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

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Presentation on theme: "Paper Structure. Introduction Paper can be thought of as a stochastic network of fibers. This is seen in the picture below."— Presentation transcript:

1 Paper Structure

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

3 Fibers are generally much longer than the thickness of the sheet, so the network can be approximated as 2-dimensional. This 2-d structure describes many paper properties, but 3d effects are still important. The 3d porous structure of paper gives it opacity, bulk and stiffness. The connectedness of these pores determines how fluids transport through a sheet. These are especially important for printing and absorbency applications.

4 We will first discuss the statistical geometry of the fiber network of paper. Next we will discuss how real paper differs from the random network. We will see that the structure of paper is nonuniform, disordered and irregular. Formation represents the structural nonuniformity at larger length scales. Fiber orientation is an important feature of real paper, controlling its behavior in many applications such as packaging and printing.

5 Two-Dimensional Network The simplest picture that describes some of the important properties of paper is a 2-d completely random network.The simplest picture that describes some of the important properties of paper is a 2-d completely random network. This can be seenThis can be seen in Figure 2. This pictureThis picture consists of straight line segments with constant length and zero width.

6 Kallness et. al. showed that handsheets have the same in-plane mechanical properties as handsheets made by laminating many thin sheets.Kallness et. al. showed that handsheets have the same in-plane mechanical properties as handsheets made by laminating many thin sheets. The 2d network explains the in-plane properties of paper.The 2d network explains the in-plane properties of paper. The randomness of the networks is significant.The randomness of the networks is significant. With the random network, all correlations between fibers are absent.With the random network, all correlations between fibers are absent. The position of each fiber is independent of other fibers.The position of each fiber is independent of other fibers. Thus, the two dimensional random network is amenable to mathematical analysis.Thus, the two dimensional random network is amenable to mathematical analysis.

7 Coverage Coverage is a useful concept in characterizing a random 2-d network of fibers.Coverage is a useful concept in characterizing a 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 fibers on any point in the plane, is given byThe average coverage c, the average number of fibers on any point in the plane, is given by c = Nl f w f /A = b/ f, where l f, w f and f are the length, width and and basis weight of fibers, respectively and b is the basis weight of the paper.

8 The coverage completely specifies the 2-d random network, when fiber properties are constant.The coverage completely specifies the 2-d random network, when fiber properties are constant. The coverage can be measured from sheet cross-sections by determining the number of bonds that intersect a reference line.The coverage can be measured from sheet cross-sections by determining the number of bonds that intersect a reference line. It gives a precise measure of the effective number of fiber layers in the sheet.It gives a precise measure of the effective number of fiber layers in the sheet. For papermaking fibers, f = 5-10 g/m 2, so that printing papers have c = 5-20 layers of fibers.For papermaking fibers, f = 5-10 g/m 2, so that printing papers have c = 5-20 layers of fibers.

9 We interpret the coverage as the effective number of layers in the sense that there are no distinct fiber layers in paper.We interpret the coverage as the effective number of layers in the sense that there are no distinct fiber layers in paper. Likewise, the ratio of paper thickness to average fiber thickness does not give a precise value for number of layers.Likewise, the ratio of paper thickness to average fiber thickness does not give a precise value for number of layers. Paper thickness decreases in wet processing and calendering, that these cannot alter the number of fiber layers.Paper thickness decreases in wet processing and calendering, that these cannot alter the number of fiber layers.

10 We assume that the local coverage values, c', are Poisson distributed for sufficiently small reference areas.We assume that the local coverage values, c', are Poisson distributed for sufficiently small reference areas. If the average coverage is c, the Poisson distribution of c' isIf the average coverage is c, the Poisson distribution of c' is P(c') = e (c'-c) /c'!, for c' 0 = 0, otherwise. P(c') is the probability of finding c' fibers covering a given point, when the average coverage is c.P(c') is the probability of finding c' fibers covering a given point, when the average coverage is c. The small reference area is assumed small on the length scale of fibers.The small reference area is assumed small on the length scale of fibers.

11 Note that from the definition of coverage, the Poisson distribution also gives a distribution of local basis weights.Note that from the definition of coverage, the Poisson distribution also gives a distribution of local basis weights. The distribution can also be interpreted in terms of the number of fiberThe distribution can also be interpreted in terms of the number of fiber centers within a unit area. The validity of theThe validity of the Poisson distribution is demonstrated in Figure 3.

12 From the definition of the Poisson distribution, the probability of finding an empty reference area is e -c.From the definition of the Poisson distribution, the probability of finding an empty reference area is e -c. Thus, we expect the frequency of pin holes to decrease with increasing basis weight as e (-b/ f ).Thus, we expect the frequency of pin holes to decrease with increasing basis weight as e (-b/ f ). The probability of an area being covered by at least one fiber is 1-e -c.The probability of an area being covered by at least one fiber is 1-e -c. This is dependent on the assumption of randomness.This is dependent on the assumption of randomness.

13 At high average coverages, the Poisson distribution is similar to a Gaussian distribution (e -c 2 ).At high average coverages, the Poisson distribution is similar to a Gaussian distribution (e -c 2 ). This is shown in Figure 4.This is shown in Figure 4. The importantThe important difference is that the Gaussian distribution contains negative values, but the Poisson distribution is meaningless for negative values.

14 Corte-Kallmes Theory Corte and Kallmes analyzed the statistical geometry of 2-d random fiber networks and found good agreement with macroscopic measurements of thin paper sheets.Corte and Kallmes analyzed the statistical geometry of 2-d random fiber networks and found good agreement with macroscopic measurements of thin paper sheets. Their theory describes the distribution of constant dimension fiber segments randomly and isotropically distributed in a plane.Their theory describes the distribution of constant dimension fiber segments randomly and isotropically distributed in a plane. Of course, real fibers do not have a constant length or width.Of course, real fibers do not have a constant length or width. However, the average properties should be enough to specify for practical purposes.However, the average properties should be enough to specify for practical purposes.

15 Consider the distribution of fiber segments.Consider the distribution of fiber segments. Fiber segments are defined as the the sections between crossings.Fiber segments are defined as the the sections between crossings. We define the segment length as the distance between the centroids.We define the segment length as the distance 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 l f /w f.The number of sections is l f /w f. If the fiber width is small, we expect the Poisson distribution to be valid.If the fiber width is small, we expect the Poisson distribution to be valid.

16 The probability of finding k adjacent sections free of other fibers isThe probability of finding k adjacent sections free of other fibers is P(0) k = e -kc Thus, the frequency of a given free segment length, l free = kw f isThus, the frequency of a given free segment length, l free = kw f is P(l free ) = e -l free c/w f This assumes that fibers cross at right angles and only at discrete locations.This assumes that fibers cross at right angles and only at discrete locations. Corte and Kallmes were able to obtain the precise equation for continuous random locations and orientations.Corte and Kallmes were able to obtain the precise equation for continuous random locations and orientations.

17 For this case, a fiber in an area A crosses another fiber, if the center of the area falls inside the area defined in Figure 5.For this case, a fiber in an area A crosses another fiber, if the center of the area falls inside the area defined in Figure 5. The correspondingThe corresponding probability is given by P c ( )=l f 2 sin /A The average over allThe average over all crossing angles is p c = 2l f 2 / w f

18 For an area containing N fibers, the average number of crossings per fiber is given byFor an area containing N fibers, the average number of crossings per fiber is given by n c =2Nl f 2 / A=2cl f / w f The crossings occur at random locations, so that the average distance, l s, between them isThe crossings occur at random locations, so that the average distance, l s, between them is l s =l f /n c = w f /2c This assumes that the fibers are very long.This assumes that the fibers are very long. Corte and Kallmes obtained the corrected (relative to Poisson) probabilityCorte and Kallmes obtained the corrected (relative to Poisson) probability P(l s )=(2c/ w f )e -2cl s / w f

19 Network Connectivity The mechanical properties of paper are controlled by the connectivity or bonding degree.The mechanical properties of paper are controlled by the connectivity or bonding degree. The network would have no cohesion if there were not enough bonds between the fibers.The network would have no cohesion if there were not enough bonds between the fibers. Thus, we define the relative bonding area (RBA) as the bonded surface divided by the total fiber surface area.Thus, we define the relative bonding area (RBA) as the bonded surface divided by the total fiber surface area. In two dimensions, we assume bonding at every fiber crossing.In two dimensions, we assume bonding at every fiber crossing.

20 For coverage c, the total surface area of fibers isFor coverage c, the total surface area of fibers is 2Nl f w f = 2cA since each fiber has two sides, top and bottom. Likewise, the paper has two sides, except at points of zero coverage.Likewise, the paper has two sides, except at points of zero coverage. Thus, the area of unbonded surface is 2A(1-e -c ) and thereforeThus, the area of unbonded surface is 2A(1-e -c ) and therefore RBA=[2cA-2A(1-e -c )]/2cA =1-(1-e -c )/c To slide 39

21 The expression on the previous slide represents the average degree of bonding of the top and bottom of fibers.The expression on the previous slide represents the average degree of bonding of the top and bottom of fibers. The corresponding RBAs for one sided and two sided bonding areThe corresponding RBAs for one sided and two sided bonding are given by B1=(1-e -c )/c-e -c B2=(1-2/c)+(1+2/c)e -c These are all shown inThese are all shown in Figure 6.

22 The preceding treatment is approximate.The preceding treatment is approximate. Real paper is three dimensional, with a finite thickness and pores available in the z direction.Real paper is three dimensional, with a finite thickness and pores available in the z direction. These pores reduce the bonding degree from the two dimensional estimate.These pores reduce the bonding degree from the two dimensional estimate. The 2-d picture is valid for low coverage or high fiber flexibility.The 2-d picture is valid for low coverage or high fiber flexibility. An estimate for the maximum basis weight for two-dimensional paper is twice the average basis weight of a single fiber or 2 f =10-20 g/m 2.An estimate for the maximum basis weight for two-dimensional paper is twice the average basis weight of a single fiber or 2 f =10-20 g/m 2.

23 Percolation At low coverages, a connected fiber network forms only if a sufficient number of bonds are formed per fiber.At low coverages, a connected fiber network forms only if a sufficient number of bonds are formed per fiber. This occurs at the percolation threshold, the minimum number of bonds needed to connect the network.This occurs at the percolation threshold, the minimum number of bonds needed to connect the network. Below the percolation threshold, the network consists of several disjointed pieces.Below the percolation threshold, the network consists of several disjointed pieces. This is a concept from percolation theory, which has found many applications in science, including electrical connectivity and porous media.This is a concept from percolation theory, which has found many applications in science, including electrical connectivity and porous media.

24 Computer simulations of 2d random networks have determined a percolation threshold for coverage ofComputer simulations of 2d random networks have determined a percolation threshold for coverage of c c ~5.7w f /l f Thus, with coverage below c c, the network is not connected.Thus, with coverage below c c, the network is not connected. At c c, one crucial fiber exists the removal of which splits the network into two parts.At c c, one crucial fiber exists the removal of which splits the network into two parts. Obviously, any real paper sheet must be far above this limit.Obviously, any real paper sheet must be far above this limit. For most papermaking fibers, c c <.1.For most papermaking fibers, c c <.1.

25 Thus, usual coverage values of 5-20 are much higher.Thus, usual coverage values of 5-20 are much higher. It is possible to prepare a thin paper of basis weight 2.5g/m 2, with c=.5, which is still above the percolation threshold.It is possible to prepare a thin paper of basis weight 2.5g/m 2, with c=.5, which is still above the percolation threshold. In terms of the number of number of bonds per fiber, the percolation threshold occurs atIn terms of the number of number of bonds per fiber, the percolation threshold occurs at n cc ~11.4/ =3.6 This is significantly greater than 2, the absolute minimum number for a connected chain, because of the two dimensional nature of the network and the possibility of dangling ends. This is significantly greater than 2, the absolute minimum number for a connected chain, because of the two dimensional nature of the network and the possibility of dangling ends.

26 The corresponding critical RBA isThe corresponding critical RBA is RBA c = c c /2~2.8w f /l f Clearly RBAc <<1, for ordinary papermaking fibers.Clearly RBAc <<1, for ordinary papermaking fibers. The remaining issue isThe remaining issue is reinforcement fibers. Sometimes, the primary fibersSometimes, the primary fibers may screen the reinforcement fibers from bonding to one another (or vice/versa). See Figure 7.See Figure 7.

27 Three-Dimensional Network The 3d pore structure of paper controls the density and optical properties directly.The 3d pore structure of paper controls the density and optical properties directly. It controls the mechanical properties indirectly through the RBA.It controls the mechanical properties indirectly through the RBA. The pore geometry is complex, because of the intertwined network of fibers and the partial flexibility of the fibers.The pore geometry is complex, because of the intertwined network of fibers and the partial flexibility of the fibers. We will call on results from computer simulations of particle packing for interpretation.We will call on results from computer simulations of particle packing for interpretation.

28 We can measure the real pore size distribution with a mercury porosimeter or a gas phase BET instrument.We can measure the real pore size distribution with a mercury porosimeter or a gas phase BET instrument. These tell how the pores are distributed by size, but not by position.These tell how the pores are distributed by size, but not by position. The 3d network is either layered or feltedThe 3d network is either layered or felted The distinction is based on how the fibers are entangled, which determines the z directional properties of the paper.The distinction is based on how the fibers are entangled, which determines the z directional properties of the paper.

29 Statistical Pore Geometry At low basis weights, a paper sheet is essentially 2-dimensional, because a mechanical contact can form between two fibers whenever their projections cross.At low basis weights, a paper sheet is essentially 2-dimensional, because a mechanical contact can form between two fibers whenever their projections cross. Some sheet area may be completely empty, with no coverage.Some sheet area may be completely empty, with no coverage. Recall that the frequency,, of these vacancies is given byRecall that the frequency,, of these vacancies is given by = e -c = e -c This is appreciable when c<2.This is appreciable when c<2.

30 If the basis weight of paper increases, the standard deviation of local thickness increases as the square root of coverage, according to the Poisson Distribution.If the basis weight of paper increases, the standard deviation of local thickness increases as the square root of coverage, according to the Poisson Distribution. Fibers must be bent more and more to make contact with other fibers.Fibers must be bent more and more to make contact with other fibers. At some point this bending is noAt some point this bending is no longer possible and some empty space opens up in the z-direction between fibers. Figure 8 shows two cases ofFigure 8 shows two cases of pore space in a handsheet. to slide 77

31 The porosity (fractional void volume) is higher when the local basis weight is lower than the average.The porosity (fractional void volume) is higher when the local basis weight is lower than the average. The porosity depends on how pores form as the basis weight grows.The porosity depends on how pores form as the basis weight grows. The results of aThe results of a simulation are shown in Figure 9. The simulationThe simulation supports the linear relationship p=p (c-c 0 ) for c>c 0

32 In this relationship, p and c 0 are constants, which depend on the stiffness of the fibers, as well as on process variables such as wet pressing and calendering.In this relationship, p and c 0 are constants, which depend on the stiffness of the fibers, as well as on process variables such as wet pressing and calendering. c 0 gives the minimum number of fiber layers necessary for pore formation.c 0 gives the minimum number of fiber layers necessary for pore formation. For real papermaking fibers, theFor real papermaking fibers, the coverage threshold is c 0 ~2-10, depending on flexibility and thickness to width ratio. Figure 10 shows qualitatively howFigure 10 shows qualitatively how the cross-section dimensions affect pore number.

33 The probability of a pore on a given unit area of a sheet is small when the fiber thickness is small.The probability of a pore on a given unit area of a sheet is small when the fiber thickness is small. The pore sizes of a paper sheet can be determined by mercury or nitrogen measurements.The pore sizes of a paper sheet can be determined by mercury or nitrogen measurements. The 3d network of pores can be envisioned as a collection of ellipsoidal pores, with narrow throats.The 3d network of pores can be envisioned as a collection of ellipsoidal pores, with narrow throats. Light diffraction measurements in 3d can yield the average shape of these ellipsoidal pores.Light diffraction measurements in 3d can yield the average shape of these ellipsoidal pores. The out of plane eccentricity (MD/ZD) ~ , while the in plane value ~ The out of plane eccentricity (MD/ZD) ~ , while the in plane value ~

34 Permeability measurements can yield the approximate pore throat size.Permeability measurements can yield the approximate pore throat size. Figure 11 shows how typical pore and the width of their distribution decreases with increasing fiber flexibility.Figure 11 shows how typical pore and the width of their distribution decreases with increasing fiber flexibility. Beating is a commonBeating is a common method of increasing the flexibility of chemical pulp fibers. The increasing SR numberThe increasing SR number indicates the accumulation of the beating action.

35 The distributions in the figure resemble a log- normal distribution (with the exception of the binary mixture).The distributions in the figure resemble a log- normal distribution (with the exception of the binary mixture). Thus, the logarithm of the pore radius is approximately Gaussian.Thus, the logarithm of the pore radius is approximately Gaussian. The free span lengths can also characterize pore sizes.The free span lengths can also characterize pore sizes. The free span length is equivalent to the free segment length l free.The free span length is equivalent to the free segment length l free. Recall that l free obeys an exponential distribution.Recall that l free obeys an exponential distribution.

36 The free span length in the thickness direction can be measured from sheet cross-sections.The free span length in the thickness direction can be measured from sheet cross-sections. This gives a distribution for the local lengths in the pore space.This gives a distribution for the local lengths in the pore space. This is not the same as the size distribution ofThis is not the same as the size distribution of entire pores. The local height of a poreThe local height of a pore doesn't determine the shape of the three dimensional pore. Measurements of the localMeasurements of the local pore height distribution are shown in Figure 12.

37 These measurements that the pore heights also satisfy an exponential distribution.These measurements that the pore heights also satisfy an exponential distribution. The deviations are at the limit of shallow pore space, where the cross-sectional shape and uncollapsed lumen of fibers control.The deviations are at the limit of shallow pore space, where the cross-sectional shape and uncollapsed lumen of fibers control. The shape of the pore height distributions should be similar for different fibers, with the length scale depending of fiber properties and sheet properties.The shape of the pore height distributions should be similar for different fibers, with the length scale depending of fiber properties and sheet properties.

38 Relative Bonded Area Recall that the RBA is the bonded surface area divided by the total surface area.Recall that the RBA is the bonded surface area divided by the total surface area. In the 2d picture the top and bottom surfaces control the bonding degree.In the 2d picture the top and bottom surfaces control the bonding degree. At high basis weights, the top and bottom of the sheet have reduced effect.At high basis weights, the top and bottom of the sheet have reduced effect. The RBA increases with increasing basis weight in the 2d picture.The RBA increases with increasing basis weight in the 2d picture.

39 In the real 3d network the pore structure limits the growth of RBA with basis weight.In the real 3d network the pore structure limits the growth of RBA with basis weight. The equation for the RBA from slide 20 is generalized toThe equation for the RBA from slide 20 is generalized toslide 20slide 20 RBA=1-(1+p)(1- )/c where p is the number of pores and =e -c. where p is the number of pores and =e -c. This equation still ignores the z-directional projection of fiber surfaces.This equation still ignores the z-directional projection of fiber surfaces. The maximum fiber surface available for bonding is still 2l f w f.The maximum fiber surface available for bonding is still 2l f w f.

40 At constant basis weight, the RBA depends on the cross-sectional dimensions and flexibility of fibers as shown in Figure 13.At constant basis weight, the RBA depends on the cross-sectional dimensions and flexibility of fibers as shown in Figure 13. Sheet consolidation inSheet consolidation in wet pressing and drying also contribute. In practice, RBA isIn practice, RBA is controlled by pulp type, beating level and wet pressing. The pore sizeThe pore size distribution is also important, causing the RBA to decrease for large pores.

41 Measurement of RBA The data in Figure 13 came directly form cross- sectional images.The data in Figure 13 came directly form cross- sectional images. Preparation and measurement of these cross- sections are tedious.Preparation and measurement of these cross- sections are tedious. Indirect methods are usual; in the measurement of RBA.Indirect methods are usual; in the measurement of RBA. One can obtain the free surface of paper at the molecular level from gas adsorption (Micromeritics TriStar in 2730).One can obtain the free surface of paper at the molecular level from gas adsorption (Micromeritics TriStar in 2730). This gives the bonding degree irrespective of orientation.This gives the bonding degree irrespective of orientation.

42 One method of determining the RBA uses the Kubelka-Munk light scattering coefficient, S[m 2 /kg], of paper.One method of determining the RBA uses the Kubelka-Munk light scattering coefficient, S[m 2 /kg], of paper. This gives the optically free surface area per unit mass that has to be normalized by the surface area of unbonded fibers.This gives the optically free surface area per unit mass that has to be normalized by the surface area of unbonded fibers. This normalization is important and unreliable values of RBA are obtained without it.This normalization is important and unreliable values of RBA are obtained without it. The light scattering method relies on the fact that a fiber surface element appears bonded if there is another fiber surface at a distance smaller than half the wavelength of light.The light scattering method relies on the fact that a fiber surface element appears bonded if there is another fiber surface at a distance smaller than half the wavelength of light.

43 This doesn't guarantee that the two fibers are bonded chemically, since the bonding distance is shorter.This doesn't guarantee that the two fibers are bonded chemically, since the bonding distance is shorter. When applied to paper sheets ofWhen applied to paper sheets of different beating levels, gas adsorption area and light scattering coefficient were seen to be linearly related to one another, as shown in Figure 14. Thus, S can be used toThus, S can be used to represent changes in the bonded area of paper.

44 The remaining problem for calculation RBA is to determine the light scattering coefficient for completely unbonded fibers, S 0.The remaining problem for calculation RBA is to determine the light scattering coefficient for completely unbonded fibers, S 0. Then the RBA is given byThen the RBA is given by RBA=1-S/S 0 In beating trials, one frequently uses for S 0, the value at which tensile strength versus S extrapolates to zero.In beating trials, one frequently uses for S 0, the value at which tensile strength versus S extrapolates to zero. However, this value is unreliable, because;However, this value is unreliable, because; 1.The surface area may change in beating, as is the case for mechanical pulps and high yield chemical pulps.

45 2.Tensile strength can change in beating for reasons other than RBA. 3.With decreasing RBA, tensile strength disappears at the percolation threshold, well before RBA reaches zero. Light scattering from unbonded fibers, cross- sectional images or some other correlation with true network geometry is necessary to determine S 0.Light scattering from unbonded fibers, cross- sectional images or some other correlation with true network geometry is necessary to determine S 0. This is generally not possible, and thus, RBA cannot be reliably determined using light scattering from paper.This is generally not possible, and thus, RBA cannot be reliably determined using light scattering from paper.

46 The density of paper gives another indirect, qualitative measure of RBA.The density of paper gives another indirect, qualitative measure of RBA. Considering how changes in pore height distribution directly effect effect RBA, we can assume that RBA depends linearly on density according to:Considering how changes in pore height distribution directly effect effect RBA, we can assume that RBA depends linearly on density according to: RBA=( - 0 )/ RBA=( - 0 )/ where 0 and are positive constants. However, there is no verification of the validity of this equation.However, there is no verification of the validity of this equation. The constants are not known.The constants are not known.

47 Layered and Felted Sheet Structure Papermaking fibers are 1-2 orders of magnitude longer than a typical sheet is thick.Papermaking fibers are 1-2 orders of magnitude longer than a typical sheet is thick. Thus, most of the fiber length must be aligned in the plane of the paper sheet.Thus, most of the fiber length must be aligned in the plane of the paper sheet. The arrangement of fibers in the z-direction can be layered or felted.The arrangement of fibers in the z-direction can be layered or felted. A layered network forms if the fibers land on the wire one after another.A layered network forms if the fibers land on the wire one after another. The fibers form an ordered sequence in the vertical direction.The fibers form an ordered sequence in the vertical direction. In a felted structure there is no clear sequence.In a felted structure there is no clear sequence.

48 In order to characterize the layering of a sheet, one considers the vertical positions of a fiber in successive cross-sections of the sheet.In order to characterize the layering of a sheet, one considers the vertical positions of a fiber in successive cross-sections of the sheet. In each cross-section, the fiber is assigned an ordering number, S, according to its position in the z direction.In each cross-section, the fiber is assigned an ordering number, S, according to its position in the z direction. These numbers are normalized by dividing by the total number of fibers in each cross-section.These numbers are normalized by dividing by the total number of fibers in each cross-section. The vertical order, h, of each fiber is the average over the length of the fiber:The vertical order, h, of each fiber is the average over the length of the fiber: h=1/l f 0 l f Sdl

49 A small value of h means the center of mass of the fiber is close to the bottom side of the sheet.A small value of h means the center of mass of the fiber is close to the bottom side of the sheet. A large value of h=1 means that it is close to the top side.A large value of h=1 means that it is close to the top side. The probability distribution of h for all fibers characterize the degree ofThe probability distribution of h for all fibers characterize the degree of layering in the sheet. In a layered structure, theIn a layered structure, the values of h have a uniform distribution from 0-1, while in a felted structure some values are more common than others as shown in Figure 15.

50 A typical layered network is formed in a handsheet mold, when using low pulp consistencies.A typical layered network is formed in a handsheet mold, when using low pulp consistencies. A felted structure is formed at high consistencies, or under pulsating drainage.A felted structure is formed at high consistencies, or under pulsating drainage. At high consistency, 3d fiber aggregates, or flocs, form in the suspension and then are squeezed in the planar sheet.At high consistency, 3d fiber aggregates, or flocs, form in the suspension and then are squeezed in the planar sheet. Fourdrinier and hybrid formers yield varying degrees of felting.Fourdrinier and hybrid formers yield varying degrees of felting. Gap formers produce a more layered paper structure.Gap formers produce a more layered paper structure.

51 A felted sheet structure should give better out of plane strength than a layered structure, because fibers are stronger than bonds.A felted sheet structure should give better out of plane strength than a layered structure, because fibers are stronger than bonds. Thus, high consistency forming promotes out-of- plane strength.Thus, high consistency forming promotes out-of- plane strength. The effect of fiber entanglements are shown in Figure 16.The effect of fiber entanglements are shown in Figure 16. If one follows the loopIf one follows the loop of fibers in the right hand portion of the figure, moving up at the intersection, one ends up below the starting point.

52 Formation In addition to fibers, paper consists of fiber fragments, mineral fillers and chemical additives.In addition to fibers, paper consists of fiber fragments, mineral fillers and chemical additives. In the web formation process, they all settle stochastically onto the wire.In the web formation process, they all settle stochastically onto the wire. Paper formation is the resulting nonuniform distribution of particles.Paper formation is the resulting nonuniform distribution of particles. More precisely, formation is the variability of the basis weight of paper.More precisely, formation is the variability of the basis weight of paper. Such variation can be easily seen with the naked eye, for some sheets.Such variation can be easily seen with the naked eye, for some sheets.

53 The basis weight variation depends on:The basis weight variation depends on: 1.The randomness of single fiber distribution. 2.Fiber interactions. 3.Flocculation - increases the variability of basis weight. 4.And hydrodynamic forces in the web forming process. –Turbulence can decrease basis weight by breaking flocs. –Hydrodynamic smoothing improves sheet uniformity.

54 The nonuniform basis weight distribution affects many properties of paper.The nonuniform basis weight distribution affects many properties of paper. Formation effects:Formation effects: 1.Print unevenness - resulting from local porosity. 2.Tensile strength. 3.Cockling In the case of strength and cockling, local basis weight variations are not separate from the effects of local fiber orientation and dried-in strains.In the case of strength and cockling, local basis weight variations are not separate from the effects of local fiber orientation and dried-in strains.

55 Characterization A useful definition of formation is the small scale basis weight variation in the plane of the paper sheet.A useful definition of formation is the small scale basis weight variation in the plane of the paper sheet. This provides for simple measurement and unambiguous connection to paper structure.This provides for simple measurement and unambiguous connection to paper structure. Other terms used include mass formation, mass distribution or the distribution of mass density.Other terms used include mass formation, mass distribution or the distribution of mass density.

56 Measurements The traditional judgment of formation involves looking through a paper sheet, which has led people to call the visual impression formation.The traditional judgment of formation involves looking through a paper sheet, which has led 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 to give results that correlate with the visual impression.Many optical formation testers have calibrations to give results that correlate with the visual impression. Such testers may give misleading results, in the sense that visual appearance is not equivalent to structural nonuniformity.Such testers may give misleading results, in the sense that visual appearance is not equivalent to structural nonuniformity. Functional properties depend only on the latter.Functional properties depend only on the latter.

57 Measurement of formation is almost always indirect.Measurement of formation is almost always indirect. The measured values must be calibrated to basis weight.The measured values must be calibrated to basis weight. A good method is to use -radiation for which the transmitted intensity decays exponentially with basis weight and the absorption coefficient is independent of furnish, for -sources that emit no -rays.A good method is to use -radiation for which the transmitted intensity decays exponentially with basis weight and the absorption coefficient is independent of furnish, for -sources that emit no -rays. C 14, Pr 147 or Kr 85 are pure -sources.C 14, Pr 147 or Kr 85 are pure -sources. X-rays can be used, but attenuation is greater for fillers than fibers.X-rays can be used, but attenuation is greater for fillers than fibers.

58 Quantification It makes sense to describe the formation of paper in terms of the standard deviation of basis weight, b.It makes sense to describe the formation of paper in terms of the standard deviation of basis weight, b. The specific formation f N is defined byThe specific formation f N is defined by f N = b / b where b is the average basis weight. Note the dimensions of f N are square root of basis weight.Note the dimensions of f N are square root of basis weight. The coefficient of variationThe coefficient of variation COV(b)= b /b is dimensionless.

59 Papers of different basis weight may be ranked differently if ranked by b, f N or COV(b).Papers of different basis weight may be ranked differently if ranked by b, f N or COV(b). In addition to amplitude of variation, formation has spatial characteristics.In addition to amplitude of variation, formation 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 17) characterize the length scale.The microscopic and specific perimeter (Figure 17) characterize the length scale. The specific perimeterThe specific perimeter is defined by the boundary line between areas where basis weight is above or below the mean value.

60 The specific perimeter is the perimeter of the borderline divided by the total area.The specific perimeter is the perimeter of the borderline divided by the total area. Microscale is the averageMicroscale is the average separation in a given direction between the borderlines. The power spectrum (Figure 18)The power spectrum (Figure 18) theoretically contains all of the information about the spatial variation of basis weight. The spectrum shows how theThe spectrum shows how the variance, b 2, distributes to different wavelengths.

61 Deviations from the theoretical shape indicate that the formation is more cloudy or grainy than the random fiber network.Deviations from the theoretical shape indicate that the formation is more cloudy or grainy than the random fiber network. The dashed curve shows how flocculation increases fluctuations at long wavelengths.The dashed curve shows how flocculation increases fluctuations at long wavelengths. The microscopic and average floc orientation can be derived from the spectrum.The microscopic and average floc orientation can be derived from the spectrum. The latter requires a two dimensional spectrum, rather than the 1 dimensional spectrum of Figure 18.The latter requires a two dimensional spectrum, rather than the 1 dimensional spectrum of Figure 18.

62 Formation of Random Fiber Network The theory of formation in fiber networks derives from the studies of Corte and Kallmes in the 1960s.The theory of formation in fiber networks derives from the studies of Corte and Kallmes in the 1960s. Free drainage is assumed, so that every fiber settles onto the mat randomly and independently.Free drainage is assumed, so that every fiber settles onto the mat randomly and independently. This implies the Poisson statistics.This implies the Poisson statistics. For a given basis weight, the number of fibers in an A isFor a given basis weight, the number of fibers in an A is N=bA/l f f where f is the fiber coarseness (m/l).

63 If we set m f =l f f, then we haveIf we set m f =l f f, then we have N=bA/m f For a Poisson distribution, the variance is N and the standard deviation of basis weight isFor a Poisson distribution, the variance is N and the standard deviation of basis weight is b = Nm f /A b = Nm f /A Thus, the specific formation becomesThus, the specific formation becomes f N = (m f /A) where A is interpreted as the square of the spatial resolution of the measurement. To slide 66

64 For this case, the only property that effects formation is the fiber mass as seen in Figure 19.For this case, the only property that effects formation is the fiber mass as seen in Figure 19. The effect of fiber mass on handsheet formation can be detected if stock consistency is sufficiently low to prevent flocculation.The effect of fiber mass on handsheet formation can be detected if stock consistency is sufficiently low to prevent flocculation. This discussion hasThis discussion has assumed that all fibers have the same mass and length and that the sheet is isotropic.

65 Dodson has shown that a distribution of fiber masses or orientations does not significantly alter b.Dodson has shown that a distribution of fiber masses or orientations does not significantly alter b. The important thing is the spatial resolution of measurement,The important thing is the spatial resolution of measurement, relative to the fiber length and width. The measured bThe measured b decreases as the window size increases as shown in Figure 20.

66 Forming Mechanisms The actual formation deviates from the theoretical value given on slide 63.The actual formation deviates from the theoretical value given on slide 63.slide 63slide 63 This results from four mechanismsThis results from four mechanisms 1.Hydrodynamic smoothing. 2.Flocculation. 3.Shear flow. 4.Turbulence. Hydrodynamic smoothing results from the fact that suspension flow during drainage can improve formation.Hydrodynamic smoothing results from the fact that suspension flow during drainage can improve formation.

67 The flow rate through the settled fiber mat is highest where the flow resistance is smallest, in areas of lowest basis weight.The flow rate through the settled fiber mat is highest where the flow resistance is smallest, in areas of lowest basis weight. Thus, suspension flows toward areas of low basis weight.Thus, suspension flows toward areas of low basis weight. This is the mechanism of hydrodynamic smoothing.This is the mechanism of hydrodynamic smoothing. As a result, handsheets have better formation than completely random networks.As a result, handsheets have better formation than completely random networks.

68 Fiber flocculation is the main factor in base sheet formation.Fiber flocculation is the main factor in base sheet formation. Consider clumps, or flocs, of several fibers as the basic constituents of paper.Consider clumps, or flocs, of several fibers as the basic constituents of paper. The standard deviation of basis weight increases roughly as the n f, where n f is the average number of fibers in the floc.The standard deviation of basis weight increases roughly as the n f, where n f is the average number of fibers in the floc. Dotson showed that an estimate for nf isDotson showed that an estimate for nf is N f =( bmeasured / brandom ) 2, when using a measurement window of A=l f 2.

69 Suspension consistency, C, directly influences flocculation.Suspension consistency, C, directly influences flocculation. High consistency gives more flocculation because of the high probability of fibers encountering one another.High consistency gives more flocculation because of the high probability of fibers encountering one another. Kerekes and Schell used dimensional analysis to estimate the consistency effect on flocculation.Kerekes and Schell used dimensional analysis to estimate the consistency effect on flocculation. They considered the crowding numberThey considered the crowding number n crowd = Cl f 2 /6 f where f is the fiber coarseness (mass/length).

70 The crowding number gives the average number of fibers in a sphere of diameter equal to the fiber length.The crowding number gives the average number of fibers in a sphere of diameter equal to the fiber length. High values give poor formation.High values give poor formation. Formation has been shown to increase exponentially with n crowd for a single machine.Formation has been shown to increase exponentially with n crowd for a single machine. Kerekes and Schell found a threshold for flocculation for n crowd ~60.Kerekes and Schell found a threshold for flocculation for n crowd ~60. With 2 mm fibers having coarseness of.2 mg/m, the threshold occurs at C=5kg/m 3 (5% in mass concentration).With 2 mm fibers having coarseness of.2 mg/m, the threshold occurs at C=5kg/m 3 (5% in mass concentration).

71 Formation depends on turbulence in the headbox and in the wire section, because turbulence on a suitable length scale disrupts flocs.Formation depends on turbulence in the headbox and in the wire section, because turbulence on a suitable length scale disrupts flocs. For Paper machine consistencies, turbulence is necessary to prevent flocculation.For Paper machine consistencies, turbulence is necessary to prevent flocculation. On the wire, adjusting dewatering elements controls turbulence.On the wire, adjusting dewatering elements controls turbulence.

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

73 The effects of jet to wire speed difference on formation are shown in Figures 22 and 23.The effects of jet to wire speed difference on formation are shown in Figures 22 and 23. The speed difference creates a shear fieldThe speed difference creates a shear field in the suspension that breaks flocs and causes turbulence.

74 The effect of turbulence can be positive or negative.The effect of turbulence can be positive or negative. On fourdriniers and hybrid formers, the optimum formation occurs near zero speed difference (Figure 22).On fourdriniers and hybrid formers, the optimum formation occurs near zero speed difference (Figure 22). On the other hand, for gap formers, the optimum is typically at a small speed difference.On the other hand, for gap formers, the optimum is typically at a small speed difference. On all former types, formation is poor if the speed difference is too large.On all former types, formation is poor if the speed difference is too large. Shear forces, between suspension and wire can become so large that the settled mat ruptures.Shear forces, between suspension and wire can become so large that the settled mat ruptures.

75 Paper Properties versus Formation Here, we discuss the effect of formation on print unevenness, or mottle.Here, we discuss the effect of formation on print unevenness, or mottle. Local basis weight variation affects print density through local porosity and associated permeability.Local basis weight variation affects print density through local porosity and associated permeability. Later we will discuss the effect on tensile strength and cockling.Later we will discuss the effect on tensile strength and cockling. For these, formation couples with local fiber orientation and dried in strains.For these, formation couples with local fiber orientation and dried in strains.

76 Print mottle results from spatial variation in surface roughness and from nonuniform ink penetration associated with spatial variation of local permeability.Print mottle results from spatial variation in surface roughness and from nonuniform ink penetration associated with spatial variation of local permeability. The details vary for different print processes, but these effects remain for all processes.The details vary for different print processes, but these effects remain for all processes. High roughness reduces the effective contact area between the image carrier and the paper, thereby varying the ink film thickness.High roughness reduces the effective contact area between the image carrier and the paper, thereby varying the ink film thickness. High (optical) print density (proportional to ink film thickness) results from low ink penetration into the sheet.High (optical) print density (proportional to ink film thickness) results from low ink penetration into the sheet.

77 Likewise, low density results from high penetration into the sheet, which is associated with high permeability of the paper.Likewise, low density results from high penetration into the sheet, which is associated with high permeability of the paper. Thus, we expect mottle, variation in ink density, to result from local variations in permeability.Thus, we expect mottle, variation in ink density, to result from local variations in permeability. Generally, local variations in basis weight, will reflect local variations in permeability.Generally, local variations in basis weight, will reflect local variations in permeability. Recall Figure 8, where the cross-section clearly suggest larger permeability in the thinner sections.Recall Figure 8, where the cross-section clearly suggest larger permeability in the thinner sections.Figure 8Figure 8

78 Generally, calendering has a profound effect on both roughness and permeability.Generally, calendering has a profound effect on both roughness and permeability. As seen in Figure 24, the correlation between local print density and local basis weight variation increasesAs seen in Figure 24, the correlation between local print density and local basis weight variation increases with increasing calendering pressure. The correlationThe correlation becomes stronger at high ink coverage, for the highest calenderingimpression.

79 Other factors can mask the effect of formation on print mottle, where comparing between different paper machines and printing devices.Other factors can mask the effect of formation on print mottle, where comparing between different paper machines and printing devices. Figures 25 and 26 show how a weak correlation with formation may be seen with different machines, while a strong correlation exists when comparing from the sameFigures 25 and 26 show how a weak correlation with formation may be seen with different machines, while a strong correlation exists when comparing from the samemachine. The correlationThe correlation is evident for both, but more pronounced for Figure 26.

80 Illusory Effects Sometimes the apparent correlation between formation and other properties.Sometimes the apparent correlation between formation and other properties. For example, fluid permeability and opacity are nonlinear functions of basis weight.For example, fluid permeability and opacity are 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 and permeability increase as the formation gets worse as shown in Figure 27.

81 In fact formation induced small scale variations in opacity are more harmful than a small decrease in average opacity at good formation.In fact formation induced small scale variations in opacity are more harmful than a small decrease in average opacity at good formation. Similarly, bending stiffness increases with poor formation.Similarly, bending stiffness increases with poor formation. This results from the nonlinear dependence of bending stiffness on formation and the fact that heavier areas contribute proportionately than lighter areas.This results from the nonlinear dependence of bending stiffness on formation and the fact that heavier areas contribute proportionately than lighter areas.

82 The Bendtsen roughness, which is sensitive to larger scale roughness because it uses a hard measuring head, shows a strong correlation with formation.The Bendtsen roughness, which is sensitive to larger scale roughness because it uses a hard measuring head, shows a strong correlation with formation. On the other hand, the Parker Print Surf roughness, which is sensitive to micro roughness because it uses a soft measuring head, shows no correlation with formation.On the other hand, the Parker Print Surf roughness, which is sensitive to micro roughness because it uses a soft measuring head, shows no correlation with formation. Other properties, such as compression strength and Kodak bending stiffness, show high variability because of poor formation.Other properties, such as compression strength and Kodak bending stiffness, show high variability because of poor formation.

83 Fiber Orientation For machine made papers, the fibers tend to align with the machine direction.For machine made papers, the fibers tend to 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 this orientation.Here, we consider the basic mechanisms of this orientation. The fiber orientation index and the fiber orientation angle are usually used to characterize the in-plane orientation distribution.The fiber orientation index and the fiber orientation angle are usually used to characterize the in-plane orientation distribution.

84 The orientation index gives the anisotropy, or eccentricity, of the distribution.The orientation index gives the anisotropy, or eccentricity, of the distribution. It is 1 for an isotropic sheet, such as a handsheet and increases for increasing anisotropy.It is 1 for an isotropic sheet, such as a handsheet and increases for increasing anisotropy. The orientation angle indicates how much the symmetry axis of the distribution deviates form the machine direction.The orientation angle indicates how much the symmetry axis of the distribution deviates form the machine direction.

85 The fiber orientation of paper directly effects the in-plane mechanical properties and dimensional stability of paper.The fiber orientation of paper directly effects the in-plane mechanical properties and dimensional stability of paper. The anisotropic properties also depend on drying shrinkage and wet straining of the paper web.The anisotropic properties also depend on drying shrinkage and wet straining of the paper web. Generally, a low orientation index is best for paper performance, since this makes paper properties similar in all in-plane directions.Generally, a low orientation index is best for paper performance, since this makes paper properties similar in all in-plane directions. The fiber orientation effects the tensile strength in the machine and cross directions.The fiber orientation effects the tensile strength in the machine and cross directions.

86 The variation of fiber orientation through the thickness of the sheet is also important for paper performance.The variation of fiber orientation through the thickness of the sheet is also important for paper performance. Orientation difference between the sheet surface and middle layers affects bending stiffness.Orientation difference between the sheet surface and middle layers affects bending stiffness. The difference between the surfaces (two sidedness) causes curl.The difference between the surfaces (two sidedness) causes curl. The fibers also have a small component in the z direction.The fibers also have a small component in the z direction. This is completely different from the in plane orientation.This is completely different from the in plane orientation.

87 Laminar Shear on the Wire Several hydrodynamic forces affect the distribution of fiber orientations in paper.Several hydrodynamic forces affect the distribution of fiber orientations in paper. The most important is the velocity difference between the suspension jet and the wire.The most important is the velocity difference between the suspension jet and the wire. Other significant effects are:Other significant effects are: 1.The acceleration and deceleration of the suspension flow in the headbox and wire section, respectively. 2.The random disturbances from turbulence.

88 The speed difference between the jet and wire creates a z-directional velocity gradient, or shear field.The speed difference between the jet and wire creates a z-directional velocity gradient, or shear field. The shear field rotates theThe shear field rotates the fibers toward the machine direction. A large speed differenceA large speed difference yields a significant fiber orientation in the machine direction. This is reflected in theThis is reflected in the anisotropy of in-plane paper properties as seen in Figure 28. to slide 103

89 The distribution of fiber orientation is related to the velocity difference between the suspension and wire.The distribution of fiber orientation is related to the velocity difference between the suspension and wire. To understand this relationship, we consider an idealized mode.To understand this relationship, we consider an idealized mode. We assume fibers are straight and stiff, their consistency is low and the flow is laminar.We assume fibers are straight and stiff, their consistency is low and the flow is laminar. With laminar flow, the suspension velocity relative to the wire is in the machine direction.With laminar flow, the suspension velocity relative to the wire is in the machine direction. In the vertical direction, it decreases continuously to zero at the wire.In the vertical direction, it decreases continuously to zero at the wire.

90 During drainage, one end of a fiber comes down and adheres to the already filtered mat.During drainage, one end of a fiber comes down and adheres to the already filtered mat. Figure 29 illustrates the viscous drag exerted on a fiber due to the flow.Figure 29 illustrates the viscous drag exerted on a fiber due to the flow. The rate of change of the fiber orientation is given byThe rate of change of the fiber orientation is given by d /dt=- v s sin d /dt=- v s sin where v s is the velocity difference in the machine direction and is a constant is a constant proportional to the viscosity.

91 The solution to this differential equation isThe solution to this differential equation is tan( /2)=tan( 0 /2)e - v s t where 0 is the initial value of the angle. This solution applies while the fiber rotates freely before it contacts the mat.This solution applies while the fiber rotates freely before it contacts the mat. If 0 is assumed to have an isotropic distribution and t is taken to be a characteristic time before the fiber contacts the mat, then we obtain the elliptic distribution:If 0 is assumed to have an isotropic distribution and t is taken to be a characteristic time before the fiber contacts the mat, then we obtain the elliptic distribution: f( )=1/ (1-q 2 )/(1+q 2 -2qcos(2 )) where q=tanh 2 ( v s ), being a constant.

92 This distribution is shown in Figure 30 for two different values of q.This distribution is shown in Figure 30 for two different values of q. An alternate derivationAn alternate derivation leads to an elliptical distribution, but with a different q given by q=2tan 2 [tan -1 (2 v s )/2] Other distributions areOther distributions are often used to characterize measurements. The most common alternative is the von Mises distribution.The most common alternative is the von Mises distribution.

93 Any fiber orientation distribution that is symmetric about the machine direction can be expressed as a Fourier seriesAny fiber orientation distribution that is symmetric about the machine direction can be expressed as a Fourier series f( )=1/ [1+ a n cos(2n )] The elliptical distribution has simple Fourier coefficients, a n =2q n.The elliptical distribution has simple Fourier coefficients, a n =2q n. These are important, since the MD/CD ratio of elastic modulus of tensile strength is given byThese are important, since the MD/CD ratio of elastic modulus of tensile strength is given by R=E MD /E CD =(6+4a 1 +a 2 )/(6-4a 1 +a 2 ) This ratio, R, serves as an orientation index that measures the degree of orientation anisotropy.This ratio, R, serves as an orientation index that measures the degree of orientation anisotropy. to slide 127

94 When using this index, it is important to remember that drying creates anisotropic internal stresses that also effect R.When using this index, it is important to remember that drying creates anisotropic internal stresses that also effect R. Figure 31 shows how R changes with the suspension-to-wireFigure 31 shows how R changes with the suspension-to-wire speed difference v s using the elliptical distribution. Comparison withComparison with Figure 28 indicates that this model overestimates R at large v s. to slide 99to slide 100

95 Effect of Fiber Properties The derivation in the previous section is independent of fiber length.The derivation in the previous section is independent of fiber length. The equation for the angular motion is independent of fiber length if the shear field dv s /dz is constant.The equation for the angular motion is independent of fiber length if the shear field dv s /dz is constant. Yet it is believed that long fibers orient more readily than short fibers.Yet it is believed that long fibers orient more readily than short fibers. There must be some dependence on fiber length for short fibers, but experiments on Formette Dynamique sheet mold suggest that there is no dependence for long enough fibers.There must be some dependence on fiber length for short fibers, but experiments on Formette Dynamique sheet mold suggest that there is no dependence for long enough fibers.

96 For real papermaking condition, the effect of fiber length may differ form that of the laminar model.For real papermaking condition, the effect of fiber length may differ form that of the laminar model. The suspension is generally not dilute and hence fiber interactions are important.The suspension is generally not dilute and hence fiber interactions are important. Fiber orientation must depend on fiber stiffness and flocculation and their coupling to turbulence.Fiber orientation must depend on fiber stiffness and flocculation and their coupling to turbulence. The effect of turbulence undoubtedly depends on fiber length, because it changes the effective Reynolds number.The effect of turbulence undoubtedly depends on fiber length, because it changes the effective Reynolds number. The sheets for the Formette Dynamique are free of flocculation because the suspension is sprayed on a rotating wire in thin layers.The sheets for the Formette Dynamique are free of flocculation because the suspension is sprayed on a rotating wire in thin layers.

97 Fiber curl also influences the relationship between paper anisotropy and fiber orientation.Fiber curl also influences the relationship between paper anisotropy and fiber orientation. Curly fibers give less anisotropy than straight fibers.Curly fibers give less anisotropy than straight fibers. Fiber interactions and flocculation resist the rotation of fibers.Fiber interactions and flocculation resist the 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 be described in terms of hindered rotation.The curl of fibers as a function of orientation can be described in terms of hindered rotation.

98 Other Hydrodynamic Effects Fiber orientation depends on other hydrodynamic effects other than laminar shear.Fiber orientation depends on other hydrodynamic effects other than laminar shear. One factor is the anisotropic orientation distribution in the jet as it emerges from the headbox.One factor is the anisotropic orientation distribution in the jet as it emerges from the headbox. This is caused by the converging slice channel that accelerates the suspension.This is caused by the converging slice channel that accelerates the suspension. Without turbulence, the elliptical orientation distribution hasWithout turbulence, the elliptical orientation distribution has q=q jet =(k-1)/(k+1) where k=(A in /A out ) 1/2, with A in/out being the areas of in and out channels, respectively.

99 As a result of the anisotropic jet, the initial fiber orientation distribution is no longer isotropic.As a result of the anisotropic jet, the initial fiber orientation distribution is no longer isotropic. Therefore, the final fiber orientation distribution is not elliptical.Therefore, the final fiber orientation distribution is not elliptical. Curve II in Figure 31 shows how the distribution changes as a result of the jet anisotropy.Curve II in Figure 31 shows how the distribution changes as a result of the jet anisotropy.Figure 31Figure 31 Another factor omitted form the laminar shear analysis is turbulence.Another factor omitted form the laminar shear analysis is turbulence. Turbulence destroys jet anisotropy from the suspension.Turbulence destroys jet anisotropy from the suspension.

100 Thus with turbulence, the anisotropic jet affects fiber orientation only on the wire side on a Fourdrinier and on the two surfaces in a gap former.Thus with turbulence, the anisotropic jet affects fiber orientation only on the wire side on a Fourdrinier and on the two surfaces in a gap former. In addition, turbulence smears out global average fiber orientation by inciting random fluctuations in the local speed difference.In addition, turbulence smears out global average fiber orientation by inciting random fluctuations in the local speed difference. The effect of turbulence on the orientation distribution can be qualitatively described as an average of elliptical distributions, f( +,v s + v s ) over and v s.The effect of turbulence on the orientation distribution can be qualitatively described as an average of elliptical distributions, f( +,v s + v s ) over and v s. Figure 31 shows that turbulence during drainage limits R for large velocity differences.Figure 31 shows that turbulence during drainage limits R for large velocity differences.Figure 31Figure 31

101 Pulsating drainage generates turbulence in the headbox and on the wire.Pulsating drainage generates turbulence in the headbox and on the wire. Disintegration of flocs and other viscous forces rapidly dampen small scale turbulence.Disintegration of flocs and other viscous forces rapidly dampen small scale turbulence. Small eddies merge and coalesce into larger vortices.Small eddies merge and coalesce into larger vortices. Large scale fluctuations survive longer than short ones.Large scale fluctuations survive longer than short ones. The turbulence generated by drainage elements is important on slow Fourdrinier machines and turbulence in the headbox is important for high speed gap formers.The turbulence generated by drainage elements is important on slow Fourdrinier machines and turbulence in the headbox is important for high speed gap formers.

102 Small scale turbulence is necessary during drainage to prevent flocculation.Small scale turbulence is necessary during drainage to prevent flocculation. Thus, intensive turbulence might be expected to improve paper formation and reduce fiber orientation anisotropy.Thus, intensive turbulence might be expected to improve paper formation and reduce fiber orientation anisotropy. In reality, it is not so simple.In reality, it is not so simple. The relationship between local fiber orientation and basis weight depends on details of the forming section of the paper machine.The relationship between local fiber orientation and basis weight depends on details of the forming section of the paper machine.

103 The only feature missing from the fiber orientation mechanism is the asymmetry between rush (v s >0) and drag (v s 0) and drag (v s <0) in Figure 28. Figure 28 Figure 28 There must be something that breaks the symmetry between the two cases.There must be something that breaks the symmetry between the two cases. It may be due to deceleration of suspension during drainage in rush and drag conditions.It may be due to deceleration of suspension during drainage in rush and drag conditions. This increases the effective speed difference in drag and decreases it in rush.This increases the effective speed difference in drag and decreases it in rush. This is consistent with the asymmetry seen in Figure 28.This is consistent with the asymmetry seen in Figure 28. Figure 28 Figure 28

104 In gap formers, the deceleration of suspension speed is easy to understand from the nip pressure.In gap formers, the deceleration of suspension speed is easy to understand from the nip pressure. The Bernoulli equationThe Bernoulli equation 1/2 v 2 =- p implies that a pressure pulse changes the absolute velocity of the suspension. On Fourdriniers, drainageOn Fourdriniers, drainage pulses cause deceleration as shown in Figure 32 for moderate rush.

105 Viscous forces push the suspension speed toward the wire speed during drainage.Viscous forces push the suspension speed toward the wire speed during drainage. This is consistent with the suspension speed not decreasing in drag conditions in Figure 32.This is consistent with the suspension speed not decreasing in drag conditions in Figure 32. Systematic changes in the suspension speed and turbulence during drainage also contribute to the two-sidedness of the fiber orientation.Systematic changes in the suspension speed and turbulence during drainage also contribute to the two-sidedness of the fiber orientation.

106 Orientation Angle If the suspension flows on the wire exactly in the machine direction then the orientation must be symmetric, f( )=f(- ), relative to the machine direction.If the suspension flows on the wire exactly in the machine direction then the orientation must be symmetric, f( )=f(- ), relative to the machine direction. Thus, the average orientation direction =o.Thus, the average orientation direction =o. In reality, the average orientation angle is nonzero.In reality, the average orientation angle is nonzero. The orientation angle defines the effective symmetry axis, or the maximum in the orientation distribution with f( - )=f( - ).The orientation angle defines the effective symmetry axis, or the maximum in the orientation distribution with f( - )=f( - ). is also called the misalignment angle. is also called the misalignment angle.

107 The direction along which fibers tend to align during drainage determines the fiber orientation angle of the paper.The direction along which fibers tend to align during drainage determines the fiber orientation angle of the paper. It depends on the direction of suspension flow relative to the wire.It depends on the direction of suspension flow relative to the wire. We assume that theWe assume that the suspension velocity has a transverse component, as shown in Figure 33. This may come from aThis may come from a small misalignment in the suspension velocity.

108 Since the speed of the suspension, v s, is close to the speed of the wire, v w, the misalignment angle can be very large.Since the speed of the suspension, v s, is close to the speed of the wire, v w, the misalignment angle can be very large. In the absence of fluctuations, the misalignment angle is given byIn the absence of fluctuations, the misalignment angle is given by =tan -1 (v CD /v s ) =tan -1 (v CD /v s ) where v CD is the transverse component of the suspension velocity. If the magnitude of v s goes to zero, then the misalignment can approach +/- 90 degrees.If the magnitude of v s goes to zero, then the misalignment can approach +/- 90 degrees.

109 The actual dependence of misalignment angle on Jet to speed ratio is shown in Figure 34 for a hybrid former.The actual dependence of misalignment angle on Jet to speed ratio is shown in Figure 34 for a hybrid former. The figure indicatesThe figure indicates that the orientation angle goes through zero at v s =0, indicating that v CD goes to zero faster than v s for this machine. The excursions to large positive or negative values reflect the sensitivity to small changes in v s or v CD near v s =0.The excursions to large positive or negative values reflect the sensitivity to small changes in v s or v CD near v s =0.

110 CD Variations For all machine made paper, there are large scale variations in fiber orientation angle across the web.For all machine made paper, there are large scale variations in fiber orientation angle across the web. The actual average value is not as important as the profile of variations in the CD direction.The actual average value is not as important as the profile of variations in the CD direction. This is because problems in paper performance, such as curl or cockling, occur when the orientation angle varies too rapidly across the web.This is because problems in paper performance, such as curl or cockling, occur when the orientation angle varies too rapidly across the web. For fiber orientation index, the average value is of greater importance.For fiber orientation index, the average value is of greater importance.

111 Large scale variations in across the paper web result from uneven suspension flow across a wide machine.Large scale variations in across the paper web result from uneven suspension flow across a wide machine. Cross directional flow components are easily created inside the converging slice channel of the headbox.Cross directional flow components are easily created inside the converging slice channel of the headbox. This nonuniform flow in the headbox in turn generates variations in basis weight profile that may be even more detrimental.This nonuniform flow in the headbox in turn generates variations in basis weight profile that may be even more detrimental. The local slice opening can be adjusted to produce a more uniform basis weight profile, since a larger opening corresponds to higher basis weight.The local slice opening can be adjusted to produce a more uniform basis weight profile, since a larger opening corresponds to higher basis weight.

112 However, this adjustment may cause cross directional flows and fiber orientation deviations in surrounding areas of the web.However, this adjustment may cause cross directional flows and fiber orientation deviations in surrounding areas of the web. The shrinkage profile resulting from uneven drying of the paper web is a common cause of variation in fiber orientation angle across the web.The shrinkage profile resulting from uneven drying of the paper web is a common cause of variation in fiber orientation angle across the web. The edge areas shrink more than the the central areas.The edge areas shrink more than the the central areas. The higher shrinkage increases the local basis weight on the edges.The higher shrinkage increases the local basis weight on the edges.

113 If the slice opening in the central jets are increased to compensate for the edges, an S- shaped orientationIf the slice opening in the central jets are increased to compensate for the edges, an S- shaped orientation angle profile results.angle profile results. This is shown inThis is shown in Figure 35. Local dilution of theLocal dilution of the suspension in the headbox allows the decoupling of the orientation angle profile from the basis weight profile.

114 Fiber Orientation Distribution The fiber orientation in paper has two aspects.The fiber orientation in paper has two aspects. 1.Fiber orientation in the plane of the sheet can vary through the thickness of the sheet. 2.Fibers may z-directional orientation. These two aspects are totally independent and effect the paper properties in different ways.These two aspects are totally independent and 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 fiber length is aligned in the plane of the sheet.Because of their high aspect ratio, most of the fiber length is aligned in the plane of the sheet.

115 The z direction orientation angle,, relative to the plane is limited byThe z direction orientation angle,, relative to the plane is limited by tan -1 (d/l f ) d/l f tan -1 (d/l f ) d/l f where d is the paper thickness. Even though is small, this isEven though is small, this is still described as z directional fiber orientation. This is misleading, since onlyThis is misleading, since only fiber segments can have appreciable orientation in the z-direction as shown in Figure 36.

116 Thus, the z-directional fiber orientation is actually fiber segment orientation.Thus, the z-directional fiber orientation is actually fiber segment orientation. It is closely related to undulations of fibers in the z-direction.It is closely related to undulations of fibers in the z-direction. For machine made paper, there is a nontrivial asymmetry in the z-directional strength.For machine made paper, there is a nontrivial asymmetry in the z-directional strength. If measured by pealing in the MD or CD, there is little difference between the two.If measured by pealing in the MD or CD, there is little difference between the two. However the MD value often depends on whether the delamination is run in the MD or the -MD.However the MD value often depends on whether the delamination is run in the MD or the -MD.

117 This is shown in Figure 37 this difference between the up-stream and down-stream values.This is shown in Figure 37 this difference between the up-stream and down-stream values. This results fromThis results from an asymmetry in the z-directional orientation angle in the MD-CD plane. The cause of thisThe cause of this asymmetry is unknown.

118 Two-Sidedness Consider the variation of the orientation through the sheet thickness.Consider the variation of the orientation through the sheet thickness. Such variation is caused by changes in the flow state of the suspension, especially the level of turbulence and orienting shear during drainage.Such variation is caused by changes in the flow state of the suspension, especially the level of turbulence and orienting shear during drainage. The prevailing flow conditions when a particular layer drains determines the layer structure.The prevailing flow conditions when a particular layer drains determines the layer structure. Image analysis techniques can detect the layered fiber orientation.Image analysis techniques can detect the layered fiber orientation. This gives structural information about the paper and gives a clue to the sheet forming dynamics.This gives structural information about the paper and gives a clue to the sheet forming dynamics.

119 The layered fiber orientation structures are sensitive to former type.The layered fiber orientation structures are sensitive to former type. Figures 38 and 39 show characteristic z- directional profiles for three former types.Figures 38 and 39 show characteristic z- directional profiles for three former types.

120 The degree of fiber orientation is plotted as a function of cumulative basis weight from the bottom to the top.The degree of fiber orientation is plotted as a function of cumulative basis weight from the bottom to the top. The figures show drag, v s 0 conditions.The figures show drag, v s 0 conditions. For the Fourdriner, the fiber orientation index changes systematically from top to bottom.For the Fourdriner, the fiber orientation index changes systematically from top to bottom. These machines show strong two-sidedness in fiber orientation.These machines show strong two-sidedness in fiber orientation. This z-directional trend of the fiber orientation in rush and drag is consistent with a decreasing suspension velocity during drainage.This z-directional trend of the fiber orientation in rush and drag is consistent with a decreasing suspension velocity during drainage.

121 Near the top surface, viscous forces cause the suspension velocity to approach that of the wire, with the resulting decrease in anisotropy.Near the top surface, viscous forces cause the suspension velocity to approach that of the wire, with the resulting decrease in anisotropy. At the bottom surface, the anisotropy is low because of turbulence during formation.At the bottom surface, the anisotropy is low because of turbulence during formation. For the gap forming case, the layered fiber orientation is symmetric.For the gap forming case, the layered fiber orientation is symmetric. The jet impinges into a curved gap created by two wires that converge around a forming roll.The jet impinges into a curved gap created by two wires that converge around a forming roll. The vacuum level inside the forming roll affects the top side of the paper.The vacuum level inside the forming roll affects the top side of the paper.

122 Wire tension creates a static pressure in the gap that controls dewatering on the bottom.Wire tension creates a static pressure in the gap that controls dewatering on the bottom. The deceleration of the jet in the gap explains the local minimum of anisotropy in the center of the sheet of the sheet under rush conditions.The deceleration of the jet in the gap explains the local minimum of anisotropy in the center of the sheet of the sheet under 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 is again due to the high rate of dewatering and high level of turbulence.The low level of anisotropy on both surfaces is again due to the high rate of dewatering and high level of turbulence.

123 For the hybrid former, 2/3 of the sheet starting from the bottom side dewater under Fourdrinier conditions.For the hybrid former, 2/3 of the sheet starting from the bottom side dewater under Fourdrinier conditions. The hybrid unit drains the other one-third of the sheet at high rate, low turbulence and moderate shear.The hybrid unit drains the other one-third of the sheet at high rate, low turbulence and moderate shear. The magnitude of these effects depend on the type of hybrid unit.The magnitude of these effects depend on the type of hybrid unit.

124 Measurement Techniques Accurate measurement of the distribution of fiber orientation in paper can be obtained only if the fibers can be made distinguishable by apply some stain or color to the fibers.Accurate measurement of the distribution of fiber orientation in paper can be obtained only if the fibers can be made distinguishable by apply some stain or color to the fibers. This is illustrated inThis is illustrated in Figure 40. Image analysisImage analysis techniques can be used to determine the orientation distribution of labeled fibers.

125 Another method is to split the sheet into thin, essentially transparent, layers and determine the orientation of the edges of the fibers.Another method is to split the sheet into thin, essentially transparent, layers and determine 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 be used for this case as well. Full sheet propertiesFull sheet properties are obtained by averaging over all of the layers of the original sheet.

126 Regardless of the measurement method, it must be recalled that definition of a unique distribution of fiber orientations is not possible, because the fibers are not perfectly straight.Regardless of the measurement method, it must be recalled that definition of a unique distribution of fiber orientations is not possible, because the fibers are not perfectly straight. The orientation of fiber segments might be more important for paper properties than the orientation of complete fibers.The orientation of fiber segments might be more important for paper properties than the orientation of complete fibers. Since sheet splitting and marking of fibers is difficult to do cleanly and representatively, indirect methods are often necessary.Since sheet splitting and marking of fibers is difficult to do cleanly and representatively, indirect methods are often necessary. This gives some estimate or quantification of fiber orientation effects.This gives some estimate or quantification of fiber orientation effects.

127 If one treats any measured signal as the response of an ellipsoid of anisotropy, then the orientation index is the ratio of major and minor axes of the ellipsoid.If one treats any measured signal as the response of an ellipsoid of anisotropy, then the orientation index is the ratio of major and minor axes of the ellipsoid. The Average orientation angle is the angle between the major axis and the machine direction.The Average orientation angle is the angle between the major axis and the machine direction. This orientation index should be approximately equal to the ratio R of moduli given on slide 93, which we have called the "orientation index".This orientation index should be approximately equal to the ratio R of moduli given on slide 93, which we have called the "orientation index".slide 93slide 93 Generally orientation indices differ, depending on the measurement method.Generally orientation indices differ, depending on the measurement method.

128 As we already seen, the MD/CD ratio of tensile strength is a good measure of orientation index.As we already seen, the MD/CD ratio of tensile strength is a good measure of orientation index. However, it is difficult to determine the actual angle from this measurement, especially since the angle is usually relative small, < 10 degrees.However, it is difficult to determine the actual angle from this measurement, especially since the angle is usually relative small, < 10 degrees. Measurement of elastic modulus from ultrasound is a useful and quick method to monitor changes in fiber orientation, when varying forming conditions.Measurement of elastic modulus from ultrasound is a useful and quick method to monitor changes in fiber orientation, when varying forming conditions. This analysis suffers from interference effects from drying stresses.This analysis suffers from interference effects from drying stresses. This method overestimates the fiber orientation index at the edges of the web.This method overestimates the fiber orientation index at the edges of the web.

129 Dielectric permittivity or microwave transmittance can be used similarly to ultrasound.Dielectric permittivity or microwave transmittance can be used similarly to ultrasound. The result is sensitive to moisture content of the paper.The result is sensitive to moisture content of the paper. Optical fiber orientation measurements use reflection or transmission of light.Optical fiber orientation measurements use reflection or transmission of light. When a narrow beam of light is directed towards paper, an ellipsoidal spot appears on the back side, reflecting the anisotropy of the paper.When a narrow beam of light is directed towards paper, an ellipsoidal spot appears on the back side, reflecting the anisotropy of the paper. These ellipsoids can be averaged over a larger area to obtain representative results.These ellipsoids can be averaged over a larger area to obtain representative results.


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