.1990] as a function of time during controlled air flow rate and temperature results showed clearly that water removal as a function of time differs in the earlywood and latewood regions. Three—dimensional imaging in whole body MRI scanners [Hall and Rajanayagam 1986, Flibotte et al.1990] was used to investigate the water signal distribution using a single echo. The echo time in MRI is limited to about 20 ms, therefore only a fraction of the water was imaged. The water components from the cell wall, the latewood lumens and the ray cells, having short T2 times, are not imaged by these methods. 1.4 Overview of Thesis The general materials and methods are given in Chapter 2. The second moment of the protons in solid wood and T2 of the cell wall water, for low moisture contents, are presented in Chapter 3. A technique for fitting multi-exponential data and a technique of selectively imaging water environments are demonstrated in Chapter 4. The role of diffusion in mixing water environments and the resulting effects on relaxation are investigated in Chapters 5, 6 and 7. In Chapter 5 the lumen water T2 relaxation is shown to follow a diffusion-Bloch equation with a surface relaxation sink at the cell wall. From this theory, the distribution of lumen sizes can be recovered from the lumen water T2 decay. In Chapter 6 a two region diffusion model is applied to describe the T2 relaxation of both the lumen and the cell wall water. The surface sink parameter is defined in terms of cell wall water diffusion and cell wall water relaxation. From this theory, an original method of calculating the cell wall diffusion rate from the T2 Chapter I. Introduction^ 20 relaxation and the distribution of cell sizes is developed. Numerical simulation of this two region model is developed. Finally, in Chapter 7 the T1 relaxation of wood is investigated and the influence of water diffusion and cross relaxation of water with the protons in solid wood is determined. Chapter 2 General Materials and Methods 2.1 Samples Samples were cut with a bandsaw from log slices, most to dimensions of 0.5 x 0.5 x 1 cm with the long axis parallel to the longitudinal tracheids of the wood. The imaged samples were cut to dimensions of 0.4 x 0.4 x 1 cm to fit in the smaller imaging probe. Unless described otherwise, all samples were were cut from logs of green moisture content and below the FSP, water was never added to avoid hysteresis effects in moisture content. Samples were allowed to equilibrate for 30 minutes before each NMR measurement, and were generally analysed at 26°C. Six species of wood have been used in various studies presented in this thesis. Samples of white spruce, alder, fir, lodgepole pine heartwood, redwood sapwood, and western redcedar sapwood have been utilized. 2.2 SEM Images Cross—sectional and tangential microtomed surfaces of the samples were dried, gold sputter coated, and imaged by a Cambridge Stereoscan 250 Scanning Electron Microscope (SEM) operating at 20 kV. The micrographs of the tracheid and ray cells were digitized into a 512 x 512 pixel array with 256 grey levels with a Kontron IBAS 2000 Image analyser to give the cell distribution as a function of cell radius. The cell lumen radius was calculated based on its measured digitized area assuming circular 21 22 Chapter 2. General Materials and Methods^ geometry. The radius was corrected for shrinkage to give the green sample radius [Haygreen and Bowyer 1982]. A cubic spline interpolation of 100 points through the original radius histogram was used for the analysis. 2.3 SEM Moisture Content Measurement In order that our SEM results relate directly to the NMR results from lumen water, the wood samples must be maximally hydrated. The maximum MC attainable in a given wood sample can be estimated from the SEM measurements of percent cross—sectional area of lumen, L, and cell wall, W. That is, WaterMass^LV d water 0.3WVdwood 100%^ 100%. OvendryWoodMass^WVdwood M C max = ^ (2.1) V is the sample volume which cancels out of the calculation. The densities dwater 1.0 g/cm 3 and dwood = 1.5 g/cm 3 [Haygreen and Bowyer 1982] are used and the fibre saturation point (FSP), i.e. the moisture content of cell wall water, is assumed to be 30%. 2.4 NMR Equipment Proton NMR measurements were carried out on a modified Bruker SXP 4-100 NMR spectrometer operating at 90 MHz with a lips receiver dead time. The 90° pulse was typically 2.50/ts and the 180° pulse was typically 5.00/ts. Temperature was varied using a Bruker B—ST 100/700 temperature controller. Data acquisition and analysis were carried out on a system including a MicroVAX II and a National 32016 computer, a Nicolet 2090 digital oscilloscope, and a locally built pulse programmer [Sternin 1985]. Recently the National computer and the Nicolet oscilloscope have been replaced with a IBM compatible computer, a Rapid Systems A/D board and a "Sync" card. Chapter 2. General Materials and Methods^ 23 2.5 NMR M2 and Moisture Content Measurements The free induction decay (FID), the NMR signal following a single 90° pulse, was analysed to give the second moment (M 2 ) and NMR moisture content (NMR MC). The signal of the solid wood decays rapidly in about 30,us, and is easily distinguishable from the water signal, which extends over hundreds of milliseconds. To refocus the dephasing of the water signal due to inhomogeneity of the magnet, 180° pulses with 100,us spacing were used, so this is not a true FID after the first 180° pulse at 50ps. Also, alternate scans were preceeded by a 180° pulse and subtracted from the cumulative data memory to reduce the dead time. Two hundred scans were averaged to give a high signal to noise ratio, typically about 100 to 1. A repetition time of 8 s (> 5T1 where T1 is the spin—lattice relaxation time) was used between pulse sequences. The FID signal from dipolar—coupled protons of our samples fits a moment expansion equation [Abragam 1961] of the form: S(t) = (SO L0)(1 - 2 t2 M4^ 6 t6 ••-) LO. 4! 6! - (2.2) where M2, M4 and M6 are the second, forth and sixth moments of the lineshape, S o is the total NMR signal at t = 0, and (S 0 — L 0 ) is the total solid wood signal at t = 0. At short times, the water protons in wood are assumed to acid only a constant L o to the signal, which is the amplitude at t = 0 of a linear fit to the water signal. After the dead time of the spectrometer, 30 points from about lips to 26,as following the centre of the 90° pulse were fit to the above equation, out to the N/6 term only. Higher order terms were negligible for these 30 points. A non—linear function optimization program minimizing X 2 was implemented [.James and Roos 1975]. The moisture content (MC), corresponding to the ovendry definition as the mass Chapter 2. General Materials and Methods^ 24 ratio of water to solid wood, is defined from the FID by L(0) * Pw" d * 100% NMR MC —^ S(0) — L(0) ,owater where p wood is the number of protons per grain of wood, Pwater (2.3) is the number of protons per gram of water, and the ratio of the two is 0.56 for white spruce, and about the same for most species [Fengel, Wegener 1984]. 2.6 T2 Relaxation The echo heights from a Carr—Purcell—Meiboom—Gill (CPMG) sequence [Carr and Purcell 1954, Meiboom and Gill 1958], with echo spacing typically 200ps, give a relaxation decay curve of the water in the wood. The CPMG sequence is represented by 000 — 2 — (18090 — r)„ — TR.^ (2.4) A repetition time of 10 s (at least 5 times T 1 ) was used for all samples. Four points were averaged per echo and 200 scans were signal averaged. About 100 echoes were used corresponding to times, from the 90° pulse, increasing approximately geometrically from the initial echo to the time at which the echo height had decayed to less than 1% of its initial amplitude. For single exponential relaxation, the envelope of the echos of the CPMG pulse train follows the curve S(t) sme t/T2 ^ - (2.5) where S(0) is proportional to the number of protons and t is the time from the 90° pulse. The envelope of the echos of the CPMG pulse train for wood is described as a multiexponential decay and can be represented by the sum of several components S(t) =^S, e —t / T22^(2.6) Chapter 2. General Materials and Methods ^ where 8(0) is proportional to the moisture content, T2i 25 are the component relaxation times, and the number of possible exponential components can range from one to several hundred. In this thesis, I solve for T2 times in two ways: (a) nonlinear solutions for a small number of discrete exponential component times and amplitudes, and (b) linear solutions for amplitudes at a large number (100) of specified relaxation times. Whittall and MacKay [1989] discussed the application of several algorithms to solve Eq. (2.6) for the relaxation components and I use their implementation of the nonnegative least— squares (NNLS) algorithm of Lawson and Hanson [1974]. A direct application of NNLS to minimize the least—squares misfit results in spectra composed of a few isolated T2 components. When the true spectrum is more likely to be continuous, for example, reflecting the continuous distribution of lumen sizes, I use NNLS to minimize a linear combination of the misfit and the first differences of the spectrum [Whittall and MacKay 1989]. Chapter 3 Second Moment and Cell Wall Water T2 1 3.1 Summary Nuclear magnetic resonance lineshape second moments of the protons in solid wood and spin—spin relaxation times, T2 of the cell wall water in lodgepole pine heartwood have , been measured at 30°C for a range of moisture contents, mainly in the hygroscopic range. The second moment of the protons in ovenclry wood was found to be about 23% lower than the rigid lattice calculation, indicative of a rigid structure with some anisotropic molecular motion of the polymeric constituents. Above 5% MC, the second moment decreased by a further 13 to 16% implying a "loosening" of the molecules in the solid with increased moisture content. The T 2 of the cell wall water increased with moisture content, and provided no evidence of separate hydration and solution water fractions as predicted by isotherm theories. The 1 11 NMR measured fibre saturation point of 27% agreed with the value calculated by the Hailwood—Horrobin isotherm model. 3.2 Introduction In this chapter, the second moment of the proton 1 H NMR lineshape from the solid wood and the spin—spin relaxation time (T2 ) of the cell wall water were measured as a function of moisture content below the FSP in lodgepole pine heartwood. The 1 This chapter closely follows a previously submitted paper. [Araujo et a1.1993b]. 26 Chapter 3. Second Moment and Cell Wall Water T2^ 27 objective was to see if the microscopic dynamic properties of the solid wood and water as monitored by 1 H NMR give any new insight into wood—water interactions in the hygroscopic range. 3.3 Materials and Methods Three specimens 5 x 5 x 10 mm in longitudinal dimension were cut from a green piece of lodgepole pine heartwood (Finns contorta Dougl. var. latifolia Engelm.). The specimens were oven dried at 103 ± 2°C until constant weight, placed in 180 mm long and 10 mm od diameter glass NMR tubes and into a conditioning chamber where the temperature and relative humidity could be maintained constant. There they were conditioned, to constant weight, at relative humidities of 30, 45, 60, 75, 85 and 95%, and at 30°C. After NMR measurements, at 30°C, the same specimens were impregnated with water by a full—cell method and then were left to dry slowly, at ambient temperature, until their moisture content was approximately 34%. In this way a moisture content above the FSP could be obtained. Next, the highest hydration was obtained by soaking 12 hours in distilled water at 26°C. A set of twenty matched wafers 25 x 25 x 4 mm thick in the longitudinal direction, were cut from the same green piece of lodgepole pine and conditioned in the chamber with the NMR specimens, simultaneously. The mean equilibrium moisture content of the twenty specimens was used for the determination of the 30°C sorption isotherm. The experimental data were fit to the 1111 model, which expresses the moisture content (in percent) as a function of the relative humidity (H), using a Gauss—Newton method of minimizing the least squares deviations. 1800/ K2H ) 1800 ( K2 H M Mh + = (3.1) W 100 K 1 A 2 H^W 100 — K2 H) where K i is the equilibritun constant of the reaction between water and dry wood Chapter 3. Second Moment and Cell Wall Water T2^ 28 substance, K2 is the equilibrium constant of the condition between water vapour and dissolved water, and W is the grams of dry wood per mole of sorption sites. The 1800/W ratio corresponds to the moisture content of the wood when there is one molecule of water on each sorption site. From the parameters K l K2 and W, the , relative amounts of the two types of cell wall water were determined. The free induction decay (FID), the 1 I-1 NMR signal following a single 90° pulse, was analysed to give a second moment (M 2 ) and the NMR moisture content. The echo heights from a CPMG sequence, with echo spacing of 200/ts and including 8 echos of spacing 100/ts collected separately, gave the relaxation decay of the water in the wood. 3.4 Results and Discussion The adsorption isotherm for lodgepole pine heartwood at 30°C obtained in this study is shown in Fig. 3.1. The experimental points as well as the curves calculated by the HH model, which correspond to the waters of solution (type I) and hydration (type III), are shown. The estimated values for W, .1 -C1 and K2 are 390, 20 and 0.83, respectively. The calculated moisture content when there is one molecule of water on each sorption site is 4.6%, and the predicted FSP which is the moisture content at 100% humidity is 26.9%. The polymeric molecules of solid wood are well known to have a fairly rigid structure. The '11 NMR spectrum, which is dominated by the dipolar interactions between neighbouring protons, is broad and featureless. Spectra of this type are quantitatively characterized by their spectral moments, in particular, M2. For a rigid solid, the measured second moment, M2 is equal to the rigid lattice M27. i9id which can be calculated from knowledge of the spatial distribution of protons in the sample, as follows: M2 = 3 (1 — 3 cos 2 t9ik ) 2 + 1) ,, 6 — 74 h 2 -4/ 4 'jk (3.2) Chapter 3. Second Moment and Cell Wall Water T2^ 25 20 — 15 10 — 5— Mh 20 40^60 H (%) ^ 80 100 Figure 3.1: Isotherm data for lodgepole pine heartwood at 30°C, HH model fit (solid curve), isotherm of hydration (M1) and isotherm of solution (M3 ). The percentage deviation in the moisture contents, from the twenty matched wafers, is at most 3%. 29 Chapter 3. Second Moment and Cell Wall Water T2^30 where the vector 77:ik describes the relative positions of two protons, which is at an angle 6jk to the applied magnetic field H o , and r ik is the distance between the protons. The expression reduces to the following, when one averages over all angles for a sample of random orientations, 3 /V/2 = - 7 4 h 2 /(/ -I5 ) 1 (3.3) k rjk In the presence of molecular motion, the 1 H NMR spectrum is narrowed and the measured M2 is lower than M2rigid . For molecules undergoing isotropic motion, for example, the water, M• is zero. The 1 H NMR spectrum is the Fourier transform of the FID. A typical FID of protons in wood is shown in Fig. 3.2. The fast decaying part is from the solid wood and the slow decaying component is from the water in wood. The fit of Eq. (2.2) to the solid wood signal is shown in the inset of Fig. 3.2. The resulting M2 values from the fit are shown in Fig. 3.3, for increasing moisture content, for the three samples. Our M2 results can be compared to rigid lattice calculations for cellulose and hemicellulose which are about 7.3 x 10 9 s -2 [MacKay et al. 1985]. Lignin is expected to have a similar M2 value, hence the average 1VI2 for ovendry wood of 5.6 x 10 9 s -2 indicates a very rigid structure for wood on the 1 1-I NMR time scale of 10 -5 s. This reduction by about 23% from the rigid lattice value is due to anisotropic motions of the polymeric constituents of wood. In all three samples we observe a further decrease in M2 of 13 to 16% in the moisture content range 5 to 13% , as sununarized in Table 3.1. This indicates that the solid wood molecules in our lodgepole pine samples underwent a "loosening" process at moisture contents in the range of 5 to 13%. This "loosening" can be interpreted as a change in the anisotropic molecular motions of the polymeric constituents of the solid wood which occurs as water molecules are added at some level above 5%. The moment measurements alone do not enable us to determine which type of molecules, Chapter 3. Second Moment and Cell Wall Water T2^ 31 S O 5 10 15 20 25 30 Lo _ Time (,us) IIIIIIIII O^200^400^600^800 Time (,us) Figure 3.2: A typical free induction decay showing a baseline, the fast decaying solid wood signal, and the slower decaying water signal. The gaps in the data are where the high power RF 180° pulses where applied. The inset is of the solid wood signal and the fit to the moments expansion. Table 3.1: Transition moisture content and average M2 's with standard deviations. Sample Transition Pre—transition Post—transition Decrease in MC (%)^M2 (10 9 S -2 )^M2 (10 9 S -2 )^M2 (%) 1 8.1 + 0.9 5.70 + 0.08 4.8 ± 0.2 16 2 5.7 + 1.3 5.6 + 0.3 4.9 + 0.1 13 3 13.4 + 1.8 5.5 + 0.2 4.6 + 0.1 16 Chapter 3. Second Moment and Cell Wall Water T2^ 6.0 1 5.5 5.0 4.5 f ^5.5 N I 2 • °'0^5.0 • N 4.5 if 5.5 3 5.0 f 4.5 4.0 0 20^40^60^80^100^120 M (%) Figure 3.3: Second moment of protons in solid wood versus moisture content for the lodgepole pine heartwood samples 1, 2, and 3. 32 Chapter 3. Second Moment and Cell Wall Water T2^33 e.g. cellulose, hemicellulose or lignin, undergo more motion. The accuracy of the M2 depends on the estimation of L o , the water signal amplitude. Since this NMR moisture content is in very good agreement with the ovendry moisture content, as shown in Fig. 3.4, we have confidence in the assignment of L o and the M2 values. Furthermore, since we can measure moisture content accurately by 0 0 ^ 20^40^60^80 ^ 100 ^ 1 1-1 NMR , 120 M Figure 3.4: Moisture content (ovendry method) versus NMR moisture content of lodgepole pine heartwood. Linear fit slope is 0.95 and the correlation coefficient is 0.992. The error bars are smallest for the low moisture contents. Also shown are the fraction of water that is in the cell wall (triangles) and the fraction of water that is free in the lumens (squares), as calculated from the discrete T2 distributions. all the water in the lodgepole pine samples must contribute to the isotropic signal with T2 greater than about 200/ts. This means that all the water molecules in the lodgepole pine samples reorient isotropically in the 1 H NMR time scale of 10 -5 s. Chapter 3. Second Moment and Cell Wall Water T2^ 34 Figure 3.5 shows the NNLS discrete and smooth T2 distributions for water in lodgepole pine at a moisture content of 11.3%. For all moisture contents, the components -e-) a) O 10 1^1^1^1^1^ 1^1^1^1^1 8 E 6 CD 4 2 0^ 10^10 -2 T 2 s ) 10^10 -1 Figure 3.5: Typical discrete and smooth T2 distributions of water in lodgepole pine at a moisture content of 11.3%. For the smooth distribution the area under the curve is 10 times the total moisture content. with T2 < 2.0 ins were assigned to cell wall water and those with T2 > 2.2 ms were assigned to free water. Slight changes in this boundary between cell wall and lumen water had little effect on the results. It is tempting to assign the two cell wall water peaks in the discrete T2 distribution in Fig. 3.5 to two individual cell wall water components. However, while the sum of the two components increases monotonically with moisture content as shown in Fig. 3.4, their relative amplitude varied randomly with moisture content and they were replaced by a single broad peak in the smooth T2 distribution shown in Fig. 3.5. We therefore believe that cell wall water in lodgepole pine has a broad unimodal distribution of T2 times. Figure 3.4 shows the fraction of water assigned to the cell wall and the lumen as a function of moisture content. Above ^ Chapter 3. Second Moment and Cell Wall Water T2^ 35 the FSP, the moisture content of the cell wall water fraction is the FSP and is found to be 26.7 ± 1.5%, which agrees with the FSP predicted by the HH model of 26.9%. The average of the cell wall water T2 peaks is shown in Fig. 3.6 to increase from about 0.2 ms to a plateau value of about 1 ms as the moisture content goes above the FSP. A similar moisture content dependence has been seen for cellulose [Froix and 1.2 ^ 1.00.80.6 - 0.40.2— N NN1 . 0 — 0.8 — 0.6 0.4 — 2 —i cs 0.2 1.0 a) C.) 0.8 — 0.6 — 0.4 — 3^ 0.2 . 0.0 ^ 0 20^40^60^80^100 ^ - 120 M (%) Figure 3.6: The cell wall water T2 averaged from the discrete T2 distributions, versus moisture content for the lodgepole pine heartwood samples 1, 2, and 3. , Nelson 1975], white spruce [Riggin et al. 1979] and northern white cedar [Hsi et al. 1977]. As discussed above, the spin—spin relaxation measurements of the water do not distinguish more than one reservoir of cell wall water. This means that, if two cell Chapter 3. Second Moment and Cell Wall Water T2^ 36 wall water fractions exist in the lodgepole pine samples, ie. the water of hydration and water of solution of the HH sorption theory, then they undergo exchange on a time scale faster than the T2 relaxation time. The self diffusion coefficient for cell wall water in wood at moisture contents of 2 to 10% has been measured to be 1 x 10 -8 to 13 x 10 -8 cm 2 /s [Stamm 1959]. Using R 2 = 6Dr, we obtain a diffusion distance for water in wood of about 80 to 280 A during 10 -5 s. Hence, diffusion much greater than the molecular scale of a few Angstr6ms prohibits any information on individual cell wall water reservoirs by 1 H NMR . 3.5 Concluding Remarks This work indicates that, as water is added to lodgepole pine heartwood, the solid cell wall structure undergoes a loosening process for moisture contents above about 5%. This observation is commensurate with sorption theories which treat the first water to be added to wood as water of hydration. However, our measurements also indicate that water of hydration cannot be distinguished from water of solution, since they undergo exchange on a time scale faster than the cell wall water T2 time. Furthermore, even at moisture contents below 5%, the water molecules in the cell wall undergo rapid isotropic reorientation on the 10 -5 s time scale, which enables the water signal to be separated from the solid signal to give an accurate NMR moisture content. The NMR FSP value agreed with the FSP calculated from the extrapolation of the HH isotherm model to 100% humidity. Chapter 4 T2 Techniques for Characterization of Water in Wood 1 4.1 Summary Two new proton magnetic resonance techniques, relaxation spectra and relaxation selective imaging, have been used to investigate the distribution of water in samples of normal white spruce sapwood, heartwood, and juvenile wood as well as two rehydrated heartwood samples containing incipient decay and compression wood respectively. It is demonstrated that the spin-spin (T2) relaxation behavior in wood is best presented as a continuous spectrum of relaxation times. Spectra of T2 for white spruce show separate peaks corresponding to the different water environments. Cell wall water gives a peak with a T2 value of about 1 ms and lumen water gives a distribution of T2 values in the range of 10 to 100 ms. The lumen water T2 value is a function of the wood cell radius. Consequently, the different cell lumen radii distributions for spruce sapwood, juvenile wood, and compression wood are readily distinguishable by the shape of their T2 spectra. Water environments which are separable on a T2 spectrum may be imaged separately. Imaging has been carried out in one dimension for cell wall water and lumen water of a spruce sapwood sample at four different moisture contents ranging from 100% to 17%. For the first time, we demonstrate that above the fibre saturation point the moisture density profile of the cell wall water is largely independent of moisture content. The feasibility and utility of using these techniques for internal scanning of logs and lumber is discussed. These techniques should provide new insights into the 1 This chapter closely follows a previously published paper. [Araujo 37 of a1.1992] Chapter 4. T2 Techniques for Characterization of Water in Wood^38 wood drying process. 4.2 Introduction The aim of the present study is to demonstrate using white spruce [Picea glauca (Moench.) Voss.] the utility of two new 1 11 NMR techniques for the characterization of water in wood. A technique for the analysis of multi—exponential relaxation in terms of a continuous distribution of relaxation times [Whittall and MacKay 1989] has been applied here to T2 studies of water in spruce. Because T2 values are a function of cell size [Brownstein and Tarr 1979] and wood generally possesses a continuous distribution of cell sizes, this approach is more appropriate for the study of T2 relaxation in wood than the common technique of fitting to a limited sum of discrete T2 components [Menon et a/.1987]. Distributions of T2 have been obtained for samples of normal white spruce sapwood, heartwood, and juvenile wood as well as a rehydrated white spruce heartwood sample containing compression wood and incipient decay. A technique of selectively imaging water environments on the basis of T2 is demonstrated. One—dimensional images of the cell wall water and the lumen water in white spruce sapwood have been obtained separately at a range of moisture contents from 100% to 17%. This method of imaging cell wall and lumen water separately at a series of moisture contents should provide new insight into the mechanisms of wood drying. 4.3 Materials and Methods 4.3.1 Samples Samples of sapwood, heartwood, and juvenile wood were taken from a log slice of normal white spruce. White spruce heartwood samples, one normal, one containing incipient decay (indicated by discolouration on the sample surface), and another containing Chapter 4. T2 Techniques for Characterization of Water in Wood^39 compression wood, were taken from a block of wood containing all three types. This block was rehydrated with distilled water by two cycles of a vacuum pressure treatment at 39 mm of mercury for 2 hours, followed by a water pressure treatment at 100 psi for 2 hours, and then stored for one month in distilled water. The sapwood sample was cut to 0.4 x 0.4 x 1 cm in order to fit in the 9 mm ID cylindrical coil used for the one—dimensional imaging studies. All other samples were cut to dimensions of 0.5 x 0.5 x 1 cm, and were analysed in a slightly larger coil. For all samples, the long axis was parallel to the longitudinal tracheids of the wood. For the drying study, the samples were placed in a vacuum oven at 55°C for 1 to 5 minutes. The lowest moisture content (17%) of the sapwood sample was achieved by leaving the sample exposed to the atmosphere for 1 hour after being dried in the vacuum oven for 2 minutes. All samples were allowed to equilibrate for 30 minutes before being analysed at 24°C. 4.3.2 One—dimensional Imaging One—dimensional imaging was obtained using a constant 19.4 gauss/cm field gradient along the direction of the main magnetic field. The CPMG pulse sequence was used with data collection starting at the top of the last echo (See Appendix A). The Fourier transform of the FID signal following the last CPMG echo corresponds to a one— dimensional projection of the moisture density distribution in the wood. The length of the CPMG train was varied so that data collection started at either a short time Ts = 400/ts or long time = 6Orns after the 90° pulse. In general, 200 scans were accumulated except for the lowest moisture content image (17%) where 600 scans were averaged to increase the signal to noise ratio. Chapter 4. T2 Techniques for Characterization of Water in Wood ^40 4.4 Results and Discussion 4.4.1 Fits of T2 Relaxation Data for White Spruce For water in a cylindrical lumen, where the T2 relaxation occurs mainly at the lumen surface, it has been shown [Brownstein and Tarr 1979] that the self diffusion of the water molecules to the lumen surface determines the relaxation behavior. The exact solution to this problem [Brownstein and Tarr 1979] consists of a sum of exponential modes giving a series of T2 values, where most of the intensity can be attributed to the first mode of this series. This diffusion model predicts that for a lumen with small radius, the water molecules diffuse to the surface rapidly on the 1 11 NMR time scale of a few milliseconds and the T2 relaxation time is proportional to the volume to surface ratio, which scales as the lumen radius. Also, for a lumen with a larger radius, the T2 relaxation time is determined by the rate at which lumen water molecules diffuse to the lumen surface and T2 scales as the square of the radius. Since the 1 1-I NMR signal is proportional to the number of protons, the distribution of T2 relaxation times for lumen water should reflect the distribution of cell lumen radii by volume (Fig. 4.1). It then follows that it is more appropriate to analyze the CPMG decay curve, S(t), in terms of a continuous distribution of T2 values rather than the more conventional interpretation in terms of a fixed number of discrete components. Previously, the CPMG curve for water in wood has been represented by discrete T2 components where the number of components is fixed at 1, 2 or 3 [Menon et a/.1989]. Here we introduce the continuous T2 analysis for spruce wood where the number of components is unrestricted. Figure 4.2 shows a fit of this continuous T2 spectrum to the CPMG data. The chi squared misfit (x 2 ) is excellent and the value of X 2 /N = 1.1 where the expected value is x 2 /N = 1.0, [Whittall and MacKay 1989] indicating that noise has not been incorporated into the fit. Figure 4.3 shows the continuous T2 spectra Chapter 4. T2 Techniques for Characterization of Water in Wood^41 0.20 JL) 7) ' a t 0.15 0 0.20 ^- a) 0.10 I 0.10 Z a) a.) 15 0.05 - " a) a, O 0.05 - b C1)^0.15 O ■ O 0.10 - > 0.10 a 0.05 - Tv. 0.05- 1) c d a)^0.15 a Ca 0.00 0^5^10^5^20^25 Lumen Radius (/Lm) 0.00 ^ 0^5 ^ ^ 10^15^20 ^ 25 Lumen Radius (gm) Figure 4.1: Relative numbers of a) cells, and b) protons as a function of cell lumen radius for the white spruce sapwood sample, and c) cells, and d) protons for the white spruce heartwood compression wood sample. Estimated from Scanning Electron Micrographs. Chapter 4. T2 Techniques for Characterization of Water in Wood^42 10 2 N s‘ 100 .00 .05 .10 .15^.20 .25 .30 Time (s) Figure 4.2: CPMG data and the fit from a continuous T2 spectrum for the white spruce sapwood sample at 100% moisture content. 2 10 4^102^102^10° T 2 (S) Figure 4.3: T2 results as a continuous curve and as a discrete fit of 2 exponential components (spikes) for the white spruce sapwood sample at moisture contents of a) 100%, b) 86%, c) 59%, and d) 17%. (Amplitudes of the discrete fit have been divided by a factor of 10). Chapter 4. T2 Techniques for Characterization of Water in Wood^43 of the sapwood sample for four moisture contents, as measured by the ovendry method. The T2 spikes result from a fit to two discrete T2 components using a nonlinear x2- minimization [James and Roos 1975]. These two components correspond approximately to the cell wall and the lumen water, except for the lowest moisture content where the single component corresponds to cell wall water alone. The discrete component fits were used in the one—dimensional imaging study which will be described later. The peaks in the continuous spectra of Fig. 4.3 can be related to the anatomical location of water in wood. The peak corresponding to the lowest T2 values is relatively constant at higher moisture contents, but decreases when the moisture content is below the fibre saturation point (FSP) of about 30%. This peak is therefore associated with the water in the cell walls. Above the FSP the two peaks corresponding to the higher T2 values decrease in amplitude with moisture content, and are associated with the cell lumen water. The continuous T 2 spectrum for 17% MC actually shows 0.3% MC of lumen water at about 100 ins. Figure 4.4 shows the continuous T2 spectra of the juvenile wood and heartwood samples, which were taken from the same log slice as the sapwood sample. The moisture contents, as measured from a FID, of the freshly cut samples of juvenile wood and heartwood were 29% and 30% respectively. These T2 spectra show that practically all the water is cell wall water. The centres of the cell wall water T2 peaks for heartwood and juvenile wood are located at about the same T2 times as for the sapwood sample (Fig. 4.3). We note the presence of low intensity peaks at higher T2 times for both the heartwood and juvenile wood. These correspond to the small amounts of lumen water in the samples. The lower T2 value for the juvenile wood sample is likely a consequence of the smaller lumen sizes in juvenile wood. Figure 4.5 shows the continuous T2 spectra of samples of heartwood, compression wood, and heartwood with incipient decay at fully rehydrated moisture contents of Chapter 4. T2 Techniques for Characterization of Water in Wood^44 Figure 4.4: Continuous T2 spectra of the water in white spruce juvenile wood (solid line) and heartwood (dashed line). 178%, 66%, and 173% respectively, as measured from FIDs, such that all available lumen space in the wood was occupied by water. The cell wall water peaks are all centred near 1 ms, but the compression wood peak has a slightly lower value, which suggests a difference in the cell walls of the compression wood from normal wood [Haygreen and Bowyer 1982]. The amplitude of the lumen T2 component is much less for the compression wood sample than the other two heartwood samples, indicating that compression wood has fewer large cells. The cell distribution in Figure 4.1c shows that compression wood does have fewer large cell lumens than normal sapwood (Fig. 4.1a). The presence of incipient decay is not readily distinguishable in the T2 spectrum. 4.4.2 One—dimensional Imaging Total Water Distribution Proton magnetic resonance is used here for one—dimensional imaging of water in the wood with a resolution of about 0.1 mm. Figure 4.6 is a one—dimensional image of the Chapter 4. Ti Techniques for Characterization of Water in Wood ^45 lo T 2 (s 10^ 10 1 0° ) Figure 4.5: Continuous T2 spectra of the water in rehydrated samples of white spruce heartwood compression wood (solid line), heartwood (long—clashed line), and heartwood with incipient decay (short—clashed line). water distribution in white spruce sapwood, at four moisture contents, measured by the ovendry method, superimposed onto an SEM image of the sample. The range of moisture contents from 100% to 17% is representative of the range of moisture contents in wood kiln drying for spruce wood. The sample was aligned with the growth ring boundaries perpendicular to the field gradient direction, so that the structure in the image corresponds to the growth rings of the sample. A higher rate of drying at the edges of the sample gives rise to a rounding off of the profiles. The profiles are expressed in terms of moisture density, which is the mass of water per unit volume of wood. The total area under the image profile is proportional to the mass of water. The moisture density can be calculated from the image profile using the moisture content, the ovendry mass of the wood, and the dimensions of the sample. The moisture density is always less than 1 g/cm 3 , the density of free water, since the solid wood occupies some of the volume. Chapter 4. T2 Techniques for Characterization of Water in Wood ^46 0.7 M C (.) 0.5 CT) •••-•-' 0.4 0.3 CD 0.2 0.0 -0.1 -4 -2^ 0^ 2 4 Position (mm) Figure 4.6: Radial Moisture Density Profile of the total water distribution in the white spruce sapwood sample at four moisture contents superimposed onto a SEM image of the sample. Chapter 4. T2 Techniques for Characterization of Water in Wood ^47 Figure 4.6 shows that at high moisture contents, moisture density is greater in the earlywood side of each growth ring. This difference arises from the fact that the earlywood tracheid lumens can hold more water, because they are much larger than the latewood tracheid lumens. As the moisture content decreases from 100% to 86% to 59%, the water dries from the earlywood side at a higher rate, so that at 59% the distribution has more water in the latewood side of each growth ring. The cell walls are more dense in the latewood tratheids and so contain more of the cell wall water. The T2 analysis shows that the final water distribution at 17% moisture content, which has a more uniform distribution, is practically all cell wall water (See Fig. 4.3). Cell Wall and Lumen Water Profiles One-dimensional images of the distribution of water in wood have previously been separated into cell wall water and lumen water images in a fully hydrated sample [Merlon et a/.1989], and here they are separated for a range of moisture contents. The images are separated on the basis of the T2 distribution of the sample. Figure 4.3 shows that in terms of T2 values, the continuous T 2 spectra are grouped into two distinct parts. The lower T2 values are attributed to cell wall water and are represented by the lowest T2 component of the discrete component fit, Tr'. This lowest T2 spike also includes about 4.6% MC of lumen water. The peaks with higher T2 values make up most of the lumen water and are represented by the higher discrete T2 component, Vni" The two 1 H NMR signals that give the images are collected after the last echo of CPMG sequences, one short sequence of length 7 8 , and one long sequence of length - Ti , where Ts < Tf w < r1 . The image signals are represented by F(Ts) e-(T.,)/TP'metz Fcw e _( T,)/ T2w F(Ti)^Flumen e -(T1)/T2'" -- " ^ (4.1) (4.2) Chapter 4. 2'2 Techniques for Characterization of Water in Wood^48 where Fcw and Fiume, are the image signals for the cell wall water and lumen water, respectively. Equation (4.1) is the total water signal for r i, much less than Tf w . Equation (4.2) is the lumen water signal since the signal from the cell wall water is practically zero at this time, for r1 = 60 ms and T2 K' about 1 ms. A multiple of Eq. (4.1) subtracted from Eq. (4.2) leaves only the cell wall water signal, Fc w. As discussed earlier, it is the Fourier transform of these signals that are the one—dimensional images. Figure 4.7 is the image of the cell wall water at the four moisture contents. Above the fibre saturation point (FSP), this image should scale as the solid wood density which is higher in the latewood [Menon et a1.1989]. It is not surprising that the cell 0.6 ^ —0.1 —4^ —2^ I o Position (mm) 2 4 Figure 4.7: Radial Moisture Density Profile of the cell wall water distribution in the white spruce sapwood sample at four moisture contents superimposed onto a SEM image of the sample. wall water profile is not a function of moisture content, above the FSP, but we believe Chapter 4. T2 Techniques for Characterization of Water in Wood ^49 this is the first direct demonstration of this fact. It should be noted that while it is not obvious how to relate the total water or the cell lumen water images to the SEM image, the correspondence of the SEM image to the cell wall water is obvious. An exception is the lower moisture content cell wall water image in which the latewood cell wall water peaks have dried out of the sample leaving an almost featureless distribution. Using the correlation of the higher MC cell wall water images to the SEM image, the total water and cell lumen water images are correlated to the SEM image. Figure 4.8 is the image of the lumen water at the four moisture contents. The -0.1 -4^ -2^ 0 ^ Position (mm) 2 ^ 4 Figure 4.8: Radial Moisture Density Profile of the lumen water distribution in the white spruce sapwood sample at four moisture contents superimposed onto a SEM image of the sample. image shows that on the earlywood side of each growth ring the moisture density is much higher, and the earlywood lumen water leaves at a higher rate than the latewood Chapter 4. T2 Techniques for Characterization of Water in Wood^50 lumen water. The lowest moisture content image shows a component in the centre of the image indicating the presence of a small amount of lumen water, which was predicted by the continuous T2 spectrum and shown to have a T2 of 100 ms and a moisture content of 0.3%. Potential of 1 H NMR for Internal Scanning of White Spruce The results of this work enable us to assess the potential of proton magnetic resonance methods for internal scanning of normal and abnormal white spruce. Normal sapwood could be readily distinguished from normal heartwood and juvenile wood since the latter two contain almost no lumen water. Incipient decay seems to be indistinguishable from normal wood. However, compression wood is readily distinguishable from normal wood when the lumens are hydrated, because of the very different size distribution of the cell lumen radii. We are also able to compare our studies on white spruce with former work on western red cedar [Flibotte et a/.1990] in order to assess the potential value of 1 1-/ NMR for species differentiation. Several differences between the two species are evident. The MC of green white spruce sapwood is much lower than that of green western red cedar sapwood and this difference is readily measurable by 1 H NMR. The T2 values for water in spruce lumens were generally lower than those for water in cedar lumens, reflecting smaller cell lumen sizes in the spruce. The T2 value for cell wall water in spruce (about lms) was similar for sapwood, heartwood, and juvenile wood, but for cedar the T2 value for cell wall water in the sapwood (about 4 to 7 ins) was a factor of two or more greater than that (about 1 ins) for cell wall water in the heartwood and juvenile wood. The difference in the cell wall water T2 for these two samples shows some promise for species differentiation in the sapwood. This study was carried out on a solid state 1 H NMR spectrometer on small wood Chapter 4. T2 Techniques for Characterization of Water in Wood^51 samples (smaller than 1 cm 3 ). For the potential application of these techniques to lumber and whole logs, large bore magnetic resonance imaging (MRI) facilities would be required. In fact several wood MRI studies have already been carried out [Hall and Rajanayagm 1986, Wang and Chang 1986, Flibotte et a1.1990]. MRI facilities are currently sensitive only to hydrogen nuclei which have T2 values greater than a few ms, that is, only the lumen water. Cell wall water in white spruce can not be imaged on current large bore MRI facilities. The present study employed a one—dimensional imaging technique to produce moisture content profiles radially across the growth rings of small wood samples. Obviously, a two—dimensional image would be more valuable for the study of water in lumber and whole logs. In principle, the small sample work carried out here could have been extended to two dimensions with the addition of two more orthogonal gradient coils for slice selection and phase encoding. However, the practical consideration of applying the gradients sufficiently fast to select a slice, and to impose the spacial encoding makes it impossible, so far, to produce a two—dimensional image of the cell wall water in white spruce. 4.5 Concluding Remarks Our goal in this work has been to demonstrate the application of new proton magnetic resonance techniques for the study of water in wood. The discovery that T2 relaxation times of lumen water can be related to lumen size has considerably expanded the utility of 1 H NMR for wood studies. The fact that there exists a broad range of T2 values corresponding to a range of water environments in wood requires that the T2 relaxation measurements cover a wide range of times and that the analysis be suitable for multi— exponential decay curves. We show that the T2 relaxation behavior in wood is well Chapter 4. T2 Techniques for Characterization of Water in Wood^52 represented as a continuous spectrum consisting of a munber of peaks. The area under each peak corresponds to the amount of moisture in a particular water environment and the T2 value indicates the nature of the environment. We have learned, from our experience with white spruce and western red cedar [Merlon et a/.1987, 1989], that T2 values below about 10 Ins correspond to cell wall water. Peaks with T2 values greater than 10 ms result from lumen water. The shape of the lumen water T2 spectrum (Figs. 4.3, 4.4, 4.5) reflects the radius distribution of water—filled cell lumens in the wood sample. Once the distribution of water environments in a wood sample is defined by the T2 spectrum, it is possible to derive the spatial distribution of each water environment by selectively imaging each component. In the present study, we have done this in one dimension for the cell wall and the lumen water in white spruce at four moisture contents. We believe that this type of study will be valuable in the detailed investigation of the wood drying process [Quick et a/.1990]. Chapter 5 Diffusion Model of Compartmentalized Water 1 5.1 Summary Spin—spin relaxation measurements were carried out on water in redwood sapwood, white spruce sapwood and white spruce compression wood samples and interpreted using a theory which modeled the lumen water T2 relaxation in terms of the cell radius distribution, the bulk water diffusion coefficient and a surface relaxation parameter. The three samples possessed different cell lumen radius distributions as measured by scanning electron microscopy. For the redwood sample, 1 1-I NMR measurements were made for 7 temperatures between 4 and 55°C over which the average lumen water 7'2 decreased from 174 to 105 ins. For all measurements, agreement between theory and experiment was excellent. In the slow diffusion regime, the theory predicted higher order T2 relaxation modes. Experimental evidence was found for the existence of these relaxation modes in the redwood results at low temperatures. Using this diffusion model, T2 relaxation decay data was fit to give a cell size distribution, comparable to scanning electron microscope results, when the bulk water diffusion coefficient and the surface relaxation parameter were known. 1 Part of this chapter closely follows, with some modifications, part of a previously published paper. [Araujo et a/.1993] 53 Chapter 5. Diffusion Model of Compartmentalized Water^ 54 5.2 T2 Relaxation of Compartmentalized Water: Lumen Water in Wood 5.2.1 Introduction The primary goal of this section is to explain quantitatively the spin—spin relaxation mechanism for wood cell lumen water in terms of a model in which the dominant source of relaxation occurs at the cell wall surface. The effectiveness of this surface relaxation in decreasing the lumen water T2 is determined by the self diffusion rate of the lumen water. We use the classical diffusion model presented by Brownstein and Tarr [1979] and experimentally verify the dependence of the spin—spin relaxation upon the diffusion coefficient of water and upon the radius of the cell lumen. We also consider the effect of deviation of the cell lumen from cylindrical symmetry. A secondary goal of this work is to discuss the interpretation of experimental spin— spin relaxation decay curves from inhomogeneous biological samples. The point is that in the presence of multiexponential relaxation, the T2 times calculated from the relaxation decay curve depend strongly upon the algorithm used for T2 estimation. The degree to which spin—spin relaxation measurements can provide meaningful information about the sample depends crucially upon how the physical mechanism for T2 is incorporated into the relaxation decay curve analysis. 5.2.2 Theory Diffusion—Bloch Equation and Boundary Conditions We define m(r, t) as the magnetization in the cell lumen at the position r and time t. Then m(r, t) satisfies the diffusion—Bloch equation with a volume relaxation T2 f „ e [Brownstein and Tarr 1979] 07n eft (r ^t) (r,, t) = D V 2 m(r, t)^ 7-72 ree (5.1) Chapter 5. Diffusion Model of Compartmentalized Water ^ 55 where D is the diffusion coefficient of bulk water. The relaxation at the cell wall has been expressed by Brownstein and Tarr [1979] as a boundary condition restricting the flux (J) out of the surface (S) to be proportional to m(r, t) • J^—D 11 • Vm(r,t)Is^M m(r, t) I s^(5.2) where M is a parameter characterizing the strength or effectiveness of the surface relaxation. This condition has a parallel in thermal conduction [Carslaw and Jaeger 1959] as a radiation condition at a boundary where heat flow from a conductor is defined to be proportional to the temperature difference across the boundary. The solution to this diffusion problem can be expressed as a sum of normal modes, m(r, t) = E A„F7L (r)e —t / T2 (n) (5.3) n=o where Fri (r) are orthogonal eigenfunctions. For infinitely long cylindrical geometry, with no z or 6 dependence, the boundary condition of Eq. (5.2) becomes —Da O m (r,t)I s = M m(r, t) (5.4) and the solution in terms of Bessel functions is, Fn (r) = Jo ri), (5.5) 1^D7/71^1 2(„)^R2^7 2f „ e (5. 6) 'Ti 1 where R is the cylinder radius. The boundary condition Eq. (5.4) defines ra Tb1L(70 _ MR Jo( 7111)^D • n from (5.7) There exists a series of solutions to Eq. (5.7). The amplitudes A,, are determined by the initial condition of constant magnetization throughout the lumen, immediately Chapter 5. Diffusion Model of Compartmentalized Water ^ 56 following the 90° pulse of a CPMG sequence. That is, 1 E A„Jo— ) • R n=0 00 m(r ,0) rn o = (5.8) Using the orthogonality of Bessel functions gives An = mo f Cell J0( 71nr R)d7 fCell J0( 7,„7. I r d,' - (5.9) ) where dr is a volume variable of integration. Figure 5.1 shows the first three terms of Eq. (5.8) and that the sum of these terms tends towards a constant function of value 7n 0 . The higher order modes relax faster, with relaxation times given by Eq. (5.6). The signal detected by 1 1-1 NMR is the total magnetization from the cell ^M(0) ./14(t) =^m(r, t)dr Cell^ E^/T2(n) n=o (5.10) where A„^f^(71„1' )^4^1 =^ M(0) Cell^R, )^714 (1 + (Thi„IMR) 2 ) • (5.11) Figure 5.2 shows I n versus MR./D. The average values of MR/D for the samples used in this study are displayed in the diagram. Relaxation may be in a fast, intermediate, or slow diffusion regime when the value of MR/D is much less than one, between one and ten, and much greater than ten, respectively. For the fast diffusion regime, only one mode, T2(0) , is required to describe lumen water relaxation; for the intermediate or slow regimes, a sum of modes, mm = 0, 1, 2... is necessary. Diffusion Model Applied to Wood This diffusion model expresses the relaxation of lumen water in wood in terms of the radius distribution, T2f „, of the bulk water, the diffusion coefficient D of the bulk water, and the surface relaxation parameter M. Figure 5.3 shows that the fundamental Chapter 5. Diffusion Model of Compartmentalized Water ^ m A mo n iT 2 0^ 0 F R Figure 5.1: The first 3 A„Jo (q„r/ R) modes. The clashed line is the sum of the first 3 modes which is close to the initial condition of constant magnetization, mo. 57 Chapter 5. Diffusion Model of Compartmentalized Water ^ 1.0 III 0.9 — 0.8 — 0.7 — = 0.6 — 0.5 — 0.4 _ 0.3 — 0 0 0 • Q. E 0 a) C.) CL. a) 0 O O n =0 0 0 (/1 0 0 0 0 cn a) 0 - C/) a) LE C_)^(-) 0 O LC) LI) 0 cp. 0.2 — n =1 0.1 — n =2 0.0 0 1^ 1.0 MR/D 10.0 50.0 Figure 5.2: I7 amplitudes as a function of MR/D. Single to multiple mode range is covered by samples presented. The average R is used for each sample. 58 ^ Chapter 5. Diffusion Model of Compartmentalized Water ^ 59 relaxation mode T2 ( 0) is proportional to R for small radii (< lOpm) and proportional to R 2 for larger radii (> 100pm). The higher order modes dependence for radii less than 100,um. T2free limits the T2 T2(1), T2 ( 2 ) exhibit R 2 for all modes for radii larger than 100,um. ---- 1 0 0-= - o 0^-1 10= _ 107-2 , 10 -3^0 10^10^10^10 Cylindrical Cell Radius (um) Figure 5.3: The diffusion model gives T2 („) versus cell radius (short dashed) and the limiting effect of a finite T2free (solid) (M = 1.40 x 10 -4 m/s, D = 2.2 x 10-9m2/s, T2free = 1.4s). Figure 5.4 shows, for water in a cylindrical lumen of radius 30,um, how the fundamental mode T2 ( 0) depends upon the diffusion coefficient. For the temperature range 4 to 55 ° C, the free water diffusion coefficient varies from 1.1 x 10' to 4.4 x 10 -9 m 2 /s [Simpson and Carr 1958], predicting a factor of 1.6 decrease in the fundamental mode T2 ( 0 ). Figure 5.5 shows how the T2( „ ) modes depend on M using the diffusion coefficient of room temperature bulk water and the average radius for redwood sapwood of 30,am. A large M value corresponds to a strong surface sink and for small M the T2 ( 0 ) is limited by T2free. ^ Chapter 5. Diffusion Model of Compartmentalized Water ^ 0.4 0.3— n=0 0 0.2— --- 0.1— n=1 0.0 .--_____^--^ n=2 i ^ I ^ I ^ i^ I 0^1^2^3^4^5^6 Diffusion Constant D (10 -9 m 2 /s) Figure 5.4: The diffusion model gives T 2( „ ) versus diffusion coefficient (short dashed) and the limiting effect of a finite T2 free (solid) (M = 1.40 x 10 -4 m/s, R = 30/GM, T2f,.„ = 1.4s). 10 1 T2 free o o - 10 n=0 n=1 n=2 10 -3 -3 10 -6^10-5^10^10 Surface Sink Parameter M (m/s) Figure 5.5: The diffusion model gives T2 („) versus surface sink parameter (short clashed) and the limiting effect of a finite T2free (solid) (D = 2.2 x 10 -9 m 2 /s, R = 30/tm, T2free = 1.4s). 60 Chapter 5. Diffusion Model of Compartmentalized Water^ 61 Wood Cell Geometry Dependence The theory presented above applies for cylindrical lumens but wood cell lumens are often shaped like rounded squares or rectangles in cross-section. We therefore consider the effect of cell geometry on 7'2 (See Appendix B). For a square or rectangular lumen cross-section, Eqs. (5.1) and (5.2) are expressed in cartesian coordinates and the solution, Eq. (5.3), involves cosine rather than Bessel functions. We consider a cell with a cross-sectional area of 2826,um 2 , which for circular geometry implies a diameter of 60.0pm, and for square geometry implies 53.2,am sides. Table 5.1 presents the relaxTable 5.1: Geometric dependence of the fundamental T2 mode, T2(o ) (for a fixed surface sink parameter, M), and of M (for a fixed T2 ( o )), for cells with a cross-sectional area of 2826/tm 2 (T2free = 1.4s and D = 2.2 x 10 -9 m 2 /s). Geometry M = 1.40 x 10 -4 m/s T2(o ) = 0.147 s Circle Square Decrease T2(o) = 0.147 s T2(o) = 0.137 s 7% M = 1.40 x 10 -4 m/s M = 1.24 x 10 -4 m/s 11% ation time of the fundamental T2 mode, T2 ( o) , given that the surface sink parameter M = 1.40 x 10 -4 m/s, or alternately presents the estimated surface sink parameter given that T2 ( 0 ) = 0.147s. We note from Table 5.1 that the effect of square cell geometry is an 11% decrease in M for fixed T2 or a 7% decrease in T2 for fixed M. Lumen water T2 values are decreased because the minimum distance to a wall is shorter for the square than the circle. Since the differences between the two geometries are quite small and could be accounted for by a small change in M, we have chosen to treat all wood cell lumens as cylinders. Chapter 5. Diffusion Model of Compartmentalized Water^ 62 5.2.3 Materials and Methods Samples Samples of sapwood were cut from log slices of redwood, and white spruce. A compression wood sample of white spruce heartwood was cut from a 10 x 5 x 3 cm block of wood which was rehydrated with distilled water by two cycles of a vacuum pressure treatment at 39 mm of mercury for 2 hours, followed by a water pressure treatment at 100 psi for 2 hours, and then stored for one month in distilled water. The spruce samples were analysed at 26°C, and a range of temperatures was examined from 4 to 55°C for the redwood sample. Maximum hydration was maintained by soaking in distilled water between NMR measurements. In the CPMG sequence, a T of 200/ts was used for the redwood and white spruce compression wood samples, and a longer T of 400fis was employed for the white spruce sapwood since a longer cell wall water T2 was detected. 5.2.4 Results and Discussion Scanning electron microscopy was used to estimate the distribution of cell lumen radii for each sample. In Fig. 5.6 these distributions are displayed in terms of relative volume which is proportional to the 1 1-I NMR signal from each cell lumen. To ensure reasonable statistics, at least 1000 cells were counted for each distribution. Spin—spin relaxation decay curves were obtained for the redwood sapwood sample at 7 temperatures between 4 and 55°C. A representative CPMG decay curve obtained at 18°C is plotted in Fig. 5.7. We note from Fig. 5.7 that the CPMG curve can not be fit by a single exponential component, in fact the logarithmic plot exhibits a curve of almost continuously varying slope. In Fig. 5.8, smooth T2 relaxation plots are shown for the redwood sample at each of the 7 temperatures. Amplitude weighted average Chapter 5. Diffusion Model of Compartmentalized Water^ aJ b Q) 1:4 1 ,„ 0 10 20 30 4:0 50 Radius Gu,n i) - Figure 5.6: Radius distributions in terms of relative volume in cells acquired from SEM images for a) redwood sapwood, b) white spruce sapwood, and c) white spruce heartwood compression wood samples. 63 Chapter 5. Diffusion Model of Compartmentalized Water^ E 1 ^ i^ i 0.2^0.4^0.6^0.8 Time (s) 1 .0 Figure 5.7: The CPMG decay curve of redwood at 18°C, the decay curve of the lumen water predicted by the diffusion model (solid line), and the smooth T2 fit of the CPMG data (dashed line). The inset shows the cell wall water component, at short times, which is not included in the diffusion model. The inset uses a linear amplitude scale. 64 Chapter 5. Diffusion Model of Compartmentalized Water^ 4.0 ° C \\I 11.0 ° C 34.0°C ----% ' 42.0 °C 18.0°C - , 55.0 °C 26.5°C , ,-- T2 (s) le 10 -4 1 -3 - -2 10^10^10 T2 (s) Figure 5.8: Diffusion model T2 relaxation plots (solid) and smooth T2 solution (dashed) are shown for a range of temperatures for the redwood sample. 65 Chapter 5. Diffusion Model of Compartmentalized Water^ 66 lumen water T2 times for each temperature are displayed in Fig. 5.9. 0.18 (/) 0 x 0 0.16 9 N 0.14 0) 0 <t 0.12 0.10 0.08 0 0^10 20 30 40 50 60 Temperature (°C) Figure 5.9: Average T2 of lumen water for the model (crosses) and the smooth T2 solution (squares) for the redwood sample. The variation of the lumen water diffusion coefficient for free water from 1.1 x 10 -9 to 4.4 x 10 -9 m 2 /s between 4 and 55°C is reflected in the shift to shorter T2 at higher temperatures in Figs. 5.8 and 5.9. Figure 5.10 shows the smooth T2 relaxation plots for spruce sapwood and spruce compression wood. The objective of this study was to show that the relaxation of lumen water in wood is well described by the Brownstein and Tarr diffusion model presented in the theory section. Several issues must be considered before we compare experiment and theory. Chapter 5. Diffusion Model of Compartmentalized Water^ a / ^t ^t /^t i^t ^i ^i / - V^ /^\^ t /^\^ \^ /^ /^t /-\\ / \ \^/ 10 -4^ 10 -3^ 10 -2^ T 2 (s) 10 -1^ 10° Figure 5.10: Diffusion model 772 plots (solid) from the corresponding radius distributions and the smooth T2 solutions (clashed) for a) white spruce sapwood, and b) rehydrated white spruce heartwood compression wood samples. 67 Chapter 5. Diffusion Model of Compartmentalized Water^ 68 Radius Distribution Radius distributions obtained by scanning electron microscopy are relevant to lumen water 1 H NMR measurements only if all wood cell lumens are filled with water. We therefore compared our 1 1-1 NMR measurements of moisture content with that predicted from the SEM radius distributions using Eq. (2.1) . One should not expect perfect agreement since the SEM distributions were derived from small surface regions of the samples whereas the 1 1-1 NMR measurements originate from the entire volume. For the redwood, the average measured MC was 310 + 60% while the SEM analysis predicted 316% for a fully hydrated sample. For the spruce, the measured MC's were 111 and 75% compared to SEM estimates of 100 and 66% for the sapwood and compression wood samples respectively. We are therefore confident that Fig. 5.6 is representative of the true cell radius distributions. Separating Signals of Lumen and Cell Wall Water Our analysis must distinguish the lumen water 1 H NMR signal from the cell wall water signal. We note in Figs. 5.8 and 5.10 that the peaks on the left side of each plot may be assigned to the cell wall water and that there is practically no overlap with the lumen water signal assigned to higher T2 's. Therefore, when fitting our lumen water T2 model to the CPMG relaxation decay curves, we used only those data at times greater than a cutoff time where the contribution to the signal from cell wall water should have decayed by a factor of 50 or to the noise level. We note on Fig. 5.7 that the cell wall water signal from redwood at 18°C decayed to the noise level well before 20 ms. For the redwood sample, we used a cutoff of 20 ins for measurements acquired below 34°C and 60 ins for those acquired at and above 34°C. A cutoff of 30 ms was used for the white spruce samples. Chapter 5. Diffusion Model of Compartmentalized Water^ 69 To check the cell wall water cutoff times, a series of lumen water fits, which are discussed later, were carried out using a decreasing number of data points, N, corresponding to increasing the start time from the first echo. The value of X 2 /N was monitored as a function of N. As N decreased, the value of x 2 /N was initially large, then it decreased to a limiting value as contributions from the cell wall water were left out of the fit. For starting times later than the cell wall water cutoff times, the value of x 2 /N was independent of N for all samples. Fitting the lumen water relaxation decay curves Smooth T2 relaxation models generally provide excellent fits to relaxation decay curves from multiexponential CPMG decay curves [Whittall and MacKay 1989, Kroeker and Henkelman 1986] (and see Chapter 4); much better than discrete fits to one, two or three exponential components. However there are two problems with using smooth T2 relaxation plots for description of spin—spin relaxation of lumen water in wood. Firstly, smooth T2 models yield somewhat ambiguous results, i.e. there exist a large number of T2 plots with different shapes which will fit the relaxation decay curves almost equally well using an acceptable statistical criterion such as x 2 [Whittall and MacKay 1989]. Secondly, smooth 7'2 models provide only indirect information relating to water in wood. Although the shapes of the lumen water T2 plots reflect the wood cell distribution, they are complicated by the existence of a complex functional dependence on the radius (Fig. 5.3) and the existence of the T2 modes expressed by Eq. (5.6). Therefore, quantitative comparisons between theoretical models and experimental results in a multiexponential spin—spin relaxation study should be made with the relaxation decay curve rather than with either T2 times or T2 relaxation plots which are both biased by the algorithm used for 7' 2 calculation. Using the Brownstein and Tarr model [1979] of Eq. (5.10), we obtained fits to Chapter 5. Diffusion Model of Compartmentalized Water^ 70 the redwood and spruce CPMG decay curves equivalent to those from the smooth T2 model but which contain more explicit information about the system. The radius distribution from Fig. 5.6 and the diffusion coefficient of free water at the appropriate temperature were used in these fits. Note that there are two sums: One over the radius distribution and the other over the T2 modes of the solution. Only the first 4 modes were included, since higher modes were of negligible amplitude and of much shorter T2. The fit minimized x 2 using a simplex routine [Press et a/.1988]. Several initial starting vertices were used to find the global minimum. The X 2 of the fit with the Brownstein and Tarr model must necessarily be worse than or equal to that for the best smooth T2 model since the diffusion model fitting function was also made up of exponential terms but had to be consistent with the known radius distribution. There were four variable parameters in the diffusion model: The total amplitude and the baseline offset of the CPMG data, the free water relaxation time T2 f ree and the surface sink M. The amplitude was well defined by the fitting procedure. The baseline offset was relatively small (less than 2% of the maximum amplitude) and was correlated with T2 f r ee• T2 free was limited to values below 3.0 s because longer values did not affect the fit. Redwood Lumen Water Figure 5.11 shows the T2 plot predicted for the redwood sample at 4°C using the Brownstein and Tarr model expressed in Eqs. (5.10) and (5.11), which used the radius distribution in Fig. 5.6a and the parameters from the best fit of the diffusion model to the CPMG decay curve. Of all the wood samples studied, this redwood sample had the highest value for MRID and hence the largest amplitudes of higher order modes. The clashed curve of the main figure is the fundamental mode and the solid curve is the sum of the first three modes. The inset shows the second and third modes as the dashed and solid curves, respectively. We note that the sum coincides with the Chapter 5. Diffusion Model of Compartmentalized Water^ 10 -2 T2 (s) Figure 5.11: The first mode (dashed) and the sum of the first three modes (solid) predicted for the redwood radius distribution at 4°C. The second (dashed) and third (solid) modes are plotted in the inset. The sharp undulations result from the steps in the histogram of cell sizes. 71 Chapter 5. Diffusion Model of Compartmentalized Water^ 72 fundamental mode at longer T2 but deviates for T2 less than 70 ms where there is substantial contribution from the higher modes. The sharp undulations in the T2 plot reflect steps in the histogram of cell sizes in Fig. 5.6a. Table 5.2 shows the values of M and T2f 7.„ required to fit Eq. (5.10) to the CPMG Table 5.2: Fit of the diffusion model to the CPMG decay curves for the redwood sample. Temp. Four Modes (°C) M( m /s) 4.0 11.0 18.0 26.5 34.0 42.0 55.0 1.38 x 10 -4 1.55 x 10 -4 1.66 x 10' 1.43 x 10 -4 1.52 x 10 -4 1.58 x 10' 1.78 x 10' T2free ( s ) x 2 /N 1.1 1.7 3.0 1.4 1.8 1.6 2.5 1.86 1.87 1.99 1.66 1.82 2.03 2.80 One Mode M (m/s) T2free(s) x 2 /N 2.36 x 10 -4 1.96 x 10' 1.79 x 10 -4 1.63 x 10' 1.57 x 10' 1.58 x 10 -4 1.81 x 10 -4 2.9 3.0 3.0 3.0 2.2 1.6 2.9 13.2 19.1 23.3 6.17 1.97 2.14 2.78 Ratio x 2 's 7.1 10.2 11.7 3.7 1.1 1.1 1.0 decay curves for the redwood sapwood at the 7 temperatures. For these fits x 2 /N was around 2 which is somewhat larger than the mean of 1 expected for a x 2 random variable with N degrees of freedom. We note that the smooth T2 plots yielded X 2 /N values of mean near 1.3. We believe that this increase from 1.3 to 2 is to be expected considering that the diffusion model contained the relatively crude SEM radius histogram obtained from sample surfaces. The importance of including the higher order modes in the CPMG fitting function is demonstrated by the increase in X 2 /N by up to a factor of 12 for the lower temperature samples when only the first mode, T 2 ( 0 ), was included in the fit. Above 26°C, the increase in x 2 /N for the single mode fit was not so pronounced since the larger diffusion rate moves the sample closer to the fast diffusion limit where higher order modes are negligible. Therefore, the fit of the four mode solution is obviously better than the one mode solution, and we believe these results strongly support the existence of the higher order T2 modes. Chapter 5. Diffusion Model of Compartmentalized Water^ 73 In Fig. 5.8, the Brownstein and Tarr model T2 plots (solid lines) are compared with the smooth T2 plots obtained from the CPMG data (dashed lines). The two lumen water T2 curves look qualitatively similar but quantitatively are quite different. The structure of the smooth T2 solution tends toward smoothed but localized peaks. The average lumen water T2 calculated from each curve are compared in Fig. 5.9 and show good agreement over almost a factor of 4 in MR/D in the intermediate regime where the higher modes are significant. Spruce Samples The white spruce sapwood and compression wood samples were chosen because they have cell lumen radius distributions of different shapes (Fig. 5.6b, c). In particular, the white spruce compression wood radius distribution was skewed to the smaller cell sizes. In Fig. 5.10 the smooth and model T2 plots are compared for the spruce samples. Table 5.3 displays the parameter values for the fits of the diffusion model to the CPMG Table 5.3: White Spruce Results Sapwood Compression M (X T2free X 2 /N model fit x 2 /N smooth Ave. T2 model fit Ave. T2 smooth 1.37 1.6 s 0.9 0.8 57 Ins 59 ins 1.26 2.5 s 3.0 1.4 40 ins 33 ms data. While the x 2 /N value for the diffusion model fit to the spruce sapwood data was close to that from the smooth T2 solution, the x 2 /N value for the model fit to the spruce compression wood data increased by a factor of 2. In Fig. 5.10, we note that the smooth solution for the compression wood sample is substantially broader than Chapter 5. Diffusion Model of Compartmentalized Water^ 74 the model T2 plot. We do not know the origin of this discrepancy but speculate that either the SEM radius distribution used for the fit was not representative of the whole sample, or that the surface sink parameter, M, was a function of wood cell size. 5.2.5 Concluding Remarks A four parameter Brownstein and Tarr diffusion model, which incorporated the known lumen radius distribution, yielded fits to the CPMG decay curves with comparable x 2 /N values to those obtained using a smooth T2 model which had no constraints other than smoothness of the T 2 distribution. The different wood samples and experimental conditions provided an excellent comparison between theory and experiment for a wide range of values of R and D and the dimensionless parameter MR/D. The ratio MR/D changes the nature of the solutions and the values of R and D influence the T2 and amplitudes. We therefore believe that the Brownstein and Tarr diffusion model adequately accounts for experimental measurements of T2 relaxation of lumen water in wood. To our knowledge, this is the first demonstration of the presence of higher order relaxation modes in a biological system; modes have been observed previously in a model system of cylindrical capillary fibres [Bronskill et a/.1990]. We now believe that we are justified in using this diffusion model to estimate the radius distribution directly from the CPMG decay curves [Whittall 1991]. Such a procedure requires values for D, T2 f „ " and M. The bulk water diffusion coefficient, D, is known a priori [Simpson and Carr 1958]. T2free is not expected to vary much and is between one and three seconds for wood samples at ambient temperature. The values for /V/ found in this study did not vary greatly from sample to sample. In the following chapter, a physical interpretation of M is introduced, relating it to the cell wall water T2 and the cell wall water diffusion coefficient. In a theoretical paper, Belton and Hills [1987] solved a similar diffusion model, 75 Chapter 5. Diffusion Model of Compartmentalized Water ^ describing spacially separated regions with diffusive exchange, using the Laplace transform method to give the analytical i ll NMR spectra solutions. They considered one dimensional geometry, allowing for regions of differing resonance frequency, bulk transverse relaxation rate and diffusion coefficient in a variety of examples, to model heterogeneous and biological systems. Relaxation models based upon water diffusion have recently been applied to 1H NMR studies of porous media including spin—lattice (T 1 ) relaxation in rocks [Davies and Packer 1990, Kenyon et a/.1989] and T1 and T2 relaxation in glasses [D'Orazio et a/.1989]. Using an isolated sphere model for pores in rock cores, Davies and Packer [1990] implemented the Brownstein and Tarr diffusion model in order to relate distributions of T 1 to the pore size distribution. In a T1 and T2 study of silica glasses, D'Orazio, et a/. [1989] accounted for diffusion from pore to pore by considering the measured relaxation rate as representing a weighted average over the volume traversed by a water molecule over the period of the 1 H NMR experiment. This model, which assumes fast diffusion inside the pore but slower diffusion from pore to pore, yields an average surface to volume ratio which corresponds to a pore size distribution only if water diffuses through homogeneous regions of the sample during the period of the 1 H NMR measurement. Mendelson [1990] used a percolation model to show that in the fast diffusion limit, the relaxation time of water in porous media was proportional to the volume to surface ratio of the system and in the slow diffusion limit the relaxation time was related to a pore dimension. There are a number of fundamental differences between porous materials and wood which influence the 1 H NMR relaxation of lumen water. Firstly, in most porous materials water is not restricted to a lumen of known geometry but is free to diffuse in torturous paths from pore to pore throughout the structure. Secondly, in rocks, spin— spin relaxation measurements are very difficult to carry out quantitatively presumably Chapter 5. Diffusion Model of Compartmentalized Water^ 76 due to signal losses caused by water diffusion in magnetic susceptibility induced field gradients; hence most studies have considered T 1 rather than T2. Such susceptibility induced signal losses are negligible in water filled wood samples. Finally, the mechanism of the surface relaxation sink in porous media is different from that in wood since water cannot penetrate the cell walls of rocks or glass. We believe that this work is of more general application than to lumen water in wood. We have defined the role of lumen water diffusion in the spin—spin relaxation process. Wood is a particularly simple biological system because the diffusion process slows down by at least one order of magnitude at the cell wall. In many other biological samples, for example, human tissue, the barrier to diffusion outside the cell lumen may be much smaller; requiring a more sophisticated model for spin—spin relaxation of lumen water. Since many diseases involve specific processes at the cellular level, the diagnostic capability of medical magnetic resonance could be improved by developing techniques, like the one presented here, which are sensitive to structures of cellular dimensions. Chapter 5. Diffusion Model of Compartmentalized Water ^ 77 5.3 Recovering Compartment Sizes from NMR Relaxation Data The objective of this section is to obtain a quantitative measurement of wood cell dimensions of order 10 microns and larger using the Brownstein and Tarr diffusion model presented in the first part of this chapter. For a distribution of cylindrical cells, the T2 relaxation decay curve is y(t) = f oo c(R)71 - R 2 h pM(t , R) dR (5.12) where c(R) is the number of cells with radius R, M(t,R) is the multiexponential decay given by Eq. (5.10), and 7-R 2 hp converts from number of cells to signal intensity using the assumed constant cylinder height h. and signal density p. The inverse problem is to recover the compartment size histogram c(R) by using the nonnegative least-squares algorithm, NNLS, to minimize x 2 for a set of cell radii R, and first derivative smoothing constraint [Whittall 1991]. Comparison of radius distribution curve generated by scanning electron microscope to that generated by NMR relaxation is presented for five wood samples. The solutions are acquired with variable parameters M, T2free, Df ree and starting time for lumen water decay to exclude fast decaying cell wall water signal. The Dfree is set to the known diffusion coefficients for bulk water [Simpson and Carr 1958]. Parameters M and T2free were set to those listed in Table 5.2 for the redwood distributions and Table 5.3 for the spruce distributions, and start times were the same as discussed in Section 5.2.4. For the alder and Douglas fir, parameters are set with T2free of 1.5 s, start time of 30 ms, and M of 0.6 x 10' m/s. A set of 100 linearly spaced allowed radii from 1 to 100 itm was used. The NMR radius distributions of redwood, shown in Fig. 5.12, correspond to the SEM distributions for NMR relaxation decays collected at temperatures 4.0 to 55.0°C. All distributions show a latewood and earlywood bimodal distribution. Figures 5.13 Chapter 5. Diffusion Model of Compartmentalized Water^ Figure 5.12: Redwood radius distribution from NMR relaxation data (solid lines) for temperatures 4.0 to 55.0°C. Compared to SEM radius distribution (histogram). 78 Chapter 5. Diffusion Model of Compartmentalized Water^ Figure 5.13: White spruce radius distribution from NMR relaxation data (solid lines) for a) white spruce sapwood, and b) white spruce compression wood. Compared to SEM radius distribution (histogram). 79 Chapter 5. Diffusion Model of Compartmentalized Water^ 80 a b r 0^10^20^30^40^50 Radius (p,m) Figure 5.14: Radius distributions from NMR relaxation data (solid lines) of a) alder sapwood, and b) Douglas fir sapwood. Compared to SEM radius distribution (histogram). and 5.14 show distributions at room temperature. The white spruce sapwood has a bimodal distribution. The alder sapwood, white spruce compression wood and Douglas fir sapwood all have SEM distributions skewed to the small radii. The NNLS solutions from the NMR relaxation decays tend towards localized peaks and represent the skewness as a large and small peak. The SEM distribution of the alder sapwood shows 2.2% of the cells having larger radii from 18 to 37 /cm (average 28 /cm) and the NMR distributions show a similar component with 2.7% of the cells at 20 ,um. The radius distribution can be calculated from the T2 relaxation of lumen water in wood which is comparable to the SEM radius distribution information. To solve for the radius distribution the surface relaxation parameter, M, is needed, which will be determined in the next chapter. Chapter 6 Diffusion Model of Two Regions of Compartmentalized Water 6.1 Summary Spin-spin relaxation of water in wood is described by a model which treats relaxation of lumen water as a magnetization exchange with cell wall water. The diffusion-Bloch equations are solved for two regions: Free water in the cell lumens and water in the cell walls. The resulting theory relates the surface sink for lumen water relaxation to the spin-spin relaxation time and the diffusion coefficient of the cell wall water. The cell wall water diffusion coefficient may therefore be estimated from experimental measurement of the spin-spin relaxation times and relative populations of lumen and cell wall water. Such estimates for cell wall water diffusion in a maximally hydrated redwood sapwood sample ranged from 0.92 x 10 -6 cm 2 /s at 4°C to 5.89 x 10 -6 cm 2 /s at 55°C. The activation energy for cell wall water diffusion in redwood sapwood in this temperature range was found to be 6700 cal/mol, about 40% higher than the free water value of 4767 cal/mol. Numerical simulations of the two region diffusion model were developed. The lumen water T2 was found to be independent of the simulated cell wall thickness, simplifying to a surface relaxation as modeled with the surface relaxation parameter in the one region model. The simulated effect of exchange on the FSP measurement was found to be an over estimate compared to experimental results. 'Part of this chapter closely follows, with some modifications, part of a previously published paper. [Araujo et a1.1993] 81 Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water ^82 6.2 Determination of the Cell Wall Water Diffusion Coefficient in Wood from T2 Relaxation Measurements 6.2.1 Introduction In the preceeding chapter, I discussed the spin-spin relaxation of lumen water in wood in terms of a diffusion model introduced by Brownstein and Tarr [1979]. This model treated the relaxation of lumen water at the cell walls as a surface sink characterized by a variable parameter, M. The Brownstein and Tarr model provided an excellent fit to lumen water relaxation results, however the incorporation of the surface sink parameter, M, which had no direct physical interpretation, was somewhat unsatisfactory. In this chapter, I introduce a spin-spin relaxation model in which water in wood is explicitly assumed to exist in two regions: i) Lumen water characterized by bulk water values for diffusion coefficient, D 1 , and bulk water spin-spin relaxation time, T21 and ii) cell wall water characterized by a cell wall water diffusion coefficient, D2 and the measured cell wall water spin-spin relaxation time, T22. The magnetization behavior of cell wall and lumen water is treated simultaneously using diffusion-Bloch equations for each region and joint boundary conditions. Two different ways are used to analyse the differential equations: i) The Laplace transform and ii) a less rigorous, but more intuitive approach. The goal is to obtain a clearer understanding of the effect of cell wall water T2 relaxation on lumen water T2 relaxation; i.e. to have a physical interpretation for the surface sink parameter M of the Brownstein and Tarr diffusion relaxation model for water in wood. I demonstrate how M is related to the cell wall water diffusion coefficient, D2, and use the NMR. results of the previous section to determine D2 in redwood sapwood as a function of temperature. Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^83 Theory The relaxation model has been applied to one dimensional, cylindrical and spherical lumen geometries. For simplicity, the one dimensional model is presented here (See appendices C and D). I consider two regions of water concentration, as shown in Fig. 6.1, with different diffusion coefficients T21, T22 D1, D2, different spin—spin relaxation times and magnetization m i (x, t), m 2 (x, t). Exchange between the two regions will ^ —R^0 R ^ Region 1 Region 2 Figure 6.1: Two region diffusion problem with different volume relaxation sinks and diffusion coefficients. occur due to flux of magnetization out of the lumen at the boundary. The diffusion— Bloch equations for these magnetizations are: Dmi^a2in^mi(x, t) . ^ (x, t) = D, 07 0 i = 1, 2. at^ Ox2 ' 172, (6.1) The initial condition of uniform magnetization following the 90° pulse of a CPMG sequence gives mi(x, 0) ^ m 2 (x , 0) = 71120 = mo; 0 <^<R K m o ;^> R ^ (6.2) where K is the partition coefficient which is the ratio of water concentration in the cell wall to water concentration in the lumen. The water concentration in the lumen is Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water ^84 just the water density which is 1.0g/m1 and equal to d„„ d FSP/100 in the cell wall by the definition of the fibre saturation point (FSP) [Sian 1984], where dwoo d 1.5g/ml [Haygreen and Bowyer 1982]. The solution to Eq. (6.1) is simplified if weimpose an alternative initial condition: mi(x, 7) = mo m2(x, r) = 0^(6.3) for T T22. > T22 which is valid when the lumen water relaxation times are much longer than This assumption generally holds for T 2 relaxation of water in wood. At the boundary x = R the magnetization must satisfy a partition balance equation and the flux must be continuous, ^Km i (R,t) = rn 2 (R,t),^ — D1 Om ^ (R, t) = — D2 2 (R, t). O x^Ox (6.4) (6.5) Initial value problems can be solved using the Laplace transform [Luikov 1968]. In this case, the transform would reduce the problem from a partial differential equation to an ordinary differential equation so that derivatives arise in only one variable, x. Defining the Laplace transform, E, as .C{m i (x,t)}^mi(x,t)e—st dt = ^ (6.6) and applying the Laplace transform to Eq. (6.1) gives a2 7.fhi s^, s) —^0) = D i ^ (x s) Ox 2 s) , T2i = 1, 2^(6.7) and the boundary conditions Eqs. (6.4) and (6.5) become Kth i (R, s) = ih 2 (R, s)^ Ofh2 ^ (R s). D1 ^ (R s) = D2 Dx^Dx , (6.8) (6.9) Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^85 The general solution to Eq. (6.7) is ,^ai(s) cosh(O i x)^-yi (s) sinh(dix)^ m=07'22 (6.10) sT2i + 1 where oz i , Oi and ryz are undetermined constants. The solutions using the boundary conditions of Eqs. (6.8) and (6.9) and requiring that rh i (x, s) are symmetric about x = 0 and rn 2 (x, s) is bounded for large x, are: s) rnioT2i 8721 + 1 A(s) cosh(O i x) A( s) 2 r [ exp(01 x)^exp( O i x)] 7'21 sT2 1 + 1 M10 (6.11) 77 1 20T22 M 2 (x, s)^= B(s) exp(— / 3 2 1x1)^,T, (6.12) 22 + 1 where A(s) irt2oT22 sT2 2 + 1 B(s) Ii sT21 + 1 D2 [Kr-ni(R,^) . sm h(l3 i R) I K cosh(O i R) 71120T22 exp(13 2 R) 87'22 + 1 1 (6.13) (6.14) and sT2i + 1 . Ni 1, 2. (6.15) The solutions m i (x,t) and m 2 (x,t) are the inverse Laplace transform of these functions in i (x, s) and 7iz 2 (x, s) respectively. For large x, 7712(x, S) - 71120 S 1/T22 (6.16) which gives 7i 2 (x, t) = m20 exp(—t/T22 ) as expected. I have chosen to not invert Eqs. (6.11) and (6.12) to give ni l t) and 7/1 2 (x, t) for all x since the inverse Laplace transform for this expression is very difficult to calculate analytically and is unnecessary for the purpose of this section. The resulting analytic solution would be unwieldy and Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water ^86 more cumbersome than the Brownstein and Tarr solution for lumen water relaxation presented in the preceeding section. Brownstein and Tarr [1979] defined a boundary condition at x = R, Eq. (5.4, in which the radiation or flux of magnetization out of the lumen is assumed to be proportional to M, the surface sink strength: —Di as (R t) = M m i (R,t).^ (6.17) We can use m i (x , t) and m 2 (x, t) from our two region problem to find an expression for M. Assuming that M is a constant, the Laplace transform of Eq. (6.17) is simply — am ax (R s) M fit i (R, 8). (6.18) Using Eqs. (6.8, 6.9, 6.12) and the initial conditions of Eq. (6.3) we obtain 1 M=K D 2 (s (6.19) M is not a constant as assumed above but depends upon s. Eq. (6.6) shows that at short times t, m(x, t) contributes to m(x, s) at all s, whereas at long times, m(x, t) only contributes to rh(x, 8) for small s. Consequently, the range s > 1/T22 corresponds to short times only, and therefore we suspect that for t < T22, the Brownstein and Tarr radiation condition of Eq. (6.17) may not be valid. However, for .s << 1/T22 , M is approximately constant and given by M K D2 (6.20) T22 Since lumen water relaxation times in wood are generally much longer than cell wall water relaxation times, Eqs. (6.17) and (6.20) are expected to hold for water in wood. The preceeding theory was for a symmetric one dimensional lumen; the solutions for cylinders and spheres are the same in the limit of VD 2 T22 << < R, which holds for wood. , Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water ^87 Intuitive Derivation of M I present here a more intuitive treatment of the relationship between lumen and cell wall water relaxation times in wood. I assume that lumen water has the diffusion rate of free water and a relatively long T2 of 1 to 3 s. I assume that cell wall water has a diffusion rate at least ten times slower than that of free water and possesses a much shorter T2 of 0.5 to 5 ins. Furthermore, upon entry to one reservoir from another, water undergoes a discontinuous change in spin—spin relaxation rate and in diffusion rate. I consider the flux at the boundary of the two regions in Fig. 6.1, ana l — D 1 ^ (R, t) = Din2 (R, t) -D2 Ain . ax^ax^ Ax D2 To estimate Am/Ax, look at changes in M and x in time T22 (6.21) at the boundary, Am = —(1 — 1/e)Km 1 (R,t), since the magnetization of water in the cell wall decays with relaxation time of T22 from its initial value Km i (R, t). Also, I approximate Ax as the root mean square distance perpendicular to the surface that a cell wall water molecule diffuses in this time, which is Ax = VD 2 T22 . Substitution these expressions into Eq. (6.21) and the condition Eq. (6.17), gives M = (1 — 1/e)K D2 122 oc K D2 T22 (6.22) 6.2.2 Results and Discussion In Fig. 6.2 I show Carr—Purcell—Meiboom—Gill (CPMG) decay curves originating from cell wall water in fully hydrated redwood sapwood at 7 temperatures from 4 to 55°C. These curves were obtained by subtracting the lumen water contribution from the experimental CPMG relaxation decay curves originating from all the water in the redwood samples Chapter 5. The lumen water contribution was calculated from the Brownstein and Tarr relaxation model using the known radius distribution, fitted values for M, the Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water ^88 10 2 _^ a) CI'• k k, ,,,,,_,,„, :"--,,, 5;1) 461144." 0 %ftftw.404.0.......... 4 °Z N,^ -- * ***•••****** 6,x^ 45Aill x< 00000 000 ** .. ...LA 0 0000 0 AA 000 A AA A 00 00 t x,s< WX 4UC ill< 0 A AALAYL, g 3<"),,,X 10 ° .00 XX xX x Cal ,,,,V3 git^xX x .01 Time (s) .02 Figure 6.2: Cell wall water decay curves of redwood sapwood (4°C box, 11°C triangle, 18°C plus, 26°C cross, 34°C circle, 42°C filled triangle, 55°C filled circle) bulk water T2 time, and the bulk water diffusion coefficient as discussed in the preceeding section. In Fig. 6.3, I display T2 relaxation plots for the cell wall water of redwood sapwood derived from the relaxation decay curves of Fig. 6.2 using a non—negative least squares algorithm [Whittall and MacKay 1989]. Table 6.1 lists the moisture content (MC) and FSP of the wood sample, the average cell wall water T2 the M values and , the calculated values for D2 as a function of temperature. The FSP values in Table 6.1 were the product of the fraction of cell wall water in the relaxation decay curve and the moisture content of the redwood sample. Although the T2 plots in Fig. 6.3 exhibit more than one peak, a single, amplitude weighted, average T2 was calculated because there was no justification for including more than one component of cell wall water. The T2 plots for 34, 42 and 55°C exhibited a 23 ms T2 component with amplitudes corresponding to 6, 5, and 15% MC respectively which were not included in the cell Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^89 4.0 ° C 18.0^- 26.5 °C - 10 -2 io -1 10 ° ° C- 42.0 °C - 55.0^- A 10 -4 0 -3 10 -2 10 - ' 12 T2 (s) Figure 6.3: 34.0 10 ° (S) relaxation plots of cell wall water decay curves from redwood sapwood. T2 Table 6.1: Cell wall water T2, fibre saturation point, moisture content, surface sink parameter and cell wall water diffusion coefficient for redwood sapwood at temperatures 4 to 55°C. Temp. T22 4.0°C 1.37 ins 11.0°C 1.26 ms 18.0°C 1.23 ins 26.5°C 1.88 ms 34.0°C 1.99 ins 42.0°C 1.98 ms 55.0°C 1.97 ms FSP 35.5% 30.2% 33.7% 28.0% 27.4% 26.3% 21.7% MC M(m/s) D2(cm2/s) 451% 1.38 x 10 -4 0.92 x 10 -6 328% 1.55 x 10 -4 1.48 x 10 -6 354% 1.66 x 10 -4 1.33 x 10 -6 291% 1.43 x 10 -4 2.18 x 10 -6 315% 1.52 x 10' 2.72 x 10 -6 279% 1.58 x 10 -4 3.18 x 10 -6 247% 1.78 x 10 -4 5.89 x 10-6 Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^90 wall water T2 calculation. This component comprises from 2 to 6% of the total water and I do not know its origin. The values for M in Table 6.1 were derived assuming a cylindrical lumen in Chapter 5. The estimated cell wall water diffusion coefficient, D2 using Eq. (20), is plotted in Fig. 6.4 as a function of inverse temperature. Two estimates are shown, one with the Temperature ( ° C) ^ 60 50 40 30^20^10 0 3.0 I I I^1 1 3. 1^3.2^3. 3^3.4^3. 5^3.6^3.7 1000/T(° K) Figure 6.4: Calculated cell wall water diffusion coefficient for redwood sapwood using FSP from Table 6.1 (crosses) or a linear fit to these FSP values (boxes). Fits of the diffusion coefficients using the FSP from Table 6.1 (solid line) or data using a linear fit of the FSP (clashed line). experimental FSP values from Table 6.1, and the other using FSP = 35.6 — 0.24T(°C) which is a linear fit to the experimental FSP values. Using this linear fit decreases the effect of the experimental errors in the FSP measurement on the values of D2. Other researchers have observed a similar linear temperature dependence of the FSP [Stamm and Loughborough 1934]. Water diffusion in wood is a thermally activated process which should exhibit a Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^91 temperature dependence according to the Arrhenius equation: D 2 = D oe —E a /RT (6.23) where D o is a constant and E a is the activation energy for cell wall water diffusion, R is the universal gas constant of 1.98 cal/(mole °K) and T is the absolute temperature. When Eq. (6.23) was fitted to the results in Fig. (6.4) using nonlinear x 2 minimization [James and Roos 1975], Ea and D o were 6877(250) cal/mol and 0.23(0.08) cm 2 /s with the experimental FSP data and 6611(210) cal/mol and 0.14(0.06) cm 2 /s with the linear fit to the experimental FSP values. The numbers in parentheses indicate confidence limits of 68.3% assuming x 2 equals the number of degrees of freedom. In Eq. (6.23) Ea was assumed to be independent of temperature or moisture content; when a linear temperature dependence (E a = E0 + CT) was incorporated there was no improvement in the fit. Our measurements of D2 are unique for a couple of reasons: i) They were equilibrium diffusion measurements, with no applied gradients of moisture content, temperature, or relative humidity and no net change in weight or dimension of the wood sample, and ii) they were measured in fully hydrated wood samples. It is difficult to imagine any other way to measure cell wall water diffusion in a fully hydrated sample. Pulsed field gradient NMR measurements of cell wall water diffusion would be difficult to perform due to the short T2 of cell wall water and the fact that cell wall water is only about 10% of the total water in fully hydrated wood samples. The only direct cell wall water diffusion measurements I am aware of in the literature are from Stamm [Stamm 1959,Stamm 1960] who measured water absorption and swelling in thin wood samples with metal filled lumens at moisture contents below the FSP. Tangential and radial diffusion coefficients were found to be two to three times smaller than longitudinal diffusion at moisture contents below the FSP [Stamm 1960], Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water ^92 presumably due to resistance to swelling of adjacent cells. I compare my results to the longitudinal values since in equilibrium, water diffusion should not be restrained by fibre swelling. Using Stamm's results [Stamm 1959,Stamm 1960], Skaar [Skaar 1988] estimate values for Ea and D o for longitudinal cell wall water diffusion to be Ea = 9600 — 170MC 6.95MC 2 — 0.160MC 3 (cal/mol) (6.24) and 0.19 cm 2 /s respectively. Substitution of the FSP values from Table 1 into Eq. (6.24) would yield Ea values from 5200 at 4°C to 7550 at 55°C with a mean of about 6500 cal/mol. Our value for Ea of 6700 cal/mol for the entire temperature range is close to the above mean and substantially greater than the activation energy for free water diffusion of 4767(49) cal/mol calculated from the bulk water diffusion values used in this study [Simpson and Carr 1958]. It is interesting that for moisture contents below the FSP, the activation energy for cell wall water diffusion was found to be a function of moisture content (Eq. (6.24)). For the fully hydrated wood sample investigated here, Ea was found to be independent of the moisture content of the cell wall water at the FSP which varied from 35.6% to 21.7% across the temperature range studied. 6.2.3 Concluding Remarks I conclude that spin—spin relaxation of lumen water in wood can be understood quantitatively using the Brownstein and Tarr diffusion model and interpreting the wood cell surface relaxation sink as a magnetization exchange between lumen and cell wall water. From the T2 relaxation results, I have determined cell wall water diffusion coefficients in approximate agreement with values obtained by other methods. To our knowledge, this constitutes the first experimental measurement of cell wall water diffusion in a fully hydrated wood sample. Our results indicate that the cell wall water diffusion coefficient is about one order of magnitude smaller than that of free water and that the Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^93 activation energy for cell wall water diffusion in fully hydrated wood is approximately 40% larger than that for free water diffusion. Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water ^94 6.3 Numerical T2 Simulations 6.3.1 Numerical Method The two region problem in the previous section was not solved analytically for the magnetization of the water in the lumen and cell wall. A numerical solution of this problem can be derived and used to investigate the limitations of the Brownstein and Tarr assumption of the surface relaxation condition, Eq. (5.2), and to investigate the effect of exchange on the determination of the cell wall and lumen components from the spin-spin relaxation decay curve. To numerically simulate the diffusion and relaxation of spin magnetization of water in two regions with differing diffusion and relaxation parameters, the diffusion—Bloch equation, 492m ni(x , t) (x,t)^D (x t)^ at^axe^T2 (6.25) must be discretized. To discretize the derivatives of a function u(x, t) at (x, t) consider the Taylor series expansions of the function [Ames 1977]. The Taylor series of u(x Ax, t) about (x,t) is u(x Ax, t) = u(x,t)-F Ax (x, ax t) + (Ax )2 492 u 2! ax (Ax) 3 a3 u + 31 ax3 (x, t) 0[(Ax)1 (x ' (6.26) From the above Taylor series one has a simple first-order approximation Ou 1 ax '^h (u^ — ui,j) + 0(h) (6.27) A double subscript notation is used so that u 2 , j is the discretized value of u(x, t) at the i th step in x and the j th step in time, t. The variable h Ax is the step size in x and the variable k = At is the step size in t. The 0(h) represents the truncation error and without it one has the forward difference approximation. The Taylor series ^ Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water ^95 A^ (j+1)k A D free T free - jk D^T cw cw / Q.) 3k 2k (j= 1 ) 0 ^ 0 a h 2h 3h wh^ih Position (x) Figure 6.5: Discrete steps in position and time. of u(x — Ax, t) about (x, t) is u(x — ox, t) = u(x,t) — Ax — au (x,t) (Ax)2a2u ( ax^2!^ x 2 ' x ' t) (Ax) 3 a3 u ax3 (x, t) ORAx) 4 } 3! (6.28) and gives another first order approximation referred to as the backwards difference approximation au 1, = —^— u i _ i , j ) + 0(h) •^ '3 ax ' 3^h (6.29) To obtain a second-order approximation, the difference of the previous two Taylor series, Eqs. (6.26,6.28), is used au^(Ox)3 03u u(x Ax ,t) — u(x — Ax, t) = 2 Ax -5--; (x,t)+ ^ 3! ax^, t) 0[(Ax) 5 16.30) to give 1 i+Li — u1_1 u1-1,j)0(h 2 ) 2h yu (6.31) Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^96 and without the 0(h 2 ) truncation error, this solution is referred to as the centered difference approximation. In a similar manner, the second derivative is a2 u 1 2tti,j^tli+Lj) a x 2 12, 3 = 11 2 (7-11-1 ,j^ ^0(h2). (6.32) To discretize a partial differential equation, one uses the centered difference for the space derivatives and the forward difference for the time derivatives. Solve for u i , j+i , resulting in a "marching" ahead in time method. For example, consider the diffusion equation u t = u s ,; , where units have been chosen so that D = 1. Define u i j as the , exact solution with no truncation error, and Ift , i as the discrete approximation. The diffusion equation gives 1 ) = h2 (Ui +1, 3 —^U2-1,3) (6.33) The solution, with p= k/11 2 , is Ut,j +1 =^+ (1 — 2p)Ui,j^pUi,j^(6.34) and for u t , , the exact solution, is u t ,J +1 = p u i _ i , + (1 — 2P)uij^p tti j + 0[k 2 + kh 2 ]^(6.35) , emphasizing the local truncation error. Initial conditions give the solution for t = 0, the boundary conditions define Ui ,j at the boundary, and Ui, j+i defines all other Discretization error decreases as h and k are decreased, but the round-off error may increase. The stability and convergence of the solution can be assured by considering a function z i , j , which is the difference of the exact solution u i , ‘; and the approximated solution Ui , j . = ui,j + 1 — Ui, 7 +1 = p z i _ Lj + (1 — 2p)z io^0[k2 kh 2 ]^(6.36) Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^97 If 0 < p < 1/2, then the coefficients with the z i , i 's are nonnegative and sum to one, so that izi,j+i I 5_ p izi-i „it + (1 - 2 Azi,a1^zi-Fi^A[k2^kh 2 ] 5_ 'HI A[k 2 ^ (6.37) where 114 =^Izi,j1. Since 11411 = 0 (u U at t = 0 as both are defined by the initial conditions), therefore, < A[k 2 kh 2 ] 11z 2 11 <^+ A [k2^kh 2 ]^A2 [k 2^kh 2 ] < Al [k 2 kV]^ (6.38) where A is the upper bound of the truncated terms u tt and u„„ The error, z i , j , tends to zero as h and k tend to zero, and converges to u i , j . So for this example of the diffusion equation, the stability condition is 0 < k/h 2 < 1/2. The solution for m i , j+i which follows Eq. (6.25), the diffusion-Bloch equation as in section 6.2, is as follows: kD _1 • + 771 2 0+.^ ^= h2 711'2^0 (1 2kD^k^kD -^ ) m ij^—h 2tn i+i h 2^ (6.39) where D D free ; 0 <^<a D,; a<x<b Tfree ; 0 < x < a Tc.„, ; a <x<b (6.40) (6.41) where a is the position of the boundary and b is the outer boundary of the region, and T can be either the transverse or longitudinal relaxation rate. The initial conditions Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water ^98 define m o , for j = 1 at t = 0. Ino;^0 < x < a =^ K mo ; < x < b (6.42) where K is the partition coefficient between the water concentrations in the cell wall and the lumen. The solution is symmetric about x = 0 requiring that am/axi o = 0 and using the forward difference discretization this gives m 1 ,.; = m 2 ,j, for i = 1 at x = 0. The condition at the outer boundary can be considered as a symmetry constraint or as a restriction of the flux to be zero, so that at x = b one also requires that am/Ox lb = 0, and using a backwards difference discretization gives m b i ii+hi = ni b / kJ , where i = b/h+1 is the outer boundary x = b. Continuity of flux at the boundary, x = a, leads to am , D freeax — = Dcw am , ax^ I a+ (6.43) The boundary at x = a is labelled i = w = a/ h 1/2 on the free water side, and labelled i = w 1 on the cell wall side. The backward difference is used to discretize the flux on the free water side at i = w, and the forward difference is used on the cell wall side at i = w 1 to give amw,J+1^(771w,j+1 — mw-1,J+1 —D free^D aX cw —D aMw+1,7 +1 ax free =^Dcw ) (mw+2,j+i — mw+i,j+i) (6.44) (6.45) The flux is continuous across the boundary, Eq. (6.43), and the partition balance equation gives m w+ i,j +i = K ra w , i+1 . These expressions result in the following "marching" forward in time definitions for 711, w, +1 and mw+i,j+i: mw+1, 3 +1 D freeMtu-1,j+1^Dawnlw+2,j+1 D f ree K Dcw (6.46) K mw,i+i (6.47) ^ Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water ^99 The stability condition for this problem works out to be that k < Th 2 /(2DT h 2 ), which reduces to k < h 2 /(2D f ree ) for typical values for water in wood. A similar solution is found for the diffusion—Bloch equation in cylindrical coordinates: ^(kD^ 2kD^k ^ h2^2lir k^ )^+ 112^ (kD^k rni+1,j (6.48) 2hr) T) rni ' 3^h2 + — where r = (i — 1)h is the position variable. The same initial and boundary conditions, and the same stability condition hold. 6.3.2 Numerical Applications Numerically created T2 decay curves, sampled at 129 times from 0.2 ms to 370 ms geometrically spaced, are fit using NNLS. Table 6.2 displays the results of simulations for latewood type geometries with small radius cells and thick walls, and earlywood type geometries with larger radius cells and thin walls. The simulation parameters are Table 6.2: Numerical T2 Simulations. (b-a) 1 pm 2 pm 3 pm 4µm 5 pm 6 pm Latewood, a = 7 ,um T2 lumen T2 CW Amplitude CW (b-a) 40 ins 0% MC 1 pm 35 ins 1.2 ins 9% MC 2 yin 34 ins 1.8 ins 16% MC 3 pm 34 ins 1.9 ins 19% MC 4µm 34 ins 1.9 ins 21% MC 5 pm 34 ms 1.9 ms 23% MC Earlywood, a = 15 pm T2 lumen T2 CW Amplitude CW 103 ms^-^0% MC 91 ms^0.8 ms^12% MC 90 ins^1.2 ms^18% MC 90 ms^1.5 ins^21% MC 90 ms^1.6 ms^23% MC Df„, = 2.2 pm 2 /ms, Dcw = 0.2 p1I1 2 /ms, T2 f „ e = 1.5 s, T2c w 2.0 ms, time step size is k=0.002 ms, position step size is h=0.1 pm, and the initial condition for the cell wall water is the total amplitude in the wall equal to 30% MC. NNLS is used with 100 geometrically spaced T2 times from 0.1 ins to 1 s. Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^100 The results of the simulation show that the T2 of the lumen water is independent of cell wall thickness for walls thicker than 2 pm for both small and large cells. These results confirm again that water in wood lumens follow a diffusion-Bloch equation with a surface relaxation at the walls [Brownstein and Tarr 1979] as presented in Chapter 5. The simulations also showed that exchange between the lumen and cell wall water was significant and resulted in a decrease in amplitude assigned to cell wall water from the NNLS T2 fit. Wood density is much higher in the latewood region and it is these cells that contribute to the FSP measurement most significantly. Even for a typical latewood cell with 5 pm thick wall, the simulation resulted in a cell wall T2 component 9% MC less than the initial condition of 30% MC. A penetration depth can be defined from the following ratio: Mcwsi„, Scale[b 2 — (a + p) 2 ]7r ratio — FSP Scale(b2 — a 2 )7 ( 6.49) to give a penetration depth p = [ /b 2 (1 — ratio) + ratio a 2 — a]. For a typical latewood cell with 5 pm thick wall, the predicted penetration depth is p = 1.8 ,am. Also, the penetration depth can be calculated from p ti V4Dcw T2cw to give p = 1.3 /..tm. ap Lumen Cell Wall Figure 6.6: Lumen and Cell Wall Water Exchange. I could correct for this decreased cell wall component amplitude and measure an accurate FSP, however NMR measurements of FSP throughout this thesis and in other Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water ^101 studies are not far from the expected FSP values. Figure 6.7 shows the amplitude of the cell wall component for western hemlock sapwood as the total moisture content is decreased. At the highest MC the cell wall component is 26.5% and near the FSP 35 IIIIIIIII C 30 • 0 0 0 25 20 1111[1111 ^ 20^30^40^50^60^70^80^90^100 110 ^ 120 Total Moisture Content (%) Figure 6.7: NMR Measurement of FSP for western hemlock sapwood [Ser 1993]. the cell wall component is 32.0% MC, and this difference of 5.5% is most likely due to exchange of cell wall water and lumen water. The latewood cell dimensions of western hemlock are similar to that of the above example, and therefore the simulation predicts an underestimate of 9% MC in the FSP. It appears that wet walls from partially full cells may contribute to the cell wall T2 component and complicate the correction for the exchange of cell wall water with lumen water. Chapter 7 Spin-lattice Relaxation and Cross Relaxation 7.1 Summary The spin-lattice relaxation of western redcedar sapwood has been investigated, for moisture contents from 216% to 1%, through three techniques of applying a modified inversion recovery sequence to give a) T1 of separate solid and liquid signals from the free induction decay signal, b) T1 components from latewood and earlywood regions from the one dimensional image across the growth rings, and c) T 1 — T2 plots from the CPMG sequence decay signal. The results indicated that on the T 1 time scale of 100 ms all proton environments are mixed by diffusion of the water, so that the T 1 of the water in the lumen and the cell wall and the protons of the solid of the cell wall all have one average value. Three T 1 times were identified, 0.55 s for the fully hydrated earlywood, a faster T1 of 0.17 to 0.40 s for the fully hydrated latewood, and 0.11 to 0.20 s for the cell wall water and protons in the solid wood for when the cell lumens are empty. Also, cross relaxation of the water and the protons in solid wood was measured for moisture contents 38% and 26%, and found to be 1.1 ins, supporting a fast exchange model for spin-lattice relaxation of cell wall water and the protons in solid wood. The cross relaxation of the protons in solid wood and the cell wall water was found to be the dominant mechanism for T2 relaxation of the cell wall water. 102 Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 103 7.2 Introduction Three modified inversion recovery (MIR) techniques were applied to investigate the spin-lattice relaxation of wood, for moisture contents from 216% to 1%. First, the T dependence of the free induction decay (FID), collected following the MIR sequence, was analysed to give the T 1 of the protons in solid wood separate from the water. Second, the T dependence of the one dimensional image (or projection) of the moisture density across the growth rings, collected following the MIR sequence, was analysed to give the T1 of the latewood and earlywood regions separately. Third, the 7 dependence of Carr-Purcell-Meiboom-Gill (CPMG) sequence, collected following the MIR sequence, was analysed to give the T1 — T2 plots separating the signal of the cell wall water from the lumen water. Also, the cross relaxation of the water and the protons in solid wood was measured for moisture contents 38% and 26%. The aim of this study was to interpret the T1 components in terms of wood morphology and to determine the influence of water diffusion and cross relaxation on the T1 times. 7.3 Materials and Methods 7.3.1 Samples One western redcedar sapwood sample, which was cut to 0.5 x 0.5 x 1.0 cm, contained eight complete growth rings, and is labeled Cedarl. Another western redcedar sapwood sample from the same logslice, used for imaging, which was cut to 0.4 x 0.4 x 1.0 cm, contained three wide complete growth rings and one partial ring of mainly latewood, and is labeled Cedar2. Chapter 7. Spin-lattice Relaxation and Cross Relaxation ^ 104 7.3.2 Spin-lattice Relaxation of Wood and Water in Wood A MIR sequence was used as follows: 90, — TR 180, — T- 90, — TR^ (7.1) where the second signal was subtracted from first to give a positive signal that decays to zero at long T. For a system with only single exponential relaxation, this sequence gives SW = 2M0 exp(-7/T1 ). One hundred scans were averaged with a recycle time TR of 5 s (TR > 5T1 ). Thirty T values were used from 1 to 3000 ins, geometrically chosen in this range. The water signal is an average of 20 points from 60 to 70,as in the FID where the solid signal had totally decayed to zero. The difference of the signal of the average of 20 points from 17 to 27ps in the FID and the water signal is proportional to the solid signal. The water and solid decay curves were fit using NNLS with a linear set of 100 T1 times from 0.001 to 2 s. 7.3.3 Spin-lattice Relaxation of the One Dimensional Water Image One dimensional imaging was applied to avoid multi-exponential decays due to inhomogeneity in sample lumen size. The following MIR sequence was used 90, — TE — 180, — TE — Echo—TR 180, — T- 90, — TE — 180, — TE — Echo — TR^(7.2) with a constant field gradient of 20.4 Gauss/cm along the direction of the main magnetic field. TE was 100/Ls and the 180° pulse refocused the signal to give an echo which was Fourier transformed to give an one dimensional image, or projection, of the moisture density distribution in the wood. The gradient was oriented so that the spread in Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 105 spacial information is across the growth rings, eliminating the inhomogeneity in the signal due to the distribution of lumen sizes. The image was collected for 30 7 values from 1 to 3000 ms, geometrically chosen in this range, and then was fit using NNLS to give a T1 image. A linear set of 100 T1 times from 0.001 to 2 s was used. 7.3.4 Two Dimensional T2 — T1 Dependence of Water in Wood A CPMG sequence was collected after the MIR sequence. T (180 y — TE)„ — TR 90, — 2— T 180, — — 90, — l— (180 y — TE)„ — TR^(7.3) with a TE of 200fts. The signal was collected for 20 7 values from 3 to 3000 ms, geometrically chosen in this range. Of the 700 echos of the CPMG sequence, 20 echo amplitudes were used to give the T2 decay curve chosen geometrically from 0.2 ms to 250 ms. For a system with a single T2 and a single T1 , the signal decays as S(t) So exp(-7/T1 ) exp( - echo /T2 )7 where t ech , = nTE. The array of T1 — T2 decay data was fit, giving T1 and T2 simultaneously, using NNLS [Whittall 1992] with a linear set of 20 T1 times from 0.01 to 2 s and a geometrically spaced set of 20 T2 times from 0.001 to 0.7 s. 7.3.5 Cross Relaxation of Protons in Solid Wood and Water The exchange of protons in the solid and liquid was measured with a cross relaxation sequence [Goldman and Shen 1966] as follows: 90, — t 1 — 90_, — T — 90, — TR 90, — t 1 — 90, — — 90, — TR^ (7.4) Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ where t 1 = 100ps allowed the transverse signal of the solid to dephase and the 106 7 cross relaxation time allowed the protons in water and solid to exchange so that the solid signal reappears. The sequence, adding the first trace and subtracting the second, eliminated T1 effects from the solid signal, but not the water. The water signal is an average of 20 points from 70 to Nits in the FID where the solid signal has totally decayed to zero. The difference of the signal of the average of 20 points from 17 to 27,as in the FID and the water signal is proportional to the solid signal. 7.4 Results 7.4.1 Spin-lattice Relaxation of Wood and Water in Wood Separate spin-lattice relaxation decays of the protons in solid and water can be calculated from the T dependence of the FID following a MIR sequence, as shown in Fig. 7.1 for the wood sample Cedarl. The signal from 60 to 70,us is all water and the T depen- dence of average amplitude over this part of the FID gives the spin-lattice decay of the water. The signal from 17 to 27,as is averaged and the water decay is subtracted to leave the relaxation decay of the solid signal. The spin-spin relaxation of the cell wall water part of the water signal taken at 60 to 70,us has been neglected and contributes a 4% systematic error to the solid signal. In not extrapolating the solid signal FID to zero, with a second moment expansion, I have assumed that the lineshape of the solid is independent of an average 1171 of T. In fact, the second moment fit of the solid FID at 118% MC gave (4.4 + 0.6) x 10 9 s -2 for five T times of 1 to 7 ins and also for 50 to 200 ms. The NNLS T1 fit is shown in Fig. 7.2 for the liquid and the solid signal at eight moisture contents as the sample is being dried. The water amplitudes have been scaled Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 107 1.0 0.8 - tau (ms) 10 25 100 200 300 400 500 750 0.2 - 1000 2000 3000 0.0 f -10^0^10^20^30^40 Time Figure 7.1: 50^60 70^80^90^100 (fLLS) dependence of FID in inversion recovery experiment for cedar sample Cedarl. 7 so that they acid up to the total moisture content. The amplitude of the solid components have been scaled in the same way as the water to represent the population of protons in the solid, which does not change with moisture content. (See the definition of the NMR MC from Chapter 2). The T1 plot for 199% MC shows that the water decays with two distinct T1 times of 0.73 s and 0.33 s. As the sample is dried to 38% MC, the amplitude of the slow T 1 component decreases to zero, and the T1 time of the fast component decreases to 0.11 s. At 199% MC the solid also decays with two distinct T1 times of 0.55 s and 0.11 s. As the sample is dried to 38% MC, the amplitude of the slow T 1 solid component decreases and the amplitude of the fast T1 solid component increases. Below 38% MC the solid has only one T1 component at 0.11 s, which is independent of MC until it increases to 0.60 s at 1% MC. At 199% MC, the average T 1 of the water signal corresponds to the Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 100. 199% 25% 80. 80. 60. 60. 40. 40. 20. 20. 100. a) c) 100. 118% 100. 80. 80. 60. 60. 40. 40. 20. 20. 15% - 7.= ... E 100. 78% 100. 80. 80. 60. 60. 40. 40. 20. 20. 100. 38% 100. 80. 80. 60. 60. 40. 40. 20. 20. 0. 0 0^0.2^0.4 0 0.6^0.8^1.0^0 0^0.2 T, (s) 10% T1 1% 0.4 0.6^0.8 1.0 (s) Figure 7.2: T1 plots for liquid (solid lines) and solid (clashed lines) signals for cedar sample Cedarl. For each plot, the total amplitude of the liquid components is scaled to correspond to the moisture content, and the same scaling makes the total amplitude of the solid components to be 50. (Total Liquid = LOx Scale, Scale = 100% 0.5/(S0 - LO), Total Solid = 50.) 108 Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 109 average T1 of the water and solid signal at early times in the FID, which indicates that the faster T1 times of the solid compared to the water times are due to systematic error in the calculation. 7.4.2 Spin-lattice Relaxation of the One Dimensional Water Image A one dimensional image of the moisture density, across the growth rings, of the cedar sample Cedar2 has been acquired following the MIR experiment and the typical 7 dependence of the image is shown in Fig. 7.3. Figure 7.4 shows amplitude images E tau (ms) -6 10 25 C-) 0) '— 100 200 .5 - 300 400 500 750 1000 2000 3000 V) 0 .0 4 Position (nirn) Figure 7.3: dependence of amplitude images in modified inversion recovery experiment for cedar sample Cedar2 at 171% MC. r and T1 images for 4 MC's. The amplitude images show the water coming out of the earlywood regions first before water dries from the latewood region. All but the highest MC image show peaks in the latewood region. The high MC T 1 image shows T1 times of about 0.40 s in the latewood regions and 0.57 s in the earlywood regions. The low Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 110 MC T1 image gives the cell wall water T 1 at about 0.20 s. For this low MC, there appears to be a difference between latewood and earlywood T1 , but the amplitude is low in the earlywood region, so the signal to noise ratio is low. In general, the NNLS analysis gave single exponential components for the decay at each position in the image. NNLS results sometimes result in split peaks from noisy data and these were averaged and presented as one component [Whittall and MacKay 1989]. The occurrence of split peaks was independent of position in the image. In all of the images there is a noisy point at about the 1 mm position, which should be neglected. 7.4.3 Two Dimensional T2 - T1 Dependence of Water in Wood Figures 7.5 and 7.6 display contour plots of the NNLS solution to the CPMG decay following a MIR experiment for the cedar sample Cedarl. Above 74% MC, the plots show two distinct T1 groupings at 0.10 to 0.15 s, and at 0.30 to 0.55 s. Each T 1 group is spread in T2 in a typical T2 plot of lumen and cell wall water where the cell wall water is in the 1 ms region and and the lumen water is in the 10 to 100 ms regions. For 216% MC, both the lumen and cell wall water signals are divided into the two T 1 components. The main change in the 129% MC plot is that all the cell wall water has a T1 of the fast component. The 74% MC plot shows two main peaks, a cell wall water component with T1 of 0.10 s and a lumen water component with a slower T1 of 0.33 s. The sample contains only cell wall water for moisture contents below the FSP with a T2 of 1.7 ms and a T1 of 0.11 s, as shown in Fig. 7.6. 7.4.4 Cross Relaxation of Protons in Solid Wood and Water From the reappearance of the solid signal in the cross relaxation experiment, the cross relaxation time was calculated for two moisture contents near the FSP. The difference of the average FID signal of 20 points from 17 to 27its and the average of 20 points Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ -2^-1^0^1 ^ Position (mm) 2 Figure 7.4: Amplitude and T1 images for cedar sample Cedar2 at 4 moisture contents; 171% (long dashed), 140% (clotted), 96% (solid), 23% (short dashed). 111 Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 10 0 Total MC 216% 153% 10' II lb-19%^20% 10 — 2 0.0 ^ 0.2 ^ 0.4^0.6 T i (s) ^ 0.8 ^ 1.0 10 0 Total MC 129% 1 0 -1 52% N I— 30%^ "-^21% 10 -2 26% 3 10 0. 0 ^ I 0.2 0.4^0.6 T 1 (s) 0.8 ^ 1.0 1 00 Total MC 74% 0 -1 4%^28% In I—N 10 -2 6% 47 30% 10 2% 3 0.0^0.2 0.4^0.6 T I (s) 0.8 ^ 1.0 Figure 7.5: T1 -T2 plots for moisture contents 216%, 129% and 74% for cedar sample Cedarl. 112 Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 10 ° Total MC 32% 10 ' - rn 10 -2 10 a 0.0 ^ 0.2 0.4^0.6 T 1 (s) 0.8 ^ 1.0 10 ° Total MC 26% 10 -1 12' 10 -2 10 3 0.0 ^ 0.2 ^ 0.4^0.6 T i (s) ^ 0.8 ^ 1.0 Figure 7.6: T1 -T2 plots for moisture contents 32% and 26% for cedar sample Ceclarl. 113 Chapter 7.^Spin-lattice Relaxation and Cross Relaxation^ 114 25000 20000 - 15000 - Cross Relaxation Time 10^itzs 10000 - 200 as ' 500 fis 1000 ,u,s 5000 - 0 - -5000 -50^0^50^100^150 ^ 200 ^ 250 ^ 300 Time (kis) Figure 7.7: The FID following the cross relaxation sequence for cedar sample Cedar]. at 32% MC. from 70 to 80/is is proportional to the solid signal and follows the form y = a + b(1 — exp(-7-/T,,)), where a is the correction for the decay of the liquid signal and b is proportional to the amplitude of the solid signal. In Fig. 7.8, the signal has been scaled to give y = 1 — exp(-7 /T„) for moisture contents 32% and 26%. Tom, is found to be - 0.66+0.10 ms for 32% MC and 0.83+0.27 ins for 26% MC. The measured Tin is related to the Tcr time as follows; 1 ^Ns^1 Ns + Ncw T^ Tcr^ ni (7.5) where Ns /(Ns Ncw ) is the probability of a proton being in the solid, and I have used 1/Tcr = p a k a = p b k b [Zimmerman and Brittin 1957], where p a and p b are the probabilities of a proton in environment a or b, respectively, and 1/k for the solid environment is the measured T. When Nc w is scaled to be the cell wall moisture content, then Ns is 56 (See NMR MC in Chapter 2). Tcr is calculated to be 1.0+0.15 ms Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ —0.2 ^ 0^1^2^3^4 5 6 Tau (ms) Figure 7.8: Reappearance of the solid signal following the cross relaxation sequence for moisture contents 32%(crosses) and 26%(boxes) for cedar sample Cedarl. Shown are the 32% MC fit (solid line) and the 26% MC fit (clashed line) 115 Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 116 for 32% MC and 1.2 + 0.4 Ins for 26% MC and the average is 1.1 + 0.2 ms. The effects of cross relaxation on T2 will be considered in the discussion. 7.5 Discussion For moisture contents below FSP, the water and solid signals exhibit one T 1 component at 0.11s for both proton environments, suggesting that protons in cell wall water and in solid wood are in fast exchange, and the cross relaxation time is measured to be 1.1 ms which is fast on the T1 time scale of 100 ins. Because of the fast exchange of the protons in solid and water, I do not know the intrinsic T 1 of the protons in solid but only the average, and since the average appears to be independent of the MC even below the FSP, the intrinsic T1 of the protons in solid is most likely 0.11 s. The increase in T 1 of the protons in solid at 1% MC likely corresponds to structural changes to the wood with the loss of moisture. It is known that wood shrinks significantly with the removal of water and the M2 (from Chapter 3) was found to increase at low MC indicating less motion. Above the FSP, both the water and the solid signals have two T 1 components. The fast solid T1 component is similar to the component below the FSP where there is only cell wall water, but the other solid T 1 component decays slower. Application of a gradient field across the growth rings direction gives one dimensional images of the water with the signal from the cells in the latewood region separated from that of the earlywood region. The spin-lattice decay of the signal at each position in the image was single exponential, and therefore the multi-exponential decay of the water part of the FID is due to the sample cell inhomogeneity. The one dimensional image shows that the water in the latewood cells is the faster T1 component and the water in the earlywood cells is the slower T 1 component. Below the FSP, the cell wall water has the Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 117 fastest T1 of 0.20 s for Cedar2 and 0.11 s for Cedarl. The 216% MC T1 — T2 plot shows two distinct groups of T 1 at 0.15 s, and 0.52 s. The cell wall water is separated from the lumen water by a faster T2 time of 1 ms. The 0.15 s T1 is the latewood water and therefore the cell wall water component at this T1 time is from the latewood cell walls. The 0.52 s T 1 is the earlywood water and therefore the cell wall water component at this T 1 time is from the earlywood cell walls. On the T1 time scale of 100 ms, the exchange of the cell wall water and the lumen water of a cell is fast and results in only one average component of T 1 . Therefore, since the cell wall water is in fast exchange with the solid and the lumen water, the measured T1 is an average over all protons in the cell. The cell wall water and protons in solid have a T1 of 0.11 s, and the free water in the lumens has an intrinsic T1 of the 1 to 3 s. The earlywood cells with thin walls and large lumens have more lumen water than the latewood cells with thick walls and small lumens, and therefore the protons in the earlywood cells have a slower spin-lattice relaxation than the protons in the latewood cells. Figure 7.9: Cross section of a cylindrical cell. Within a cell, the protons in cell wall water and solid exchange magnetization by cross-relaxation with T„ = 1.1 ins, and the cell wall water and the lumen water Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 118 exchange by diffusion. A fast exchange model is simply applied as follows: 1 _ Ns + Ncw ^NL T1^N Ti cw^N Tifree (7.6) where Ti. cw is the T1 of the protons in solid and the cell wall water, Tyree is the free water T1 , N = Ns + Ncw NL , Ns is the number of protons in solid wood in the cell wall, Ncw is the number of protons as cell wall water and N.L, is the number of protons as free water in the lumen. I will calculate Ns, Ncw and XL, for a fully hydrated cylindrical cell shown in Fig. 7.9, where a and b are the lumen and cell radii, respectively. NCW FSP Pwater dwood 2 1 H/molec. 18 g/mole 0.0500 NA 100% VI"" NA 1.5g/m1 0.3 Vwall (7.7) Vwall^ Ns^Pwood dwood Vwall = 0.56 Pwater dwood Vwall 2 1 H/molec. = 0.56 ^ NA 1.5g/m1 18 g/mole = 0.0933 NA NL Vwall (7.8) Vwall^ = Pwater dwater V111771en 2 1 H/molec. ^ 18 g/mole = 0.1111 NA NA 1.0g/m1 Viu men Vlunzen^ where p is the proton density per unit mass, and the ratio PwoodPwater (7.9) = 0.56 is used [Fengel and Wegener 1984], which was used to define the NMR MC in Chapter 2. d is the mass density per unit volume [Haygreen and Bowyer 1982] and NA is Avragados number Chapter 7. Spin-lattice Relaxation and Cross Relaxation ^ 119 giving molecules per mole. For a cylindrical cell of length 1, and cross section dimensions as in Fig. 7.9, the volumes of the cell wall and lumen are defined as Vuall = 71- 1(b 2 — a 2 ) and Vlumen = 7/a 2 . The expression for T1 reduces to the following: 1 T1 1 Tl 1^ [0.1433(b2 — a 2 ) —^0.1433b 2 — 0.0333a 2 L^Ticw (12^1^0.7753] 1 0.1111a 2 1 Tifree (7.10) (7.11) a2 — 0.2324 [ 111cw^ilfree where a = b/a. Table 7.1 shows the T1 calculation using Tww of 0.11 s, Tifree of 3 s and typical cell dimensions of western redcedar [Panshin et al. 1964]. This fast exchange model Table 7.1: Fast Exchange Model of T 1 for typical cedar cells. Cell Type Latewood Earlywood Lumen Radius (a) 7,am 18,am Cell Wall Thickness (b-a) 5fim 2,am Diffusing Time Lumen Cell Wall 6 ms 31 ms 37 ms S ins Predicted T1 0.15 s 0.42 s predicts that on the basis of differences in cell diameters and cell wall thicknesses between earlywood cells and latewood cells, the T 1 times of these cell types will be different. Since the latewood cells have small lumens and thick walls, T 1 of latewood is faster than that of earlywood, which have larger lumens and thin walls. The diffusing time is the estimate of the time for water to cross into the next region and is calculated from t a 2 /(4D fr „) for the lumen, and t (b — a) 2 /(4D cw ) for the cell wall. Diffusion values of D fr „ = 2.2 iim 2 /s and D cw = 0.2 fim 2 /s where used (Chapter 6). From these diffusing times shown in Table 7.1, I estimate that on the T1 time scale of 100 ms that all water molecules diffuse through the entire cell, and sample both relaxation in the wall and lumen. The protons in solid wood cross relax with the cell wall water and in this way attain the same relaxation time, and influence Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 120 the relaxation time. I have neglected any diffusion between cells, assuming that all neighbouring cells are identical, which is reasonable since the change from earlywood to latewood is gradual except at the winter boundary of each growth ring. Since T1 can be expressed in terms of a = b/a as in Eq. (7.11), I can derive a from T1 as follows: 1 a 2 = [0.2324 —^( rn^— 0.7753 .1 r rn^ 3 )] [1 ^ 11CW^1 f ree^-L 1CW -1 (7.12) The T1 data from the T1 — T2 plots gives Tww = 0.11 s below the FSP, T1 for latewood water was 0.15 s, and T1 of the earlywood water was 0.52 s. From this data, I calculate b/a = 1.7 for the latewood signal and b/a = 1.1 for the earlywood signal. From the cell dimensions given in Table 7.1, I calculate corresponding ratios of 2.0 for the latewood and 1.1 for the earlywood, which are in excellent agreement with the T 1 data calculations. The fast exchange model explains the two T1 components in terms of cell density differences in latewood and earlywood regions. In general, as the MC is decreased the amplitude of the earlywood T1 components decrease before that of the latewood, as shown in the amplitude image of Fig. 7.4. For the highest MC of the T 1 — T2 plots there is an earlywood cell wall component which disappears when the MC decreases. Also in Fig. 7.2, the earlywood solid signal disappears when the MC decreases. As wood dries the cell wall water does not decrease in any region until the MC is below the FSP as seen in the cell wall water moisture profiles, which were separated from the lumen water on the basis of T2 as shown in Fig. 4.7. Therefore, I can conclude that the cell wall solid and water components in empty earlywood cells, above the FSP, have T1 times similar to that expected below the FSP. These empty lumen, cell wall signals are indistinguishable from the fully hydrated latewood cell signals, since they are not separated in T1 times significantly. Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 121 In Fig. 7.5, the 216% MC T1 — T2 plot shows a 20% MC component with a T1 of 0.45 s, which indicates that it is signal from a thin walled earlywood cell, and a T2 of 15 ms, which indicates that it is signal from a small diameter cell such as latewood cells. A typical SEM of cedar [Panshin el al. 1964] shows that the earlywood does contain a significant number of small radius, thin walled cells. 7.6 Numerical T1 Simulations A diffusion model similar to the T2 diffusion model (Chapter 6), for water in two regions, is applied to T1 with cell wall water exchanging with the protons in solid wood in the cell wall region. The following partial differential equations describe the z-magnetization of wood and water in wood approximated by cylindrical geometry. The variable m(r, t) is the z-magnetization of the free water in the lumens when r < a, and is the zmagnetization of the cell wall water when a < r < b. 47., t) is the z-magnetization of the protons in solid wood in the cell wall. The lumen radius is a and the cell wall is at a < r < b. Om at Om (7 t) (r t) at ' = DcwV 2 m(r,t) + t) 712(X^t) D f „e N r I. s(r,t) 771,(X^t) Ti free j < a (7.13) m(r,t) 71,cw m(r,t)1 • Tc,. ^' Ns^Ncw a<r <b s(r,t)^N ini(r,t)^s(7-,t)1 TL5^Ter [ Ncw^Ns a<r<b (7.14) (7.15) where Tif ,„, 711' cw and 7Y s. are the relaxation time of free water, the intrinsic relaxation times of protons in cell wall water and in solid wood if no exchange was present, respectively. The cross relaxation time of the exchange of protons in solid and cell wall Chapter 7. Spin lattice Relaxation and Cross Relaxation^ - 122 water magnetizations is T„. D f ree and D cw are the diffusion coefficients of free and cell wall water, respectively. The numerical solution is developed (See Appendix E), and the results are shown in Fig. 7.10 for cylindrical cells with earlywood and latewood cell dimensions as in Table 7.1. For a cell type, all proton environments attain the same 1o 3 Harlywood urn et ? SO/jd cfry 10 water 1 10 2 10° 0 100^200^300 400 ( m s) Figure 7.10: Simulations of spin-lattice relaxation in cylindrical cells with earlywood and latewood cell dimensions as in Table 7.1. relaxation rate, which is faster for the latewood cells than the earlywood cells. The simulated spin-lattice decay is non-exponential for r times smaller than about 20/is Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 123 for the earlywood cells and 3,us for the latewooci cells. This non-exponential behavior is not observed in the measured decay curves. The spin-lattice decay is shown for the solid signal in Fig. 7.11 for three moisture contents. For 199% and 78% moisture contents, where significant numbers of cells are fully hydrated, the simulations predict a non-exponential decay of the solid signal, but the measured decays are monotomically decreasing. For 38% MC, cell lumens are almost completely empty and it has been assumed that the solid and cell wall water have the same intrinsic T 1 , as discussed in the previous section, and therefore the decay is strictly exponential. 7.7 Cross Relaxation and T2 The cross relaxation time of Tcr = 1.1 ms corresponds to fast exchange between the cell wall water and the solid on the T1 time scale of 100 ms, but on the T2 time scale of T2* = 30,us for the solid this is slow exchange. In Chapter 3 it was found that since moisture content could be measured accurately by NMR, there is no exchange of the solid and cell wall water on the T2 time scale. The following equation describes the x-magnetization of cell wall water. cam cat _ +Nrs m Ncw TL,w^T„. Ns [ (7.16) Since the solid dephases with T: = 30ps, and T21 c w is assumed to be much slower than T„, the resulting water spin-spin relaxation is T2C W N lVC W r cw T„ (7.17) For the measured cross relaxation time, Eq. (7.17) predicts a T2c w of 0.3 ms which is faster than the observed times of 1 to 2 ins, but predicts a decrease in the measured cell wall water T2 with decreasing moisture content, as observed for the lodgepole Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 1.0 •• 0.9 199% ,■• 0.8 0.7 78% 38% 0.9 0.8 0.7 0.6 0 10 20 30 40 50 -r (nis) Figure 7.11: Spin-lattice decay of the protons in solid wood for cedar sample Cedarl, at three moisture contents. 124 Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 125 pine shown in Fig. 3.6. Analysis of the MC dependence of the lodgepole pine, using Eq. (7.17) where N c w and Ns are known, gives a T„ =3.1 + 0.1 ms. 7.8 Concluding Remarks Western redcedar sapwood is found to have two components of T 1 for the water and the same two components for the solid when the wood is fully hydrated. These components correspond to a fast exchange of all protons: lumen water, cell wall water and protons in solid wood in the cell wall. The inhomogeneity in the sample of lumen radius and cell wall thickness gives an earlywood component of about 0.55 s and a faster latewood component of 0.15 to 0.40 s. When the MC is below the FSP, only one T 1 component, of 0.11 to 0.20 s, from the solid and cell wall water exists. Indication that cell wall water in cells with empty lumens, even at MCs above the FSP, have the same Ti as cell wall water below the FSP was found. The mixing of proton environments is accomplished by water diffusion and the cross relaxation of the protons in solid wood and the cell wall water with a cross relaxation time of 1.1 ms, which is fast on the T1 time scale of 100 ms. The cross relaxation of the protons in solid wood and the cell wall water was found to be the dominant mechanism for T2 relaxation of the cell wall water. Bibliography Abragam, A. (1961). Principles of Nuclear Magnetism, Clarendon Press, Oxford. Ames, W.F. (1977). 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Vetterling, (1988) Numerical recipes : the art of scientific computing, Cambridge University Press, USA. Riggin, M.T., A.R. Sharp, R. Kaiser and M.H. Schneider, J. Appl. Polym. Sci., 23, 3147 (1979) Ser, T., B.Sc. Thesis, Physics Department, University of British Columbia (1993) Sharp, A.R., M.T. Riggin, R. Kaiser and M.H. Schneider, Wood and Fiber, 10, 74-81.(1978) Siau, J.F., (1984) Transport Processes in Wood, Springer-Verlag, Berlin, New York. Simpson, W.T., Wood and Fiber. 5, 41-49.(1973) Simpson, J.H., H.Y. Carr, Phys. Rev., 111, 1201 (1958). Skaar, C., (1988) Wood-water relations, Springer-Verlag, Berlin, New York. Bibliography^ 130 Slichter, C.P., (1980) Principles of Magnetic Resonance, Springer—Verlag Berlin Heidelberg, New York Stamm, A.J., Forest Products J., 9, 27 (1959). Stamm, A.J., Forest Products J., 10, 524 (1960). Stamm, A.J.,(1964), Wood and Cellulose Science. The Ronald Press Co., New York. Stamm, A.J., and W.K. Loughborough, J. Phys. Chem., 39, 121 (1934). Stejskal, E.O., and J.E. Tanner,J. Chem. Phys., 42, 288-292 (1965) Sternin, E., Rev. Sc. Instrum., 56, 2043-2049 (1985). Quick, J.J., J.R.T. Hailey and A. L. MacKay, Wood Fiber Sci., 22, 404-412 (1990) Wang, P.C., and S.T. Chang. 1986. Wood and Fiber Sci. 18: 308-314 Whittall, K.P., and A.L. MacKay. J. Mag. Reson., 84, 134-153 (1989). Whittall, K.P., J. Magn. Reson., 94, 486 (1991). Whittall, K.P., Personal Communications (1992). Zimmerman, J.R. and W.E. Brittin, J.Phys.Chem. 61, 1328-1333 (1957) Appendix A Field Gradient NMR There are two types of diffusion concerned with the diffusion of the magnetization vector produced by the spins of a sample. One is spin—diffusion which can be detected in a solid where the molecules are in fixed positions. The other is self—diffusion where the molecules which carry the magnetization diffuse throughout the sample. The measurement of self—diffusion, or equivalently molecular diffusion, is measured in NMR using a field gradient. This method is more direct than tracer diffusion studies which contaminate the sample and measure the diffusion of the tracer, not the diffusion of the molecules being studied. The idea behind using a field gradient is that the spins of the sample are given slightly different Larmor frequencies depending on the their position in the gradient field. A 7 pulse can refocus magnetization to give an echo only if the spin has not diffused to a different location during the experiment. So the diffusion of the molecules results in an attenuation of the echo amplitude[Fuskushima and Roeder 1981]. A single echo experiment can be used with a linear field gradient to measure the diffusion constant D of the spin carrier of the sample [Slichter 1980]. 7r (A.1) 212DG273) In ( M = —^ Mo^T72^3 (A.2) A Carr—Purcell (CPMG) pulse sequence is used to minimize the effects of diffusion due to an applied gradient, inhomogeneities in the applied magnet field Ho , or gradients 131 Appendix A. Field Gradient NMR ^ 132 induced from succeptibility difference created at air/water interfaces. For a linear field gradient the CPMG echo amplitude is as follows: 2 — — (7r — 2r)„^ (A.3) 27rr^272DG27-37i)^ ln^= ( A.4 ) 3 MO^T2 ^( where M is the magnetization amplitude of the n th echo. The difference in these two echo sequences is that for the single echo experiment 27 - is the time the echo amplitude M is measured at, where as for the CPMG experiment the time the echo amplitude is measured at is at t = 2nr and the magnetization has been refocused at every 7 intervals so that phase loss due to the diffusion takes place independently in each interval. A calculation of the ratio of amplitudes of the single echo and CPMG sequences using values from wood studies demonstrates the difference between these sequences. ln( ' mg ) = 7 2DG2t (t2 3 echo^ 4 r 2) (A.5) = 2.67 x 10 4 (gauss • s) - ' D = 2.2 x 10 -5 cm 2 /3 G = 19.4gauss/cm t = 60ms rcp „ig = 400ps where G is the value of the applied gradient used in Chapter 4. These values give a ratio of the order 10 20 for the CPMG to single echo magnetization. The effect of the CPMG sequence to reduce phase lost due to diffusion is demonstrated to be very good. In Chapter 4, one dimensional imaging using a constant gradient was also applied to the wood being studied. The images were interpreted using previously acquired T2 relaxation data. It was assumed that the diffusion effect to the magnetization was Appendix A. Field Gradient NMR ^ 133 negligible and the following calculation of the ratio of the magnetization of the CPMG sequence with and without a gradient shows that the assumption was valid. ^M ^nogr d^2 In ur gra ) — --„7 2 DG 2 7-2 t (A.6) ad^t) With the values used above, where t = 60 ms was a typical time that an image would be measured, Eq. (A.6) has the value 0.04. Eq. (A.4) for the magnetization of a CPMG sequence with a gradient shows that the above calculated term should be compared to t/T2 to decide if the term can be neglected. With T2 in the range of 1-200 ms the term should be small compared to values of 0.2-60., so that the attenuation due to diffusion term is at most 10% which is an acceptable error contribution for an imaging experiment. For measuring diffusion, a pulsed field gradient method has several advantages over a continuous gradient method. One advantage is that the time between r.f. pulses is constant and the duration of the gradient is altered so that only one measurement of the magnetization without a gradient needs to be taken, where for the continuous gradient method a measurement without a gradient is needed for each change in T. Another advantage is to do with the electronics of the r.f. pulses. In the presence of a gradient the free induction decay signal (FID), or an echo signal is shorter in time because of the inhomogeneous field produced by the gradient and thus giving a very large linewidth in frequency. A r.f. pulse would have to be very short in time so that all parts of the magnetization experience the same r.f. power. The idea behind the pulsed gradient method is for the gradient to be off when an r.f. pulse is applied. A Stejskal and Tanner sequence [1965] starts with a 7r/2 r.f. pulse putting the magnetization along the y—axis of the rotating frame. At an arbitrary time t i after the pulse the gradient is applied for a time 6. At time T a 7r r.f. pulse about the y—axis flips the spins to refocus the phases. At time t 1 + 0 the gradient is applied for the Appendix A. Field Gradient NMR ^ 134 same time 8 as the first gradient pulse and undoes the effects of the first gradient pulse, unless of course the spins have diffused to a different part of the sample where the field is slightly different, giving it a different Larmor frequency and preventing the proper refocusing. The echo signal to be measured occurs at 2r. The echo amplitude is given as follows: ln ( Mm .0 G ) - - ( 277,2 PF 7 2 D8 2 (A - -6 ) \^31 G2 ) (A.7) The expression is more complicated if a continuous gradient G o is also present [Fuskushiman and Roeder 1981]. The duration over which diffusion is measured is (A — 8/3) or simply 0 for 0 Appendix B Diffusion Model for Rectangular Geometry The diffusion—Bloch equation for infinitely long rectangular cells, with dimensions x = +R s and y = +R y , no z dependence and a volume relaxation T2free is ^Din^0277.1^02m^m(x, y,z,t) . ^(x, y, z, t^D ^ (x, y, z, t) ) = ^ ay2 (x, y, z, t)) at^‘ax2^ T2free (B.1) where m(x, y, z, t) is the magnetization in the cell lumen at the position (x, y, z) and time t, and D is the diffusion coefficient of bulk water. The boundary condition for the flux (J) out of the surface is s —D n Vnils = —D n am, am, — ay am az ) 1•Ji s = M ax (B.2) where M is a parameter characterizing the strength or effectiveness of the surface relaxation. For the surfaces with x = +R x , the surface normals are n = (+1, 0, 0), and for the surfaces with y = +R y , the surface normals are CI = (0, +1, 0) so that the boundary conditions are — D a711 is=±R. (B.3) Din M m l y =±R y ay Iy=±Ry = (B.4) aX ix=±1T? = M 771 The solution to this diffusion problem can he expressed as a sum of normal modes, E E A„,„ F„(x)G,„(y) e—t/T2(7.) 00 00 in(x, y, z, t) = 71=0 711=0 135 (B.5) ^ Appendix B. Diffusion Model for Rectangular Geometry ^ 136 where Fri (x) and G„,(y) are orthogonal eigenfunctions and are satisfied by the cosine function ^F,i(x) cos ( 7 " x.-)^ Rs = (B.6) G„(y) = cos (" mY^(B.7) -14 with 1^2^2^ 1 = D (717i + P711 ) + ^ T2(nm)^R1^fq^212 free' ^ (B.8) The boundary conditions Eq. (B.3,B.4) define 7i„ from q„ tan(7)„) — M Rs D .^ (B.9) and /c m from tan(p,„) = ^it, MR Y^ (B.10) The amplitudes A„„, are determined by the initial condition of constant magnetization thoughout the lumen, immediately following the 90° pulse of a CPMG sequence. That is, 00 00 m(x, y, z ,0) = m o = EE A n ,„ cos (-71- 7 ) cos( ItniY ) Rs /^Ry (B.11) n=0 m=0^ Using the orthogonality of cosine functions gives ff,i cos (T,„ x Rs ) dx f RR cos (u m y R y ) dy s m0 R^ f^COS2 (77„x R s )dx f_R;ty cos 2 (a 7n y R y ) dy The signal detected by NMR is the total magnetization from the cell J 00 M(t) =^m(x, y, z, t) dx dy dz where ell^ = -A4 ( 0 ) E E^e—ti712(..) 00 71=0 772,--.0 (B.12) Appendix B. Diffusion Model for Rectangular Geometry ^ where 2^sin2(7171) In 7 g, (1 + (D / M R x ) sin 2 (70) 2 ^sin2(1i,n) — iqn (1^(D/ MR y )sin 2 (it ni )) • 137 Appendix C Two Regions with Cylindrical Geometry We consider two regions of water concentration with diffusion coefficient D 1 , spin—spin relaxation time T21 and magnetization rn i (r, t) for r < R, and D2, T22 and m 2 (r, t), respectively, for r > R. Exchange between the two regions will occur due to flux of magnetization across the boundary. The diffusion—Bloch equation, for infinitely long cylindrical cells with no z or B dependence is 0771, ^a2m,^ ^(r^ , t) = D, ( ^ (r, t) at^( + 1 anti (r, t)^ 37.2^7' ar^T2i t) ;i (C.1) = 1, 2. where r = (r,19,z) in cylindrical coordinates with time t. The initial condition of uniform magnetization following the 90° pulse of a CPMG sequence gives in i (r, 0) = rn 10 = mo; r < R m, 2 (r, 0)^71120 KM(); r > R ^ (C.2) where K is the partition coefficient. The solution is simplified if we impose an alternative initial condition: mi(r, r) = rri o , in 2 (r, r) = 0 ^ (C.3) At the boundary r = R the magnetization must satisfy a partition balance equation and the flux must be continuous, — Di Km i (R, t) = m 2 (R, t),^ (C.4) am ^a7n2 Or (R t) = — (C.5) i D2 138 ^ (R , t). ^ Appendix C. Two Regions with Cylindrical Geometry ^ 139 This model can be applied to water in wood, with the inner region, r < R, representing the free water in the lumens and the outter region, r > R, representing the water in the cell walls. The alternate initial condition is reasonable for water in wood where T21 > T > T22 holds. Defining the Laplace transform, ,C, as .C{m i (r,t)} = I m i (r, t) e St dt^liz i (r, s)^(C.6) and applying the Laplace transform to the diffusion-Bloch equation, Eq. (C.1), gives s Mi(r, s) — m i (r, 0) = D, ( (a 2 772 i M i (r, s) ^1 07iii ^ (r , s)) — ; Cdr (r ' s) + 7' Or^ •i = 1,2 (C.7) and the boundary conditions Eqs. (C.4,C.5) become ^s) K^s)^s)^ (C.8) rhi (R^^ (R , s).^ = D2 a D1 a 07' o (C.9) The general solution to the Laplace transformed problem is sMi i0T2i oz i (s) /0 0ir)^7i(s) Ko(dir) + ^ M i (r, s)^ T2 + (C.10) where^and y i are undetermined constants, and / 0 and K0 are modified Bessel functions, which satisfy the equation 1 y" + Ty' — 13 Y = O.^ (C.11) The series expansion of 1 0 is 2^X 4 6 X X /0(X) = 1 + 22(1!)2 ^24(2!)2^26(3!)2 (C.12) where^/0 = 1, and lim x „ /0 = oo. The series expansion of K o is Ko(x) = ( ;i) 1/2 e-T (1^+^ 9 2^• • •) 8x^2(8x) (C.13) Appendix C. Two Regions with Cylindrical Geometry ^ 140 where lim s _, 0 K0 = —oo, and^K0 = 0. The solutions using the boundary conditions of Eqs. (C.8,C.9) and requiring that fii i (r, s) are symmetric about r = 0, s) is bounded for r = 0, and 771, 2 (r, s) is bounded for large r, are: s) = A(s) /O(317') + M10 7121 sT21 + 1 ih 2 (r, s) = B(s) Ko (13 2 r) (C.14) (C.15) where sT2i + 1 i = 1, 2.^ 722 (C.16) 1 The Brownstein and Tarr [Brownstein and Tarr 1979] boundary condition, Eq. (5.4), at r = R is — D1 Or (R t) = M m i (R,t). (C.17) which is written as a function of the magnetization in the region r < R, m i (r,t), and a surface relaxation parameter, M. Assuming that M is a constant, the Laplace transform of Eq. (C.17) is simply — ai-h 1 ^ (R 8) m 7-hl(R, 8). car (C.18) Using Eqs. (C.8,C.9,C.15) we obtain M =—D 2 K = where Kax) = — 2 aih2 /Or th e T 32 ^,- K1 (132 R) KO ( 3 2 R ) (C.19) K 1 (x). This solution is identical to the one dimensional example, Eq. (6.19), in the limit that K1(02R)/ICo(02R) 1 which holds for \/D 2 T22 << R. This condition is satisfied by water in wood. ^ Appendix D Two Regions with Spherical Geometry We consider two regions of water concentration with diffusion coefficient D 1 , spin-spin relaxation time T21 and magnetization m i (r, t) for r < R, and D2 T22 and m 2 (r, t), respectively, for r > R. Exchange between the two regions will occur due to flux of magnetization across the boundary. The diffusion-Bloch equation, for spherical cells with no 0 or 0 dependence is am i (r,^mi(r, t) ^ ; zi = 1,2.^(D.1) th.2^ 7' ar^Tip at^ a772/ a2 771i (r, t)^2 D1 ^ z (r,t) = where r = (r, 0, 0) in spherical coordinates with time t. The initial condition of uniform magnetization following the 90° pulse of a CPMG sequence gives rni (r, 0) = rn 10 = 772 0 ; r < R ^m2(r, 0) = 7n20 = Km o ; T > R ^ (D .2) where K is the partition coefficient. The solution is simplified if we impose an alternative initial condition: m i (r,r)^m o , 777 2 (r,^= 0^ This simplification is reasonable for cases where T 21 > T > T22. (D.3) At the boundary r R the magnetization must satisfy a partition balance equation and the flux must be continuous, Kmi(R, t) = 772. 2 (R, t),^ ari l - D 1 ^ (R, t) = - D2 am2 ar^ar 141 (D.4) (R, t).^(D.5) ^ Appendix D. Two Regions with Spherical Geometry ^ 142 Defining the Laplace transform,^as L{mi(r,t)} =^mi(r,t)e-st dt = fiz i (r,^) ^ (D.6) and applying the Laplace transform to the diffusion-Bloch equation, Eq. (D.1), gives (a2rhi^2 577-ti^s) s 75-4,(r, .^) - m i (r, 0) = D, ^ (r , s) ^(r, s) a r 2^ r Or^T2i i^1, 2^(D.7) and the boundary conditions Eqs. (D.4,D.5) become Kiii i (R,^) = M 2 (R,^)^ Di (D.8) afil2 57 /1 (R s) = D2 (R s).^ (D.9) Or Or , ' The general solution to the Laplace transformed problem is sinh(Ar)^cosh(A r) rnioT2i ih i (r, s) = a i (s)^+ 7,(s)^+^(D.10) 7.^r^sT2i + 1 where a i , /3 and -yi are undetermined constants. Since, lim,_, 0 sinh(x)/x = 1 and lim x _,0 cosh(x)/x = oo, therefore, the solutions using the boundary conditions of Eqs. (D.8,D.9) and requiring that ih i (r, s) are symmetric about r = 0, iii i (r, s) is bounded for r = 0, - 2 (r,.^) is hounded for large r, are: and th sinh(O i r)^inioTn M i (r,^) = A(s)^ ^(D.11) r^ + sT^ 2i + 1 /300) + mioT21 A(s) ( =^ exp(Oir) exp( sT21 + 1 exp( -132 r ) r7/ 2 (r, s) = B(s) ^(D.12) r where lei - s8712 , -I- 1 i = T2i 1, 2.^ (D.13) The Brownstein and Tarr [Brownstein and Tarr 1979] boundary condition, Eq. (5.4), at 1. = R is anti — D 1 ^ (R car , t)^M (D.14) Appendix D. Two Regions with Spherical Geometry ^ 143 which is written as a function of the magnetization in the region r < R, m i (r, t), and a surface relaxation parameter, M. Assuming that M is a constant, the Laplace transform of the above equation is simply — ^ (R s) = M rh, i (R,^). (D.15) ' Using Eqs. (D.8,D.9,D.12), we have M =—D 2 K ath2/ar ih 2 1\ = D2 K (02 -) • (D.16) This solution is identical to the one dimensional example, Eq. (6.19), for 1/D2 T22 << < R. Appendix E Numerical T1 Simulations The following partial differential equations describe the z-magnetization of wood and water in wood approximated by cylindrical geometry. The variable m(r, t) is the zmagnetization of the free water in the lumens when r < a, and m(r, ,t) is the zmagnetization of the cell wall water when a < r < b. s(r, , t) is the z-magnetization of the solid protons in the cell wall. The lumen radius is a and the cell wall is at a < r < b. ^ an?' (7' t) at Dm at as (r , t) t) rti(x t) Dfree V 2 ni(x, t) T1 free 7<a rn(r, ,t) D cw V 2 rn(r, t) rn^ I cw m(r, t)] a N rs (r , t) < r< b Tcr Ns^N cw s(r , t)^N m(r, ,t)^s(r , t) [ 1. (E.1) s^Tcr NCw^Ns (E.2) a < r < b^(E.3) where Ti free, Ti' cw and Tl s are the relaxation time of free water, the intrinsic relaxation times of cell wall water and the solid protons if no exchange was present, respectively. The cross relaxation time of the exchange of the solid proton and cell wall water magnetizations is T,• D f „ e and D cw are the diffusion coefficients of free and cell wall water, respectively. The number of solid protons in the cell wall is Ns and the number of protons as cell wall water is Ncw , and are calculated in Chapter 7. Discretizing these equations gives the following for the lumen magnetization when 144 Appendix E. Numerical T1 Simulations^ 7' < 145 a: = [kD free^k h2^2hr [1^ 2kD free h2 Tifkree + i kD f „ e (E.4) h2^2hr and for the cell wall water when a < r < b: rkD cw^k h2^2hr 2kDcw^1^1 )] + [1^ k( /cw + ^ m i,i Ti Ncw Tcr + k.Dcw 112 k 1^k + s, 3^ .1mi-1-1,3 + L^ h2^ 211,7. NsT, ' (E.5) and for the solid protons when a < r < b: 8 i,.i+i \i 1^1 + ^ = {1 — k( ^ Tis^NsT„.)18i'3m Ncw 71„. • (E.6) where r = (i — 1)h. A double subscript notation is used, so that m i , j is the discretized value of rn(r, t) and s i f is the discretized value of s(r, t) at the i th step in x and the i th step in time, t. The variables h and k are the step sizes in x and t, respectively. (See Section 6.3 for more on numerical methods.) The initial conditions define m i , 1 and s o , for j = 1 at t = 0. = so = K I mo;^0 < r < a (E.7) K mo; a < r < b 100% P water For' Pwood r, • MO^ a< 7' < b (E.8) as defined by NMR MC in Chapter 2 and the definition of FSP. K is the partition coefficient between the water concentrations in the cell wall and the lumen. The solution is symmetric at r = 0 requiring that Oni/Ori o = 0 and using the forward difference discretization this gives ^= m 2 , , for i = 1 at r = 0. The Appendix E. Numerical T1 Simulations^ 146 condition at the outer boundary can be considered as a symmetry constraint or as a restriction of the flux to be zero, so that at r = b one also requires that am/ar = 0, and using a backwards difference discretization gives m b i h+Li = m b i ka , where i = bl h 1 is the outer boundary r = b. Continuity of flux at the boundary, r = a, leads to am,,^am , D free^= -Ucw^+ ar^ar (E.9) The boundary at r = a is labelled i = w = a/h + 1 on the free water side, and labelled i = w 1 on the cell wall side. The backward difference is used to discretize the flux on the free water side at i = w, and the forward difference is used on the cell wall side at i = w 1 to give —D free — D cto anz w , i+i 07. — arnw+7,j+7 ar - (m.,j+i — 771,1,j+1) h (E.10) ( 7nw+2,j+1^Mw+1,j+1) (E.11) Dfree -DCW The flux is continuous across the boundary, Eq. [E.9], and the partition balance equation gives m w+ J.+ = K 77L w , j+1 . These expressions result in the following "marching" forward in time definitions for m„,, i+i and m„, +1 ,i+1 : rnw,j+1 mw+i,j-Fi D freeMw-1,j+1 DCWMw+2,j+1 D free K Dcw = K 771 w, +1 (E.12) (E.13) There are no boundary condition for s(r, t) since there are no position derivatives in the Bloch equation describing s(r, t). The stability condition for this problem works out to be that k < Th 2 /(2DT h 2 ), which reduces to k < h 2 /(2D free ) for typical values for water in wood.
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Proton magnetic resonance of wood and water in wood Araujo, Cynthia D. 1993
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Title | Proton magnetic resonance of wood and water in wood |
Creator |
Araujo, Cynthia D. |
Date Issued | 1993 |
Description | Proton nuclear magnetic resonance (11-1 NMR ) was used to investigate protons in solid wood and compartmentalized water in the wood cell walls and lumens. A lineshape second moment study found the second moment of protons in oven dry wood to be about 23% lower than the rigid lattice calculation, indicating a rigid structure with some anisotropic molecular motion of the polymeric constituents. Above 5% moisture content, the second moment decreased by a further 13 to 16% implying a "loosening" of the molecules in the solid with the increased moisture content. The T2 of the cell wall water was found to be single exponential and increased with moisture content. The 11-1 NMR measured fibre saturation point of the cell wall water agreed with the value calculated from the moisture isotherm. Two T2 techniques for characterization of water in wood are demonstrated. First a technique for analysing multi-exponential relaxation in terms of a continuous distribution of relaxation times was applied to T2 analysis of lumen water in wood. The lumen water T2 times vary as a function of the wood cell radius and are therefore expected to reflect the cell size distribution, which is continuous. A technique of selectively imaging water environments on the basis of T2 was applied for a range of moisture contents. The moisture density profile of the hound water was found to be independent of moisture content above the fibre saturation point. Spin- spin relaxation measurements of lumen water in wood were interpreted using a diffusion theory which models the lumen waterT2 relaxation in terms of the cell radius distribution, the bulk water diffusion coefficient and a surface relaxation parameter. Agreement between theory and experiment was excellent. Evidence was found for the existence of higher order T2 relaxation modes predicted in the slow diffusion regime, using a sample with rather large cell lumens and at low temperatures. Using this diffusion model, T2 relaxation decay data were fitted to give a cell size distribution, comparable to scanning electron microscope results, when the bulk water diffusion coefficient and the surface relaxation parameter were known. A two region diffusion model was considered with free water in the cell lumens and water in the cell walls. The surface relaxation parameter was found to depend on the spin-spin relaxation time and diffusion coefficient of the cell wall water. Consequently, the cell wall water diffusion coefficient may be estimated from spin—spin relaxation times and the relative populations of lumen and cell wall water. The cell wall diffusion coefficient of maximum hydrated redwood sapwood was found to be 0.2 x 10' m2/sat room temperature, and from the temperature dependence the activation energy was found to be 6700 cal/mol, about 40% higher than the free water value. Numerical simulations of the two region diffusion model were developed. The lumen water T2 was found to be independent of the simulated cell wall thickness, simplifying to a surface relaxation as modeled with the surface relaxation parameter in the one region model. The simulated effect of exchange on the fibre saturation point measurement was found to be an over estimate compared to experimental results. Three techniques were used to investigate the spin-lattice relaxation of the solid wood and the water in wood. Separate T1 measurements of the solid and water, separate T1 measurements of water in the early wood and latewood regions, and separateT1 measurements of the cell wall water and lumen water were acquired. The results indicated that, on the T1 time scale of 100 ms, all proton environments are mixed by diffusion of the water. The T1 of the water in the lumen and the cell wall and the protons of the solid were found to have the same T1, which is an average of the T1 of the three environments. The T1 was found to be dependent on the proportion of cell wall to lumen volume. Thick walled latewood cells had a lower T1 than thin walled early wood cells. Lastly, the cross relaxation of the protons in solid wood and the cell wall water was found to be the dominant mechanism for 7'2 relaxation of the cell wall water. |
Extent | 6131171 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-09-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085629 |
URI | http://hdl.handle.net/2429/1846 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1993-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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