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Proton magnetic resonance of wood and water in wood Araujo, Cynthia D. 1993

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PROTON MAGNETIC RESONANCE OF WOOD AND WATER INWOOD.Cynthia D. AraujoB. Sc. (Physics) University of British ColumbiaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJune 1993© Cynthia D. AraujoIn presenting this thesis in partial fulfilment of the requirements for an advanced degreeat the University of British Columbia, I agree that the Library shall make it freelyavailable for reference and study. I further agree that permission for extensive copyingof this thesis for scholarly purposes may be granted by the head of my department orby his or her representatives. It is understood that copying or publication of this thesisfor financial gain shall not be allowed without my written permission.Department of PhysicsThe University of British Columbia1956 Main MallVancouver, CanadaDate:AbstractProton nuclear magnetic resonance ( 1 1-1 NMR ) was used to investigate protons in solidwood and compartmentalized water in the wood cell walls and lumens. A lineshapesecond moment study found the second moment of protons in ovendry wood to beabout 23% lower than the rigid lattice calculation, indicating a rigid structure withsome anisotropic molecular motion of the polymeric constituents. Above 5% moisturecontent, 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 cellwall 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 thevalue calculated from the moisture isotherm.Two T2 techniques for characterization of water in wood are demonstrated. First atechnique for analysing multi-exponential relaxation in terms of a continuous distribu-tion of relaxation times was applied to T2 analysis of lumen water in wood. The lumenwater T2 times vary as a function of the wood cell radius and are therefore expected toreflect the cell size distribution, which is continuous. A technique of selectively imagingwater environments on the basis of T2 was applied for a range of moisture contents. Themoisture density profile of the hound water was found to be independent of moisturecontent above the fibre saturation point. Spin- spin relaxation measurements of lumenwater 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 coefficientand a surface relaxation parameter. Agreement between theory and experiment wasexcellent. Evidence was found for the existence of higher order T2 relaxation modesiipredicted in the slow diffusion regime, using a sample with rather large cell lumens andat low temperatures. Using this diffusion model, T2 relaxation decay data were fitted togive a cell size distribution, comparable to scanning electron microscope results, whenthe 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 andwater in the cell walls. The surface relaxation parameter was found to depend on thespin-spin relaxation time and diffusion coefficient of the cell wall water. Consequently,the cell wall water diffusion coefficient may be estimated from spin—spin relaxationtimes and the relative populations of lumen and cell wall water. The cell wall diffusioncoefficient of maximum hydrated redwood sapwood was found to be 0.2 x 10' m 2 /sat room temperature, and from the temperature dependence the activation energy wasfound to be 6700 cal/mol, about 40% higher than the free water value.Numerical simulations of the two region diffusion model were developed. The lumenwater T2 was found to be independent of the simulated cell wall thickness, simplifying toa surface relaxation as modeled with the surface relaxation parameter in the one regionmodel. The simulated effect of exchange on the fibre saturation point measurementwas found to be an over estimate compared to experimental results.Three techniques were used to investigate the spin-lattice relaxation of the solidwood and the water in wood. Separate T1 measurements of the solid and water, sep-arate T1 measurements of water in the earlywood and latewood regions, and separateT1 measurements of the cell wall water and lumen water were acquired. The resultsindicated that, on the T1 time scale of 100 ms, all proton environments are mixed bydiffusion of the water. The T1 of the water in the lumen and the cell wall and theprotons of the solid were found to have the same T1 , which is an average of the T1 ofthe three environments. The T1 was found to be dependent on the proportion of cellwall to lumen volume. Thick walled latewood cells had a lower T 1 than thin walledearlywood cells. Lastly, the cross relaxation of the protons in solid wood and the cellwall water was found to be the dominant mechanism for 7'2 relaxation of the cell wallwater.ivPAGINATION 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 ContentsAbstract^ iiList of Tables^ viList of Figures^ ixAcknowledgement1 Introduction 11.1 Motivation ^ 11.2 Wood and Water in Wood ^ 21.2.1^Structure of Wood 21.2.2^Cell Wall Composition and Structure ^ 81.2.3^Water in Wood ^ 121.3 1 1-I NMR and Wood 141.3.1^1 11 NMR Relaxation Theory^ 141.3.2^Review NMR work of water in wood ^ 171.4 Overview of Thesis ^ 192 General Materials and Methods 212.1 Samples ^ 212.2 SEM Images 212.3 SEM Moisture Content Measurement ^ 222.4 NMR Equipment ^ 22ii3^2.5^NMR M2 and Moisture Content Measurements ^2.6^T2 Relaxation ^Second Moment and Cell Wall Water T2 12324263.1 Summary ^ 263.2 Introduction 263.3 Materials and Methods 273.4 Results and Discussion ^ 283.5 Concluding Remarks 364 T2 Techniques for Characterization of Water in Wood 2 374.1 Summary ^ 374.2 Introduction 384.3 Materials and Methods ^ 384.3.1^Samples  384.3.2^One-dimensional Imaging ^ 394.4 Results and Discussion ^ 404.4.1^Fits of T2 Relaxation Data for White Spruce ^ 404.4.2^One-dimensional Imaging ^ 444.5 Concluding Remarks ^ 515 Diffusion Model of Compartmentalized Water 3 535.1 Summary ^ 535.2 772 Relaxation of Compartmentalized Water: Lumen Water in Wood 545.2.1^Introduction ^ 541 This chapter closely follows a previously submitted paper. [Araujo et a1.1993b].2 This 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]iii5.2.2^Theory ^ 545.2.3^Materials and Methods ^ 625.2.4^Results and Discussion 625.2.5^Concluding Remarks ^ 745.3 Recovering Compartment Sizes from NMR Relaxation Data ^ 776 Diffusion Model of Two Regions of Compartmentalized Water 4 816.1 Summary ^ 816.2 Determination of the Cell Wall Water Diffusion Coefficient in Woodfrom 12 Relaxation Measurements ^ 826.2.1^Introduction ^ 826.2.2^Results and Discussion ^ 876.2.3^Concluding Remarks 926.3 Numerical T2 Simulations ^ 946.3.1^Numerical Method 946.3.2^Numerical Applications ^ 997 Spin-lattice Relaxation and Cross Relaxation 1027.1 Summary ^ 1027.2 Introduction 1037.3 Materials and Methods 1037.3.1^Samples ^ 1037.3.2^Spin-lattice Relaxation of Wood and Water in Wood ^ 1047.3.3^Spin-lattice Relaxation of the One Dimensional Water Image ^. . 1047.3.4^Two Dimensional 12 - T1 Dependence of Water in Wood^. . . . 1054 Part of this chapter closely follows, with some modifications, part of a previously published paper.[Araujo et a1.1993]iv7.3.5 Cross Relaxation of Protons in Solid Wood and Water ^ 1057.4 Results ^  1067.4.1 Spin-lattice Relaxation of Wood and Water in Wood ^ 1067.4.2 Spin-lattice Relaxation of the One Dimensional Water Image^1097.4.3 Two Dimensional T2 - T, Dependence of Water in Wood^1107.4.4 Cross Relaxation of Protons in Solid Wood and Water ^ 1107.5 Discussion ^  1167.6 Numerical T1 Simulations ^  1217.7 Cross Relaxation and T2  1237.8 Concluding Remarks ^  125Bibliography^ 126A Field Gradient NMR^ 131B Diffusion Model for Rectangular Geometry^ 135C Two Regions with Cylindrical Geometry^ 138D Two Regions with Spherical Geometry^ 141E Numerical T1 Simulations^ 144List of Tables1.1 Cell Wall Composition in Wood ^ 93.1 Transition moisture content and average M 2 's with standard deviations. 315.1 Geometric dependence of the fundamental T2 mode, T2(o) (for a fixedsurface sink parameter, M), and of M (for a fixed T2 (o)), for cells with across-sectional area of 2826,am 2 (T2free = 1.4s and D = 2.2 x 10 -9m2/s). 615.2 Fit of the diffusion model to the CPMG decay curves for the redwoodsample. ^ 725.3 White Spruce Results 736.1 Cell wall water T2, fibre saturation point, moisture content, surface sinkparameter and cell wall water diffusion coefficient for redwood sapwoodat temperatures 4 to 55°C. ^ 896.2 Numerical T2 Simulations 997.1 Fast Exchange Model of T1 for typical cedar cells. ^ 119viList of Figures1.1 Earlywood (large cells) and latewood tracheid cells (small, thick walledcells) in a softwood. ^ 31.2 Ring-porous hardwood. 51.3 Diffuse-porous hardwood. ^ 61.4 Logslice of a softwood 71.5 Diagram of two cellulose repeat units. ^ 91.6 Cominom forms of lignin. ^ 101.7 SEM showing the direction of the fibrils in the Si, S2, and S3 layers ina earlywood tracheid (left side) and a latewood tracheid (right side) cellwalls. ^ 111.8 Pit pairs. 121.9 Moisture Isotherm^ 133.1 Lodgepole Pine Isotherm. ^ 293.2 Free Induction Decay. 313.3 Second Moments of Protons in Solid Wood versus Moisture Content. . . 323.4 Moisture Content (ovendry method) versus NMR Moisture Content oflodgepole pine heartwood^ 333.5 T2 Distributions of Water in Lodgepole Pine at a Moisture Content of11.3%^ 343.6 Lodgepole Pine cell wall T2 versus Moisture Content^ 35vii4.1 Relative numbers of cells and protons as a function of cell lumen radiusfor the white spruce sapwood and heartwood compression wood samples. 414.2 CPMG data and the fit from a continuous T2 spectrum.   424.3 T2 results as a continuous curve and as a discrete fit of 2 exponentialcomponents. ^  424.4 Continuous T2 spectra of the water in white spruce juvenile wood andheartwood. ^  444.5 Continuous T2 spectra of the water in rehydrated samples of white spruceheartwood compression wood, heartwood, and heartwood with incipientdecay. ^  454.6 Radial Moisture Density Profile of the total water distribution. ^ 464.7 Radial Moisture Density Profile of the cell wall water distribution.^484.8 Radial Moisture Density Profile of the lumen water distribution.^49^5.1 The first 3 An ,10 (71„r1R) modes   575.2 In amplitudes as a function of MR/D^  585.3 The diffusion model gives T2(„ ) versus cell radius^  595.4 The diffusion model gives T2(„ ) versus diffusion coefficient^ 605.5 The diffusion model gives T2(„ ) versus surface sink parameter^ 605.6 Radius distributions of redwood sapwood, white spruce sapwood, andwhite spruce heartwood compression wood samples. ^ 635.7 The CPMG decay curve of redwood at 18°C. ^  645.8 T2 relaxation plots for the redwood sample.  655.9 Average T2 of lumen water for the redwood sample.^  665.10 T2 plots for white spruce sapwood, and rehydrated white spruce heart-wood compression wood samples^   67viii5.11 The first 3 modes and the sum of the first 3 modes predicted for theredwood radius distribution at 4°C. ^  715.12 Redwood Radius Distribution from NMR  785.13 White Spruce Radius Distributions from NMR^  795.14 Alder and Douglas Fir Radius Distributions from NMR^ 806.1 Two region diffusion problem^  836.2 Cell wall water decay curves of redwood sapwood. ^ 886.3 T2 relaxation plots of cell wall water decay curves from redwood sapwood. 896.4 Calculated cell wall water diffusion coefficient for redwood sapwood.^. 906.5 Discrete steps in position and time. ^  956.6 Lumen and Cell Wall Water Exchange.  100^6.7 NMR Measurement of FSP    1017.1 r dependence of FID in inversion recovery experiment^ 1077.2 T1 plots for liquid and solid signals. ^  1087.3 r dependence of amplitude images in modified inversion recovery exper-iment. ^  1097.4 Amplitude and T1 images at 4 moisture contents^  1117.5 T1 -T2 plots for high moisture contents^   1127.6 T1 -T2 plots for low moisture contents.  1137.7 The FID following the cross relaxation sequence. ^  1147.8 Reappearance of the solid signal following the cross relaxation sequence ^ 1157.9 Cross section of a cylindrical cell^   1177.10 Simulations of spin-lattice relaxation in cylindrical cells. ^ 1227.11 Spin-lattice decay of the protons in solid wood. ^  124ixAcknowledgementI thank my supervisor Alex MacKay for his enthusiasm and guidance, Myer Bloom forbeing my professor for many years and sharing his lab, Elliott Burnell for his involve-ment 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 havemade this thesis possible and enjoyable. I also thank my first family, Mom, Dad andShelly for their encouragement. I will always be thankful to my husband Dan for hisunderstanding, caring and patience.Chapter 1Introduction1.1 MotivationThe interpretation of 1 H NMR relaxation of water in biological systems is complicatedby a number of factors including heterogeneity, chemical exchange, paramagnetic so-lutes, compartmentalization, magnetic susceptibility, and diffusion [Mathur—De Vre1979, Belton and Ratcliffe 1985]. Although it has much of the complexity of generalbiological systems, wood is a better candidate for quantitative 1 11 NMR studies sinceits regular and robust structure makes it easy to work with, and simplifies the interpre-tation. Proton magnetic resonance is a useful tool for wood science since signals fromthe solid, cell wall water and lumen water can be distinguished to give the moisturecontent and allow investigation of the separate proton environments and the interac-tion between these environments. In this thesis, I have been able to i) show that thefibre saturation point of the cell wall water can be measured accurately, ii) image thecell wall moisture profile for a full range of moisture contents, iii) recover cell lumensize distributions iv) measure the cell wall water diffusion coefficient and activationenergy, 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 iden-tification. Diffusion of water in wood has played an important role in mixing waterenvironments and the effect of diffusion on the relaxation times has been modeled inthis thesis. Understanding diffusion as a mechanism of mixing water environments on1Chapter 1. Introduction^ 2the 1 H NMR time scale is necessary for interpretation of relaxation data of biologicalsystems [Belton and Ratcliffe 1985], and the qualities of wood make it ideal for diffu-sion studies. Diffusion of water in wood is similar to that in human tissue, however thewood geometry is simple and measurable by scanning electron microscopy and the cellwalls 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 rele-vant biological systems have rarely proven to be diagnostic [Bottomley et a/.1987]. Themethods and approach developed for water in wood can be applied in tissue to studytissue pathology, and the state of water.1.2 Wood and Water in Wood1.2.1 Structure of WoodTrees are divided into two categories: Hardwoods and softwoods, based on the char-acteristics of their seeds. Softwoods have needle like leaves and are commonly calledevergreens. The seeds of softwoods are basically unprotected, usually inside scaly cones,and are therefore often referred to as conifers. Hardwoods have broad leaves, whichgenerally change color and drop in the autumn in temperate zones: Hardwoods produceseeds within acorns, pods or other fruiting bodies. The woods formed by softwoodsand hardwoods contain different types of cells. Hardwoods are not necessarily moredense or harder than softwoods, as their names imply [Haygreen and Bowyer 1982].In softwoods, there are three main reservoirs for lumen water: earlywood and late-wood tracheids, which are oriented longitudinally along the tree trunk, and ray cellswhich are oriented radially. The earlywood tracheids, which grow during the springand early summer, are larger in radius than the latewood tracheids, which grow inChapter 1. Introduction^ 3Figure 1.1: Earlywood and latewood tracheids in redwood, asoftwood.x85.[Haygreen and Bowyer 1982]Chapter 1. Introduction^ 4late summer and autumn. The latewood tracheids have thick cell walls and give thelatewood region a higher solid wood density. The alternation of these cell types givesthe appearance of annual growth rings, as shown in Fig. 1.1. Tracheids vary in sizefrom 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 radiiof about 8 pm. The tracheid cell walls are one to several microns thick and provide adiffusion barrier for water, since the translational diffusion coefficient of cell wall wateris 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 andlongitudinal parenchyma cells which are oriented longitudinally along the tree trunk,and ray cells which are oriented radially. Short, large diameter cells joined end toend with perforated unrestricted holes are called vessels or pores and are unique tohardwoods. Vessels give rise to two types of hardwoods, as shown in Figs. 1.2, and1.3. Ring-porous hardwoods have smaller vessels in the latewood and diffuse-poroushardwoods 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. Longitudinalparenchyma cells are thin walled storage units, divided by cross walls. Ray cells arefound either alone or in groups up to 30 cells wide. Ray cells make up about 17% ofthe total lumen volume and sometimes more than 30%. Ray cells are distorted fromradial orientation in the vicinity of large vessels.Figure 1.4 shows a cross-section log slice of a softwood. A log slice of a hardwoodis similar, except that often the growth rings are less distinct. Cell division occurs inan outer layer called the cambium. The cells become part of the bark or part of thewood. The newer cells are referred to as sapwood, and in a freshly cut or green samplethese cells are filled with water. In the heartwood region all cells have died, losing theirnuclei and protoplasm. Cells in the heartwood have lower moisture content, especiallyChapter 1. Introduction^ 5Figure 1.2: Ring-porous hardwood, northern red oak. x 35. [Haygreen andBowyer 1982]Chapter 1. Introduction^ 6Figure 1.3: Diffuse-porous hardwood, yellow poplar. x 80.[Haygreen andBowyer 1982]LATEWOODPITHEARLY WOODANNUALINCREMENT( GROWTH RING)Chapter 1. Introduction^ 7Figure 1.4: Cross section of tree showing various wood zones. [Bramhalland Wellwood 1976]Chapter 1. Introduction^ 8in softwoods, and are usually high in extractives, which are the decomposition productsof starches and sugars giving fats, waxes, oils, resins, gums, tannin, and aromatic andcolouring materials [Haygreen and Bowyer 1982]. The sapwood is thought of as thecells storing and conducting water and minerals, and the heartwood gives the treemechanical support. A juvenile region, the first 5-20 central growth rings, containscells from early stages of the tree's existence. These cells are shorter and have fewerlatewood cells than mature wood cells, although there is no abrupt division to maturewood. The high proportion of thin walled cells having low wood density in juvenilewood gives it lower strength than mature wood.Reaction wood forms in both softwoods and hardwoods. In softwoods, reactionwood forms on the bottom side of leaning stems and branches and is called compressionwood. Compression wood cells are typically 30% shorter, have 10% less cellulose and 8-9% more lignin and hemicellulose. Compression wood has a high proportion of latewoodand often forms wide growth rings so that growth is eccentric about the pith. Inhardwoods, reaction wood forms on the top side of leaning stems and branches and iscalled tension wood. Tension wood has fewer and smaller vessels, fewer ray cells, andthick walled, small lumen fibers. The secondary cell wall layer is almost pure celluloseand loosely connected to the primary wall. Tension wood also has wide growth ringsso that growth is eccentric about the pith. One cause of reaction wood formation hasbeen shown to be gravity, not stress [Haygreen and Bowyer 1982], and it is the effect ofgravity on the distribution of growth stimulators (auxins) which is thought to inducereaction wood formation.1.2.2 Cell Wall Composition and StructureWood is mainly composed of cellulose, hemicellulose and lignin [Haygreen and Bowyer1982]. Cellulose is a long straight chain of glucose molecules. Figure 1.5 shows twoChapter .1. Introduction^ 9repeat units of cellulose, and 5000 to 10000 units form cellulose making it at least5/cm long and 8A in diameter [Aspinall 1983]. In wood 60-70% of the cellulose ishydrogen bonded to form crystalline cellulose. A single cellulose chain may extend intoseveral crystalline and amorphous regions and in this way restrict the swelling of thestructure. Hemicelluloses are low molecular weight branch chain polymers of variousC1 H20 H^ CH2OHIC 0\ C^0\F71 H \r--,,.^H /I1,/^H^\i'-•C‘ C^0_^C CNI\ OH H^-------] OH H^i\ 1 H 1  HC1C1CiC1H OH H OHFigure 1.5: Diagram of two cellulose repeat units.types of sugars, which form a matrix in the wood and are associated with the celluloseand the lignin. Lignin is made up of high molecular weight polymers of phenylpropaneunits, 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 asa stiffening ingredient. Lignin stiffens the structure, but it is the cellulose that gives itits strength.These composites arrange into long bundles called microfibrils of a few nanometersTable 1.1: Cell Wall Composition in WoodCellulose Hemicellulose LigninSoftwood 40-44% 20-32% 25-35%Hardwood 40-44% 15-35% 18-25%Chapter I. Introduction^ 10Softwoods1CHN, NC 7HH\c7 NOCH,Hardwoods1^C—v 1-17CH 30 \c^OCH 3OFigure 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 perpendicularto the long axis of a cell. The centre layer, S2, is the thickest and its fibrils are alignedalong the axis of the cell. The S2 layer is much thicker in the latewood cells comparedto the earlywood cells.Cells in both softwoods and hardwoods are joined by pits, either simple, borderedor half-bordered, as described in Fig. 1.8. Pit regions are thin spots in the cell wall andmay 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 isblocked, preventing normal moisture conduction through cells.Chapter 1. Introduction^ 11Figure 1.7: SEM showing the direction of the fibrils in the Si, S2, and S3layers of the cell wall [Siau 19S4].Chapter 1. Introduction^ 12Figure 1.8: Profile of various types of pit pairs.[Haygreen and Bowyer 1982]1.2.3 Water in WoodIn green wood, water inside the cell walls, is typically 25 to 30% of the mass of thesolid wood. It is well known [Siau 1984] that as wood dries, lumen water comes outfirst 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 moisturecontents but are found to change as the bound water is removed from wood. Forexample: i) The heat required to evaporate a unit weight of lumen water from woodis the heat of vaporization of free water while there is an extra amount of heat, theheat 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 boundwater is removed, but is not affected by the amount of lumen water, iii) the thermalChapter I. Introduction^ 13and electrical conductivities of wood are also altered by the removal of bound waterbut 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 thatthe cell walls are saturated and that there is no free water in the lumens. But, morepractically, the FSP is defined as the moisture content where abrupt changes in thephysical properties of wood occur [Stamm 1964].25 ^20^40^60^80^100I-1 (%)Figure 1.9: Moisture isotherm showing equilibrium moisture content (M)as a function of relative humidity (H). The isotherms of thewater of hydration (Mh ) and the water of solution (Ms ) com-ponents are shown.Cell wall water is closely associated with the cell—wall substance through hydro-gen bonding. The major part of cell wall water adsorption in wood involves replacingsolid—solid interfaces by solid—liquid—solid interfaces, and as a result adsorption of wa-ter swells wood by forming a solid solution. In the hygroscopic range (between 0%and FSP), the relationship between equilibritun moisture content of wood and relativehumidity at a given temperature is called the sorption isotherm, as shown in Fig. 1.9.Chapter 1. Introduction^ 14It 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 toconsist of two components, one strongly and the other weakly attracted by the sorptionsites [Stamm 1964, Skaar 1988]. When the sorption process is considered to be a sur-face phenomenon, the strongly bound part is called monomolecular water whereas theweakly bound one is called polymolecular water. When the sorption process is consid-ered a solution phenomenon, then they are called water of hydration (Mh ) and water ofsolution (Ms ), respectively. Most of the current sorption theories [Simpson 1973, Skaar1988] divide the sigmoid isotherm into two curves; the type I and III, correspondingto 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 theHailwood—Horrobin (HH) model [Hailwood and Horrobin 1946], belonging to the groupthat treats the sorption process as a solution phenomenon, gave the most satisfactoryresults. Numerically, the total moisture content at a particular relative humidity andtemperature will be equal to MI?, plus Ms . The water of hydration, Mh , at maximumhumidity, has been found to range between 4 and 6% for most temperate zone woodspecies [Skaar 1988].1.3 1 H NMR and Wood1.3.1 1 11 NMR Relaxation TheoryProtons 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 levelswith the Hamiltonian 7-1 = = The equilibrium state of a spin systemat a temperature T is characterized by a Boltzmann population of spins in these energyChapter I. Introduction^ 15levels such thatN_^of — aticdN+ = e kT = e kTfor spin 1/2, where E = ±1/2-yhHo , and wo = -yHo . Therefore the magnetization,defined as M = [7, is non zero and is given by Curies Law as follows;< M >— I(I + 1)72 ti, 23kT^Ho^ (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 themagnetization follows classical equations of motion.d < >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 , istiapplied. Power from an applied RF magnetic field , perpendicular to 170, with frequencyw is absorbed when the resonant condition w = wo holds. In the rotating frame offrequency coo about 170, < M > precesses through an angle co i tp = 71/1 tp , where tp isthe length of time the RF field is applied.The lineshape of absorption at wo is broadened by dipolar interactions of neigh-bouring spins in the system. For two spins, the dipolar Hamiltonian isH12^71732 h2 [I; ^. f)(1--2 n)]7'7172h2 (A-1-B+C-FD+E+F)^(1.4)7' 3wheredt = 7 < > x (1.3)A = /1 ,/2 ,(1 — 3 cost 0)B = --4(/-0-- + /T4)(1 — 3 cos t 0)Chapter 1. Introduction^ 163 f T+ T1 2' + Ilz in Sill 9 cos Be -2 d'23D =^2 1- 12Z + z In Sill 9 cos Oe i `kIE = __4(/+/-+) sin 2 0e -2i csI 22-- ) sin2 0e2i g9F = - -(/-/^(1.5)4 where r is the separation distance between the two spins, and h is the unit vector in thedirection joining them and making angles 0, and 0 with respect to the main magneticfield H0 , in spherical coordinates. The A and B terms give the secular or truncatedHamiltonian7-(d) = 1^3172h (1^fi3 cos 2 8)(3/1J2z - • 12)r 2 (1.6)which only gives transitions between states having the same Zeeman energy, and arethe only terms used in the first order perturbation theory. C and D terms give energytransitions of tiw0 , and E and F terms give energy transitions of 2t1w 0 .Motion averages the dipolar interaction in the limit where M2 7,2 << 1. M2 is thesecond moment of the dipolar broadened lineshape in the absence of motion, and 7-, isa characteristic correlation time of the motions. Relaxation is described by Redfield'stheory [Abragam 1961,Slichter 1980]. For spins on the same molecule, so that r isconstant, and assiuning that the relative orientation varies isotropically, one hasT11^41;2=^1 it4r6 [J(wo) 4J(2w0)]1 y4 h216v6 [6J(0) 10J(co0 ) 4J(2w0)]rn/2The spectral density, J(w), describes the fluctuating part of the Hamiltonian, and is thethe Fourier transform of the correlation function. The correlation function is commonlyassumed to be exponential, as follows.T(t).F*(t +^r ) =^ (1.9)(1.7)(1.8)Chapter I.^Introduction 17The relaxation rates are now written as1^3 74h2^1 4(1.10)Tc[1 + 444 1Tl^10^r6^COdTc2^11^3 74h2 5(1.11)Tc[3 + 1 + 42(.4) (Wi20^r6^1 + WciTc2For short correlation time, such that co ol-, << 1, one has1^1= 114-2 7-C (1.12)Tr^T2In 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 tomacromolecules. The relaxation rates of water are affected by the strength of the localmagnetic interaction, i.e. the dipole-dipole coupling, and by the motions. The motionof the hydration water molecules is restricted and anisotropic due to the bonding to themacromolecules. Also the motion of the macromolecule influences the motion of thehydration water. Typically, the motions must be described by more than one Tc or evena distribution of 7-, times. These correlation times are generally longer than that of bulkwater. 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, andtherefore show slightly lower rates of rotation than bulk water. Consequently, themeasured relaxation times, T1 and T2, are faster than for bulk water, and in generalT2 < T1 [Mathur-De Vre 1979].1.3.2 Review NMR work of water in woodThe wide-line 1 11 NMR absorption spectrum [Nanassy 1973,Nanassy 1974] of wood ex-hibits a narrow water line superimposed on a broad line from hydrogens of the solidChapter 1. Introduction^ 18wood fibre. The narrow line amplitude scaled to the amount of water, but was broaderthan 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 sepa-rable [Sharp et a1.1978,Riggin et a1.1979,Menon et al.1987]. The solid component canbe fit to a second moment expansion to give an absolute moisture content measurementby 1 H NMR [Menon et a/.1987]. Furthermore, the water signal could be separated onthe 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, andcomparison with anatomical data from SEM images confirmed the relative amplitudeof lumen water components, and the ratio of T2 times scaled as the radius of cell typeas predicted by a fast exchange model of free lumen water in exchange with a fractionof water on the cell wall surface [Menon et al.1987]. Further investigation of westernred cedar [Flibotte et a/.1990] showed that the solid wood signal has a second momentof about 5 x 10 9 s -2 which increased by about 20 % at low moisture contents belowthe FSP. Heartwood and juvenile wood were found to have substantially less waterand shorter cell wall water T2 relaxation times than the sapwood, which would enablethe sapwood-heartwood boundary to be distinguished in a cross sectional image of acedar log, but the heartwood- juvenile wood boundary would be more difficult to dis-cern. Decayed wood was found to have high moisture contents and so would be easilyidentifiable, especially if in heartwood or juvenile wood.Pulsed field gradient methods on water in wood [MacGregor et al.1983, Peemoelleret a/.1985] showed that water in wood lumens undergoes restricted diffusion in all di-rections, even in the long direction along the cell lumens. Water squeezed from the woodwas 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 one-dimensional images of cell wall and lumen water separately. One-dimensional images ofChapter 1. Introduction^ 19earlywood, 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 etal.1990] as a function of time during controlled air flow rate and temperature resultsshowed clearly that water removal as a function of time differs in the earlywood andlatewood regions. Three—dimensional imaging in whole body MRI scanners [Hall andRajanayagam 1986, Flibotte et al.1990] was used to investigate the water signal distri-bution using a single echo. The echo time in MRI is limited to about 20 ms, thereforeonly 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 thesemethods.1.4 Overview of ThesisThe general materials and methods are given in Chapter 2. The second moment ofthe protons in solid wood and T2 of the cell wall water, for low moisture contents, arepresented in Chapter 3. A technique for fitting multi-exponential data and a techniqueof selectively imaging water environments are demonstrated in Chapter 4. The roleof diffusion in mixing water environments and the resulting effects on relaxation areinvestigated in Chapters 5, 6 and 7. In Chapter 5 the lumen water T2 relaxation isshown to follow a diffusion-Bloch equation with a surface relaxation sink at the cellwall. From this theory, the distribution of lumen sizes can be recovered from the lumenwater T2 decay. In Chapter 6 a two region diffusion model is applied to describe theT2 relaxation of both the lumen and the cell wall water. The surface sink parameteris defined in terms of cell wall water diffusion and cell wall water relaxation. Fromthis theory, an original method of calculating the cell wall diffusion rate from the T2Chapter I. Introduction^ 20relaxation and the distribution of cell sizes is developed. Numerical simulation of thistwo region model is developed. Finally, in Chapter 7 the T1 relaxation of wood isinvestigated and the influence of water diffusion and cross relaxation of water with theprotons in solid wood is determined.Chapter 2General Materials and Methods2.1 SamplesSamples were cut with a bandsaw from log slices, most to dimensions of 0.5 x 0.5 x 1 cmwith the long axis parallel to the longitudinal tracheids of the wood. The imaged sam-ples were cut to dimensions of 0.4 x 0.4 x 1 cm to fit in the smaller imaging probe. Unlessdescribed otherwise, all samples were were cut from logs of green moisture content andbelow 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, andwestern redcedar sapwood have been utilized.2.2 SEM ImagesCross—sectional and tangential microtomed surfaces of the samples were dried, goldsputter coated, and imaged by a Cambridge Stereoscan 250 Scanning Electron Mi-croscope (SEM) operating at 20 kV. The micrographs of the tracheid and ray cellswere digitized into a 512 x 512 pixel array with 256 grey levels with a Kontron IBAS2000 Image analyser to give the cell distribution as a function of cell radius. The celllumen radius was calculated based on its measured digitized area assuming circular21Chapter 2. General Materials and Methods^ 22geometry. The radius was corrected for shrinkage to give the green sample radius[Haygreen and Bowyer 1982]. A cubic spline interpolation of 100 points through theoriginal radius histogram was used for the analysis.2.3 SEM Moisture Content MeasurementIn order that our SEM results relate directly to the NMR results from lumen water, thewood samples must be maximally hydrated. The maximum MC attainable in a givenwood sample can be estimated from the SEM measurements of percent cross—sectionalarea of lumen, L, and cell wall, W. That is,WaterMass^LV d water 0.3WVdwoodM C max = ^ 100%^ 100%.OvendryWoodMass WVdwood(2.1)V is the sample volume which cancels out of the calculation. The densities dwater1.0 g/cm3 and dwood = 1.5 g/cm3 [Haygreen and Bowyer 1982] are used and the fibresaturation point (FSP), i.e. the moisture content of cell wall water, is assumed to be30%.2.4 NMR EquipmentProton NMR measurements were carried out on a modified Bruker SXP 4-100 NMRspectrometer operating at 90 MHz with a lips receiver dead time. The 90° pulsewas typically 2.50/ts and the 180° pulse was typically 5.00/ts. Temperature was variedusing a Bruker B—ST 100/700 temperature controller. Data acquisition and analysiswere 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 witha IBM compatible computer, a Rapid Systems A/D board and a "Sync" card.Chapter 2. General Materials and Methods^ 232.5 NMR M2 and Moisture Content MeasurementsThe free induction decay (FID), the NMR signal following a single 90° pulse, wasanalysed to give the second moment (M2 ) and NMR moisture content (NMR MC).The signal of the solid wood decays rapidly in about 30,us, and is easily distinguishablefrom the water signal, which extends over hundreds of milliseconds. To refocus thedephasing of the water signal due to inhomogeneity of the magnet, 180° pulses with100,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 cumulativedata memory to reduce the dead time. Two hundred scans were averaged to give ahigh signal to noise ratio, typically about 100 to 1. A repetition time of 8 s (> 5T1where 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 expan-sion equation [Abragam 1961] of the form:S(t) = (SO L0)(1 - 2 t2 -M4! 4^ 6t6 ••-) LO.6!(2.2)where M2, M4 and M6 are the second, forth and sixth moments of the lineshape, S o isthe total NMR signal at t = 0, and (S0 — L0 ) 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 tothe signal, which is the amplitude at t = 0 of a linear fit to the water signal. After thedead time of the spectrometer, 30 points from about lips to 26,as following the centreof the 90° pulse were fit to the above equation, out to the N/6 term only. Higher orderterms were negligible for these 30 points. A non—linear function optimization programminimizing X 2 was implemented [.James and Roos 1975].The moisture content (MC), corresponding to the ovendry definition as the massChapter 2. General Materials and Methods^ 24ratio of water to solid wood, is defined from the FID by* Pw"d * 100%L(0) NMR MC —^ (2.3)S(0) — L(0) o, waterwhere pwood is the number of protons per grain of wood, Pwater is the number of protonsper gram of water, and the ratio of the two is 0.56 for white spruce, and about thesame for most species [Fengel, Wegener 1984].2.6 T2 RelaxationThe echo heights from a Carr—Purcell—Meiboom—Gill (CPMG) sequence [Carr and Pur-cell 1954, Meiboom and Gill 1958], with echo spacing typically 200ps, give a relaxationdecay curve of the water in the wood. The CPMG sequence is represented by000 — 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 wereaveraged per echo and 200 scans were signal averaged. About 100 echoes were usedcorresponding to times, from the 90° pulse, increasing approximately geometricallyfrom 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 pulsetrain follows the curveS(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 amultiexponential decay and can be represented by the sum of several componentsS(t) =^S, e —t/ T22^(2.6)Chapter 2. General Materials and Methods^ 25where 8(0) is proportional to the moisture content, T2i are the component relaxationtimes, and the number of possible exponential components can range from one to severalhundred.In this thesis, I solve for T2 times in two ways: (a) nonlinear solutions for a smallnumber of discrete exponential component times and amplitudes, and (b) linear solu-tions for amplitudes at a large number (100) of specified relaxation times. Whittalland 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 NNLSto minimize the least—squares misfit results in spectra composed of a few isolated T2components. 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 lin-ear combination of the misfit and the first differences of the spectrum [Whittall andMacKay 1989].Chapter 3Second Moment and Cell Wall Water T2 13.1 SummaryNuclear magnetic resonance lineshape second moments of the protons in solid wood andspin—spin relaxation times, T2 , of the cell wall water in lodgepole pine heartwood havebeen measured at 30°C for a range of moisture contents, mainly in the hygroscopicrange. The second moment of the protons in ovenclry wood was found to be about23% lower than the rigid lattice calculation, indicative of a rigid structure with someanisotropic molecular motion of the polymeric constituents. Above 5% MC, the secondmoment decreased by a further 13 to 16% implying a "loosening" of the molecules inthe solid with increased moisture content. The T 2 of the cell wall water increased withmoisture content, and provided no evidence of separate hydration and solution waterfractions as predicted by isotherm theories. The 1 11 NMR measured fibre saturationpoint of 27% agreed with the value calculated by the Hailwood—Horrobin isothermmodel.3.2 IntroductionIn this chapter, the second moment of the proton 1 H NMR lineshape from the solidwood and the spin—spin relaxation time (T2 ) of the cell wall water were measuredas a function of moisture content below the FSP in lodgepole pine heartwood. The1 This chapter closely follows a previously submitted paper. [Araujo et a1.1993b].26Chapter 3. Second Moment and Cell Wall Water T2^ 27objective was to see if the microscopic dynamic properties of the solid wood and wateras monitored by 1 H NMR give any new insight into wood—water interactions in thehygroscopic range.3.3 Materials and MethodsThree specimens 5 x 5 x 10 mm in longitudinal dimension were cut from a greenpiece of lodgepole pine heartwood (Finns contorta Dougl. var. latifolia Engelm.).The specimens were oven dried at 103 ± 2°C until constant weight, placed in 180mm long and 10 mm od diameter glass NMR tubes and into a conditioning chamberwhere the temperature and relative humidity could be maintained constant. Therethey were conditioned, to constant weight, at relative humidities of 30, 45, 60, 75, 85and 95%, and at 30°C. After NMR measurements, at 30°C, the same specimens wereimpregnated with water by a full—cell method and then were left to dry slowly, atambient temperature, until their moisture content was approximately 34%. In this waya moisture content above the FSP could be obtained. Next, the highest hydration wasobtained 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 chamberwith the NMR specimens, simultaneously. The mean equilibrium moisture content ofthe 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 methodof minimizing the least squares deviations.1800/  K2H  ) 1800 (  K2H M Mh + =  (3.1)W 100 K1 A 2 H^W 100 — K2H)where Ki is the equilibritun constant of the reaction between water and dry woodChapter 3. Second Moment and Cell Wall Water T2^ 28substance, K2 is the equilibrium constant of the condition between water vapour anddissolved water, and W is the grams of dry wood per mole of sorption sites. The1800/W ratio corresponds to the moisture content of the wood when there is onemolecule of water on each sorption site. From the parameters Kl , K2 and W, therelative 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 (M2 ) and the NMR moisture content. The echoheights from a CPMG sequence, with echo spacing of 200/ts and including 8 echos ofspacing 100/ts collected separately, gave the relaxation decay of the water in the wood.3.4 Results and DiscussionThe adsorption isotherm for lodgepole pine heartwood at 30°C obtained in this study isshown in Fig. 3.1. The experimental points as well as the curves calculated by the HHmodel, 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 sorptionsite is 4.6%, and the predicted FSP which is the moisture content at 100% humidity is26.9%.The polymeric molecules of solid wood are well known to have a fairly rigid struc-ture. The '11 NMR spectrum, which is dominated by the dipolar interactions betweenneighbouring protons, is broad and featureless. Spectra of this type are quantitativelycharacterized by their spectral moments, in particular, M2. For a rigid solid, the mea-sured second moment, M2 is equal to the rigid lattice M27.i9id which can be calculatedfrom knowledge of the spatial distribution of protons in the sample, as follows:3 (1 — 3 cos 2 t9ik ) 2M2 = —474h 62-4/ + 1) (3.2),,'jk20Chapter 3. Second Moment and Cell Wall Water T2^ 292520 —1510 —5 —Mh40^60^80H (%)Figure 3.1: Isotherm data for lodgepole pine heartwood at 30°C, HH modelfit (solid curve), isotherm of hydration (M1) and isotherm ofsolution (M3 ). The percentage deviation in the moisture con-tents, from the twenty matched wafers, is at most 3%.100Chapter 3. Second Moment and Cell Wall Water T2^30where the vector 77:ik describes the relative positions of two protons, which is at an angle6jk to the applied magnetic field Ho , and rik is the distance between the protons. Theexpression reduces to the following, when one averages over all angles for a sample ofrandom orientations,/V/2 = -374 h2/(/ -I- ) 1 (3.3)5 k rjkIn the presence of molecular motion, the 1 H NMR spectrum is narrowed and the mea-sured 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 protonsin wood is shown in Fig. 3.2. The fast decaying part is from the solid wood and the slowdecaying component is from the water in wood. The fit of Eq. (2.2) to the solid woodsignal is shown in the inset of Fig. 3.2. The resulting M2 values from the fit are shownin Fig. 3.3, for increasing moisture content, for the three samples. Our M2 results canbe compared to rigid lattice calculations for cellulose and hemicellulose which are about7.3 x 109 s -2 [MacKay et al. 1985]. Lignin is expected to have a similar M2 value, hencethe average 1VI2 for ovendry wood of 5.6 x 10 9 s -2 indicates a very rigid structure forwood on the 1 1-I NMR time scale of 10 -5 s. This reduction by about 23% from the rigidlattice 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 moisturecontent range 5 to 13% , as sununarized in Table 3.1. This indicates that the solidwood molecules in our lodgepole pine samples underwent a "loosening" process atmoisture contents in the range of 5 to 13%. This "loosening" can be interpreted asa change in the anisotropic molecular motions of the polymeric constituents of thesolid wood which occurs as water molecules are added at some level above 5%. Themoment measurements alone do not enable us to determine which type of molecules,Chapter 3. Second Moment and Cell Wall Water T2^ 31SL o _O 5 10 15 20 25 30Time (,us)IIIIIIIIIO^200^400^600^800Time (,us)Figure 3.2: A typical free induction decay showing a baseline, the fast de-caying solid wood signal, and the slower decaying water signal.The gaps in the data are where the high power RF 180° pulseswhere applied. The inset is of the solid wood signal and the fitto the moments expansion.Table 3.1: Transition moisture content and average M2 's with standard deviations.Sample Transition Pre—transition Post—transition Decrease inMC (%)^M2 (10 9 S -2 )^M2 (109 S -2 )^M2 (%)1 8.1 + 0.9 5.70 + 0.08 4.8 ± 0.2 162 5.7 + 1.3 5.6 + 0.3 4.9 + 0.1 133 13.4 + 1.8 5.5 + 0.2 4.6 + 0.1 16Chapter 3. Second Moment and Cell Wall Water T2^ 326.05.55.04.51f2•NI^5.5°' ^5.00 •N 4.535.5if5.04.54.00f20^40^60^80^100^120M (%)Figure 3.3: Second moment of protons in solid wood versus moisture con-tent for the lodgepole pine heartwood samples 1, 2, and 3.0Chapter 3. Second Moment and Cell Wall Water T2^33e.g. cellulose, hemicellulose or lignin, undergo more motion.The accuracy of the M2 depends on the estimation of Lo , the water signal amplitude.Since this NMR moisture content is in very good agreement with the ovendry moisturecontent, as shown in Fig. 3.4, we have confidence in the assignment of L o and the M2values. Furthermore, since we can measure moisture content accurately by 1 1-1 NMR ,0^20^40^60^80^100^120MFigure 3.4: Moisture content (ovendry method) versus NMR moisture con-tent of lodgepole pine heartwood. Linear fit slope is 0.95 andthe correlation coefficient is 0.992. The error bars are smallestfor the low moisture contents. Also shown are the fraction ofwater that is in the cell wall (triangles) and the fraction of wa-ter that is free in the lumens (squares), as calculated from thediscrete T2 distributions.all the water in the lodgepole pine samples must contribute to the isotropic signal withT2 greater than about 200/ts. This means that all the water molecules in the lodgepolepine samples reorient isotropically in the 1 H NMR time scale of 10 -5 s.Chapter 3. Second Moment and Cell Wall Water T2^ 34Figure 3.5 shows the NNLS discrete and smooth T2 distributions for water in lodge-pole pine at a moisture content of 11.3%. For all moisture contents, the components1^1^1^1^1 1^1^1^1^1-e-) 100 ^ -210 10 Ts ) 10^10 -12Figure 3.5: Typical discrete and smooth T2 distributions of water in lodge-pole pine at a moisture content of 11.3%. For the smooth distri-bution the area under the curve is 10 times the total moisturecontent.with T2 < 2.0 ins were assigned to cell wall water and those with T2 > 2.2 ms wereassigned to free water. Slight changes in this boundary between cell wall and lumenwater had little effect on the results. It is tempting to assign the two cell wall wa-ter peaks in the discrete T2 distribution in Fig. 3.5 to two individual cell wall watercomponents. However, while the sum of the two components increases monotonicallywith moisture content as shown in Fig. 3.4, their relative amplitude varied randomlywith moisture content and they were replaced by a single broad peak in the smooth T2distribution shown in Fig. 3.5. We therefore believe that cell wall water in lodgepolepine has a broad unimodal distribution of T2 times. Figure 3.4 shows the fraction ofwater assigned to the cell wall and the lumen as a function of moisture content. Abovea)8OE 64CD2Chapter 3. Second Moment and Cell Wall Water T2^ 35the FSP, the moisture content of the cell wall water fraction is the FSP and is foundto 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 fromabout 0.2 ms to a plateau value of about 1 ms as the moisture content goes abovethe FSP. A similar moisture content dependence has been seen for cellulose [Froix and1.2 ^1.0-0.8-0.6 -0.4-0.2—NNN 1 . 0 —0.8 —0.60.4 ——ics 0.21.0a)C.) 0.8 —0.6 —0.4 —0.2 .^0.0^023^-20^40^60^80^100^120M (%)Figure 3.6: The cell wall water T2 , averaged from the discrete T2 distribu-tions, versus moisture content for the lodgepole pine heartwoodsamples 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 notdistinguish more than one reservoir of cell wall water. This means that, if two cellChapter 3. Second Moment and Cell Wall Water T2^ 36wall water fractions exist in the lodgepole pine samples, ie. the water of hydrationand water of solution of the HH sorption theory, then they undergo exchange on atime scale faster than the T2 relaxation time. The self diffusion coefficient for cell wallwater in wood at moisture contents of 2 to 10% has been measured to be 1 x 10 -8 to13 x 10 -8 cm 2 /s [Stamm 1959]. Using R2 = 6Dr, we obtain a diffusion distance forwater in wood of about 80 to 280 A during 10 -5 s. Hence, diffusion much greater thanthe molecular scale of a few Angstr6ms prohibits any information on individual cell wallwater reservoirs by 1 H NMR .3.5 Concluding RemarksThis work indicates that, as water is added to lodgepole pine heartwood, the solid cellwall structure undergoes a loosening process for moisture contents above about 5%.This observation is commensurate with sorption theories which treat the first water tobe added to wood as water of hydration. However, our measurements also indicate thatwater of hydration cannot be distinguished from water of solution, since they undergoexchange on a time scale faster than the cell wall water T2 time. Furthermore, even atmoisture contents below 5%, the water molecules in the cell wall undergo rapid isotropicreorientation on the 10 -5 s time scale, which enables the water signal to be separatedfrom the solid signal to give an accurate NMR moisture content. The NMR FSP valueagreed with the FSP calculated from the extrapolation of the HH isotherm model to100% humidity.Chapter 4T2 Techniques for Characterization of Water in Wood 14.1 SummaryTwo new proton magnetic resonance techniques, relaxation spectra and relaxation se-lective imaging, have been used to investigate the distribution of water in samples ofnormal white spruce sapwood, heartwood, and juvenile wood as well as two rehydratedheartwood samples containing incipient decay and compression wood respectively. It isdemonstrated that the spin-spin (T2) relaxation behavior in wood is best presented as acontinuous spectrum of relaxation times. Spectra of T2 for white spruce show separatepeaks corresponding to the different water environments. Cell wall water gives a peakwith a T2 value of about 1 ms and lumen water gives a distribution of T2 values inthe range of 10 to 100 ms. The lumen water T2 value is a function of the wood cellradius. Consequently, the different cell lumen radii distributions for spruce sapwood,juvenile wood, and compression wood are readily distinguishable by the shape of theirT2 spectra. Water environments which are separable on a T2 spectrum may be imagedseparately. Imaging has been carried out in one dimension for cell wall water and lu-men water of a spruce sapwood sample at four different moisture contents ranging from100% to 17%. For the first time, we demonstrate that above the fibre saturation pointthe moisture density profile of the cell wall water is largely independent of moisturecontent. The feasibility and utility of using these techniques for internal scanning oflogs and lumber is discussed. These techniques should provide new insights into the1 This chapter closely follows a previously published paper. [Araujo of a1.1992]37Chapter 4. T2 Techniques for Characterization of Water in Wood^38wood drying process.4.2 IntroductionThe 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 characteriza-tion of water in wood. A technique for the analysis of multi—exponential relaxation interms of a continuous distribution of relaxation times [Whittall and MacKay 1989] hasbeen applied here to T2 studies of water in spruce. Because T2 values are a function ofcell size [Brownstein and Tarr 1979] and wood generally possesses a continuous distri-bution of cell sizes, this approach is more appropriate for the study of T2 relaxation inwood 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 whitespruce sapwood, heartwood, and juvenile wood as well as a rehydrated white spruceheartwood sample containing compression wood and incipient decay.A technique of selectively imaging water environments on the basis of T2 is demon-strated. One—dimensional images of the cell wall water and the lumen water in whitespruce sapwood have been obtained separately at a range of moisture contents from100% to 17%. This method of imaging cell wall and lumen water separately at a seriesof moisture contents should provide new insight into the mechanisms of wood drying.4.3 Materials and Methods4.3.1 SamplesSamples of sapwood, heartwood, and juvenile wood were taken from a log slice of normalwhite spruce. White spruce heartwood samples, one normal, one containing incipientdecay (indicated by discolouration on the sample surface), and another containingChapter 4. T2 Techniques for Characterization of Water in Wood^39compression wood, were taken from a block of wood containing all three types. Thisblock was rehydrated with distilled water by two cycles of a vacuum pressure treatmentat 39 mm of mercury for 2 hours, followed by a water pressure treatment at 100 psifor 2 hours, and then stored for one month in distilled water. The sapwood samplewas cut to 0.4 x 0.4 x 1 cm in order to fit in the 9 mm ID cylindrical coil used forthe one—dimensional imaging studies. All other samples were cut to dimensions of0.5 x 0.5 x 1 cm, and were analysed in a slightly larger coil. For all samples, the longaxis 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 to5 minutes. The lowest moisture content (17%) of the sapwood sample was achievedby leaving the sample exposed to the atmosphere for 1 hour after being dried in thevacuum oven for 2 minutes. All samples were allowed to equilibrate for 30 minutesbefore being analysed at 24°C.4.3.2 One—dimensional ImagingOne—dimensional imaging was obtained using a constant 19.4 gauss/cm field gradientalong the direction of the main magnetic field. The CPMG pulse sequence was usedwith data collection starting at the top of the last echo (See Appendix A). The Fouriertransform of the FID signal following the last CPMG echo corresponds to a one—dimensional projection of the moisture density distribution in the wood. The lengthof the CPMG train was varied so that data collection started at either a short timeTs = 400/ts or long time = 6Orns after the 90° pulse. In general, 200 scans wereaccumulated except for the lowest moisture content image (17%) where 600 scans wereaveraged to increase the signal to noise ratio.Chapter 4. T2 Techniques for Characterization of Water in Wood^404.4 Results and Discussion4.4.1 Fits of T2 Relaxation Data for White SpruceFor water in a cylindrical lumen, where the T2 relaxation occurs mainly at the lumensurface, it has been shown [Brownstein and Tarr 1979] that the self diffusion of thewater molecules to the lumen surface determines the relaxation behavior. The exactsolution to this problem [Brownstein and Tarr 1979] consists of a sum of exponentialmodes giving a series of T2 values, where most of the intensity can be attributed tothe first mode of this series. This diffusion model predicts that for a lumen with smallradius, the water molecules diffuse to the surface rapidly on the 1 11 NMR time scale ofa few milliseconds and the T2 relaxation time is proportional to the volume to surfaceratio, which scales as the lumen radius. Also, for a lumen with a larger radius, the T2relaxation time is determined by the rate at which lumen water molecules diffuse tothe lumen surface and T2 scales as the square of the radius. Since the 1 1-I NMR signalis proportional to the number of protons, the distribution of T2 relaxation times forlumen 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 conventionalinterpretation in terms of a fixed number of discrete components.Previously, the CPMG curve for water in wood has been represented by discrete T2components 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 ofcomponents is unrestricted. Figure 4.2 shows a fit of this continuous T2 spectrum tothe CPMG data. The chi squared misfit (x 2 ) is excellent and the value of X 2 /N = 1.1where the expected value is x 2 /N = 1.0, [Whittall and MacKay 1989] indicating thatnoise has not been incorporated into the fit. Figure 4.3 shows the continuous T2 spectraChapter 4. T2 Techniques for Characterization of Water in Wood^410.207)'t 0.15a)a0.20JL)0 ^-0.10 I 0.10 -Za.) a)15 0.05 - "O 0.05 -a) a,C1)^0.15 b a)^0.15 dO O0.10 - > 0.10 -■1) aca0.05 - Tv. 0.05-Ca0.00 0.000^5^10^5^20^25^0^5^10^15^20^25Lumen Radius (/Lm) Lumen Radius (gm)Figure 4.1: Relative numbers of a) cells, and b) protons as a function of celllumen radius for the white spruce sapwood sample, and c) cells,and d) protons for the white spruce heartwood compressionwood sample. Estimated from Scanning Electron Micrographs.. 1 0. 0 0 .05 .25 .30.15^.2010 21 0 0Ns‘210 4^102^102^10°T2 (S)Chapter 4. T2 Techniques for Characterization of Water in Wood^42Time (s)Figure 4.2: CPMG data and the fit from a continuous T2 spectrum for thewhite spruce sapwood sample at 100% moisture content.Figure 4.3: T2 results as a continuous curve and as a discrete fit of 2 ex-ponential components (spikes) for the white spruce sapwoodsample at moisture contents of a) 100%, b) 86%, c) 59%, andd) 17%. (Amplitudes of the discrete fit have been divided bya factor of 10).Chapter 4. T2 Techniques for Characterization of Water in Wood^43of 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 approximatelyto the cell wall and the lumen water, except for the lowest moisture content where thesingle component corresponds to cell wall water alone. The discrete component fitswere 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 anatomicallocation of water in wood. The peak corresponding to the lowest T2 values is relativelyconstant at higher moisture contents, but decreases when the moisture content is belowthe fibre saturation point (FSP) of about 30%. This peak is therefore associated withthe water in the cell walls. Above the FSP the two peaks corresponding to the higherT2 values decrease in amplitude with moisture content, and are associated with the celllumen water. The continuous T2 spectrum for 17% MC actually shows 0.3% MC oflumen water at about 100 ins.Figure 4.4 shows the continuous T2 spectra of the juvenile wood and heartwoodsamples, which were taken from the same log slice as the sapwood sample. The moisturecontents, as measured from a FID, of the freshly cut samples of juvenile wood andheartwood were 29% and 30% respectively. These T2 spectra show that practically allthe water is cell wall water. The centres of the cell wall water T2 peaks for heartwoodand 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 theheartwood and juvenile wood. These correspond to the small amounts of lumen waterin the samples. The lower T2 value for the juvenile wood sample is likely a consequenceof the smaller lumen sizes in juvenile wood.Figure 4.5 shows the continuous T2 spectra of samples of heartwood, compressionwood, and heartwood with incipient decay at fully rehydrated moisture contents ofChapter 4. T2 Techniques for Characterization of Water in Wood^44Figure 4.4: Continuous T2 spectra of the water in white spruce juvenilewood (solid line) and heartwood (dashed line).178%, 66%, and 173% respectively, as measured from FIDs, such that all availablelumen space in the wood was occupied by water. The cell wall water peaks are allcentred near 1 ms, but the compression wood peak has a slightly lower value, whichsuggests 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 lessfor the compression wood sample than the other two heartwood samples, indicating thatcompression wood has fewer large cells. The cell distribution in Figure 4.1c shows thatcompression 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 ImagingTotal Water DistributionProton magnetic resonance is used here for one—dimensional imaging of water in thewood with a resolution of about 0.1 mm. Figure 4.6 is a one—dimensional image of the1 0°1010^ loT 2 (s )Chapter 4. Ti Techniques for Characterization of Water in Wood^45Figure 4.5: Continuous T2 spectra of the water in rehydrated samples ofwhite spruce heartwood compression wood (solid line), heart-wood (long—clashed line), and heartwood with incipient decay(short—clashed line).water distribution in white spruce sapwood, at four moisture contents, measured bythe ovendry method, superimposed onto an SEM image of the sample. The range ofmoisture contents from 100% to 17% is representative of the range of moisture contentsin wood kiln drying for spruce wood. The sample was aligned with the growth ringboundaries perpendicular to the field gradient direction, so that the structure in theimage corresponds to the growth rings of the sample. A higher rate of drying at theedges of the sample gives rise to a rounding off of the profiles. The profiles are expressedin terms of moisture density, which is the mass of water per unit volume of wood. Thetotal area under the image profile is proportional to the mass of water. The moisturedensity can be calculated from the image profile using the moisture content, the ovendrymass of the wood, and the dimensions of the sample. The moisture density is alwaysless than 1 g/cm 3 , the density of free water, since the solid wood occupies some of thevolume.0.70.0-0.1-4 40.2MC0.5(.)CT)•••-•-' 0.40.3CD-2^ 0^ 2Position (mm)Chapter 4. T2 Techniques for Characterization of Water in Wood^46Figure 4.6: Radial Moisture Density Profile of the total water distributionin the white spruce sapwood sample at four moisture contentssuperimposed onto a SEM image of the sample.Chapter 4. T2 Techniques for Characterization of Water in Wood^47Figure 4.6 shows that at high moisture contents, moisture density is greater inthe earlywood side of each growth ring. This difference arises from the fact that theearlywood tracheid lumens can hold more water, because they are much larger thanthe 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% thedistribution has more water in the latewood side of each growth ring. The cell wallsare 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, whichhas a more uniform distribution, is practically all cell wall water (See Fig. 4.3).Cell Wall and Lumen Water ProfilesOne-dimensional images of the distribution of water in wood have previously beenseparated 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. Theimages are separated on the basis of the T2 distribution of the sample. Figure 4.3 showsthat in terms of T2 values, the continuous T2 spectra are grouped into two distinct parts.The lower T2 values are attributed to cell wall water and are represented by the lowestT2 component of the discrete component fit, Tr'. This lowest T2 spike also includesabout 4.6% MC of lumen water. The peaks with higher T2 values make up most of thelumen 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 ofCPMG sequences, one short sequence of length 7 -8 , and one long sequence of length Ti ,where Ts < Tfw < r1 . The image signals are represented byF(Ts) e-(T.,)/TP'metz Fcw e _( T,)/T2w (4.1)F(Ti)^Flumen e -(T1)/T2'"--"^(4.2)Chapter 4. 2'2 Techniques for Characterization of Water in Wood^48where 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 Tfw . Equa-tion (4.2) is the lumen water signal since the signal from the cell wall water is practicallyzero at this time, for r1 = 60 ms and T2 K' about 1 ms. A multiple of Eq. (4.1) sub-tracted 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. Abovethe fibre saturation point (FSP), this image should scale as the solid wood densitywhich is higher in the latewood [Menon et a1.1989]. It is not surprising that the cell0.6 ^—0.1I—4^ —2^ o 2Position (mm)Figure 4.7: Radial Moisture Density Profile of the cell wall water distri-bution in the white spruce sapwood sample at four moisturecontents superimposed onto a SEM image of the sample.4wall water profile is not a function of moisture content, above the FSP, but we believeChapter 4. T2 Techniques for Characterization of Water in Wood^49this is the first direct demonstration of this fact. It should be noted that while it is notobvious 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 isthe lower moisture content cell wall water image in which the latewood cell wall waterpeaks have dried out of the sample leaving an almost featureless distribution. Usingthe correlation of the higher MC cell wall water images to the SEM image, the totalwater 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^ 0^2^4Position (mm)Figure 4.8: Radial Moisture Density Profile of the lumen water distributionin the white spruce sapwood sample at four moisture contentssuperimposed onto a SEM image of the sample.image shows that on the earlywood side of each growth ring the moisture density ismuch higher, and the earlywood lumen water leaves at a higher rate than the latewoodChapter 4. T2 Techniques for Characterization of Water in Wood^50lumen water. The lowest moisture content image shows a component in the centreof the image indicating the presence of a small amount of lumen water, which waspredicted by the continuous T2 spectrum and shown to have a T2 of 100 ms and amoisture content of 0.3%.Potential of 1 H NMR for Internal Scanning of White SpruceThe results of this work enable us to assess the potential of proton magnetic resonancemethods for internal scanning of normal and abnormal white spruce. Normal sapwoodcould be readily distinguished from normal heartwood and juvenile wood since the lattertwo contain almost no lumen water. Incipient decay seems to be indistinguishable fromnormal wood. However, compression wood is readily distinguishable from normal woodwhen the lumens are hydrated, because of the very different size distribution of the celllumen radii.We are also able to compare our studies on white spruce with former work on westernred cedar [Flibotte et a/.1990] in order to assess the potential value of 1 1-/ NMR forspecies differentiation. Several differences between the two species are evident. TheMC of green white spruce sapwood is much lower than that of green western red cedarsapwood and this difference is readily measurable by 1 H NMR. The T2 values for waterin spruce lumens were generally lower than those for water in cedar lumens, reflectingsmaller cell lumen sizes in the spruce. The T2 value for cell wall water in spruce (aboutlms) was similar for sapwood, heartwood, and juvenile wood, but for cedar the T2value for cell wall water in the sapwood (about 4 to 7 ins) was a factor of two or moregreater 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 forspecies differentiation in the sapwood.This study was carried out on a solid state 1 H NMR spectrometer on small woodChapter 4. T2 Techniques for Characterization of Water in Wood^51samples (smaller than 1 cm3). For the potential application of these techniques tolumber and whole logs, large bore magnetic resonance imaging (MRI) facilities wouldbe required. In fact several wood MRI studies have already been carried out [Halland Rajanayagm 1986, Wang and Chang 1986, Flibotte et a1.1990]. MRI facilities arecurrently 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 oncurrent large bore MRI facilities.The present study employed a one—dimensional imaging technique to produce mois-ture 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 andwhole logs. In principle, the small sample work carried out here could have been ex-tended to two dimensions with the addition of two more orthogonal gradient coils forslice selection and phase encoding. However, the practical consideration of applying thegradients sufficiently fast to select a slice, and to impose the spacial encoding makes itimpossible, so far, to produce a two—dimensional image of the cell wall water in whitespruce.4.5 Concluding RemarksOur goal in this work has been to demonstrate the application of new proton magneticresonance techniques for the study of water in wood. The discovery that T2 relaxationtimes of lumen water can be related to lumen size has considerably expanded the utilityof 1 H NMR for wood studies. The fact that there exists a broad range of T2 valuescorresponding to a range of water environments in wood requires that the T2 relaxationmeasurements 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 wellChapter 4. T2 Techniques for Characterization of Water in Wood^52represented as a continuous spectrum consisting of a munber of peaks. The area undereach peak corresponds to the amount of moisture in a particular water environmentand the T2 value indicates the nature of the environment. We have learned, from ourexperience with white spruce and western red cedar [Merlon et a/.1987, 1989], that T2values below about 10 Ins correspond to cell wall water. Peaks with T2 values greaterthan 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 thewood sample.Once the distribution of water environments in a wood sample is defined by theT2 spectrum, it is possible to derive the spatial distribution of each water environmentby selectively imaging each component. In the present study, we have done this inone dimension for the cell wall and the lumen water in white spruce at four moisturecontents. We believe that this type of study will be valuable in the detailed investigationof the wood drying process [Quick et a/.1990].Chapter 5Diffusion Model of Compartmentalized Water 15.1 SummarySpin—spin relaxation measurements were carried out on water in redwood sapwood,white spruce sapwood and white spruce compression wood samples and interpretedusing a theory which modeled the lumen water T2 relaxation in terms of the cell radiusdistribution, the bulk water diffusion coefficient and a surface relaxation parameter.The three samples possessed different cell lumen radius distributions as measured byscanning electron microscopy. For the redwood sample, 1 1-I NMR measurements weremade for 7 temperatures between 4 and 55°C over which the average lumen water 7'2decreased from 174 to 105 ins. For all measurements, agreement between theory andexperiment was excellent. In the slow diffusion regime, the theory predicted higherorder T2 relaxation modes. Experimental evidence was found for the existence of theserelaxation modes in the redwood results at low temperatures. Using this diffusionmodel, T2 relaxation decay data was fit to give a cell size distribution, comparable toscanning electron microscope results, when the bulk water diffusion coefficient and thesurface relaxation parameter were known.1 Part of this chapter closely follows, with some modifications, part of a previously published paper.[Araujo et a/.1993]53Chapter 5. Diffusion Model of Compartmentalized Water^ 545.2 T2 Relaxation of Compartmentalized Water: Lumen Water in Wood5.2.1 IntroductionThe primary goal of this section is to explain quantitatively the spin—spin relaxationmechanism for wood cell lumen water in terms of a model in which the dominant sourceof relaxation occurs at the cell wall surface. The effectiveness of this surface relaxationin decreasing the lumen water T2 is determined by the self diffusion rate of the lumenwater. We use the classical diffusion model presented by Brownstein and Tarr [1979]and experimentally verify the dependence of the spin—spin relaxation upon the diffusioncoefficient of water and upon the radius of the cell lumen. We also consider the effectof 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 thatin the presence of multiexponential relaxation, the T2 times calculated from the relax-ation decay curve depend strongly upon the algorithm used for T2 estimation. Thedegree to which spin—spin relaxation measurements can provide meaningful informa-tion about the sample depends crucially upon how the physical mechanism for T2 isincorporated into the relaxation decay curve analysis.5.2.2 TheoryDiffusion—Bloch Equation and Boundary ConditionsWe define m(r, t) as the magnetization in the cell lumen at the position r and timet. Then m(r, t) satisfies the diffusion—Bloch equation with a volume relaxation T2 f „ e[Brownstein and Tarr 1979]07n (r , ^t)(r, t) = D V2m(r, t)^ (5.1)eft 7-72 reeChapter 5. Diffusion Model of Compartmentalized Water^ 55where D is the diffusion coefficient of bulk water. The relaxation at the cell wall hasbeen expressed by Brownstein and Tarr [1979] as a boundary condition restricting theflux (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 relax-ation. 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 definedto be proportional to the temperature difference across the boundary. The solution tothis 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=owhere Fri (r) are orthogonal eigenfunctions.For infinitely long cylindrical geometry, with no z or 6 dependence, the boundarycondition of Eq. (5.2) becomes— D aOm  (r,t)I s = M m(r, t)and the solution in terms of Bessel functions is,Fn (r) = Jo ri),1^D7/^171 'Ti2(„)^R2^712f„ ewhere R is the cylinder radius. The boundary condition Eq. (5.4) defines ra n fromTb1L(70 _ MRJo( 7111)^D •There exists a series of solutions to Eq. (5.7). The amplitudes A,, are determinedby the initial condition of constant magnetization throughout the lumen, immediately(5.4)(5.5)(5.6)(5.7)Chapter 5. Diffusion Model of Compartmentalized Water^ 56following the 90° pulse of a CPMG sequence. That is,001m(r ,0) rno = E A„Jo—R ) •Using the orthogonality of Bessel functions givesAn = mo  f Cell J0( 71nr R)d7-fCell J0( 7,„7. I r ) d,'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 termstends 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 cellwhere^M(0) E^/T2(n)./14(t) =^m(r, t)drCell^ n=o(5.10)=^)^4^1A„^f^(71„1'(5.11)M(0) Cell^R, )^714 (1 + (Thi„IMR) 2 ) •Figure 5.2 shows In versus MR./D. The average values of MR/D for the samples usedin 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 oneand ten, and much greater than ten, respectively. For the fast diffusion regime, onlyone mode, T2(0) , is required to describe lumen water relaxation; for the intermediate orslow regimes, a sum of modes, mm = 0, 1, 2... is necessary.Diffusion Model Applied to WoodThis diffusion model expresses the relaxation of lumen water in wood in terms ofthe radius distribution, T2f „, of the bulk water, the diffusion coefficient D of the bulkwater, and the surface relaxation parameter M. Figure 5.3 shows that the fundamentaln=0(5.8)(5.9)FChapter 5. Diffusion Model of Compartmentalized Water^ 57Ammon iT 20 ^0 RFigure 5.1: The first 3 A„Jo (q„r/ R) modes. The clashed line is the sumof the first 3 modes which is close to the initial condition ofconstant magnetization, mo.Chapter 5. Diffusion Model of Compartmentalized Water^ 581.00.9 —0.8 —0.7 —0.6 —=0.5 —0.4 _0.3 —0.2 —0.1 —0.0• 0Q.0OO0000(/1n =0E 00 0cn 00a) a) -0C.)CL.C/)a) a) C_)^(-)0 0LE O cp.LC)LI)n =1n =250.00 1^ 1.0MR/D10.0IIIFigure 5.2: I7  amplitudes as a function of MR/D. Single to multiple moderange is covered by samples presented. The average R is usedfor each sample.---- 1 0 0-=-o0^-11 0 =_107-2,10 -3^^100^10^10^10Cylindrical Cell Radius (um)Chapter 5. Diffusion Model of Compartmentalized Water^ 59relaxation mode T2 (0) is proportional to R for small radii (< lOpm) and proportionalto R2 for larger radii (> 100pm). The higher order modes T2(1), T2 (2) exhibit R2dependence for radii less than 100,um. T2free limits the T2 for all modes for radii largerthan 100,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 x10 -4m/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 funda-mental mode T2 (0) depends upon the diffusion coefficient. For the temperature range4 to 55 ° C, the free water diffusion coefficient varies from 1.1 x 10' to 4.4 x 10 -9m2 /s[Simpson and Carr 1958], predicting a factor of 1.6 decrease in the fundamental modeT2 (0). Figure 5.5 shows how the T2(„ ) modes depend on M using the diffusion coeffi-cient of room temperature bulk water and the average radius for redwood sapwood of30,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^ 600.40.3—n=00 0.2—---0.1—n=1^.--_____^--^n=20.0 i^I I^i^I0^1 2^3 4 5^6Diffusion Constant D (10 -9m 2/s)Figure 5.4: The diffusion model gives T2(„ ) versus diffusion coefficient(short dashed) and the limiting effect of a finite T2 free (solid)(M = 1.40 x 10 -4m/s, R = 30/GM, T2f,.„ = 1.4s).10 1T2 free-oo10n=0n=1n=210 -310 -6^10-5^10^10 -3Surface 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 -9m2/s, R = 30/tm, T2free = 1.4s).Chapter 5. Diffusion Model of Compartmentalized Water^ 61Wood Cell Geometry DependenceThe theory presented above applies for cylindrical lumens but wood cell lumens areoften shaped like rounded squares or rectangles in cross-section. We therefore considerthe effect of cell geometry on 7'2 (See Appendix B). For a square or rectangular lumencross-section, Eqs. (5.1) and (5.2) are expressed in cartesian coordinates and the solu-tion, Eq. (5.3), involves cosine rather than Bessel functions. We consider a cell witha cross-sectional area of 2826,um2 , which for circular geometry implies a diameter of60.0pm, and for square geometry implies 53.2,am sides. Table 5.1 presents the relax-Table 5.1: Geometric dependence of the fundamental T2 mode, T2(o ) (for a fixed surfacesink parameter, M), and of M (for a fixed T2 (o )), for cells with a cross-sectional areaof 2826/tm 2 (T2free = 1.4s and D = 2.2 x 10 -9m 2 /s).Geometry M = 1.40 x 10 -4m/s T2(o) = 0.147 sCircle T2(o) = 0.147 s M = 1.40 x 10 -4m/sSquare T2(o) = 0.137 s M = 1.24 x 10 -4m/sDecrease 7% 11%ation time of the fundamental T2 mode, T2 (o) , given that the surface sink parameterM = 1.40 x 10 -4m/s, or alternately presents the estimated surface sink parameter giventhat T2 (0) = 0.147s. We note from Table 5.1 that the effect of square cell geometry isan 11% decrease in M for fixed T2 or a 7% decrease in T2 for fixed M. Lumen water T2values are decreased because the minimum distance to a wall is shorter for the squarethan the circle. Since the differences between the two geometries are quite small andcould be accounted for by a small change in M, we have chosen to treat all wood celllumens as cylinders.Chapter 5. Diffusion Model of Compartmentalized Water^ 625.2.3 Materials and MethodsSamplesSamples of sapwood were cut from log slices of redwood, and white spruce. A com-pression wood sample of white spruce heartwood was cut from a 10 x 5 x 3 cm blockof wood which was rehydrated with distilled water by two cycles of a vacuum pressuretreatment at 39 mm of mercury for 2 hours, followed by a water pressure treatmentat 100 psi for 2 hours, and then stored for one month in distilled water. The sprucesamples were analysed at 26°C, and a range of temperatures was examined from 4to 55°C for the redwood sample. Maximum hydration was maintained by soaking indistilled water between NMR measurements.In the CPMG sequence, a T of 200/ts was used for the redwood and white sprucecompression wood samples, and a longer T of 400fis was employed for the white sprucesapwood since a longer cell wall water T2 was detected.5.2.4 Results and DiscussionScanning electron microscopy was used to estimate the distribution of cell lumen radiifor each sample. In Fig. 5.6 these distributions are displayed in terms of relative volumewhich is proportional to the 1 1-I NMR signal from each cell lumen. To ensure reasonablestatistics, at least 1000 cells were counted for each distribution.Spin—spin relaxation decay curves were obtained for the redwood sapwood sampleat 7 temperatures between 4 and 55°C. A representative CPMG decay curve obtainedat 18°C is plotted in Fig. 5.7. We note from Fig. 5.7 that the CPMG curve can not befit by a single exponential component, in fact the logarithmic plot exhibits a curve ofalmost continuously varying slope. In Fig. 5.8, smooth T2 relaxation plots are shownfor the redwood sample at each of the 7 temperatures. Amplitude weighted averageChapter 5. Diffusion Model of Compartmentalized Water^ 63baJQ)1:41,„ 0 10 20 30 4:0 50Radius Gu,n-i)Figure 5.6: Radius distributions in terms of relative volume in cells ac-quired from SEM images for a) redwood sapwood, b) whitespruce sapwood, and c) white spruce heartwood compressionwood samples.E1^i^i0.2 0.4 0.6^0.8Time (s)1 .0Chapter 5. Diffusion Model of Compartmentalized Water^ 64Figure 5.7: The CPMG decay curve of redwood at 18°C, the decay curve ofthe lumen water predicted by the diffusion model (solid line),and the smooth T2 fit of the CPMG data (dashed line). The in-set shows the cell wall water component, at short times, whichis not included in the diffusion model. The inset uses a linearamplitude scale.Chapter 5. Diffusion Model of Compartmentalized Water^ 654.0 °C\\I11.0 °C----% '18.0°C26.5°C,-- ,34.0°C42.0 °C- ,55.0 °Cle 10 -4 1T2 (s)-3 -2 -10^10^10T2 (s)Figure 5.8: Diffusion model T2 relaxation plots (solid) and smooth T2 so-lution (dashed) are shown for a range of temperatures for theredwood sample.Chapter 5. Diffusion Model of Compartmentalized Water^ 66lumen water T2 times for each temperature are displayed in Fig. 5.9.0.180.16(/)N 0.140)0 0.12 -<t0.10 -0x0900.080^10 20 30 40 50 60Temperature (°C)Figure 5.9: Average T2 of lumen water for the model (crosses) and thesmooth T2 solution (squares) for the redwood sample.The variation of the lumen water diffusion coefficient for free water from 1.1 x 10 -9to 4.4 x 10 -9m 2 /s between 4 and 55°C is reflected in the shift to shorter T2 at highertemperatures in Figs. 5.8 and 5.9. Figure 5.10 shows the smooth T2 relaxation plotsfor spruce sapwood and spruce compression wood.The objective of this study was to show that the relaxation of lumen water in woodis well described by the Brownstein and Tarr diffusion model presented in the theorysection. Several issues must be considered before we compare experiment and theory.Chapter 5. Diffusion Model of Compartmentalized Water^ 67a/^i ^tt/ - V^ /^t/^\ i t/ \ t/ \^/^t/-\\/ \\^/10-4^10 -3^10-2^10-1^10°T 2 (s)Figure 5.10: Diffusion model 772 plots (solid) from the corresponding radiusdistributions and the smooth T2 solutions (clashed) for a) whitespruce sapwood, and b) rehydrated white spruce heartwoodcompression wood samples.Chapter 5. Diffusion Model of Compartmentalized Water^ 68Radius DistributionRadius distributions obtained by scanning electron microscopy are relevant to lumenwater 1 H NMR measurements only if all wood cell lumens are filled with water. Wetherefore compared our 1 1-1 NMR measurements of moisture content with that predictedfrom the SEM radius distributions using Eq. (2.1) . One should not expect perfectagreement since the SEM distributions were derived from small surface regions of thesamples whereas the 1 1-1 NMR measurements originate from the entire volume. For theredwood, the average measured MC was 310 + 60% while the SEM analysis predicted316% for a fully hydrated sample. For the spruce, the measured MC's were 111 and75% compared to SEM estimates of 100 and 66% for the sapwood and compressionwood samples respectively. We are therefore confident that Fig. 5.6 is representativeof the true cell radius distributions.Separating Signals of Lumen and Cell Wall WaterOur analysis must distinguish the lumen water 1 H NMR signal from the cell wall watersignal. We note in Figs. 5.8 and 5.10 that the peaks on the left side of each plot may beassigned to the cell wall water and that there is practically no overlap with the lumenwater signal assigned to higher T2 's. Therefore, when fitting our lumen water T2 modelto the CPMG relaxation decay curves, we used only those data at times greater thana cutoff time where the contribution to the signal from cell wall water should havedecayed by a factor of 50 or to the noise level. We note on Fig. 5.7 that the cell wallwater signal from redwood at 18°C decayed to the noise level well before 20 ms. Forthe redwood sample, we used a cutoff of 20 ins for measurements acquired below 34°Cand 60 ins for those acquired at and above 34°C. A cutoff of 30 ms was used for thewhite spruce samples.Chapter 5. Diffusion Model of Compartmentalized Water^ 69To check the cell wall water cutoff times, a series of lumen water fits, which arediscussed later, were carried out using a decreasing number of data points, N, cor-responding to increasing the start time from the first echo. The value of X2 /N wasmonitored as a function of N. As N decreased, the value of x2 /N was initially large,then it decreased to a limiting value as contributions from the cell wall water were leftout of the fit. For starting times later than the cell wall water cutoff times, the valueof x2 /N was independent of N for all samples.Fitting the lumen water relaxation decay curvesSmooth T2 relaxation models generally provide excellent fits to relaxation decay curvesfrom multiexponential CPMG decay curves [Whittall and MacKay 1989, Kroeker andHenkelman 1986] (and see Chapter 4); much better than discrete fits to one, two orthree exponential components. However there are two problems with using smooth T2relaxation 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 ofT2 plots with different shapes which will fit the relaxation decay curves almost equallywell 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 inwood. Although the shapes of the lumen water T2 plots reflect the wood cell distri-bution, they are complicated by the existence of a complex functional dependence onthe radius (Fig. 5.3) and the existence of the T2 modes expressed by Eq. (5.6). There-fore, quantitative comparisons between theoretical models and experimental results ina multiexponential spin—spin relaxation study should be made with the relaxation de-cay curve rather than with either T2 times or T2 relaxation plots which are both biasedby the algorithm used for 7'2 calculation.Using the Brownstein and Tarr model [1979] of Eq. (5.10), we obtained fits toChapter 5. Diffusion Model of Compartmentalized Water^ 70the redwood and spruce CPMG decay curves equivalent to those from the smoothT2 model but which contain more explicit information about the system. The radiusdistribution from Fig. 5.6 and the diffusion coefficient of free water at the appropriatetemperature were used in these fits. Note that there are two sums: One over the radiusdistribution and the other over the T2 modes of the solution. Only the first 4 modeswere 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 startingvertices were used to find the global minimum. The X 2 of the fit with the Brownsteinand Tarr model must necessarily be worse than or equal to that for the best smoothT2 model since the diffusion model fitting function was also made up of exponentialterms but had to be consistent with the known radius distribution. There were fourvariable parameters in the diffusion model: The total amplitude and the baseline offsetof the CPMG data, the free water relaxation time T2 f ree and the surface sink M. Theamplitude was well defined by the fitting procedure. The baseline offset was relativelysmall (less than 2% of the maximum amplitude) and was correlated with T2 f r ee• T2 freewas limited to values below 3.0 s because longer values did not affect the fit.Redwood Lumen WaterFigure 5.11 shows the T2 plot predicted for the redwood sample at 4°C using theBrownstein and Tarr model expressed in Eqs. (5.10) and (5.11), which used the radiusdistribution in Fig. 5.6a and the parameters from the best fit of the diffusion model tothe CPMG decay curve. Of all the wood samples studied, this redwood sample hadthe 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 isthe sum of the first three modes. The inset shows the second and third modes asthe dashed and solid curves, respectively. We note that the sum coincides with the10 -2 T2 (s)Chapter 5. Diffusion Model of Compartmentalized Water^ 71Figure 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 inthe inset. The sharp undulations result from the steps in thehistogram of cell sizes.Chapter 5. Diffusion Model of Compartmentalized Water^ 72fundamental mode at longer T2 but deviates for T2 less than 70 ms where there issubstantial contribution from the higher modes. The sharp undulations in the T2 plotreflect 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 CPMGTable 5.2: Fit of the diffusion model to the CPMG decay curves for the redwood sample.Temp.(°C)Four Modes One Mode Ratio x 2 'sM(m/s) T2free ( s ) x 2 /N M(m/s) T2free(s) x2/N4.0 1.38 x 10 -4 1.1 1.86 2.36 x 10 -4 2.9 13.2 7.111.0 1.55 x 10 -4 1.7 1.87 1.96 x 10' 3.0 19.1 10.218.0 1.66 x 10' 3.0 1.99 1.79 x 10 -4 3.0 23.3 11.726.5 1.43 x 10 -4 1.4 1.66 1.63 x 10' 3.0 6.17 3.734.0 1.52 x 10 -4 1.8 1.82 1.57 x 10' 2.2 1.97 1.142.0 1.58 x 10' 1.6 2.03 1.58 x 10 -4 1.6 2.14 1.155.0 1.78 x 10' 2.5 2.80 1.81 x 10 -4 2.9 2.78 1.0decay curves for the redwood sapwood at the 7 temperatures. For these fits x 2 /N wasaround 2 which is somewhat larger than the mean of 1 expected for a x 2 random variablewith N degrees of freedom. We note that the smooth T2 plots yielded X 2 /N values ofmean near 1.3. We believe that this increase from 1.3 to 2 is to be expected consideringthat the diffusion model contained the relatively crude SEM radius histogram obtainedfrom sample surfaces. The importance of including the higher order modes in theCPMG fitting function is demonstrated by the increase in X 2 /N by up to a factor of 12for the lower temperature samples when only the first mode, T 2 ( 0), was included in thefit. Above 26°C, the increase in x 2 /N for the single mode fit was not so pronounced sincethe larger diffusion rate moves the sample closer to the fast diffusion limit where higherorder modes are negligible. Therefore, the fit of the four mode solution is obviouslybetter than the one mode solution, and we believe these results strongly support theexistence of the higher order T2 modes.Chapter 5. Diffusion Model of Compartmentalized Water^ 73In Fig. 5.8, the Brownstein and Tarr model T2 plots (solid lines) are compared withthe smooth T2 plots obtained from the CPMG data (dashed lines). The two lumenwater T2 curves look qualitatively similar but quantitatively are quite different. Thestructure of the smooth T2 solution tends toward smoothed but localized peaks. Theaverage lumen water T2 calculated from each curve are compared in Fig. 5.9 and showgood agreement over almost a factor of 4 in MR/D in the intermediate regime wherethe higher modes are significant.Spruce SamplesThe white spruce sapwood and compression wood samples were chosen because theyhave 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 cellsizes. 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 CPMGTable 5.3: White Spruce ResultsSapwood CompressionM (XT2freeX 2 /N model fitx 2 /N smoothAve. T2 model fitAve. T2 smooth1.371.6 s0.90.857 Ins59 ins1.262.5 s3.01.440 ins33 msdata. While the x 2 /N value for the diffusion model fit to the spruce sapwood datawas close to that from the smooth T2 solution, the x 2 /N value for the model fit to thespruce compression wood data increased by a factor of 2. In Fig. 5.10, we note thatthe smooth solution for the compression wood sample is substantially broader thanChapter 5. Diffusion Model of Compartmentalized Water^ 74the model T2 plot. We do not know the origin of this discrepancy but speculate thateither the SEM radius distribution used for the fit was not representative of the wholesample, or that the surface sink parameter, M, was a function of wood cell size.5.2.5 Concluding RemarksA four parameter Brownstein and Tarr diffusion model, which incorporated the knownlumen radius distribution, yielded fits to the CPMG decay curves with comparablex2 /N values to those obtained using a smooth T2 model which had no constraintsother than smoothness of the T2 distribution. The different wood samples and experi-mental conditions provided an excellent comparison between theory and experiment fora wide range of values of R and D and the dimensionless parameter MR/D. The ratioMR/D changes the nature of the solutions and the values of R and D influence theT2 and amplitudes. We therefore believe that the Brownstein and Tarr diffusion modeladequately accounts for experimental measurements of T2 relaxation of lumen waterin wood. To our knowledge, this is the first demonstration of the presence of higherorder relaxation modes in a biological system; modes have been observed previously ina model system of cylindrical capillary fibres [Bronskill et a/.1990].We now believe that we are justified in using this diffusion model to estimate theradius distribution directly from the CPMG decay curves [Whittall 1991]. Such aprocedure 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 muchand is between one and three seconds for wood samples at ambient temperature. Thevalues for /V/ found in this study did not vary greatly from sample to sample. In thefollowing chapter, a physical interpretation of M is introduced, relating it to the cellwall water T2 and the cell wall water diffusion coefficient.In a theoretical paper, Belton and Hills [1987] solved a similar diffusion model,Chapter 5. Diffusion Model of Compartmentalized Water^ 75describing spacially separated regions with diffusive exchange, using the Laplace trans-form method to give the analytical i ll NMR spectra solutions. They considered onedimensional geometry, allowing for regions of differing resonance frequency, bulk trans-verse relaxation rate and diffusion coefficient in a variety of examples, to model het-erogeneous and biological systems.Relaxation models based upon water diffusion have recently been applied to 1 H NMRstudies of porous media including spin—lattice (T 1 ) relaxation in rocks [Davies andPacker 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] imple-mented the Brownstein and Tarr diffusion model in order to relate distributions of T 1 tothe 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 rateas representing a weighted average over the volume traversed by a water moleculeover the period of the 1 H NMR experiment. This model, which assumes fast diffusioninside the pore but slower diffusion from pore to pore, yields an average surface to vol-ume ratio which corresponds to a pore size distribution only if water diffuses throughhomogeneous 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, therelaxation time of water in porous media was proportional to the volume to surfaceratio of the system and in the slow diffusion limit the relaxation time was related to apore dimension.There are a number of fundamental differences between porous materials and woodwhich influence the 1 H NMR relaxation of lumen water. Firstly, in most porous ma-terials water is not restricted to a lumen of known geometry but is free to diffuse intorturous paths from pore to pore throughout the structure. Secondly, in rocks, spin—spin relaxation measurements are very difficult to carry out quantitatively presumablyChapter 5. Diffusion Model of Compartmentalized Water^ 76due to signal losses caused by water diffusion in magnetic susceptibility induced fieldgradients; hence most studies have considered T 1 rather than T2. Such susceptibilityinduced signal losses are negligible in water filled wood samples. Finally, the mecha-nism of the surface relaxation sink in porous media is different from that in wood sincewater cannot penetrate the cell walls of rocks or glass.We believe that this work is of more general application than to lumen water inwood. We have defined the role of lumen water diffusion in the spin—spin relaxationprocess. Wood is a particularly simple biological system because the diffusion processslows down by at least one order of magnitude at the cell wall. In many other biologicalsamples, for example, human tissue, the barrier to diffusion outside the cell lumen maybe much smaller; requiring a more sophisticated model for spin—spin relaxation of lumenwater. Since many diseases involve specific processes at the cellular level, the diagnosticcapability 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^ 775.3 Recovering Compartment Sizes from NMR Relaxation DataThe objective of this section is to obtain a quantitative measurement of wood celldimensions of order 10 microns and larger using the Brownstein and Tarr diffusionmodel presented in the first part of this chapter. For a distribution of cylindrical cells,the T2 relaxation decay curve isooy(t) = f c(R)71 -R2 h pM(t , R) dR (5.12)where c(R) is the number of cells with radius R, M(t,R) is the multiexponential decaygiven by Eq. (5.10), and 7-R2 hp converts from number of cells to signal intensity usingthe assumed constant cylinder height h. and signal density p. The inverse problem is torecover the compartment size histogram c(R) by using the nonnegative least-squaresalgorithm, NNLS, to minimize x 2 for a set of cell radii R, and first derivative smoothingconstraint [Whittall 1991].Comparison of radius distribution curve generated by scanning electron microscopeto that generated by NMR relaxation is presented for five wood samples. The solutionsare acquired with variable parameters M, T2free, Dfree and starting time for lumenwater decay to exclude fast decaying cell wall water signal. The Dfree is set to theknown diffusion coefficients for bulk water [Simpson and Carr 1958]. Parameters Mand T2free were set to those listed in Table 5.2 for the redwood distributions andTable 5.3 for the spruce distributions, and start times were the same as discussed inSection 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 allowedradii from 1 to 100 itm was used.The NMR radius distributions of redwood, shown in Fig. 5.12, correspond to theSEM distributions for NMR relaxation decays collected at temperatures 4.0 to 55.0°C.All distributions show a latewood and earlywood bimodal distribution. Figures 5.13Chapter 5. Diffusion Model of Compartmentalized Water^ 78Figure 5.12: Redwood radius distribution from NMR relaxation data (solidlines) for temperatures 4.0 to 55.0°C. Compared to SEM radiusdistribution (histogram).Chapter 5. Diffusion Model of Compartmentalized Water^ 79Figure 5.13: White spruce radius distribution from NMR relaxation data(solid lines) for a) white spruce sapwood, and b) white sprucecompression wood. Compared to SEM radius distribution (his-togram).abChapter 5. Diffusion Model of Compartmentalized Water^ 80r0^10^20^30^40^50Radius (p,m)Figure 5.14: Radius distributions from NMR relaxation data (solid lines) ofa) alder sapwood, and b) Douglas fir sapwood. Compared toSEM radius distribution (histogram).and 5.14 show distributions at room temperature. The white spruce sapwood has abimodal distribution. The alder sapwood, white spruce compression wood and Douglasfir sapwood all have SEM distributions skewed to the small radii. The NNLS solu-tions from the NMR relaxation decays tend towards localized peaks and represent theskewness as a large and small peak. The SEM distribution of the alder sapwood shows2.2% of the cells having larger radii from 18 to 37 /cm (average 28 /cm) and the NMRdistributions 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 inwood which is comparable to the SEM radius distribution information. To solve forthe radius distribution the surface relaxation parameter, M, is needed, which will bedetermined in the next chapter.Chapter 6Diffusion Model of Two Regions of Compartmentalized Water6.1 SummarySpin-spin relaxation of water in wood is described by a model which treats relaxationof lumen water as a magnetization exchange with cell wall water. The diffusion-Blochequations are solved for two regions: Free water in the cell lumens and water in thecell walls. The resulting theory relates the surface sink for lumen water relaxationto 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 experimentalmeasurement of the spin-spin relaxation times and relative populations of lumen andcell wall water. Such estimates for cell wall water diffusion in a maximally hydratedredwood sapwood sample ranged from 0.92 x 10 -6cm2/s at 4°C to 5.89 x 10 -6cm2 /s at55°C. The activation energy for cell wall water diffusion in redwood sapwood in thistemperature range was found to be 6700 cal/mol, about 40% higher than the free watervalue of 4767 cal/mol.Numerical simulations of the two region diffusion model were developed. The lumenwater T2 was found to be independent of the simulated cell wall thickness, simplifyingto a surface relaxation as modeled with the surface relaxation parameter in the oneregion model. The simulated effect of exchange on the FSP measurement was found tobe 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]81Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^826.2 Determination of the Cell Wall Water Diffusion Coefficient in Woodfrom T2 Relaxation Measurements6.2.1 IntroductionIn the preceeding chapter, I discussed the spin-spin relaxation of lumen water in woodin terms of a diffusion model introduced by Brownstein and Tarr [1979]. This modeltreated the relaxation of lumen water at the cell walls as a surface sink characterized bya variable parameter, M. The Brownstein and Tarr model provided an excellent fit tolumen 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 isexplicitly assumed to exist in two regions: i) Lumen water characterized by bulk watervalues for diffusion coefficient, D 1 , and bulk water spin-spin relaxation time, T21 andii) cell wall water characterized by a cell wall water diffusion coefficient, D2 and themeasured cell wall water spin-spin relaxation time, T22. The magnetization behaviorof cell wall and lumen water is treated simultaneously using diffusion-Bloch equationsfor each region and joint boundary conditions. Two different ways are used to analysethe differential equations: i) The Laplace transform and ii) a less rigorous, but moreintuitive approach.The goal is to obtain a clearer understanding of the effect of cell wall water T2relaxation on lumen water T2 relaxation; i.e. to have a physical interpretation forthe surface sink parameter M of the Brownstein and Tarr diffusion relaxation modelfor water in wood. I demonstrate how M is related to the cell wall water diffusioncoefficient, D2, and use the NMR. results of the previous section to determine D2 inredwood sapwood as a function of temperature.Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^83TheoryThe relaxation model has been applied to one dimensional, cylindrical and sphericallumen geometries. For simplicity, the one dimensional model is presented here (Seeappendices C and D). I consider two regions of water concentration, as shown inFig. 6.1, with different diffusion coefficients D1, D2, different spin—spin relaxation timesT21, T22 and magnetization m i (x, t), m 2 (x, t). Exchange between the two regions will—R^0^RRegion 1^Region 2Figure 6.1: Two region diffusion problem with different volume relaxationsinks 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, Ox2 07 ' 0   i = 1, 2.at 172, (6.1)The initial condition of uniform magnetization following the 90° pulse of a CPMGsequence givesmi(x, 0)^mo; 0 <^<Rm 2 (x , 0) = 71120 = K mo ;^> R (6.2)where K is the partition coefficient which is the ratio of water concentration in thecell wall to water concentration in the lumen. The water concentration in the lumen isChapter 6. Diffusion Model of Two Regions of Compartmentalized Water^84just the water density which is 1.0g/m1 and equal to d„„ dFSP/100 in the cell wall bythe definition of the fibre saturation point (FSP) [Sian 1984], where dwood 1.5g/ml[Haygreen and Bowyer 1982]. The solution to Eq. (6.1) is simplified if weimpose analternative initial condition:mi(x, 7) = mo m2(x, r) = 0^(6.3)for T > T22 which is valid when the lumen water relaxation times are much longer thanT22. This assumption generally holds for T2 relaxation of water in wood.At the boundary x = R the magnetization must satisfy a partition balance equationand the flux must be continuous,^K i (R,t) = rn 2 (R,t),^ (6.4)D1—O(R, t) = — D2 Om2 (R, t).x^Ox (6.5)Initial value problems can be solved using the Laplace transform [Luikov 1968]. Inthis case, the transform would reduce the problem from a partial differential equationto 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) givesa2 7.fhis^, s) —^0) = D i ^ (x , s)Ox 2and the boundary conditions Eqs. (6.4) and (6.5) becomes)T2i= 1, 2^(6.7)Kth i (R, s) = ih 2 (R, s)^ (6.8)D1 D^x Dx(R s) = D2 Ofh2 (R , s). (6.9),^ai(s) cosh(O ix)^-yi (s) sinh(dix)^m=07'22sT2i + 1(6.10)S 1/T22711207712(x, S) - (6.16)Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^85The general solution to Eq. (6.7) iswhere oz i , Oi and ryz are undetermined constants. The solutions using the boundaryconditions of Eqs. (6.8) and (6.9) and requiring that rh i (x, s) are symmetric aboutx = 0 and rn 2 (x, s) is bounded for large x, are:s)M2 (x, s)^=A(2s)A(s) cosh(Oi x)rrnioT2i(6.11)(6.12)[ exp(018721 +x)^exp(1M107'21B(s) 771 20T22Oi x)]sT2 1 + 1exp(— / 32 1x1)^,T,22 + 1whereA(s) irt2oT22 Ii .sm I K cosh(Oi R) 1 (6.13)sT2 2 + 1 sT21 + 1h(l3iR)D2B(s) [Kr-ni(R,^) 71120T22 exp(132 R) (6.14)87'22 + 1andsT2i + 1 .Ni 2. (6.15)1,The solutions m i (x,t) and m 2(x,t) are the inverse Laplace transform of thesefunctions in i (x, s) and 7iz 2 (x, s) respectively. For large x,which gives 7i 2 (x, t) = m20 exp(—t/T22 ) as expected. I have chosen to not invertEqs. (6.11) and (6.12) to give ni l t) and 7/1 2 (x, t) for all x since the inverse Laplacetransform for this expression is very difficult to calculate analytically and is unnecessaryfor the purpose of this section. The resulting analytic solution would be unwieldy andChapter 6. Diffusion Model of Two Regions of Compartmentalized Water^86more cumbersome than the Brownstein and Tarr solution for lumen water relaxationpresented 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 beproportional 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 expressionfor M. Assuming that M is a constant, the Laplace transform of Eq. (6.17) is simplyam — (R s) M fit i (R, 8).axUsing Eqs. (6.8, 6.9, 6.12) and the initial conditions of Eq. (6.3) we obtain(6.18)1M=K  D 2 (s (6.19)M is not a constant as assumed above but depends upon s. Eq. (6.6) shows that atshort times t, m(x, t) contributes to m(x, s) at all s, whereas at long times, m(x, t) onlycontributes to rh(x, 8) for small s. Consequently, the range s > 1/T22 corresponds toshort times only, and therefore we suspect that for t < T22, the Brownstein and Tarrradiation condition of Eq. (6.17) may not be valid. However, for .s << 1/T22 , M isapproximately constant and given byM K D2T22(6.20)Since lumen water relaxation times in wood are generally much longer than cell wallwater 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 forcylinders and spheres are the same in the limit of ,VD2 T22 << R, which holds for wood.M = (1 — 1/e)K oc K122(6.22)D2 D2T22Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^87Intuitive Derivation of MI present here a more intuitive treatment of the relationship between lumen and cellwall water relaxation times in wood. I assume that lumen water has the diffusion rateof free water and a relatively long T2 of 1 to 3 s. I assume that cell wall water has adiffusion rate at least ten times slower than that of free water and possesses a muchshorter T2 of 0.5 to 5 ins. Furthermore, upon entry to one reservoir from another, waterundergoes a discontinuous change in spin—spin relaxation rate and in diffusion rate. Iconsider the flux at the boundary of the two regions in Fig. 6.1,ana l— D1 ^ (R, t) = D2 Din2 (R, t) -D2 Ain .ax ax^Ax (6.21)To estimate Am/Ax, look at changes in M and x in time T22 at the boundary, Am =—(1 — 1/e)Km 1 (R,t), since the magnetization of water in the cell wall decays withrelaxation time of T22 from its initial value Km i (R, t). Also, I approximate Ax as theroot mean square distance perpendicular to the surface that a cell wall water moleculediffuses in this time, which is Ax = VD2T22 . Substitution these expressions intoEq. (6.21) and the condition Eq. (6.17), gives6.2.2 Results and DiscussionIn Fig. 6.2 I show Carr—Purcell—Meiboom—Gill (CPMG) decay curves originating fromcell 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 ex-perimental CPMG relaxation decay curves originating from all the water in the redwoodsamples Chapter 5. The lumen water contribution was calculated from the Brownsteinand Tarr relaxation model using the known radius distribution, fitted values for M, theCI'•k05;1) 461144."%ftftw.404.0..........°Z4k,N,^ -- ***•••******:"--,,,,, ,,_,,„,** ..00000 0006,x^ ...LA 0 0000045Aill x<4UCtill<WXx,s<g3<"),,XXX xX xCal, ,,V3 git^xX xAA 000A AA A 00 00A AALAYL,010 °.00 .02.01Time (s)10 2 _^Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^88a)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 filledtriangle, 55°C filled circle)bulk water T2 time, and the bulk water diffusion coefficient as discussed in the preceed-ing section. In Fig. 6.3, I display T2 relaxation plots for the cell wall water of redwoodsapwood derived from the relaxation decay curves of Fig. 6.2 using a non—negative leastsquares 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 andthe calculated values for D2 as a function of temperature. The FSP values in Table 6.1were the product of the fraction of cell wall water in the relaxation decay curve andthe moisture content of the redwood sample. Although the T2 plots in Fig. 6.3 exhibitmore than one peak, a single, amplitude weighted, average T2 was calculated becausethere 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 amplitudescorresponding to 6, 5, and 15% MC respectively which were not included in the cellChapter 6. Diffusion Model of Two Regions of Compartmentalized Water^894.0 °C18.0^-26.5 °C -10 -2 io -1 10 °T2 (s)34.0 °C-42.0 °C -55.0^-A 10 -4 0 -3 10 -2 10 -' 10 °1 2 (S)Figure 6.3: T2 relaxation plots of cell wall water decay curves from redwoodsapwood.Table 6.1: Cell wall water T2, fibre saturation point, moisture content, surface sinkparameter and cell wall water diffusion coefficient for redwood sapwood at temperatures4 to 55°C.Temp. T22 FSP MC M(m/s) D2(cm2/s)4.0°C 1.37 ins 35.5% 451% 1.38 x 10 -4 0.92 x 10 -611.0°C 1.26 ms 30.2% 328% 1.55 x 10 -4 1.48 x 10 -618.0°C 1.23 ins 33.7% 354% 1.66 x 10 -4 1.33 x 10 -626.5°C 1.88 ms 28.0% 291% 1.43 x 10 -4 2.18 x 10 -634.0°C 1.99 ins 27.4% 315% 1.52 x 10' 2.72 x 10 -642.0°C 1.98 ms 26.3% 279% 1.58 x 10 -4 3.18 x 10 -655.0°C 1.97 ms 21.7% 247% 1.78 x 10 -4 5.89 x 10-6I^13.I 1^3.2^3.I 3^3.4^3.1 5^3.6^3.71000/T(°K)3.0Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^90wall water T2 calculation. This component comprises from 2 to 6% of the total waterand I do not know its origin. The values for M in Table 6.1 were derived assuming acylindrical lumen in Chapter 5.The estimated cell wall water diffusion coefficient, D2 using Eq. (20), is plotted inFig. 6.4 as a function of inverse temperature. Two estimates are shown, one with theTemperature ( °C)60 50 40 30^20^10^0Figure 6.4: Calculated cell wall water diffusion coefficient for redwood sap-wood using FSP from Table 6.1 (crosses) or a linear fit to theseFSP values (boxes). Fits of the diffusion coefficients using theFSP from Table 6.1 (solid line) or data using a linear fit of theFSP (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 decreasesthe 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 aChapter 6. Diffusion Model of Two Regions of Compartmentalized Water^91temperature dependence according to the Arrhenius equation:D 2 = Doe —Ea/RT (6.23)where Do is a constant and Ea is the activation energy for cell wall water diffusion, Ris 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 Do were 6877(250) cal/mol and 0.23(0.08) cm 2/s withthe experimental FSP data and 6611(210) cal/mol and 0.14(0.06) cm 2 /s with the linearfit to the experimental FSP values. The numbers in parentheses indicate confidencelimits 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 lineartemperature dependence (E a = E0 + CT) was incorporated there was no improvementin the fit.Our measurements of D2 are unique for a couple of reasons: i) They were equilibriumdiffusion measurements, with no applied gradients of moisture content, temperature,or relative humidity and no net change in weight or dimension of the wood sample, andii) they were measured in fully hydrated wood samples. It is difficult to imagine anyother way to measure cell wall water diffusion in a fully hydrated sample. Pulsed fieldgradient NMR measurements of cell wall water diffusion would be difficult to performdue to the short T2 of cell wall water and the fact that cell wall water is only about10% of the total water in fully hydrated wood samples.The only direct cell wall water diffusion measurements I am aware of in the litera-ture are from Stamm [Stamm 1959,Stamm 1960] who measured water absorption andswelling in thin wood samples with metal filled lumens at moisture contents below theFSP. Tangential and radial diffusion coefficients were found to be two to three timessmaller than longitudinal diffusion at moisture contents below the FSP [Stamm 1960],Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^92presumably due to resistance to swelling of adjacent cells. I compare my results to thelongitudinal values since in equilibrium, water diffusion should not be restrained byfibre swelling. Using Stamm's results [Stamm 1959,Stamm 1960], Skaar [Skaar 1988]estimate values for Ea and Do for longitudinal cell wall water diffusion to beEa = 9600 — 170MC 6.95MC 2 — 0.160MC3(cal/mol) (6.24)and 0.19 cm2/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 about6500 cal/mol. Our value for Ea of 6700 cal/mol for the entire temperature range is closeto the above mean and substantially greater than the activation energy for free waterdiffusion of 4767(49) cal/mol calculated from the bulk water diffusion values used inthis study [Simpson and Carr 1958]. It is interesting that for moisture contents belowthe FSP, the activation energy for cell wall water diffusion was found to be a functionof 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 theFSP which varied from 35.6% to 21.7% across the temperature range studied.6.2.3 Concluding RemarksI conclude that spin—spin relaxation of lumen water in wood can be understood quanti-tatively using the Brownstein and Tarr diffusion model and interpreting the wood cellsurface relaxation sink as a magnetization exchange between lumen and cell wall water.From the T2 relaxation results, I have determined cell wall water diffusion coefficientsin approximate agreement with values obtained by other methods. To our knowledge,this constitutes the first experimental measurement of cell wall water diffusion in afully hydrated wood sample. Our results indicate that the cell wall water diffusioncoefficient is about one order of magnitude smaller than that of free water and that theChapter 6. Diffusion Model of Two Regions of Compartmentalized Water^93activation energy for cell wall water diffusion in fully hydrated wood is approximately40% larger than that for free water diffusion.Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^946.3 Numerical T2 Simulations6.3.1 Numerical MethodThe two region problem in the previous section was not solved analytically for themagnetization of the water in the lumen and cell wall. A numerical solution of thisproblem can be derived and used to investigate the limitations of the Brownstein andTarr assumption of the surface relaxation condition, Eq. (5.2), and to investigate theeffect of exchange on the determination of the cell wall and lumen components fromthe spin-spin relaxation decay curve.To numerically simulate the diffusion and relaxation of spin magnetization of waterin two regions with differing diffusion and relaxation parameters, the diffusion—Blochequation,(x,t)^D 492m (x t)^ni(x , t)(6.25)at^axe^T2must be discretized. To discretize the derivatives of a function u(x, t) at (x, t) considerthe Taylor series expansions of the function [Ames 1977]. The Taylor series of u(xAx, t) about (x,t) isAx )2 492 uu(x Ax, t) = u(x,t)-F Ax (x, t) + ( 2! ax (x 'ax(Ax) 3 a3 u+ 31 ax3 (x, t) 0[(Ax)1From the above Taylor series one has a simple first-order approximationOu 1ax '^h (u^— ui,j) + 0(h)(6.26)(6.27)A double subscript notation is used so that u 2 ,j is the discretized value of u(x, t) atthe i th step in x and the j th step in time, t. The variable h Ax is the step size inx and the variable k = At is the step size in t. The 0(h) represents the truncationerror and without it one has the forward difference approximation. The Taylor seriesA^ AD free T freeh 2h 3hD^Tcw cw(j+1)k  -jk/Q.) 3k -2k(j= 1 ) 0 ^0 awh^ihChapter 6. Diffusion Model of Two Regions of Compartmentalized Water^95Position (x)Figure 6.5: Discrete steps in position and time.of u(x — Ax, t) about (x, t) isu(x — ox, t) = u(x,t) — Ax —au (x,t) (Ax)2a2u (ax^2!^ x 2 'x ' t)(Ax)3 a3 uax3 (x, t) ORAx) 4 }3!(6.28)and gives another first order approximation referred to as the backwards differenceapproximationau 1 ,^ax '3^h•'3= —^— u i_ i ,j) + 0(h) (6.29)To obtain a second-order approximation, the difference of the previous two Taylorseries, Eqs. (6.26,6.28), is usedau^(Ox)3 03uu(x Ax,t) — u(x — Ax, t) = 2 Ax -5--;(x,t)+ ^ ax^, t) 0[(Ax) 5 16.30)3!to give12h yui+Li — u1_1u1-1,j)0(h 2 ) (6.31)Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^96and without the 0(h 2) truncation error, this solution is referred to as the centereddifference approximation. In a similar manner, the second derivative isa2u 1ax2 12,3 = 11 2 (7-11-1 ,j^2tti,j^tli+Lj)^0(h2). (6.32)To discretize a partial differential equation, one uses the centered difference for thespace 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 diffusionequation u t = us,; , where units have been chosen so that D = 1. Define u i ,j as theexact solution with no truncation error, and Ift ,i as the discrete approximation. Thediffusion equation gives1) = h2 (Ui+1,3 —^U2-1,3) (6.33)The solution, with p= k/11 2 , isUt,j+1 =^+ (1 — 2p)Ui,j^pUi,j^(6.34)and for u t , , the exact solution, isu 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 otherDiscretization error decreases as h and k are decreased, but the round-off error mayincrease. The stability and convergence of the solution can be assured by consideringa function zi ,j , which is the difference of the exact solution u i , ‘; and the approximatedsolution 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^97If 0 < p < 1/2, then the coefficients with the z i ,i 's are nonnegative and sum to one, sothatizi,j+i I 5_ p izi-i „it + (1 - 2 Azi,a1^zi-Fi^A[k2^kh2 ]5_ 'HI A[k 2^(6.37)where 114 =^Izi,j1. Since 11411 = 0 (u U at t = 0 as both are defined bythe initial conditions), therefore,< A[k 2 kh 2 ]11z2 11 <^+ A [k2^kh 2 ]^A2 [k2^kh 2 ]< Al [k2 kV]^ (6.38)where A is the upper bound of the truncated terms u tt and u„„ The error, zi ,j , tendsto zero as h and k tend to zero, and converges to u i ,j . So for this example of thediffusion 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 insection 6.2, is as follows:kD711' _12^0• + (12kD^k^kD(6.39)771 ^=2 0+.^h2 -^ ) mij^— tn i+ih 2 h 2whereD free ; 0 <^<aDD,; a < x < b(6.40)Tfree ; 0 < x < aTc.„, ; a < x < b(6.41)where a is the position of the boundary and b is the outer boundary of the region, andT can be either the transverse or longitudinal relaxation rate. The initial conditionsChapter 6. Diffusion Model of Two Regions of Compartmentalized Water^98define mo , for j = 1 at t = 0.Ino;^0 < x < a=^ (6.42)K mo; < x < bwhere K is the partition coefficient between the water concentrations in the cell walland the lumen. The solution is symmetric about x = 0 requiring that am/axi o = 0 andusing 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 asa 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+1is the outer boundary x = b. Continuity of flux at the boundary, x = a, leads toam am ,D free —ax , = Dcw ax^ I a+The boundary at x = a is labelled i = w = a/ h 1/2 on the free water side, andlabelled i = w 1 on the cell wall side. The backward difference is used to discretizethe flux on the free water side at i = w, and the forward difference is used on the cellwall side at i = w 1 to giveamw,J+1^(771w,j+1 — mw-1,J+1 )aX—D free^D free—D aMw+1,7 +1 =^Dcw (mw+2,j+i — mw+i,j+i)cwax(6.44)(6.45)The flux is continuous across the boundary, Eq. (6.43), and the partition balance equa-tion gives mw+ i,j+i = K raw , 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 +1D freeMtu-1,j+1^Dawnlw+2,j+1 D f ree K DcwK mw,i+i(6.46)(6.47)(6.43)Chapter 6. Diffusion Model of Two Regions of Compartmentalized Water^99The 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.A similar solution is found for the diffusion—Bloch equation in cylindrical coordi-nates:^(kD^k2kD^k (kD^k^h2^2lir )^+ ^112^T) rni '3^h2 + —2hr) rni+1,j (6.48)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 ApplicationsNumerically created T2 decay curves, sampled at 129 times from 0.2 ms to 370 msgeometrically spaced, are fit using NNLS. Table 6.2 displays the results of simulationsfor latewood type geometries with small radius cells and thick walls, and earlywoodtype geometries with larger radius cells and thin walls. The simulation parameters areTable 6.2: Numerical T2 Simulations.Latewood, a = 7 ,um Earlywood, a = 15 pm(b-a) T2 lumen T2 CW Amplitude CW (b-a) T2 lumen T2 CW Amplitude CW1 pm2 pm3 pm4µm5 pm6 pm40 ins35 ins34 ins34 ins34 ins34 ms-1.2 ins1.8 ins1.9 ins1.9 ins1.9 ms0% MC9% MC16% MC19% MC21% MC23% MC1 pm2 yin3 pm4µm5 pm103 ms^-^0% MC91 ms^0.8 ms^12% MC90 ins^1.2 ms^18% MC90 ms^1.5 ins^21% MC90 ms^1.6 ms^23% MCDf„, = 2.2 pm 2/ms, Dcw = 0.2 p1I1 2/ms, T2 f „e = 1.5 s, T2c w 2.0 ms, time stepsize is k=0.002 ms, position step size is h=0.1 pm, and the initial condition for the cellwall water is the total amplitude in the wall equal to 30% MC. NNLS is used with 100geometrically spaced T2 times from 0.1 ins to 1 s.a pLumenCell WallChapter 6. Diffusion Model of Two Regions of Compartmentalized Water^100The results of the simulation show that the T2 of the lumen water is independentof cell wall thickness for walls thicker than 2 pm for both small and large cells. Theseresults confirm again that water in wood lumens follow a diffusion-Bloch equation witha 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 waterwas significant and resulted in a decrease in amplitude assigned to cell wall water fromthe NNLS T2 fit. Wood density is much higher in the latewood region and it is thesecells that contribute to the FSP measurement most significantly. Even for a typicallatewood cell with 5 pm thick wall, the simulation resulted in a cell wall T2 component9% MC less than the initial condition of 30% MC. A penetration depth can be definedfrom the following ratio:Mcwsi„, Scale[b2 — (a + p) 2 ]7rratio    ( 6.49)FSP Scale(b2 — a 2 )7to give a penetration depth p = [ /b 2 (1 — ratio) + ratio a 2 — a]. For a typical latewoodcell with 5 pm thick wall, the predicted penetration depth is p = 1.8 ,am. Also, thepenetration depth can be calculated from p ti V4DcwT2cw to give p = 1.3 /..tm.Figure 6.6: Lumen and Cell Wall Water Exchange.I could correct for this decreased cell wall component amplitude and measure anaccurate FSP, however NMR measurements of FSP throughout this thesis and in otherChapter 6. Diffusion Model of Two Regions of Compartmentalized Water^101studies are not far from the expected FSP values. Figure 6.7 shows the amplitude ofthe cell wall component for western hemlock sapwood as the total moisture contentis decreased. At the highest MC the cell wall component is 26.5% and near the FSPIIIIIIIII•1111[111120^30^40^50^60^70^80^90^100^110^120Total 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 toexchange of cell wall water and lumen water. The latewood cell dimensions of westernhemlock are similar to that of the above example, and therefore the simulation predictsan underestimate of 9% MC in the FSP. It appears that wet walls from partially fullcells may contribute to the cell wall T2 component and complicate the correction forthe exchange of cell wall water with lumen water.35C 300002520Chapter 7Spin-lattice Relaxation and Cross Relaxation7.1 SummaryThe spin-lattice relaxation of western redcedar sapwood has been investigated, formoisture contents from 216% to 1%, through three techniques of applying a modifiedinversion recovery sequence to give a) T1 of separate solid and liquid signals from thefree induction decay signal, b) T1 components from latewood and earlywood regionsfrom the one dimensional image across the growth rings, and c) T 1 — T2 plots fromthe CPMG sequence decay signal. The results indicated that on the T 1 time scale of100 ms all proton environments are mixed by diffusion of the water, so that the T 1 ofthe water in the lumen and the cell wall and the protons of the solid of the cell wall allhave one average value. Three T1 times were identified, 0.55 s for the fully hydratedearlywood, a faster T1 of 0.17 to 0.40 s for the fully hydrated latewood, and 0.11 to0.20 s for the cell wall water and protons in the solid wood for when the cell lumens areempty. Also, cross relaxation of the water and the protons in solid wood was measuredfor moisture contents 38% and 26%, and found to be 1.1 ins, supporting a fast exchangemodel for spin-lattice relaxation of cell wall water and the protons in solid wood. Thecross relaxation of the protons in solid wood and the cell wall water was found to bethe dominant mechanism for T2 relaxation of the cell wall water.102Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 1037.2 IntroductionThree modified inversion recovery (MIR) techniques were applied to investigate thespin-lattice relaxation of wood, for moisture contents from 216% to 1%. First, the Tdependence of the free induction decay (FID), collected following the MIR sequence,was analysed to give the T1 of the protons in solid wood separate from the water.Second, the T dependence of the one dimensional image (or projection) of the moisturedensity across the growth rings, collected following the MIR sequence, was analysed togive the T1 of the latewood and earlywood regions separately. Third, the 7 dependenceof 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 fromthe lumen water. Also, the cross relaxation of the water and the protons in solidwood was measured for moisture contents 38% and 26%. The aim of this study wasto interpret the T1 components in terms of wood morphology and to determine theinfluence of water diffusion and cross relaxation on the T1 times.7.3 Materials and Methods7.3.1 SamplesOne western redcedar sapwood sample, which was cut to 0.5 x 0.5 x 1.0 cm, containedeight complete growth rings, and is labeled Cedarl. Another western redcedar sapwoodsample 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^ 1047.3.2 Spin-lattice Relaxation of Wood and Water in WoodA MIR sequence was used as follows:90, — TR180, — T- 90, — TR^ (7.1)where the second signal was subtracted from first to give a positive signal that decaysto zero at long T. For a system with only single exponential relaxation, this sequencegives SW = 2M0 exp(-7/T1 ). One hundred scans were averaged with a recycle timeTR of 5 s (TR > 5T1 ). Thirty T values were used from 1 to 3000 ins, geometricallychosen in this range. The water signal is an average of 20 points from 60 to 70,as in theFID where the solid signal had totally decayed to zero. The difference of the signal ofthe average of 20 points from 17 to 27ps in the FID and the water signal is proportionalto the solid signal. The water and solid decay curves were fit using NNLS with a linearset of 100 T1 times from 0.001 to 2 s.7.3.3 Spin-lattice Relaxation of the One Dimensional Water ImageOne dimensional imaging was applied to avoid multi-exponential decays due to inho-mogeneity in sample lumen size. The following MIR sequence was used90, — TE — 180, — TE — Echo—TR180, — 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 magneticfield. TE was 100/Ls and the 180° pulse refocused the signal to give an echo which wasFourier transformed to give an one dimensional image, or projection, of the moisturedensity distribution in the wood. The gradient was oriented so that the spread inChapter 7. Spin-lattice Relaxation and Cross Relaxation^ 105spacial information is across the growth rings, eliminating the inhomogeneity in thesignal due to the distribution of lumen sizes. The image was collected for 30 7 valuesfrom 1 to 3000 ms, geometrically chosen in this range, and then was fit using NNLS togive 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 WoodA CPMG sequence was collected after the MIR sequence.90, — T2— (180 y — TE)„ — TR180, — — 90, — Tl— (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 echoamplitudes were used to give the T2 decay curve chosen geometrically from 0.2 ms to250 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 wasfit, giving T1 and T2 simultaneously, using NNLS [Whittall 1992] with a linear set of20 T1 times from 0.01 to 2 s and a geometrically spaced set of 20 T2 times from 0.001to 0.7 s.7.3.5 Cross Relaxation of Protons in Solid Wood and WaterThe exchange of protons in the solid and liquid was measured with a cross relaxationsequence [Goldman and Shen 1966] as follows:90, — t 1 — 90_, — T — 90, — TR90, — t 1 — 90, — — 90, — TR^ (7.4)Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 106where t 1 = 100ps allowed the transverse signal of the solid to dephase and the 7 crossrelaxation time allowed the protons in water and solid to exchange so that the solidsignal 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 anaverage of 20 points from 70 to Nits in the FID where the solid signal has totallydecayed to zero. The difference of the signal of the average of 20 points from 17 to27,as in the FID and the water signal is proportional to the solid signal.7.4 Results7.4.1 Spin-lattice Relaxation of Wood and Water in WoodSeparate spin-lattice relaxation decays of the protons in solid and water can be calcu-lated from the T dependence of the FID following a MIR sequence, as shown in Fig. 7.1for 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 thewater. The signal from 17 to 27,as is averaged and the water decay is subtracted toleave the relaxation decay of the solid signal. The spin-spin relaxation of the cell wallwater part of the water signal taken at 60 to 70,us has been neglected and contributesa 4% systematic error to the solid signal. In not extrapolating the solid signal FID tozero, with a second moment expansion, I have assumed that the lineshape of the solidis independent of T. In fact, the second moment fit of the solid FID at 118% MC gavean average 1171 of (4.4 + 0.6) x 109 s -2 for five T times of 1 to 7 ins and also for 50 to200 ms.The NNLS T1 fit is shown in Fig. 7.2 for the liquid and the solid signal at eightmoisture contents as the sample is being dried. The water amplitudes have been scaled0.01.0tau (ms)10251002003004005007501000200030000.8 -0.2 --10^0^10^20^30^40f50^60 70^80^90^100Time (fLLS)Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 107Figure 7.1: 7 dependence of FID in inversion recovery experiment for cedarsample Cedarl.so that they acid up to the total moisture content. The amplitude of the solid com-ponents have been scaled in the same way as the water to represent the population ofprotons in the solid, which does not change with moisture content. (See the definitionof the NMR MC from Chapter 2).The T1 plot for 199% MC shows that the water decays with two distinct T1 timesof 0.73 s and 0.33 s. As the sample is dried to 38% MC, the amplitude of the slow T1component 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. Asthe sample is dried to 38% MC, the amplitude of the slow T 1 solid component decreasesand the amplitude of the fast T1 solid component increases. Below 38% MC the solidhas only one T1 component at 0.11 s, which is independent of MC until it increases to0.60 s at 1% MC. At 199% MC, the average T1 of the water signal corresponds to theChapter 7. Spin-lattice Relaxation and Cross Relaxation^ 108100.199%100.25%80. 80.60. 60.40. 40.20. 20.100. 100.118% 15%80. 80.60. 60.40. 40.a) 20. 20.-c)7.=...100. 78% 100. 10%E80. 80.60. 60.40. 40.20. 20.100. 100.38% 1%80. 80.60. 60.40. 40.20. 20.0. 00 0^0.2^0.4 0.6^0.8^1.0^0 0^0.2 0.4 0.6^0.8 1.0T, (s) T1 (s)Figure 7.2: T1 plots for liquid (solid lines) and solid (clashed lines) signalsfor cedar sample Cedarl. For each plot, the total amplitude ofthe liquid components is scaled to correspond to the moisturecontent, and the same scaling makes the total amplitude of thesolid components to be 50. (Total Liquid = LOx Scale, Scale= 100% 0.5/(S0 - LO), Total Solid = 50.)E - 6C-)0)'— .5 -tau (ms)1025100200300400500750100020003000V)0.04Position (nirn)Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 109average T1 of the water and solid signal at early times in the FID, which indicates thatthe faster T1 times of the solid compared to the water times are due to systematic errorin the calculation.7.4.2 Spin-lattice Relaxation of the One Dimensional Water ImageA one dimensional image of the moisture density, across the growth rings, of the cedarsample Cedar2 has been acquired following the MIR experiment and the typical 7dependence of the image is shown in Fig. 7.3. Figure 7.4 shows amplitude imagesFigure 7.3: r dependence of amplitude images in modified inversion recov-ery experiment for cedar sample Cedar2 at 171% MC.and T1 images for 4 MC's. The amplitude images show the water coming out of theearlywood regions first before water dries from the latewood region. All but the highestMC image show peaks in the latewood region. The high MC T1 image shows T1 timesof about 0.40 s in the latewood regions and 0.57 s in the earlywood regions. The lowChapter 7. Spin-lattice Relaxation and Cross Relaxation^ 110MC T1 image gives the cell wall water T1 at about 0.20 s. For this low MC, thereappears to be a difference between latewood and earlywood T1 , but the amplitude islow in the earlywood region, so the signal to noise ratio is low. In general, the NNLSanalysis 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 averagedand presented as one component [Whittall and MacKay 1989]. The occurrence of splitpeaks was independent of position in the image. In all of the images there is a noisypoint at about the 1 mm position, which should be neglected.7.4.3 Two Dimensional T2 - T1 Dependence of Water in WoodFigures 7.5 and 7.6 display contour plots of the NNLS solution to the CPMG decayfollowing a MIR experiment for the cedar sample Cedarl. Above 74% MC, the plotsshow two distinct T1 groupings at 0.10 to 0.15 s, and at 0.30 to 0.55 s. Each T 1 groupis spread in T2 in a typical T2 plot of lumen and cell wall water where the cell wallwater 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 T1components. The main change in the 129% MC plot is that all the cell wall water hasa T1 of the fast component. The 74% MC plot shows two main peaks, a cell wall watercomponent 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 aT2 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 WaterFrom the reappearance of the solid signal in the cross relaxation experiment, the crossrelaxation time was calculated for two moisture contents near the FSP. The differenceof the average FID signal of 20 points from 17 to 27its and the average of 20 pointsChapter 7. Spin-lattice Relaxation and Cross Relaxation^ 111-2^-1^0^1^2Position (mm)Figure 7.4: Amplitude and T1 images for cedar sample Cedar2 at 4 mois-ture contents; 171% (long dashed), 140% (clotted), 96% (solid),23% (short dashed).Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 112100Total MC 216%10'153%II lb--19%^20%—10 21000.0^0.2^0.4^0.6^0.8^1.0T i (s)Total MC 129%10 -1 52%NI— 30%^ "-^21%10 -226%10 31 00I0.4^0.6T 1 (s)0.0^0.2 0.8^1.0Total MC 74%0 -1InI—N10 -24%^28%6%4730%   2%10 30.0^0.2 0.4^0.6T I (s)0.8^1.0Figure 7.5: T1 -T2 plots for moisture contents 216%, 129% and 74% forcedar sample Cedarl.Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 11310°Total MC 32%10- 'rn10 -210 a0.4^0.6T 1 (s)0.0^0.2 0.8^1.010°Total MC 26%10-112'10 -210 30.0^0.2^0.4^0.6^0.8^1.0T i (s)Figure 7.6: T1 -T2 plots for moisture contents 32% and 26% for cedar sampleCeclarl.Chapter 7.^Spin-lattice Relaxation and Cross Relaxation^ 1142500020000 -15000 -Cross Relaxation Time10^itzs10000 -200 'as500 fis5000 - 1000 ,u,s0 --5000-50^0^50^100^150^200^250^300Time (kis)Figure 7.7: The FID following the cross relaxation sequence for cedar sam-ple 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 isproportional to the amplitude of the solid signal. In Fig. 7.8, the signal has been scaledto give y = 1 — exp(-7- /T„) for moisture contents 32% and 26%. Tom, is found to be0.66+0.10 ms for 32% MC and 0.83+0.27 ins for 26% MC. The measured Tin is relatedto the Tcr time as follows;1 ^Ns^1Tcr niNs + Ncw T(7.5)where Ns/(Ns Ncw ) is the probability of a proton being in the solid, and I haveused 1/Tcr = pa ka = pb kb [Zimmerman and Brittin 1957], where pa and pb are theprobabilities of a proton in environment a or b, respectively, and 1/k for the solidenvironment is the measured T. When Ncw is scaled to be the cell wall moisturecontent, then Ns is 56 (See NMR MC in Chapter 2). Tcr is calculated to be 1.0+0.15 ms5 6—0.2 ^0 1^2^3^4Tau (ms)Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 115Figure 7.8: Reappearance of the solid signal following the cross relaxationsequence for moisture contents 32%(crosses) and 26%(boxes)for cedar sample Cedarl. Shown are the 32% MC fit (solidline) and the 26% MC fit (clashed line)Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 116for 32% MC and 1.2 + 0.4 Ins for 26% MC and the average is 1.1 + 0.2 ms. The effectsof cross relaxation on T2 will be considered in the discussion.7.5 DiscussionFor moisture contents below FSP, the water and solid signals exhibit one T 1 componentat 0.11s for both proton environments, suggesting that protons in cell wall water and insolid wood are in fast exchange, and the cross relaxation time is measured to be 1.1 mswhich is fast on the T1 time scale of 100 ins. Because of the fast exchange of the protonsin solid and water, I do not know the intrinsic T 1 of the protons in solid but only theaverage, and since the average appears to be independent of the MC even below theFSP, the intrinsic T1 of the protons in solid is most likely 0.11 s. The increase in T 1of the protons in solid at 1% MC likely corresponds to structural changes to the woodwith the loss of moisture. It is known that wood shrinks significantly with the removalof water and the M2 (from Chapter 3) was found to increase at low MC indicating lessmotion.Above the FSP, both the water and the solid signals have two T1 components. Thefast solid T1 component is similar to the component below the FSP where there isonly cell wall water, but the other solid T 1 component decays slower. Application ofa gradient field across the growth rings direction gives one dimensional images of thewater with the signal from the cells in the latewood region separated from that of theearlywood region. The spin-lattice decay of the signal at each position in the imagewas single exponential, and therefore the multi-exponential decay of the water part ofthe FID is due to the sample cell inhomogeneity. The one dimensional image showsthat the water in the latewood cells is the faster T1 component and the water in theearlywood cells is the slower T1 component. Below the FSP, the cell wall water has theChapter 7. Spin-lattice Relaxation and Cross Relaxation^ 117fastest T1 of 0.20 s for Cedar2 and 0.11 s for Cedarl.The 216% MC T1 — T2 plot shows two distinct groups of T1 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. The0.15 s T1 is the latewood water and therefore the cell wall water component at this T1time is from the latewood cell walls. The 0.52 s T 1 is the earlywood water and thereforethe cell wall water component at this T1 time is from the earlywood cell walls. On theT1 time scale of 100 ms, the exchange of the cell wall water and the lumen water ofa cell is fast and results in only one average component of T 1 . Therefore, since thecell wall water is in fast exchange with the solid and the lumen water, the measuredT1 is an average over all protons in the cell. The cell wall water and protons in solidhave a T1 of 0.11 s, and the free water in the lumens has an intrinsic T1 of the 1 to3 s. The earlywood cells with thin walls and large lumens have more lumen water thanthe latewood cells with thick walls and small lumens, and therefore the protons in theearlywood cells have a slower spin-lattice relaxation than the protons in the latewoodcells.Figure 7.9: Cross section of a cylindrical cell.Within a cell, the protons in cell wall water and solid exchange magnetizationby cross-relaxation with T„ = 1.1 ins, and the cell wall water and the lumen waterChapter 7. Spin-lattice Relaxation and Cross Relaxation^ 118exchange 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 thefree water T1 , N = Ns + Ncw NL , Ns is the number of protons in solid wood inthe cell wall, Ncw is the number of protons as cell wall water and N.L, is the numberof protons as free water in the lumen. I will calculate Ns, Ncw and XL, for a fullyhydrated cylindrical cell shown in Fig. 7.9, where a and b are the lumen and cell radii,respectively.FSPPwater dwood 100% VI""2 1 H/molec.NA 1.5g/m1 0.3 Vwall18 g/mole0.0500 NA Vwall^ (7.7)Ns^Pwood dwood Vwall= 0.56 Pwater dwood Vwall2 1 H/molec.= 0.56 ^ NA 1.5g/m1 Vwall18 g/mole= 0.0933 NA Vwall^ (7.8)NL = Pwater dwater V111771en2 1 H/molec.men^ NA 1.0g/m1 Viu18 g/mole= 0.1111 NA Vlunzen^ (7.9)where p is the proton density per unit mass, and the ratio PwoodPwater = 0.56 is used[Fengel and Wegener 1984], which was used to define the NMR MC in Chapter 2. d is themass density per unit volume [Haygreen and Bowyer 1982] and NA is Avragados numberNCWChapter 7. Spin-lattice Relaxation and Cross Relaxation^ 119giving molecules per mole. For a cylindrical cell of length 1, and cross section dimensionsas in Fig. 7.9, the volumes of the cell wall and lumen are defined as Vuall = 71- 1(b2 — a2 )and Vlumen = 7/a 2 . The expression for T1 reduces to the following:1 1^ — a2 )[0.1433(b2 0.1111a2 1(7.10)—^0.1433b 2 — 0.0333a2 L^Ticw TifreeT11 1 (12^1^0.7753](7.11)a2 — 0.2324 [ 111cw^ilfreeTlwhere a = b/a.Table 7.1 shows the T1 calculation using Tww of 0.11 s, Tifree of 3 s and typicalcell dimensions of western redcedar [Panshin et al. 1964]. This fast exchange modelTable 7.1: Fast Exchange Model of T1 for typical cedar cells.Cell Type Lumen Radius Cell Wall Diffusing Time Predicted(a) Thickness (b-a) Lumen Cell Wall T1Latewood 7,am 5fim 6 ms 31 ms 0.15 sEarlywood 18,am 2,am 37 ms S ins 0.42 spredicts that on the basis of differences in cell diameters and cell wall thicknessesbetween earlywood cells and latewood cells, the T 1 times of these cell types will bedifferent. Since the latewood cells have small lumens and thick walls, T 1 of latewood isfaster 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 regionand is calculated from t a2 /(4D fr„) for the lumen, and t (b — a) 2 /(4Dcw ) forthe cell wall. Diffusion values of Dfr„ = 2.2 iim 2/s and Dcw = 0.2 fim2/s where used(Chapter 6). From these diffusing times shown in Table 7.1, I estimate that on theT1 time scale of 100 ms that all water molecules diffuse through the entire cell, andsample both relaxation in the wall and lumen. The protons in solid wood cross relaxwith the cell wall water and in this way attain the same relaxation time, and influenceChapter 7. Spin-lattice Relaxation and Cross Relaxation^ 120the relaxation time. I have neglected any diffusion between cells, assuming that allneighbouring cells are identical, which is reasonable since the change from earlywoodto 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 fromT1 as follows:0.7753 .1 r3 )] [1 ^a2 = [0.2324 —^(1rn^— rn^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 latewoodwater was 0.15 s, and T1 of the earlywood water was 0.52 s. From this data, I calculateb/a = 1.7 for the latewood signal and b/a = 1.1 for the earlywood signal. Fromthe cell dimensions given in Table 7.1, I calculate corresponding ratios of 2.0 for thelatewood and 1.1 for the earlywood, which are in excellent agreement with the T 1 datacalculations.The fast exchange model explains the two T1 components in terms of cell densitydifferences in latewood and earlywood regions. In general, as the MC is decreased theamplitude of the earlywood T1 components decrease before that of the latewood, asshown in the amplitude image of Fig. 7.4. For the highest MC of the T 1 — T2 plotsthere 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. Aswood dries the cell wall water does not decrease in any region until the MC is belowthe FSP as seen in the cell wall water moisture profiles, which were separated from thelumen water on the basis of T2 as shown in Fig. 4.7. Therefore, I can conclude that thecell wall solid and water components in empty earlywood cells, above the FSP, haveT1 times similar to that expected below the FSP. These empty lumen, cell wall signalsare indistinguishable from the fully hydrated latewood cell signals, since they are notseparated in T1 times significantly.Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 121In Fig. 7.5, the 216% MC T1 — T2 plot shows a 20% MC component with a T1 of0.45 s, which indicates that it is signal from a thin walled earlywood cell, and a T2 of15 ms, which indicates that it is signal from a small diameter cell such as latewoodcells. A typical SEM of cedar [Panshin el al. 1964] shows that the earlywood doescontain a significant number of small radius, thin walled cells.7.6 Numerical T1 SimulationsA 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 cellwall region. The following partial differential equations describe the z-magnetization ofwood 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 z-magnetization of the cell wall water when a < r < b. 47., t) is the z-magnetization ofthe protons in solid wood in the cell wall. The lumen radius is a and the cell wall is ata < r < b.Om(7 t)atOm (r t)at 't)771,(X^t)= D f „eDcwVN712(X^t)2 m(r,t)r s(r,t)< a< b(7.13)(7.14)jTi freem(r,t)71,cwm(r,t)1 •a < r+ Tc,.I.^'Ns^Ncws(r,t)^N ini(r,t)^s(7-,t)1a < r < b (7.15)TL5^Ter [ Ncw^Nswhere Tif,„, 711'cw and 7Y s. are the relaxation time of free water, the intrinsic relaxationtimes 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 wall1o 3Harlywoodurn et?SO/jdcfry water100^200^300( m s)010110 210°400Chapter 7. Spin -lattice Relaxation and Cross Relaxation^ 122water magnetizations is T„. D f ree and Dcw are the diffusion coefficients of free andcell 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 latewoodcell dimensions as in Table 7.1. For a cell type, all proton environments attain the sameFigure 7.10: Simulations of spin-lattice relaxation in cylindrical cells withearlywood and latewood cell dimensions as in Table 7.1.relaxation rate, which is faster for the latewood cells than the earlywood cells. Thesimulated spin-lattice decay is non-exponential for r times smaller than about 20/isChapter 7. Spin-lattice Relaxation and Cross Relaxation^ 123for the earlywood cells and 3,us for the latewooci cells. This non-exponential behavioris not observed in the measured decay curves. The spin-lattice decay is shown forthe solid signal in Fig. 7.11 for three moisture contents. For 199% and 78% moisturecontents, where significant numbers of cells are fully hydrated, the simulations predict anon-exponential decay of the solid signal, but the measured decays are monotomicallydecreasing. For 38% MC, cell lumens are almost completely empty and it has beenassumed that the solid and cell wall water have the same intrinsic T1 , as discussed inthe previous section, and therefore the decay is strictly exponential.7.7 Cross Relaxation and T2The cross relaxation time of Tcr = 1.1 ms corresponds to fast exchange between thecell wall water and the solid on the T1 time scale of 100 ms, but on the T2 time scaleof T2* = 30,us for the solid this is slow exchange. In Chapter 3 it was found that sincemoisture content could be measured accurately by NMR, there is no exchange of thesolid and cell wall water on the T2 time scale. The following equation describes thex-magnetization of cell wall water.cam m_ +Nrs(7.16)cat TL,w^T„. [ Ns NcwSince the solid dephases with T: = 30ps, and T21 c w is assumed to be much slower thanT„, the resulting water spin-spin relaxation islVC WT2C W Ncw  r T„ (7.17)For the measured cross relaxation time, Eq. (7.17) predicts a T2c w of 0.3 ms which isfaster than the observed times of 1 to 2 ins, but predicts a decrease in the measuredcell wall water T2 with decreasing moisture content, as observed for the lodgepole• •199% ,■•78%1.00.90.80.7Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 1240.90.80.70.6038%10 20 30 40 50-r (nis)Figure 7.11: Spin-lattice decay of the protons in solid wood for cedar sampleCedarl, at three moisture contents.Chapter 7. Spin-lattice Relaxation and Cross Relaxation^ 125pine shown in Fig. 3.6. Analysis of the MC dependence of the lodgepole pine, usingEq. 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Rev., 111, 1201 (1958).Skaar, C., (1988) Wood-water relations, Springer-Verlag, Berlin, New York.Bibliography^ 130Slichter, C.P., (1980) Principles of Magnetic Resonance, Springer—Verlag BerlinHeidelberg, New YorkStamm, 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., NewYork.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: 308-314Whittall, 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 AField Gradient NMRThere are two types of diffusion concerned with the diffusion of the magnetizationvector produced by the spins of a sample. One is spin—diffusion which can be de-tected in a solid where the molecules are in fixed positions. The other is self—diffusionwhere the molecules which carry the magnetization diffuse throughout the sample.The measurement of self—diffusion, or equivalently molecular diffusion, is measured inNMR using a field gradient. This method is more direct than tracer diffusion stud-ies which contaminate the sample and measure the diffusion of the tracer, not thediffusion of the molecules being studied. The idea behind using a field gradient isthat the spins of the sample are given slightly different Larmor frequencies depend-ing on the their position in the gradient field. A 7 pulse can refocus magnetizationto give an echo only if the spin has not diffused to a different location during theexperiment. So the diffusion of the molecules results in an attenuation of the echoamplitude[Fuskushima and Roeder 1981]. A single echo experiment can be used witha linear field gradient to measure the diffusion constant D of the spin carrier of thesample [Slichter 1980].7rIn ( M = —^212DG273) (A.2)Mo^T72^3A Carr—Purcell (CPMG) pulse sequence is used to minimize the effects of diffusion dueto an applied gradient, inhomogeneities in the applied magnet field Ho , or gradients(A.1)131Appendix A. Field Gradient NMR^ 132induced from succeptibility difference created at air/water interfaces. For a linear fieldgradient the CPMG echo amplitude is as follows:— — (7r — 2r)„^ (A.3)2ln^=^( 27rr^272DG27-37i)^A.4 )(MO^T2 3where M is the magnetization amplitude of the n th echo. The difference in these twoecho sequences is that for the single echo experiment 27 - is the time the echo amplitudeM is measured at, where as for the CPMG experiment the time the echo amplitude ismeasured at is at t = 2nr and the magnetization has been refocused at every 7 intervalsso 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 sequencesusing values from wood studies demonstrates the difference between these sequences.ln( 'mg ) = 72DG2t (t2 r2)echo^ 43(A.5)= 2.67 x 104 (gauss • s) - 'D = 2.2 x 10 -5 cm 2 /3G = 19.4gauss/cmt = 60msrcp„ig = 400pswhere G is the value of the applied gradient used in Chapter 4. These values give aratio of the order 10 20 for the CPMG to single echo magnetization. The effect of theCPMG 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 appliedto the wood being studied. The images were interpreted using previously acquired T2relaxation data. It was assumed that the diffusion effect to the magnetization wasAppendix A. Field Gradient NMR^ 133negligible and the following calculation of the ratio of the magnetization of the CPMGsequence with and without a gradient shows that the assumption was valid.^M d^2In ur gra ) — --„7 2DG2 7-2 t^nogr ad^t)(A.6)With the values used above, where t = 60 ms was a typical time that an image wouldbe measured, Eq. (A.6) has the value 0.04. Eq. (A.4) for the magnetization of a CPMGsequence with a gradient shows that the above calculated term should be comparedto t/T2 to decide if the term can be neglected. With T2 in the range of 1-200 ms theterm should be small compared to values of 0.2-60., so that the attenuation due todiffusion term is at most 10% which is an acceptable error contribution for an imagingexperiment.For measuring diffusion, a pulsed field gradient method has several advantages overa continuous gradient method. One advantage is that the time between r.f. pulsesis constant and the duration of the gradient is altered so that only one measurementof the magnetization without a gradient needs to be taken, where for the continuousgradient 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 ofa gradient the free induction decay signal (FID), or an echo signal is shorter in timebecause of the inhomogeneous field produced by the gradient and thus giving a verylarge linewidth in frequency. A r.f. pulse would have to be very short in time so thatall parts of the magnetization experience the same r.f. power. The idea behind thepulsed 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 themagnetization along the y—axis of the rotating frame. At an arbitrary time t i after thepulse the gradient is applied for a time 6. At time T a 7r r.f. pulse about the y—axisflips the spins to refocus the phases. At time t 1 + 0 the gradient is applied for theAppendix A. Field Gradient NMR^ 134same 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 fieldis slightly different, giving it a different Larmor frequency and preventing the properrefocusing. The echo signal to be measured occurs at 2r. The echo amplitude is givenas follows:ln (Mm.PF0 G ) - - ( 27,72 7 2 D82 (A - -6 ) G2 )\^31(A.7)The expression is more complicated if a continuous gradient G o is also present [Fuskushi-man and Roeder 1981]. The duration over which diffusion is measured is (A — 8/3) orsimply 0 for 0Appendix BDiffusion Model for Rectangular GeometryThe diffusion—Bloch equation for infinitely long rectangular cells, with dimensions x =+Rs and y = +Ry , 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) ^ (x, y, z, t))at ‘ax2 ay2 T2free(B.1)where m(x, y, z, t) is the magnetization in the cell lumen at the position (x, y, z) andtime t, and D is the diffusion coefficient of bulk water. The boundary condition for theflux (J) out of the surface is11 • Jis —D n Vnils = —D n am, am, amax —ay z ) s = M (B.2)where M is a parameter characterizing the strength or effectiveness of the surfacerelaxation. For the surfaces with x = +Rx , 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 theboundary conditions area711— D aX ix=±1T? = M 771 is=±R. (B.3)Dinay Iy=±Ry = M m l y=±RyThe solution to this diffusion problem can he expressed as a sum of normal modes,00 00in(x, y, z, t) = E E A„,„ F„(x)G,„(y) e—t/T2(7.)71=0 711=0(B.5)(B.4)135m(x, y, z ,0) = mo = E E A n ,„ cos (-71- 7 ) cos( ItniY )n=0 m=0^Rs /^RyUsing the orthogonality of cosine functions givesff,is cos (T,„ x Rs ) dx f RR cos (um y Ry ) dym0 R^f^COS2 (77„x Rs )dx f_R;ty cos 2 (a 7n y Ry ) dyThe signal detected by NMR is the total magnetization from the cell00 00Appendix B. Diffusion Model for Rectangular Geometry^ 136where Fri (x) and G„,(y) are orthogonal eigenfunctions and are satisfied by the cosinefunction^F,i(x) = cos ( 7 " x.-)^ (B.6)Rs^G„(y) = cos ("mY^(B.7)-14with1^2^2^1^ = D (717i + P711 ) + ^ (B.8)T2(nm)^R1^fq^212 free'The boundary conditions Eq. (B.3,B.4) define 7i„ fromM Rsq„ tan(7)„) — D .^ (B.9)and /c m from^it, tan(p,„) = M RY^ (B.10)The amplitudes A„„, are determined by the initial condition of constant magnetiza-tion thoughout the lumen, immediately following the 90° pulse of a CPMG sequence.That is,where00 00M(t) =^m(x, y, z, t) dx dy dzJ ell^ = -A4 (0) E E^e—ti712(..)(B.11)(B.12)71=0 772,-.0Appendix B. Diffusion Model for Rectangular Geometry^ 137whereIn2^sin2(7171) 7g, (1 + (D / M Rx ) sin2 (70)2 ^sin2(1i,n) —iqn (1^(D/ MRy )sin2 (itni )) •Appendix CTwo Regions with Cylindrical GeometryWe consider two regions of water concentration with diffusion coefficient D 1 , spin—spinrelaxation 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 ofmagnetization across the boundary. The diffusion—Bloch equation, for infinitely longcylindrical cells with no z or B dependence is0771,^a2m,^+ 1 anti (r, t)^t) ^(r ; i = 1, 2., t) = D, (^ (r, t)at ( 37.2^7' ar^T2i (C.1)where r = (r,19,z) in cylindrical coordinates with time t. The initial condition ofuniform magnetization following the 90° pulse of a CPMG sequence givesin i (r, 0) = rn 10 = mo; r < Rm, 2 (r, 0)^71120 KM(); r > R^(C.2)where K is the partition coefficient. The solution is simplified if we impose an alterna-tive initial condition:mi(r, r) = rrio , in 2 (r, r) = 0^(C.3)At the boundary r = R the magnetization must satisfy a partition balance equationand the flux must be continuous,Km i (R, t) = m 2 (R, t),^ (C.4)am i^a7n2— Di Or (R t) = —D2 ^ (R , t). (C.5)138Appendix C. Two Regions with Cylindrical Geometry^ 139This model can be applied to water in wood, with the inner region, r < R, representingthe free water in the lumens and the outter region, r > R, representing the water inthe cell walls. The alternate initial condition is reasonable for water in wood whereT21 > 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( (a2 772 i^1 07iii (r , s)) — M i (r, s); = 1,2s Mi(r, s) — m i (r, 0) = D,  (C.7)Cdr (r ' s) + 7' Or^ •iand the boundary conditions Eqs. (C.4,C.5) becomeK^s)^s)^ (C.8)D1 a07'rhi  (R^o^s) = D2 a^ (R , s).^ (C.9)The general solution to the Laplace transformed problem isoz i (s) /0 0ir)^7i(s) Ko(dir) +^+M i (r, s)^sT2Mi i0T2i (C.10)where^and yi are undetermined constants, and / 0 and K0 are modified Besselfunctions, which satisfy the equation1y" + Ty' — 13 Y = O.^ (C.11)The series expansion of 10 is6XX 2^X 4/0(X) = 1 + 22(1!)2^24(2!)2^26(3!)2where^/0 = 1, and lim x„ /0 = oo. The series expansion of K o isKo(x) = (;i) 1/2 e-T (1^+^9^8x^2(8x) 2^• • •)(C.12)(C.13)s) = A(s) /O(317') + sT21 + 1ih2 (r, s) = B(s) Ko (132 r)M107121 M =—D 2 K aih2 /Orth eK1 (132 R)= T 32^,- 2KO ( 32 R )(C.19)Appendix C. Two Regions with Cylindrical Geometry^ 140where lims_,0 K0 = —oo, and^K0 = 0. The solutions using the boundaryconditions 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:(C.14)(C.15)wheresT2i + 1 i = 1, 2.^ (C.16)712 2The Brownstein and Tarr [Brownstein and Tarr 1979] boundary condition, Eq. (5.4),at r = R is— D 1O(R t) = M m i (R,t).r(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 Laplacetransform of Eq. (C.17) is simplyai-h 1— ^ (R 8) m 7-hl(R, 8).carUsing Eqs. (C.8,C.9,C.15) we obtain(C.18)where Kax) = —K1 (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 \/D2 T22 << R.This condition is satisfied by water in wood.Appendix DTwo Regions with Spherical GeometryWe consider two regions of water concentration with diffusion coefficient D 1 , spin-spinrelaxation 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 ofmagnetization across the boundary. The diffusion-Bloch equation, for spherical cellswith no 0 or 0 dependence isa772/ a2 771iz (r,t) = D1 ^ (r, t)^2 am i (r,^mi(r, t) i = 1,2.^(D.1)^ ; zat^th.2^7' ar Tipwhere r = (r, 0, 0) in spherical coordinates with time t. The initial condition of uniformmagnetization following the 90° pulse of a CPMG sequence givesrni (r, 0) = rn 10 = 772 0 ; r < R^m2(r, 0) = 7n20 = Kmo ; T > R^(D .2)where K is the partition coefficient. The solution is simplified if we impose an alterna-tive initial condition:m i (r,r)^mo , 777 2 (r,^= 0^ (D.3)This simplification is reasonable for cases where T 21 > T > T22. At the boundaryr R the magnetization must satisfy a partition balance equation and the flux mustbe continuous,^Kmi(R, t) = 772. 2 (R, t),^ (D.4)ari l- D1 ^ (R, t) = - D2 am2 (R, t).^(D.5)ar ar141^Appendix D. Two Regions with Spherical Geometry^ 142Defining the Laplace transform,^asL{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) i^1, 2^(D.7)s 75-4,(r, .^) - m i (r, 0) = D, ^ (r , s) ^(r, s)ar 2 r Or T2iand the boundary conditions Eqs. (D.4,D.5) becomeKiii i (R,^) = M 2(R,^) (D.8)D i 57r/1(R' s) = D2 afiOl2 (R, s). (D.9)O r The general solution to the Laplace transformed problem isrnioT2i sinh(Ar)^cosh(A r) ih i(r, s) = a i (s)^+ 7,(s)^+^(D.10)7. r^sT2i + 1where a i , /3 and -yi are undetermined constants. Since, lim,_, 0 sinh(x)/x = 1 andlimx_,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,and -th 2 (r,.^) is hounded for large r, are:sinh(O i r)^inioTn M i (r,^) = A(s)^+ sT^1/300) + mioT21^(D.11)r^2i +A(s) (exp(Oir)= exp(sT21 + 1r7/ 2 (r, s) = B(s) exp( -132 r )^(D.12)rwheres8712 , -I- 1-lei  i = 1, 2.^ (D.13)T2iThe Brownstein and Tarr [Brownstein and Tarr 1979] boundary condition, Eq. (5.4),at 1. = R isanti— D1 ^ (R , t)^Mcar (D.14)Appendix D. Two Regions with Spherical Geometry^ 143which 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 Laplacetransform of the above equation is simply— ^ (R' s) = M rh, i (R,^).Using Eqs. (D.8,D.9,D.12), we haveM =—D2 Kath2/arih 21\= D2 K (02 -) •(D.15)(D.16)This solution is identical to the one dimensional example, Eq. (6.19), for 1/D2 T22 << R.Dfree V2 ni(x, t)D cw V 2 rn(r, t)N rs (r , t)< r< bTcr Ns^N cw7 < as(r , t)^N [ m(r, ,t)^s(r , t)1. s^Tcr NCw^Nsan?'^at (7' t)Dmat (r , t)as t)rti(x t) T1 freern(r, ,t)rn^Icwm(r, t)] a(E.1)(E.2)a < r < b^(E.3)Appendix ENumerical T1 SimulationsThe following partial differential equations describe the z-magnetization of wood andwater 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 m(r, ,t) is the z-magnetization of the cell wall water when a < r < b. s(r, , t) is the z-magnetizationof the solid protons in the cell wall. The lumen radius is a and the cell wall is ata < r < b.where Ti free, Ti'cw and Tls are the relaxation time of free water, the intrinsic relaxationtimes 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 mag-netizations is T,• D f„ e and Dcw are the diffusion coefficients of free and cell wallwater, respectively. The number of solid protons in the cell wall is Ns and the numberof protons as cell wall water is Ncw , and are calculated in Chapter 7.Discretizing these equations gives the following for the lumen magnetization when144I mo;^0 < r < aK mo; a < r < b= (E.7)(E.8)Pwood 100% so = K MO^r, • a < 7' < bP water For'Appendix E. Numerical T1 Simulations^ 1457' < a:[kD free^k= h2^2hr [1^2kD freeh2 Tifkree + i kDf„ eh2^2hrand for the cell wall water when a < r < b:(E.4)rkDcw^kh2^2hr2kDcw^1^1 k(  /cw Ncw Tcr )]m+ [1 +  ^i,iTi+ k.Dcw 112 k 1^k ^ + 211,7. .1mi-1-1,3 +L h2 NsT,s,'3^ (E.5)and for the solid protons when a < r < b:1^18i,.i+i = {1 — k(^ +^\iTis^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 discretizedvalue 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 thstep in time, t. The variables h and k are the step sizes in x and t, respectively. (SeeSection 6.3 for more on numerical methods.)The initial conditions define m i , 1 and so , for j = 1 at t = 0.as defined by NMR MC in Chapter 2 and the definition of FSP. K is the partitioncoefficient 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 theforward difference discretization this gives^= m 2 , , for i = 1 at r = 0. TheAppendix E. Numerical T1 Simulations^ 146condition at the outer boundary can be considered as a symmetry constraint or as arestriction of the flux to be zero, so that at r = b one also requires that am/ar = 0, andusing a backwards difference discretization gives m bih+Li = m bika , where i = bl h 1is the outer boundary r = b. Continuity of flux at the boundary, r = a, leads toam,,^am ,Dfree^= -Ucw^+ar ar(E.9)The boundary at r = a is labelled i = w = a/h + 1 on the free water side, and labelledi = w 1 on the cell wall side. The backward difference is used to discretize the fluxon the free water side at i = w, and the forward difference is used on the cell wall sideat i = w 1 to giveanzw ,i+i—D free 07.arnw+7,j+7— Dctoar(m.,j+i — 771,1,j+1)h— Dfree( 7nw+2,j+1^Mw+1,j+1)- -DCW(E.10)(E.11)The flux is continuous across the boundary, Eq. [E.9], and the partition balance equa-tion gives mw+ J.+ = K 77Lw ,j+1 . These expressions result in the following "marching"forward in time definitions for m„,, i+i and m„,+1 ,i+1 :rnw,j+1mw+i,j-FiD freeMw-1,j+1 DCWMw+2,j+1Dfree 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 theBloch equation describing s(r, t). The stability condition for this problem works out tobe that k < Th 2/(2DT h 2 ), which reduces to k < h 2 /(2D free ) for typical values forwater in wood.

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