Paper Structure. Introduction Paper can be thought of as a stochastic network of fibers. This is seen in the picture below.

Paper StructurePaper Structure

IntroductionIntroductionPaper can be thought of as a Paper can be thought of as a stochastic stochastic networknetwork of fibers. of fibers.

This is seen in the picture below.This is seen in the picture below.

Fibers are generally much longer than the Fibers are generally much longer than the thickness of the sheet, so the network can be thickness of the sheet, so the network can be approximated as 2-dimensional.approximated as 2-dimensional.

This 2-d structure describes many paper This 2-d structure describes many paper properties, but 3d effects are still important.properties, but 3d effects are still important.

The 3d porous structure of paper gives it The 3d porous structure of paper gives it opacity, bulk and stiffness.opacity, bulk and stiffness.

The connectedness of these pores determines The connectedness of these pores determines how fluids transport through a sheet.how fluids transport through a sheet.

These are especially important for printing and These are especially important for printing and absorbency applications.absorbency applications.

We will first discuss the statistical geometry of We will first discuss the statistical geometry of the fiber network of paper.the fiber network of paper.

Next we will discuss how real paper differs from Next we will discuss how real paper differs from the random network.the random network.

We will see that the structure of paper is We will see that the structure of paper is nonuniform, disordered and irregular.nonuniform, disordered and irregular.

Formation represents the structural Formation represents the structural nonuniformity at larger length scales.nonuniformity at larger length scales.

Fiber orientation is an important feature of real Fiber orientation is an important feature of real paper, controlling its behavior in many paper, controlling its behavior in many applications such as packaging and printing.applications such as packaging and printing.

Two-Dimensional NetworkTwo-Dimensional Network• The simplest picture that describes some of the The simplest picture that describes some of the

important properties of paper is a 2-d important properties of paper is a 2-d completely random network.completely random network.

• This can be seenThis can be seenin Figure 2.in Figure 2.

• This pictureThis pictureconsists ofconsists ofstraight linestraight linesegments withsegments withconstant lengthconstant lengthand zero width.and zero width.

• Kallness et. al. showed that handsheets have the Kallness et. al. showed that handsheets have the same in-plane mechanical properties as same in-plane mechanical properties as handsheets made by laminating many thin sheets.handsheets made by laminating many thin sheets.

• The 2d network explains the in-plane properties of The 2d network explains the in-plane properties of paper.paper.

• The randomness of the networks is significant.The randomness of the networks is significant.

• With the random network, all correlations between With the random network, all correlations between fibers are absent.fibers are absent.

• The position of each fiber is independent of other The position of each fiber is independent of other fibers.fibers.

• Thus, the two dimensional random network is Thus, the two dimensional random network is amenable to mathematical analysis.amenable to mathematical analysis.

CoverageCoverage

• Coverage is a useful concept in characterizing a Coverage is a useful concept in characterizing a random 2-d network of fibers.random 2-d network of fibers.

• Consider N fibers in a area A.Consider N fibers in a area A.

• The average coverage c, the average number of The average coverage c, the average number of fibers on any point in the plane, is given byfibers on any point in the plane, is given by

c = Nc = Nllffwwff/A = b//A = b/ff ,,

where where llff, w, wff and and ff are the length, width and and are the length, width and and

basis weight of fibers, respectively and b is the basis weight of fibers, respectively and b is the basis weight of the paper.basis weight of the paper.

• The coverage completely specifies the 2-d The coverage completely specifies the 2-d random network, when fiber properties are random network, when fiber properties are constant.constant.

• The coverage can be measured from sheet The coverage can be measured from sheet cross-sections by determining the number of cross-sections by determining the number of bonds that intersect a reference line.bonds that intersect a reference line.

• It gives a precise measure of the effective It gives a precise measure of the effective number of fiber layers in the sheet.number of fiber layers in the sheet.

• For papermaking fibers, For papermaking fibers, ff = 5-10 g/m = 5-10 g/m22, so that , so that

printing papers have c = 5-20 layers of fibers.printing papers have c = 5-20 layers of fibers.

• We interpret the coverage as the “effective” We interpret the coverage as the “effective” number of layers in the sense that there are number of layers in the sense that there are no distinct fiber layers in paper.no distinct fiber layers in paper.

• Likewise, the ratio of paper thickness to Likewise, the ratio of paper thickness to average fiber thickness does not give a average fiber thickness does not give a precise value for number of layers.precise value for number of layers.

• Paper thickness decreases in wet Paper thickness decreases in wet processing and calendering, that these processing and calendering, that these cannot alter the number of fiber layers.cannot alter the number of fiber layers.

• We assume that the local coverage values, c', We assume that the local coverage values, c', are Poisson distributed for sufficiently small are Poisson distributed for sufficiently small reference areas.reference areas.

• If the average coverage is c, the Poisson If the average coverage is c, the Poisson distribution of c' isdistribution of c' is

P(c') P(c') = e= e(c'-c)(c'-c)/c'!, for c'/c'!, for c'00

= 0, otherwise.= 0, otherwise.

• P(c') is the probability of finding c' fibers P(c') is the probability of finding c' fibers covering a given point, when the average covering a given point, when the average coverage is c.coverage is c.

• The small reference area is assumed small on The small reference area is assumed small on the length scale of fibers.the length scale of fibers.

• Note that from the definition of coverage, the Note that from the definition of coverage, the Poisson distribution also gives a distribution of Poisson distribution also gives a distribution of local basis weights.local basis weights.

• The distribution can also be interpreted in terms The distribution can also be interpreted in terms of the number of fiberof the number of fibercenters within a unitcenters within a unitarea.area.

• The validity of theThe validity of thePoisson distribution isPoisson distribution isdemonstrated indemonstrated inFigure 3.Figure 3.

• From the definition of the Poisson From the definition of the Poisson distribution, the probability of finding an distribution, the probability of finding an empty reference area is eempty reference area is e-c-c..

• Thus, we expect the frequency of pin holes Thus, we expect the frequency of pin holes to decrease with increasing basis weight as to decrease with increasing basis weight as ee(-b/(-b/ff).).

• The probability of an area being covered by The probability of an area being covered by at least one fiber is 1-eat least one fiber is 1-e-c-c..

• This is dependent on the assumption of This is dependent on the assumption of randomness.randomness.

• At high average coverages, the Poisson At high average coverages, the Poisson distribution is similar to a Gaussian distribution distribution is similar to a Gaussian distribution (e(e-c-c22).).

• This is shown in Figure 4.This is shown in Figure 4.

• The importantThe importantdifference is that thedifference is that theGaussian distributionGaussian distributioncontains negativecontains negativevalues, but thevalues, but thePoisson distributionPoisson distributionis meaningless foris meaningless fornegative values.negative values.

Corte-Kallmes TheoryCorte-Kallmes Theory

• Corte and Kallmes analyzed the statistical Corte and Kallmes analyzed the statistical geometry of 2-d random fiber networks and geometry of 2-d random fiber networks and found good agreement with macroscopic found good agreement with macroscopic measurements of thin paper sheets.measurements of thin paper sheets.

• Their theory describes the distribution of Their theory describes the distribution of constant dimension fiber segments randomly constant dimension fiber segments randomly and isotropically distributed in a plane.and isotropically distributed in a plane.

• Of course, real fibers do not have a constant Of course, real fibers do not have a constant length or width.length or width.

• However, the average properties should be However, the average properties should be enough to specify for practical purposes.enough to specify for practical purposes.

• Consider the distribution of “fiber segments”.Consider the distribution of “fiber segments”.

• Fiber segments are defined as the the sections Fiber segments are defined as the the sections between crossings.between crossings.

• We define the segment length as the distance We define the segment length as the distance between the centroids.between the centroids.

• Consider a test fiber in a large network.Consider a test fiber in a large network.

• Divide its length into square sections.Divide its length into square sections.

• The number of sections is The number of sections is llff/w/wff..

• If the fiber width is small, we expect the Poisson If the fiber width is small, we expect the Poisson distribution to be valid.distribution to be valid.

• The probability of finding k adjacent sections The probability of finding k adjacent sections free of other fibers isfree of other fibers is

P(0)P(0)kk = e = e-kc-kc

• Thus, the frequency of a given free segment Thus, the frequency of a given free segment length, length, llfreefree = kw = kwff is is

P(P(llfreefree) = e) = e--llfreefreec/wc/wff

• This assumes that fibers cross at right angles This assumes that fibers cross at right angles and only at discrete locations.and only at discrete locations.

• Corte and Kallmes were able to obtain the Corte and Kallmes were able to obtain the precise equation for continuous random precise equation for continuous random locations and orientations.locations and orientations.

• For this case, a fiber in an area A crosses For this case, a fiber in an area A crosses another fiber, if the center of the area falls inside another fiber, if the center of the area falls inside the area defined in Figure 5.the area defined in Figure 5.

• The correspondingThe correspondingprobability is given byprobability is given by

PPcc(()=)=llff22sinsin/A/A

• The average over allThe average over allcrossing angles iscrossing angles is

ppcc = 2 = 2llff22//wwff

• For an area containing N fibers, the average For an area containing N fibers, the average number of crossings per fiber is given bynumber of crossings per fiber is given by

nncc=2N=2Nllff22//A=2cA=2cllff//wwff

• The crossings occur at random locations, so that The crossings occur at random locations, so that the average distance, lthe average distance, lss, between them is, between them is

llss==llff/n/ncc==wwff/2c/2c

• This assumes that the fibers are very long.This assumes that the fibers are very long.

• Corte and Kallmes obtained the corrected Corte and Kallmes obtained the corrected (relative to Poisson) probability(relative to Poisson) probability

P(lP(lss)=(2c/)=(2c/wwff)e)e-2c-2cllss//wwff

Network ConnectivityNetwork Connectivity

• The mechanical properties of paper are The mechanical properties of paper are controlled by the connectivity or bonding controlled by the connectivity or bonding degree.degree.

• The network would have no cohesion if there The network would have no cohesion if there were not enough bonds between the fibers.were not enough bonds between the fibers.

• Thus, we define the relative bonding area (RBA) Thus, we define the relative bonding area (RBA) as the bonded surface divided by the total fiber as the bonded surface divided by the total fiber surface area.surface area.

• In two dimensions, we assume bonding at every In two dimensions, we assume bonding at every fiber crossing.fiber crossing.

• For coverage c, the total surface area of fibers isFor coverage c, the total surface area of fibers is

2N2Nllffwwff = 2cA = 2cA

since each fiber has two sides, top and bottom.since each fiber has two sides, top and bottom.

• Likewise, the paper has two sides, except at Likewise, the paper has two sides, except at points of zero coverage.points of zero coverage.

• Thus, the area of unbonded surface is 2A(1-eThus, the area of unbonded surface is 2A(1-e-c-c) ) and thereforeand therefore

RBARBA =[2cA-2A(1-e=[2cA-2A(1-e-c-c)]/2cA)]/2cA

=1-(1-e=1-(1-e-c-c)/c)/cTo slide 39

• The expression on the previous slide represents The expression on the previous slide represents the average degree of bonding of the top and the average degree of bonding of the top and bottom of fibers.bottom of fibers.

• The corresponding RBAs for one sided and two The corresponding RBAs for one sided and two sided bonding aresided bonding aregiven bygiven by

B1=(1-eB1=(1-e-c-c)/c-e)/c-e-c-c

B2=(1-2/c)+(1+2/c)eB2=(1-2/c)+(1+2/c)e-c-c

• These are all shown inThese are all shown inFigure 6.Figure 6.

• The preceding treatment is approximate.The preceding treatment is approximate.

• Real paper is three dimensional, with a finite Real paper is three dimensional, with a finite thickness and pores available in the z direction.thickness and pores available in the z direction.

• These pores reduce the bonding degree from These pores reduce the bonding degree from the two dimensional estimate.the two dimensional estimate.

• The 2-d picture is valid for low coverage or high The 2-d picture is valid for low coverage or high fiber flexibility.fiber flexibility.

• An estimate for the maximum basis weight for An estimate for the maximum basis weight for “two-dimensional” paper is twice the average “two-dimensional” paper is twice the average basis weight of a single fiber or 2basis weight of a single fiber or 2ff=10-20 g/m=10-20 g/m22..

PercolationPercolation

• At low coverages, a connected fiber network At low coverages, a connected fiber network forms only if a sufficient number of bonds are forms only if a sufficient number of bonds are formed per fiber.formed per fiber.

• This occurs at the percolation threshold, the This occurs at the percolation threshold, the minimum number of bonds needed to connect the minimum number of bonds needed to connect the network.network.

• Below the percolation threshold, the network Below the percolation threshold, the network consists of several disjointed pieces.consists of several disjointed pieces.

• This is a concept from percolation theory, which This is a concept from percolation theory, which has found many applications in science, including has found many applications in science, including electrical connectivity and porous media.electrical connectivity and porous media.

• Computer simulations of 2d random networks Computer simulations of 2d random networks have determined a percolation threshold for have determined a percolation threshold for coverage ofcoverage of

cccc~5.7w~5.7wff//llff

• Thus, with coverage below cThus, with coverage below ccc, the network is not , the network is not

connected.connected.

• At cAt ccc, one crucial fiber exists the removal of , one crucial fiber exists the removal of

which splits the network into two parts.which splits the network into two parts.

• Obviously, any real paper sheet must be far Obviously, any real paper sheet must be far above this limit.above this limit.

• For most papermaking fibers, cFor most papermaking fibers, ccc<.1.<.1.

• Thus, usual coverage values of 5-20 are much Thus, usual coverage values of 5-20 are much higher.higher.

• It is possible to prepare a thin paper of basis It is possible to prepare a thin paper of basis weight 2.5g/mweight 2.5g/m22, with c=.5, which is still above , with c=.5, which is still above the percolation threshold.the percolation threshold.

• In terms of the number of number of bonds per In terms of the number of number of bonds per fiber, the percolation threshold occurs atfiber, the percolation threshold occurs at

nncccc~11.4/~11.4/=3.6=3.6

• This is significantly greater than 2, the absolute This is significantly greater than 2, the absolute minimum number for a connected chain, minimum number for a connected chain, because of the two dimensional nature of the because of the two dimensional nature of the network and the possibility of dangling ends.network and the possibility of dangling ends.

• The corresponding critical RBA isThe corresponding critical RBA is

RBARBAcc = c = ccc/2~2.8w/2~2.8wff//llff

• Clearly RBAc <<1, for ordinary papermaking Clearly RBAc <<1, for ordinary papermaking fibers.fibers.

• The remaining issue isThe remaining issue isreinforcement fibers.reinforcement fibers.

• Sometimes, the primary fibersSometimes, the primary fibersmay screen the reinforcementmay screen the reinforcementfibers from bonding to onefibers from bonding to oneanother (or vice/versa).another (or vice/versa).

• See Figure 7.See Figure 7.

Three-Dimensional NetworkThree-Dimensional Network• The 3d pore structure of paper controls the The 3d pore structure of paper controls the

density and optical properties directly.density and optical properties directly.

• It controls the mechanical properties indirectly It controls the mechanical properties indirectly through the RBA. through the RBA.

• The pore geometry is complex, because of the The pore geometry is complex, because of the intertwined network of fibers and the partial intertwined network of fibers and the partial flexibility of the fibers.flexibility of the fibers.

• We will call on results from computer We will call on results from computer simulations of particle packing for interpretation.simulations of particle packing for interpretation.

• We can measure the real pore size We can measure the real pore size distribution with a mercury porosimeter or a distribution with a mercury porosimeter or a gas phase BET instrument.gas phase BET instrument.

• These tell how the pores are distributed by These tell how the pores are distributed by size, but not by position.size, but not by position.

• The 3d network is either layered or “felted”The 3d network is either layered or “felted”

• The distinction is based on how the fibers The distinction is based on how the fibers are entangled, which determines the z are entangled, which determines the z directional properties of the paper.directional properties of the paper.

Statistical Pore GeometryStatistical Pore Geometry

• At low basis weights, a paper sheet is At low basis weights, a paper sheet is essentially 2-dimensional, because a essentially 2-dimensional, because a mechanical contact can form between two fibers mechanical contact can form between two fibers whenever their projections cross.whenever their projections cross.

• Some sheet area may be completely empty, Some sheet area may be completely empty, with no coverage.with no coverage.

• Recall that the frequency,Recall that the frequency,, of these vacancies , of these vacancies is given byis given by

= e= e-c-c

• This is appreciable when c<2.This is appreciable when c<2.

• If the basis weight of paper increases, the If the basis weight of paper increases, the standard deviation of local thickness increases standard deviation of local thickness increases as the square root of coverage, according to the as the square root of coverage, according to the Poisson Distribution.Poisson Distribution.

• Fibers must be bent more and more to make Fibers must be bent more and more to make contact with other fibers.contact with other fibers.

• At some point this bending is noAt some point this bending is nolonger possible and somelonger possible and someempty space opens up in theempty space opens up in thez-direction between fibers.z-direction between fibers.

• Figure 8 shows two cases ofFigure 8 shows two cases ofpore space in a handsheet.pore space in a handsheet.

to slide 77

• The porosity (fractional void volume) is higher The porosity (fractional void volume) is higher when the local basis weight is lower than the when the local basis weight is lower than the average.average.

• The porosity depends on how pores form as the The porosity depends on how pores form as the basis weight grows.basis weight grows.

• The results of aThe results of asimulation are shownsimulation are shownin Figure 9.in Figure 9.

• The simulationThe simulationsupports the linearsupports the linearrelationshiprelationship

p=pp=p(c-c(c-c00) for c>c) for c>c00

• In this relationship, pIn this relationship, p and c and c00 are constants, are constants,

which depend on the stiffness of the fibers, as which depend on the stiffness of the fibers, as well as on process variables such as wet well as on process variables such as wet pressing and calendering.pressing and calendering.

• cc00 gives the minimum number of fiber layers gives the minimum number of fiber layers

necessary for pore formation.necessary for pore formation.

• For real papermaking fibers, theFor real papermaking fibers, thecoverage threshold is ccoverage threshold is c00~2-10,~2-10,

depending on flexibility anddepending on flexibility andthickness to width ratio.thickness to width ratio.

• Figure 10 shows qualitatively howFigure 10 shows qualitatively howthe cross-section dimensionsthe cross-section dimensionsaffect pore number.affect pore number.

• The probability of a pore on a given unit area of a The probability of a pore on a given unit area of a sheet is small when the fiber thickness is small.sheet is small when the fiber thickness is small.

• The pore sizes of a paper sheet can be determined The pore sizes of a paper sheet can be determined by mercury or nitrogen measurements.by mercury or nitrogen measurements.

• The 3d network of pores can be envisioned as a The 3d network of pores can be envisioned as a collection of ellipsoidal pores, with narrow throats.collection of ellipsoidal pores, with narrow throats.

• Light diffraction measurements in 3d can yield the Light diffraction measurements in 3d can yield the average shape of these ellipsoidal pores.average shape of these ellipsoidal pores.

• The out of plane eccentricity (MD/ZD) ~ 2.2-2.6, The out of plane eccentricity (MD/ZD) ~ 2.2-2.6, while the in plane value ~ 1.1-1.6.while the in plane value ~ 1.1-1.6.

• Permeability measurements can yield the Permeability measurements can yield the approximate pore throat size.approximate pore throat size.

• Figure 11 shows how typical pore and the width Figure 11 shows how typical pore and the width of their distribution decreases with increasing of their distribution decreases with increasing fiber flexibility.fiber flexibility.

• Beating is a commonBeating is a commonmethod of increasing themethod of increasing theflexibility of chemical pulpflexibility of chemical pulpfibers.fibers.

• The increasing SR numberThe increasing SR numberindicates the accumulationindicates the accumulationof the beating action.of the beating action.

• The distributions in the figure resemble a log-The distributions in the figure resemble a log-normal distribution (with the exception of the normal distribution (with the exception of the binary mixture).binary mixture).

• Thus, the logarithm of the pore radius is Thus, the logarithm of the pore radius is approximately Gaussian.approximately Gaussian.

• The free span lengths can also characterize The free span lengths can also characterize pore sizes.pore sizes.

• The free span length is equivalent to the free The free span length is equivalent to the free segment length segment length llfreefree..

• Recall that lRecall that lfreefree obeys an exponential distribution. obeys an exponential distribution.

• The free span length in the thickness direction The free span length in the thickness direction can be measured from sheet cross-sections.can be measured from sheet cross-sections.

• This gives a distribution for the local lengths in This gives a distribution for the local lengths in the pore space.the pore space.

• This is not the same as the size distribution ofThis is not the same as the size distribution ofentire pores.entire pores.

• The local height of a poreThe local height of a poredoesn't determine the shapedoesn't determine the shapeof the three dimensionalof the three dimensionalpore.pore.

• Measurements of the localMeasurements of the localpore height distribution are shown in Figure 12.pore height distribution are shown in Figure 12.

• These measurements that the pore heights These measurements that the pore heights also satisfy an exponential distribution.also satisfy an exponential distribution.

• The deviations are at the limit of shallow The deviations are at the limit of shallow pore space, where the cross-sectional pore space, where the cross-sectional shape and uncollapsed lumen of fibers shape and uncollapsed lumen of fibers control.control.

• The shape of the pore height distributions The shape of the pore height distributions should be similar for different fibers, with should be similar for different fibers, with the length scale depending of fiber the length scale depending of fiber properties and sheet properties.properties and sheet properties.

Relative Bonded AreaRelative Bonded Area

• Recall that the RBA is the bonded surface Recall that the RBA is the bonded surface area divided by the total surface area.area divided by the total surface area.

• In the 2d picture the top and bottom In the 2d picture the top and bottom surfaces control the bonding degree.surfaces control the bonding degree.

• At high basis weights, the top and bottom of At high basis weights, the top and bottom of the sheet have reduced effect.the sheet have reduced effect.

• The RBA increases with increasing basis The RBA increases with increasing basis weight in the 2d picture.weight in the 2d picture.

• In the real 3d network the pore structure limits In the real 3d network the pore structure limits the growth of RBA with basis weight.the growth of RBA with basis weight.

• The equation for the RBA from The equation for the RBA from slide 20slide 20 is is generalized togeneralized to

RBA=1-(1+p)(1-RBA=1-(1+p)(1-)/c)/c

where p is the number of pores and where p is the number of pores and =e=e-c-c..

• This equation still ignores the z-directional This equation still ignores the z-directional projection of fiber surfaces.projection of fiber surfaces.

• The maximum fiber surface available for The maximum fiber surface available for bonding is still 2bonding is still 2llffwwff..

• At constant basis weight, the RBA depends on At constant basis weight, the RBA depends on the cross-sectional dimensions and flexibility of the cross-sectional dimensions and flexibility of fibers as shown in Figure 13.fibers as shown in Figure 13.

• Sheet consolidation inSheet consolidation inwet pressing andwet pressing anddrying also contribute.drying also contribute.

• In practice, RBA isIn practice, RBA iscontrolled by pulp type,controlled by pulp type,beating level and wetbeating level and wetpressing.pressing.

• The pore sizeThe pore sizedistribution is also important, causing the RBA distribution is also important, causing the RBA to decrease for large pores.to decrease for large pores.

Measurement of RBAMeasurement of RBA• The data in Figure 13 came directly form cross-The data in Figure 13 came directly form cross-

sectional images.sectional images.

• Preparation and measurement of these cross-Preparation and measurement of these cross-sections are tedious.sections are tedious.

• Indirect methods are usual; in the measurement Indirect methods are usual; in the measurement of RBA.of RBA.

• One can obtain the free surface of paper at the One can obtain the free surface of paper at the molecular level from gas adsorption molecular level from gas adsorption (Micromeritics TriStar in 2730).(Micromeritics TriStar in 2730).

• This gives the bonding degree irrespective of This gives the bonding degree irrespective of orientation.orientation.

• One method of determining the RBA uses the One method of determining the RBA uses the Kubelka-Munk light scattering coefficient, Kubelka-Munk light scattering coefficient, S[mS[m22/kg], of paper./kg], of paper.

• This gives the optically free surface area per This gives the optically free surface area per unit mass that has to be normalized by the unit mass that has to be normalized by the surface area of unbonded fibers.surface area of unbonded fibers.

• This normalization is important and unreliable This normalization is important and unreliable values of RBA are obtained without it.values of RBA are obtained without it.

• The light scattering method relies on the fact The light scattering method relies on the fact that a fiber surface element appears bonded if that a fiber surface element appears bonded if there is another fiber surface at a distance there is another fiber surface at a distance smaller than half the wavelength of light.smaller than half the wavelength of light.

• This doesn't guarantee that the two fibers are This doesn't guarantee that the two fibers are bonded chemically, since the bonding distance bonded chemically, since the bonding distance is shorter.is shorter.

• When applied to paper sheets ofWhen applied to paper sheets ofdifferent beating levels,different beating levels,gas adsorption area and lightgas adsorption area and lightscattering coefficient were seenscattering coefficient were seento be linearly related to oneto be linearly related to oneanother, as shown in Figure 14.another, as shown in Figure 14.

• Thus, S can be used toThus, S can be used torepresent changes in therepresent changes in thebonded area of paper.bonded area of paper.

• The remaining problem for calculation RBA is to The remaining problem for calculation RBA is to determine the light scattering coefficient for determine the light scattering coefficient for completely unbonded fibers, Scompletely unbonded fibers, S00..

• Then the RBA is given byThen the RBA is given by

RBA=1-S/SRBA=1-S/S00

• In beating trials, one frequently uses for SIn beating trials, one frequently uses for S00, the , the

value at which tensile strength versus S value at which tensile strength versus S extrapolates to zero.extrapolates to zero.

• However, this value is unreliable, because;However, this value is unreliable, because;

1.1.The surface area may change in beating, as The surface area may change in beating, as is the case for mechanical pulps and high is the case for mechanical pulps and high yield chemical pulps.yield chemical pulps.

2.2. Tensile strength can change in beating for Tensile strength can change in beating for reasons other than RBA.reasons other than RBA.

3.3. With decreasing RBA, tensile strength With decreasing RBA, tensile strength disappears at the percolation threshold, well disappears at the percolation threshold, well before RBA reaches zero.before RBA reaches zero.

• Light scattering from unbonded fibers, cross-Light scattering from unbonded fibers, cross-sectional images or some other correlation with sectional images or some other correlation with true network geometry is necessary to true network geometry is necessary to determine Sdetermine S00..

• This is generally not possible, and thus, RBA This is generally not possible, and thus, RBA cannot be reliably determined using light cannot be reliably determined using light scattering from paper.scattering from paper.

• The density of paper gives another indirect, The density of paper gives another indirect, qualitative measure of RBA.qualitative measure of RBA.

• Considering how changes in pore height Considering how changes in pore height distribution directly effect effect RBA, we can distribution directly effect effect RBA, we can assume that RBA depends linearly on density assume that RBA depends linearly on density according to:according to:

RBA=(RBA=(--00)/)/

where where 00 and and are positive constants. are positive constants.

• However, there is no verification of the validity However, there is no verification of the validity of this equation.of this equation.

• The constants are not known.The constants are not known.

Layered and Felted Sheet StructureLayered and Felted Sheet Structure• Papermaking fibers are 1-2 orders of magnitude Papermaking fibers are 1-2 orders of magnitude

longer than a typical sheet is thick.longer than a typical sheet is thick.

• Thus, most of the fiber length must be aligned in Thus, most of the fiber length must be aligned in the plane of the paper sheet.the plane of the paper sheet.

• The arrangement of fibers in the z-direction can The arrangement of fibers in the z-direction can be layered or felted.be layered or felted.

• A layered network forms if the fibers land on the A layered network forms if the fibers land on the wire one after another.wire one after another.

• The fibers form an ordered sequence in the The fibers form an ordered sequence in the vertical direction.vertical direction.

• In a felted structure there is no clear sequence.In a felted structure there is no clear sequence.

• In order to characterize the layering of a sheet, In order to characterize the layering of a sheet, one considers the vertical positions of a fiber in one considers the vertical positions of a fiber in successive cross-sections of the sheet.successive cross-sections of the sheet.

• In each cross-section, the fiber is assigned an In each cross-section, the fiber is assigned an ordering number, S, according to its position in ordering number, S, according to its position in the z direction.the z direction.

• These numbers are normalized by dividing by These numbers are normalized by dividing by the total number of fibers in each cross-section.the total number of fibers in each cross-section.

• The vertical order, h, of each fiber is the The vertical order, h, of each fiber is the average over the length of the fiber:average over the length of the fiber:

h=1/h=1/llf 0f 0

llffSdSdll

• A small value of h means the center of mass of A small value of h means the center of mass of the fiber is close to the bottom side of the sheet.the fiber is close to the bottom side of the sheet.

• A large value of h=1 means that it is close to the A large value of h=1 means that it is close to the top side.top side.

• The probability distribution of h for all fibers The probability distribution of h for all fibers characterize the degree ofcharacterize the degree oflayering in the sheet.layering in the sheet.

• In a layered structure, theIn a layered structure, thevalues of h have a uniformvalues of h have a uniformdistribution from 0-1, while in adistribution from 0-1, while in afelted structure some valuesfelted structure some valuesare more common than othersare more common than othersas shown in Figure 15.as shown in Figure 15.

• A typical layered network is formed in a A typical layered network is formed in a handsheet mold, when using low pulp handsheet mold, when using low pulp consistencies.consistencies.

• A felted structure is formed at high A felted structure is formed at high consistencies, or under pulsating drainage.consistencies, or under pulsating drainage.

• At high consistency, 3d fiber aggregates, or At high consistency, 3d fiber aggregates, or flocs, form in the suspension and then are flocs, form in the suspension and then are squeezed in the planar sheet.squeezed in the planar sheet.

• Fourdrinier and hybrid formers yield varying Fourdrinier and hybrid formers yield varying degrees of felting.degrees of felting.

• Gap formers produce a more layered paper Gap formers produce a more layered paper structure.structure.

• A felted sheet structure should give better out of A felted sheet structure should give better out of plane strength than a layered structure, because plane strength than a layered structure, because fibers are stronger than bonds.fibers are stronger than bonds.

• Thus, high consistency forming promotes out-of-Thus, high consistency forming promotes out-of-plane strength.plane strength.

• The effect of fiber entanglements are shown in The effect of fiber entanglements are shown in Figure 16.Figure 16.

• If one follows the loopIf one follows the loopof fibers in the rightof fibers in the righthand portion of thehand portion of thefigure, moving up at thefigure, moving up at theintersection, one ends up below the starting intersection, one ends up below the starting point.point.

FormationFormation• In addition to fibers, paper consists of fiber In addition to fibers, paper consists of fiber

fragments, mineral fillers and chemical fragments, mineral fillers and chemical additives.additives.

• In the web formation process, they all settle In the web formation process, they all settle stochastically onto the wire.stochastically onto the wire.

• Paper formation is the resulting nonuniform Paper formation is the resulting nonuniform distribution of particles.distribution of particles.

• More precisely, formation is the variability of the More precisely, formation is the variability of the basis weight of paper.basis weight of paper.

• Such variation can be easily seen with the Such variation can be easily seen with the naked eye, for some sheets.naked eye, for some sheets.

• The basis weight variation depends on:The basis weight variation depends on:

1.1. The randomness of single fiber distribution.The randomness of single fiber distribution.

2.2. Fiber interactions.Fiber interactions.

3.3. Flocculation - increases the variability of Flocculation - increases the variability of basis weight.basis weight.

4.4. And hydrodynamic forces in the web And hydrodynamic forces in the web forming process.forming process.

– Turbulence can decrease basis weight by Turbulence can decrease basis weight by breaking flocs.breaking flocs.

– ““Hydrodynamic smoothing” improves Hydrodynamic smoothing” improves sheet uniformity.sheet uniformity.

• The nonuniform basis weight distribution The nonuniform basis weight distribution affects many properties of paper.affects many properties of paper.

• Formation effects:Formation effects:

1.1. Print unevenness - resulting from local Print unevenness - resulting from local porosity.porosity.

2.2. Tensile strength.Tensile strength.

3.3. CocklingCockling

• In the case of strength and cockling, local In the case of strength and cockling, local basis weight variations are not separate from basis weight variations are not separate from the effects of local fiber orientation and dried-in the effects of local fiber orientation and dried-in strains.strains.

CharacterizationCharacterization

• A useful definition of formation is the small A useful definition of formation is the small scale basis weight variation in the plane of scale basis weight variation in the plane of the paper sheet.the paper sheet.

• This provides for simple measurement and This provides for simple measurement and unambiguous connection to paper unambiguous connection to paper structure.structure.

• Other terms used include mass formation, Other terms used include mass formation, mass distribution or the distribution of mass mass distribution or the distribution of mass density.density.

MeasurementsMeasurements• The traditional judgment of formation involves The traditional judgment of formation involves

looking through a paper sheet, which has led looking through a paper sheet, which has led people to call the visual impression formation.people to call the visual impression formation.

• A better term for this property is “look-through”.A better term for this property is “look-through”.

• Many optical formation testers have calibrations Many optical formation testers have calibrations to give results that correlate with the visual to give results that correlate with the visual impression.impression.

• Such testers may give misleading results, in the Such testers may give misleading results, in the sense that visual appearance is not equivalent sense that visual appearance is not equivalent to structural nonuniformity.to structural nonuniformity.

• Functional properties depend only on the latter.Functional properties depend only on the latter.

• Measurement of formation is almost always Measurement of formation is almost always indirect.indirect.

• The measured values must be calibrated to The measured values must be calibrated to basis weight.basis weight.

• A good method is to use A good method is to use -radiation for which -radiation for which the transmitted intensity decays exponentially the transmitted intensity decays exponentially with basis weight and the absorption coefficient with basis weight and the absorption coefficient is independent of furnish, for is independent of furnish, for -sources that emit -sources that emit no no -rays.-rays.

• CC1414, Pr, Pr147147 or Kr or Kr8585 are pure are pure -sources.-sources.

• X-rays can be used, but attenuation is greater X-rays can be used, but attenuation is greater for fillers than fibers.for fillers than fibers.

QuantificationQuantification

• It makes sense to describe the formation of paper It makes sense to describe the formation of paper in terms of the standard deviation of basis weight, in terms of the standard deviation of basis weight, bb..

• The specific formation fThe specific formation fNN is defined by is defined by

ffNN==bb//bb

where b is the average basis weight.where b is the average basis weight.

• Note the dimensions of fNote the dimensions of fNN are square root of basis are square root of basis

weight.weight.

• The coefficient of variationThe coefficient of variation

COV(b)=COV(b)=bb/b/b

is dimensionless.is dimensionless.

• Papers of different basis weight may be ranked Papers of different basis weight may be ranked differently if ranked by differently if ranked by bb, f, fNN or COV(b). or COV(b).

• In addition to amplitude of variation, formation In addition to amplitude of variation, formation has spatial characteristics.has spatial characteristics.

• The most important of these is length scale.The most important of these is length scale.

• The microscopic and specific perimeter (Figure The microscopic and specific perimeter (Figure 17) characterize the length scale.17) characterize the length scale.

• The specific perimeterThe specific perimeteris defined by theis defined by theboundary line betweenboundary line betweenareas where basisareas where basisweight is above orweight is above orbelow the mean value.below the mean value.

• The specific perimeter is the perimeter of the The specific perimeter is the perimeter of the borderline divided by the total area.borderline divided by the total area.

• Microscale is the averageMicroscale is the averageseparation in a given directionseparation in a given directionbetween the borderlines.between the borderlines.

• The power spectrum (Figure 18)The power spectrum (Figure 18)theoretically contains all of thetheoretically contains all of theinformation about the spatialinformation about the spatialvariation of basis weight.variation of basis weight.

• The spectrum shows how theThe spectrum shows how thevariance, variance, bb

22, distributes to, distributes to

different wavelengths.different wavelengths.

• Deviations from the theoretical shape indicate Deviations from the theoretical shape indicate that the formation is more “cloudy” or “grainy” that the formation is more “cloudy” or “grainy” than the random fiber network.than the random fiber network.

• The dashed curve shows how flocculation The dashed curve shows how flocculation increases fluctuations at long wavelengths.increases fluctuations at long wavelengths.

• The microscopic and average floc orientation The microscopic and average floc orientation can be derived from the spectrum.can be derived from the spectrum.

• The latter requires a two dimensional The latter requires a two dimensional spectrum, rather than the 1 dimensional spectrum, rather than the 1 dimensional spectrum of Figure 18.spectrum of Figure 18.

Formation of Random Fiber NetworkFormation of Random Fiber Network• The theory of formation in fiber networks derives The theory of formation in fiber networks derives

from the studies of Corte and Kallmes in the from the studies of Corte and Kallmes in the 1960s.1960s.

• Free drainage is assumed, so that every fiber Free drainage is assumed, so that every fiber settles onto the mat randomly and settles onto the mat randomly and independently.independently.

• This implies the Poisson statistics.This implies the Poisson statistics.

• For a given basis weight, the number of fibers in For a given basis weight, the number of fibers in an A isan A is

N=bA/N=bA/llffff

where where ff is the fiber coarseness (m/ is the fiber coarseness (m/ll).).

• If we set mIf we set mff==llffff, then we have, then we have

N=bA/mN=bA/mff

• For a Poisson distribution, the variance is N and For a Poisson distribution, the variance is N and the standard deviation of basis weight isthe standard deviation of basis weight is

bb==NmNmff/A/A

• Thus, the specific formation becomesThus, the specific formation becomes

ffNN==(m(mff/A)/A)

where A is interpreted as the square of the spatial where A is interpreted as the square of the spatial resolution of the measurement.resolution of the measurement.

To slide 66

• For this case, the only property that effects For this case, the only property that effects formation is the fiber mass as seen in Figure 19.formation is the fiber mass as seen in Figure 19.

• The effect of fiber mass on handsheet formation The effect of fiber mass on handsheet formation can be detected if stock consistency is can be detected if stock consistency is sufficiently low to prevent flocculation.sufficiently low to prevent flocculation.

• This discussion hasThis discussion hasassumed that allassumed that allfibers have the samefibers have the samemass and length andmass and length andthat the sheet isthat the sheet isisotropic.isotropic.

• Dodson has shown that a distribution of fiber Dodson has shown that a distribution of fiber masses or orientations does not significantly masses or orientations does not significantly alter alter bb..

• The important thing is the spatial resolution of The important thing is the spatial resolution of measurement,measurement,relative to the fiberrelative to the fiberlength and width.length and width.

• The measured The measured bb

decreases as thedecreases as thewindow sizewindow sizeincreases as shownincreases as shownin Figure 20.in Figure 20.

Forming MechanismsForming Mechanisms

• The actual formation deviates from the The actual formation deviates from the theoretical value given on theoretical value given on slide 63slide 63..

• This results from four mechanismsThis results from four mechanisms

1.1. Hydrodynamic smoothing.Hydrodynamic smoothing.

2.2. Flocculation.Flocculation.

3.3. Shear flow.Shear flow.

4.4. Turbulence.Turbulence.

• Hydrodynamic smoothing results from the fact Hydrodynamic smoothing results from the fact that suspension flow during drainage can that suspension flow during drainage can improve formation.improve formation.

• The flow rate through the settled fiber mat is The flow rate through the settled fiber mat is highest where the flow resistance is smallest, highest where the flow resistance is smallest, in areas of lowest basis weight.in areas of lowest basis weight.

• Thus, suspension flows toward areas of low Thus, suspension flows toward areas of low basis weight.basis weight.

• This is the mechanism of hydrodynamic This is the mechanism of hydrodynamic smoothing.smoothing.

• As a result, handsheets have better formation As a result, handsheets have better formation than completely random networks.than completely random networks.

• Fiber flocculation is the main factor in base Fiber flocculation is the main factor in base sheet formation.sheet formation.

• Consider clumps, or flocs, of several fibers as Consider clumps, or flocs, of several fibers as the basic constituents of paper.the basic constituents of paper.

• The standard deviation of basis weight The standard deviation of basis weight increases roughly as the increases roughly as the nnff, where n, where nff is the is the

average number of fibers in the floc.average number of fibers in the floc.

• Dotson showed that an estimate for nf isDotson showed that an estimate for nf is

NNff=(=(bmeasuredbmeasured//brandombrandom))22,,

when using a measurement window of A=lwhen using a measurement window of A=lff22..

• Suspension consistency, C, directly influences Suspension consistency, C, directly influences flocculation.flocculation.

• High consistency gives more flocculation High consistency gives more flocculation because of the high probability of fibers because of the high probability of fibers encountering one another.encountering one another.

• Kerekes and Schell used dimensional analysis Kerekes and Schell used dimensional analysis to estimate the consistency effect on to estimate the consistency effect on flocculation.flocculation.

• They considered the crowding numberThey considered the crowding number

nncrowdcrowd==ClClff22/6/6ff

where where ff is the fiber coarseness (mass/length). is the fiber coarseness (mass/length).

• The crowding number gives the average The crowding number gives the average number of fibers in a sphere of diameter equal number of fibers in a sphere of diameter equal to the fiber length.to the fiber length.

• High values give poor formation.High values give poor formation.

• Formation has been shown to increase Formation has been shown to increase exponentially with nexponentially with ncrowdcrowd for a single machine. for a single machine.

• Kerekes and Schell found a threshold for Kerekes and Schell found a threshold for flocculation for nflocculation for ncrowdcrowd~60.~60.

• With 2 mm fibers having coarseness of .2 mg/m, With 2 mm fibers having coarseness of .2 mg/m, the threshold occurs at C=5kg/mthe threshold occurs at C=5kg/m33 (5% in mass (5% in mass concentration).concentration).

• Formation depends on turbulence in the Formation depends on turbulence in the headbox and in the wire section, because headbox and in the wire section, because turbulence on a suitable length scale turbulence on a suitable length scale disrupts flocs.disrupts flocs.

• For Paper machine consistencies, For Paper machine consistencies, turbulence is necessary to prevent turbulence is necessary to prevent flocculation.flocculation.

• On the wire, adjusting dewatering elements On the wire, adjusting dewatering elements controls turbulence.controls turbulence.

• For Fourdriner machines, accessible formation For Fourdriner machines, accessible formation improvements are small, but hybrid and gap improvements are small, but hybrid and gap formers yield much greater formation formers yield much greater formation improvements as shown in Figure 21.improvements as shown in Figure 21.

• The effects of jet to wire speed difference on The effects of jet to wire speed difference on formation are shown in Figures 22 and 23.formation are shown in Figures 22 and 23.

• The speed difference creates a shear field The speed difference creates a shear field in the suspension that in the suspension that breaks flocs and breaks flocs and

causes causes turbulence.turbulence.

• The effect of turbulence can be positive or The effect of turbulence can be positive or negative.negative.

• On fourdriniers and hybrid formers, the optimum On fourdriniers and hybrid formers, the optimum formation occurs near zero speed difference formation occurs near zero speed difference (Figure 22).(Figure 22).

• On the other hand, for gap formers, the optimum On the other hand, for gap formers, the optimum is typically at a small speed difference.is typically at a small speed difference.

• On all former types, formation is poor if the On all former types, formation is poor if the speed difference is too large.speed difference is too large.

• Shear forces, between suspension and wire can Shear forces, between suspension and wire can become so large that the settled mat ruptures.become so large that the settled mat ruptures.

Paper Properties versus FormationPaper Properties versus Formation

• Here, we discuss the effect of formation on print Here, we discuss the effect of formation on print unevenness, or mottle.unevenness, or mottle.

• Local basis weight variation affects print density Local basis weight variation affects print density through local porosity and associated through local porosity and associated permeability.permeability.

• Later we will discuss the effect on tensile Later we will discuss the effect on tensile strength and cockling.strength and cockling.

• For these, formation couples with local fiber For these, formation couples with local fiber orientation and dried in strains.orientation and dried in strains.

• Print mottle results from spatial variation in Print mottle results from spatial variation in surface roughness and from nonuniform ink surface roughness and from nonuniform ink penetration associated with spatial variation of penetration associated with spatial variation of local permeability.local permeability.

• The details vary for different print processes, but The details vary for different print processes, but these effects remain for all processes.these effects remain for all processes.

• High roughness reduces the effective contact High roughness reduces the effective contact area between the image carrier and the paper, area between the image carrier and the paper, thereby varying the ink film thickness.thereby varying the ink film thickness.

• High (optical) print density (proportional to ink High (optical) print density (proportional to ink film thickness) results from low ink penetration film thickness) results from low ink penetration into the sheet.into the sheet.

• Likewise, low density results from high Likewise, low density results from high penetration into the sheet, which is associated penetration into the sheet, which is associated with high permeability of the paper.with high permeability of the paper.

• Thus, we expect mottle, variation in ink density, Thus, we expect mottle, variation in ink density, to result from local variations in permeability.to result from local variations in permeability.

• Generally, local variations in basis weight, will Generally, local variations in basis weight, will reflect local variations in permeability.reflect local variations in permeability.

• Recall Recall Figure 8Figure 8, where the cross-section clearly , where the cross-section clearly suggest larger permeability in the thinner suggest larger permeability in the thinner sections.sections.

• Generally, calendering has a profound effect on Generally, calendering has a profound effect on both roughness and permeability.both roughness and permeability.

• As seen in Figure 24, the correlation between As seen in Figure 24, the correlation between local print density and local basis weight local print density and local basis weight variation increasesvariation increaseswith increasingwith increasingcalendering pressure.calendering pressure.

• The correlationThe correlationbecomes stronger atbecomes stronger athigh ink coverage, forhigh ink coverage, forthe highestthe highestcalendering calendering impression.impression.

• Other factors can mask the effect of formation Other factors can mask the effect of formation on print mottle, where comparing between on print mottle, where comparing between different paper machines and printing devices.different paper machines and printing devices.

• Figures 25 and 26 show how a weak correlation Figures 25 and 26 show how a weak correlation with formation may be seen with different with formation may be seen with different machines, while a strong correlation exists machines, while a strong correlation exists when comparing from the samewhen comparing from the samemachine.machine.

• The correlationThe correlationis evident foris evident forboth, but moreboth, but morepronounced forpronounced forFigure 26.Figure 26.

Illusory EffectsIllusory Effects

• Sometimes the apparent correlation between Sometimes the apparent correlation between formation and other properties.formation and other properties.

• For example, fluid permeability and opacity are For example, fluid permeability and opacity are nonlinear functions of basis weight.nonlinear functions of basis weight.

• Both are very large at low basis weight.Both are very large at low basis weight.

• Thus, opacity andThus, opacity andpermeability increasepermeability increaseas the formation getsas the formation getsworse as shown inworse as shown inFigure 27.Figure 27.

• In fact formation induced small scale In fact formation induced small scale variations in opacity are more harmful than variations in opacity are more harmful than a small decrease in average opacity at a small decrease in average opacity at good formation.good formation.

• Similarly, bending stiffness increases with Similarly, bending stiffness increases with poor formation.poor formation.

• This results from the nonlinear dependence This results from the nonlinear dependence of bending stiffness on formation and the of bending stiffness on formation and the fact that heavier areas contribute fact that heavier areas contribute proportionately than lighter areas.proportionately than lighter areas.

• The Bendtsen roughness, which is sensitive to The Bendtsen roughness, which is sensitive to larger scale roughness because it uses a hard larger scale roughness because it uses a hard measuring head, shows a strong correlation measuring head, shows a strong correlation with formation.with formation.

• On the other hand, the Parker Print Surf On the other hand, the Parker Print Surf roughness, which is sensitive to micro roughness, which is sensitive to micro roughness because it uses a soft measuring roughness because it uses a soft measuring head, shows no correlation with formation.head, shows no correlation with formation.

• Other properties, such as compression Other properties, such as compression strength and Kodak bending stiffness, show strength and Kodak bending stiffness, show high variability because of poor formation.high variability because of poor formation.

Fiber OrientationFiber Orientation• For machine made papers, the fibers tend to For machine made papers, the fibers tend to

align with the machine direction.align with the machine direction.

• We refer to this anisotropy as fiber orientation.We refer to this anisotropy as fiber orientation.

• Here, we consider the basic mechanisms of Here, we consider the basic mechanisms of this orientation.this orientation.

• The The fiber orientation indexfiber orientation index and the and the fiber fiber orientation angleorientation angle are usually used to are usually used to characterize the in-plane orientation characterize the in-plane orientation distribution.distribution.

• The orientation index gives the anisotropy, The orientation index gives the anisotropy, or eccentricity, of the distribution.or eccentricity, of the distribution.

• It is 1 for an isotropic sheet, such as a It is 1 for an isotropic sheet, such as a handsheet and increases for increasing handsheet and increases for increasing anisotropy.anisotropy.

• The orientation angle indicates how much The orientation angle indicates how much the symmetry axis of the distribution the symmetry axis of the distribution deviates form the machine direction.deviates form the machine direction.

• The fiber orientation of paper directly effects the The fiber orientation of paper directly effects the in-plane mechanical properties and dimensional in-plane mechanical properties and dimensional stability of paper.stability of paper.

• The anisotropic properties also depend on The anisotropic properties also depend on drying shrinkage and wet straining of the paper drying shrinkage and wet straining of the paper web.web.

• Generally, a low orientation index is best for Generally, a low orientation index is best for paper performance, since this makes paper paper performance, since this makes paper properties similar in all in-plane directions.properties similar in all in-plane directions.

• The fiber orientation effects the tensile strength The fiber orientation effects the tensile strength in the machine and cross directions.in the machine and cross directions.

• The variation of fiber orientation through the The variation of fiber orientation through the thickness of the sheet is also important for thickness of the sheet is also important for paper performance.paper performance.

• Orientation difference between the sheet Orientation difference between the sheet surface and middle layers affects bending surface and middle layers affects bending stiffness.stiffness.

• The difference between the surfaces (two The difference between the surfaces (two sidedness) causes curl.sidedness) causes curl.

• The fibers also have a small component in the z The fibers also have a small component in the z direction.direction.

• This is completely different from the in plane This is completely different from the in plane orientation.orientation.

Laminar Shear on the WireLaminar Shear on the Wire

• Several hydrodynamic forces affect the Several hydrodynamic forces affect the distribution of fiber orientations in paper.distribution of fiber orientations in paper.

• The most important is the velocity difference The most important is the velocity difference between the suspension jet and the wire.between the suspension jet and the wire.

• Other significant effects are:Other significant effects are:

1.1. The acceleration and deceleration of the The acceleration and deceleration of the suspension flow in the headbox and wire suspension flow in the headbox and wire section, respectively.section, respectively.

2.2. The random disturbances from turbulence.The random disturbances from turbulence.

• The speed difference between the jet and wire The speed difference between the jet and wire creates a z-directional velocity gradient, or creates a z-directional velocity gradient, or shear field.shear field.

• The shear field rotates theThe shear field rotates thefibers toward the machinefibers toward the machinedirection.direction.

• A large speed differenceA large speed differenceyields a significant fiberyields a significant fiberorientation in the machineorientation in the machinedirection.direction.

• This is reflected in theThis is reflected in theanisotropy of in-plane paper properties as seen anisotropy of in-plane paper properties as seen in Figure 28.in Figure 28. to slide 103

• The distribution of fiber orientation is related to The distribution of fiber orientation is related to the velocity difference between the suspension the velocity difference between the suspension and wire.and wire.

• To understand this relationship, we consider an To understand this relationship, we consider an idealized mode.idealized mode.

• We assume fibers are straight and stiff, their We assume fibers are straight and stiff, their consistency is low and the flow is laminar.consistency is low and the flow is laminar.

• With laminar flow, the suspension velocity With laminar flow, the suspension velocity relative to the wire is in the machine direction.relative to the wire is in the machine direction.

• In the vertical direction, it decreases In the vertical direction, it decreases continuously to zero at the wire.continuously to zero at the wire.

• During drainage, one end of a fiber comes down During drainage, one end of a fiber comes down and adheres to the already filtered mat.and adheres to the already filtered mat.

• Figure 29 illustrates the viscous drag exerted on Figure 29 illustrates the viscous drag exerted on a fiber due to the flow.a fiber due to the flow.

• The rate of change of the fiber orientation is The rate of change of the fiber orientation is given bygiven by

dd/dt=-/dt=-vvsssinsin

where vwhere vss is the velocity is the velocity

difference in thedifference in themachine direction andmachine direction and is a constantis a constantproportional to theproportional to theviscosity.viscosity.

• The solution to this differential equation isThe solution to this differential equation is

tan(tan(/2)=tan(/2)=tan(00/2)e/2)e--vvsstt

where where 00 is the initial value of the angle. is the initial value of the angle.

• This solution applies while the fiber rotates freely This solution applies while the fiber rotates freely before it contacts the mat.before it contacts the mat.

• If If 00 is assumed to have an isotropic distribution is assumed to have an isotropic distribution

and t is taken to be a characteristic time before and t is taken to be a characteristic time before the fiber contacts the mat, then we obtain the the fiber contacts the mat, then we obtain the elliptic distribution:elliptic distribution:

f(f()=1/)=1/(1-q(1-q22)/(1+q)/(1+q22-2qcos(2-2qcos(2))))

where q=tanhwhere q=tanh22((vvss), ), being a constant. being a constant.

• This distribution is shown in Figure 30 for two This distribution is shown in Figure 30 for two different values of q.different values of q.

• An alternate derivationAn alternate derivationleads to an ellipticalleads to an ellipticaldistribution, but with adistribution, but with adifferent q given bydifferent q given by

q=2tanq=2tan22[tan[tan-1-1(2(2vvss)/2])/2]

• Other distributions areOther distributions areoften used tooften used tocharacterize measurements.characterize measurements.

• The most common alternative is the von Mises The most common alternative is the von Mises distribution.distribution.

• Any fiber orientation distribution that is symmetric Any fiber orientation distribution that is symmetric about the machine direction can be expressed as about the machine direction can be expressed as a Fourier seriesa Fourier series

f(f()=1/)=1/[1+[1+aanncos(2ncos(2n)])]

• The elliptical distribution has simple Fourier The elliptical distribution has simple Fourier coefficients, acoefficients, ann=2q=2qnn..

• These are important, since the MD/CD ratio of These are important, since the MD/CD ratio of elastic modulus of tensile strength is given byelastic modulus of tensile strength is given by

R=ER=EMDMD/E/ECDCD=(6+4a=(6+4a11+a+a22)/(6-4a)/(6-4a11+a+a22))

• This ratio, R, serves as an orientation index that This ratio, R, serves as an orientation index that measures the degree of orientation anisotropy.measures the degree of orientation anisotropy.

to slide 127

• When using this index, it is important to When using this index, it is important to remember that drying creates anisotropic remember that drying creates anisotropic internal stresses that also effect R.internal stresses that also effect R.

• Figure 31 shows how R changes with the Figure 31 shows how R changes with the suspension-to-wiresuspension-to-wirespeed difference vspeed difference vss

using the ellipticalusing the ellipticaldistribution.distribution.

• Comparison withComparison withFigure 28 indicatesFigure 28 indicatesthat this modelthat this modeloverestimates R atoverestimates R atlarge vlarge vss.. to slide 99 to slide 100

Effect of Fiber PropertiesEffect of Fiber Properties

• The derivation in the previous section is The derivation in the previous section is independent of fiber length.independent of fiber length.

• The equation for the angular motion is The equation for the angular motion is independent of fiber length if the shear field independent of fiber length if the shear field dvdvss/dz is constant./dz is constant.

• Yet it is believed that long fibers orient more Yet it is believed that long fibers orient more readily than short fibers.readily than short fibers.

• There must be some dependence on fiber There must be some dependence on fiber length for short fibers, but experiments on length for short fibers, but experiments on Formette Dynamique sheet mold suggest that Formette Dynamique sheet mold suggest that there is no dependence for long enough fibers.there is no dependence for long enough fibers.

• For real papermaking condition, the effect of fiber For real papermaking condition, the effect of fiber length may differ form that of the laminar model.length may differ form that of the laminar model.

• The suspension is generally not dilute and hence The suspension is generally not dilute and hence fiber interactions are important.fiber interactions are important.

• Fiber orientation must depend on fiber stiffness Fiber orientation must depend on fiber stiffness and flocculation and their coupling to turbulence.and flocculation and their coupling to turbulence.

• The effect of turbulence undoubtedly depends on The effect of turbulence undoubtedly depends on fiber length, because it changes the effective fiber length, because it changes the effective Reynolds number.Reynolds number.

• The sheets for the Formette Dynamique are free The sheets for the Formette Dynamique are free of flocculation because the suspension is of flocculation because the suspension is sprayed on a rotating wire in thin layers.sprayed on a rotating wire in thin layers.

• Fiber curl also influences the relationship Fiber curl also influences the relationship between paper anisotropy and fiber orientation.between paper anisotropy and fiber orientation.

• Curly fibers give less anisotropy than straight Curly fibers give less anisotropy than straight fibers.fibers.

• Fiber interactions and flocculation resist the Fiber interactions and flocculation resist the rotation of fibers.rotation of fibers.

• Thus, fibers may bend during drainage.Thus, fibers may bend during drainage.

• The curl of fibers as a function of orientation can The curl of fibers as a function of orientation can be described in terms of hindered rotation.be described in terms of hindered rotation.

Other Hydrodynamic EffectsOther Hydrodynamic Effects• Fiber orientation depends on other hydrodynamic Fiber orientation depends on other hydrodynamic effects other than laminar shear.effects other than laminar shear.

• One factor is the anisotropic orientation distribution One factor is the anisotropic orientation distribution in the jet as it emerges from the headbox.in the jet as it emerges from the headbox.

• This is caused by the converging slice channel that This is caused by the converging slice channel that accelerates the suspension.accelerates the suspension.

• Without turbulence, the elliptical orientation Without turbulence, the elliptical orientation distribution hasdistribution has

q=qq=qjetjet=(k-1)/(k+1)=(k-1)/(k+1)

where k=(Awhere k=(Ainin/A/Aoutout))1/21/2, with A, with Ain/outin/out being the areas of in being the areas of in

and out channels, respectively.and out channels, respectively.

• As a result of the anisotropic jet, the initial fiber As a result of the anisotropic jet, the initial fiber orientation distribution is no longer isotropic.orientation distribution is no longer isotropic.

• Therefore, the final fiber orientation distribution Therefore, the final fiber orientation distribution is not elliptical.is not elliptical.

• Curve II in Curve II in Figure 31Figure 31 shows how the distribution shows how the distribution changes as a result of the jet anisotropy.changes as a result of the jet anisotropy.

• Another factor omitted form the laminar shear Another factor omitted form the laminar shear analysis is turbulence.analysis is turbulence.

• Turbulence destroys jet anisotropy from the Turbulence destroys jet anisotropy from the suspension.suspension.

• Thus with turbulence, the anisotropic jet affects Thus with turbulence, the anisotropic jet affects fiber orientation only on the wire side on a fiber orientation only on the wire side on a Fourdrinier and on the two surfaces in a gap Fourdrinier and on the two surfaces in a gap former.former.

• In addition, turbulence smears out global In addition, turbulence smears out global average fiber orientation by inciting random average fiber orientation by inciting random fluctuations in the local speed difference.fluctuations in the local speed difference.

• The effect of turbulence on the orientation The effect of turbulence on the orientation distribution can be qualitatively described as an distribution can be qualitatively described as an average of elliptical distributions, f(average of elliptical distributions, f(++,v,vss++vvss) )

over over and and vvss..

• Figure 31Figure 31 shows that turbulence during shows that turbulence during drainage limits R for large velocity differences.drainage limits R for large velocity differences.

• Pulsating drainage generates turbulence in the Pulsating drainage generates turbulence in the headbox and on the wire.headbox and on the wire.

• Disintegration of flocs and other viscous forces Disintegration of flocs and other viscous forces rapidly dampen small scale turbulence.rapidly dampen small scale turbulence.

• Small eddies merge and coalesce into larger Small eddies merge and coalesce into larger vortices.vortices.

• Large scale fluctuations survive longer than Large scale fluctuations survive longer than short ones.short ones.

• The turbulence generated by drainage elements The turbulence generated by drainage elements is important on slow Fourdrinier machines and is important on slow Fourdrinier machines and turbulence in the headbox is important for high turbulence in the headbox is important for high speed gap formers.speed gap formers.

• Small scale turbulence is necessary during Small scale turbulence is necessary during drainage to prevent flocculation.drainage to prevent flocculation.

• Thus, intensive turbulence might be Thus, intensive turbulence might be expected to improve paper formation and expected to improve paper formation and reduce fiber orientation anisotropy.reduce fiber orientation anisotropy.

• In reality, it is not so simple.In reality, it is not so simple.

• The relationship between local fiber The relationship between local fiber orientation and basis weight depends on orientation and basis weight depends on details of the forming section of the paper details of the forming section of the paper machine.machine.

• The only feature missing from the fiber The only feature missing from the fiber orientation mechanism is the asymmetry orientation mechanism is the asymmetry between “rush” (vbetween “rush” (vss>0) and “drag” (v>0) and “drag” (vss<0) in <0) in

Figure 28Figure 28..

• There must be something that breaks the There must be something that breaks the symmetry between the two cases.symmetry between the two cases.

• It may be due to deceleration of suspension It may be due to deceleration of suspension during drainage in rush and drag conditions.during drainage in rush and drag conditions.

• This increases the effective speed difference in This increases the effective speed difference in drag and decreases it in rush.drag and decreases it in rush.

• This is consistent with the asymmetry seen in This is consistent with the asymmetry seen in Figure 28Figure 28..

• In gap formers, the deceleration of suspension In gap formers, the deceleration of suspension speed is easy to understand from the nip speed is easy to understand from the nip pressure.pressure.

• The Bernoulli equationThe Bernoulli equation1/21/2vv22=-=-pp

implies that a pressureimplies that a pressurepulse changes pulse changes thetheabsolute velocity of theabsolute velocity of thesuspension.suspension.

• On Fourdriniers, drainageOn Fourdriniers, drainagepulses cause decelerationpulses cause decelerationas shown in Figure 32 foras shown in Figure 32 formoderate rush.moderate rush.

• Viscous forces push the suspension speed Viscous forces push the suspension speed toward the wire speed during drainage.toward the wire speed during drainage.

• This is consistent with the suspension This is consistent with the suspension speed not decreasing in drag conditions in speed not decreasing in drag conditions in Figure 32.Figure 32.

• Systematic changes in the suspension Systematic changes in the suspension speed and turbulence during drainage also speed and turbulence during drainage also contribute to the two-sidedness of the fiber contribute to the two-sidedness of the fiber orientation.orientation.

Orientation AngleOrientation Angle• If the suspension flows on the wire exactly in the If the suspension flows on the wire exactly in the

machine direction then the orientation must be machine direction then the orientation must be symmetric, f(symmetric, f()=f(-)=f(-), relative to the machine ), relative to the machine direction.direction.

• Thus, the average orientation direction <Thus, the average orientation direction <>=o.>=o.

• In reality, the average orientation angle is In reality, the average orientation angle is nonzero.nonzero.

• The orientation angle defines the effective The orientation angle defines the effective symmetry axis, or the maximum in the symmetry axis, or the maximum in the orientation distribution with f(orientation distribution with f(-<-<>)=f(<>)=f(<>->-).).

• <<> is also called the misalignment angle.> is also called the misalignment angle.

• The direction along which fibers tend to align The direction along which fibers tend to align during drainage determines the fiber orientation during drainage determines the fiber orientation angle of the paper.angle of the paper.

• It depends on the direction of suspension flow It depends on the direction of suspension flow relative to the wire.relative to the wire.

• We assume that theWe assume that thesuspension velocitysuspension velocityhas a transversehas a transversecomponent, as showncomponent, as shownin Figure 33.in Figure 33.

• This may come from aThis may come from asmall misalignment in the suspension velocity.small misalignment in the suspension velocity.

• Since the speed of the suspension, vSince the speed of the suspension, vss, is close , is close

to the speed of the wire, vto the speed of the wire, vww, the misalignment , the misalignment

angle can be very large.angle can be very large.

• In the absence of fluctuations, the misalignment In the absence of fluctuations, the misalignment angle is given byangle is given by

<<>=tan>=tan-1-1(v(vCDCD/v/vss))

where vwhere vCDCD is the transverse component of the is the transverse component of the

suspension velocity.suspension velocity.

• If the magnitude of vIf the magnitude of vss goes to zero, then the goes to zero, then the

misalignment can approach +/- 90 degrees.misalignment can approach +/- 90 degrees.

• The actual dependence of misalignment angle on The actual dependence of misalignment angle on Jet to speed ratio is shown in Figure 34 for a Jet to speed ratio is shown in Figure 34 for a hybrid former.hybrid former.

• The figure indicatesThe figure indicatesthat the orientationthat the orientationangle goes throughangle goes throughzero at vzero at vss=0, indicating=0, indicating

that vthat vCDCD goes to zero goes to zero

faster than vfaster than vss for this for this

machine.machine.

• The excursions to large positive or negative The excursions to large positive or negative values reflect the sensitivity to small changes in values reflect the sensitivity to small changes in vvss or v or vCDCD near v near vss=0.=0.

CD VariationsCD Variations

• For all machine made paper, there are large For all machine made paper, there are large scale variations in fiber orientation angle across scale variations in fiber orientation angle across the web.the web.

• The actual average value is not as important as The actual average value is not as important as the profile of variations in the CD direction.the profile of variations in the CD direction.

• This is because problems in paper performance, This is because problems in paper performance, such as curl or cockling, occur when the such as curl or cockling, occur when the orientation angle varies too rapidly across the orientation angle varies too rapidly across the web.web.

• For fiber orientation index, the average value is For fiber orientation index, the average value is of greater importance.of greater importance.

• Large scale variations in <Large scale variations in <> across the paper > across the paper web result from uneven suspension flow across web result from uneven suspension flow across a wide machine.a wide machine.

• Cross directional flow components are easily Cross directional flow components are easily created inside the converging slice channel of created inside the converging slice channel of the headbox.the headbox.

• This nonuniform flow in the headbox in turn This nonuniform flow in the headbox in turn generates variations in basis weight profile that generates variations in basis weight profile that may be even more detrimental.may be even more detrimental.

• The local slice opening can be adjusted to The local slice opening can be adjusted to produce a more uniform basis weight profile, produce a more uniform basis weight profile, since a larger opening corresponds to higher since a larger opening corresponds to higher basis weight.basis weight.

• However, this adjustment may cause cross However, this adjustment may cause cross directional flows and fiber orientation deviations directional flows and fiber orientation deviations in surrounding areas of the web.in surrounding areas of the web.

• The shrinkage profile resulting from uneven The shrinkage profile resulting from uneven drying of the paper web is a common cause of drying of the paper web is a common cause of variation in fiber orientation angle across the variation in fiber orientation angle across the web.web.

• The edge areas shrink more than the the central The edge areas shrink more than the the central areas.areas.

• The higher shrinkage increases the local basis The higher shrinkage increases the local basis weight on the edges.weight on the edges.

• If the slice opening in the central jets are If the slice opening in the central jets are increased to compensate for the edges, an S-increased to compensate for the edges, an S-shaped orientationshaped orientation

• angle profile results.angle profile results.

• This is shown inThis is shown inFigure 35.Figure 35.

• Local dilution of theLocal dilution of thesuspension in thesuspension in theheadbox allows theheadbox allows thedecoupling of thedecoupling of theorientation angleorientation angleprofile from the basisprofile from the basisweight profile.weight profile.

Fiber Orientation DistributionFiber Orientation Distribution

• The fiber orientation in paper has two aspects.The fiber orientation in paper has two aspects.

1.1. Fiber orientation in the plane of the sheet Fiber orientation in the plane of the sheet can vary through the thickness of the sheet.can vary through the thickness of the sheet.

2.2. Fibers may z-directional orientation.Fibers may z-directional orientation.

• These two aspects are totally independent and These two aspects are totally independent and effect the paper properties in different ways.effect the paper properties in different ways.

• First we consider the z-directional orientation.First we consider the z-directional orientation.

• Because of their high aspect ratio, most of the Because of their high aspect ratio, most of the fiber length is aligned in the plane of the sheet.fiber length is aligned in the plane of the sheet.

• The z direction orientation angle, The z direction orientation angle, , relative to , relative to the plane is limited bythe plane is limited by

tantan-1-1(d/l(d/lff))d/ld/lff

where d is the paper thickness.where d is the paper thickness.

• Even though Even though is small, this is is small, this isstill described as z directionalstill described as z directionalfiber orientation.fiber orientation.

• This is misleading, since onlyThis is misleading, since onlyfiber segments can havefiber segments can haveappreciable orientation in theappreciable orientation in thez-direction as shown in Figurez-direction as shown in Figure36.36.

• Thus, the z-directional fiber orientation is Thus, the z-directional fiber orientation is actually fiber segment orientation.actually fiber segment orientation.

• It is closely related to “undulations” of fibers in It is closely related to “undulations” of fibers in the z-direction.the z-direction.

• For machine made paper, there is a nontrivial For machine made paper, there is a nontrivial asymmetry in the z-directional strength.asymmetry in the z-directional strength.

• If measured by pealing in the MD or CD, there is If measured by pealing in the MD or CD, there is little difference between the two.little difference between the two.

• However the MD value often depends on However the MD value often depends on whether the delamination is run in the MD or the whether the delamination is run in the MD or the -MD.-MD.

• This is shown in Figure 37 this difference This is shown in Figure 37 this difference between the “up-stream” and “down-stream” between the “up-stream” and “down-stream” values.values.

• This results fromThis results froman asymmetry inan asymmetry inthe z-directionalthe z-directionalorientation angleorientation anglein the MD-CDin the MD-CDplane.plane.

• The cause of thisThe cause of thisasymmetry isasymmetry isunknown.unknown.

Two-SidednessTwo-Sidedness• Consider the variation of the orientation through Consider the variation of the orientation through

the sheet thickness.the sheet thickness.

• Such variation is caused by changes in the flow Such variation is caused by changes in the flow state of the suspension, especially the level of state of the suspension, especially the level of turbulence and orienting shear during drainage.turbulence and orienting shear during drainage.

• The prevailing flow conditions when a particular The prevailing flow conditions when a particular layer drains determines the layer structure.layer drains determines the layer structure.

• Image analysis techniques can detect the Image analysis techniques can detect the layered fiber orientation.layered fiber orientation.

• This gives structural information about the paper This gives structural information about the paper and gives a clue to the sheet forming dynamics.and gives a clue to the sheet forming dynamics.

• The layered fiber orientation structures are The layered fiber orientation structures are sensitive to former type.sensitive to former type.

• Figures 38 and 39 show characteristic z-Figures 38 and 39 show characteristic z-directional profiles for three former types.directional profiles for three former types.

• The degree of fiber orientation is plotted as a The degree of fiber orientation is plotted as a function of cumulative basis weight from the function of cumulative basis weight from the bottom to the top.bottom to the top.

• The figures show “drag”, vThe figures show “drag”, vss < 0 and “rush”, v < 0 and “rush”, vss>0 >0

conditions.conditions.

• For the Fourdriner, the fiber orientation index For the Fourdriner, the fiber orientation index changes systematically from top to bottom.changes systematically from top to bottom.

• These machines show strong two-sidedness in These machines show strong two-sidedness in fiber orientation.fiber orientation.

• This z-directional trend of the fiber orientation in This z-directional trend of the fiber orientation in rush and drag is consistent with a decreasing rush and drag is consistent with a decreasing suspension velocity during drainage.suspension velocity during drainage.

• Near the top surface, viscous forces cause the Near the top surface, viscous forces cause the suspension velocity to approach that of the wire, suspension velocity to approach that of the wire, with the resulting decrease in anisotropy.with the resulting decrease in anisotropy.

• At the bottom surface, the anisotropy is low At the bottom surface, the anisotropy is low because of turbulence during formation.because of turbulence during formation.

• For the gap forming case, the layered fiber For the gap forming case, the layered fiber orientation is symmetric.orientation is symmetric.

• The jet impinges into a curved gap created by The jet impinges into a curved gap created by two wires that converge around a forming roll.two wires that converge around a forming roll.

• The vacuum level inside the forming roll affects The vacuum level inside the forming roll affects the top side of the paper.the top side of the paper.

• Wire tension creates a static pressure in the Wire tension creates a static pressure in the gap that controls dewatering on the bottom.gap that controls dewatering on the bottom.

• The deceleration of the jet in the gap The deceleration of the jet in the gap explains the local minimum of anisotropy in explains the local minimum of anisotropy in the center of the sheet of the sheet under the center of the sheet of the sheet under rush conditions.rush conditions.

• These layers are the ones that form last.These layers are the ones that form last.

• The low level of anisotropy on both surfaces The low level of anisotropy on both surfaces is again due to the high rate of dewatering is again due to the high rate of dewatering and high level of turbulence.and high level of turbulence.

• For the hybrid former, 2/3 of the sheet For the hybrid former, 2/3 of the sheet starting from the bottom side dewater under starting from the bottom side dewater under Fourdrinier conditions.Fourdrinier conditions.

• The hybrid unit drains the other one-third of The hybrid unit drains the other one-third of the sheet at high rate, low turbulence and the sheet at high rate, low turbulence and moderate shear.moderate shear.

• The magnitude of these effects depend on The magnitude of these effects depend on the type of hybrid unit.the type of hybrid unit.

Measurement TechniquesMeasurement Techniques• Accurate measurement of the distribution of Accurate measurement of the distribution of

fiber orientation in paper can be obtained only if fiber orientation in paper can be obtained only if the fibers can be made distinguishable by apply the fibers can be made distinguishable by apply some stain or color to the fibers.some stain or color to the fibers.

• This is illustrated inThis is illustrated inFigure 40.Figure 40.

• Image analysisImage analysistechniques can betechniques can beused to determine theused to determine theorientation distributionorientation distributionof labeled fibers.of labeled fibers.

• Another method is to split the sheet into thin, Another method is to split the sheet into thin, essentially transparent, layers and determine essentially transparent, layers and determine the orientation of the edges of the fibers.the orientation of the edges of the fibers.

• This is shown in Figure 41.This is shown in Figure 41.

• Image analysis can beImage analysis can beused for this case asused for this case aswell.well.

• Full sheet propertiesFull sheet propertiesare obtained byare obtained byaveraging over all ofaveraging over all ofthe layers of thethe layers of theoriginal sheet.original sheet.

• Regardless of the measurement method, it must Regardless of the measurement method, it must be recalled that definition of a unique be recalled that definition of a unique distribution of fiber orientations is not possible, distribution of fiber orientations is not possible, because the fibers are not perfectly straight.because the fibers are not perfectly straight.

• The orientation of fiber segments might be more The orientation of fiber segments might be more important for paper properties than the important for paper properties than the orientation of complete fibers.orientation of complete fibers.

• Since sheet splitting and marking of fibers is Since sheet splitting and marking of fibers is difficult to do cleanly and representatively, difficult to do cleanly and representatively, indirect methods are often necessary.indirect methods are often necessary.

• This gives some estimate or quantification of This gives some estimate or quantification of fiber orientation effects.fiber orientation effects.

• If one treats any measured signal as the If one treats any measured signal as the response of an ellipsoid of anisotropy, then the response of an ellipsoid of anisotropy, then the orientation index is the ratio of major and minor orientation index is the ratio of major and minor axes of the ellipsoid.axes of the ellipsoid.

• The Average orientation angle is the angle The Average orientation angle is the angle between the major axis and the machine between the major axis and the machine direction.direction.

• This orientation index should be approximately This orientation index should be approximately equal to the ratio R of moduli given on equal to the ratio R of moduli given on slide 93slide 93, , which we have called the "orientation index".which we have called the "orientation index".

• Generally orientation indices differ, depending Generally orientation indices differ, depending on the measurement method.on the measurement method.

• As we already seen, the MD/CD ratio of tensile As we already seen, the MD/CD ratio of tensile strength is a good measure of orientation index.strength is a good measure of orientation index.

• However, it is difficult to determine the actual However, it is difficult to determine the actual angle from this measurement, especially since angle from this measurement, especially since the angle is usually relative small, < 10 degrees.the angle is usually relative small, < 10 degrees.

• Measurement of elastic modulus from Measurement of elastic modulus from ultrasound is a useful and quick method to ultrasound is a useful and quick method to monitor changes in fiber orientation, when monitor changes in fiber orientation, when varying forming conditions.varying forming conditions.

• This analysis suffers from interference effects This analysis suffers from interference effects from drying stresses.from drying stresses.

• This method overestimates the fiber orientation This method overestimates the fiber orientation index at the edges of the web.index at the edges of the web.

• Dielectric permittivity or microwave Dielectric permittivity or microwave transmittance can be used similarly to transmittance can be used similarly to ultrasound.ultrasound.

• The result is sensitive to moisture content of the The result is sensitive to moisture content of the paper.paper.

• Optical fiber orientation measurements use Optical fiber orientation measurements use reflection or transmission of light.reflection or transmission of light.

• When a narrow beam of light is directed towards When a narrow beam of light is directed towards paper, an ellipsoidal spot appears on the back paper, an ellipsoidal spot appears on the back side, reflecting the anisotropy of the paper.side, reflecting the anisotropy of the paper.

• These ellipsoids can be averaged over a larger These ellipsoids can be averaged over a larger area to obtain representative results.area to obtain representative results.