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Paper Structure

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

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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.

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**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.

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**Two-Dimensional Network**

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

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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 randomness of the networks is significant. With the random network, all correlations between fibers are absent. The position of each fiber is independent of other fibers. Thus, the two dimensional random network is amenable to mathematical analysis.

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Coverage Coverage is a useful concept in characterizing a random 2-d network of fibers. Consider N fibers in a area A. The average coverage c, the average number of fibers on any point in the plane, is given by c = Nlfwf/A = b/bf , where lf, wf and bf are the length, width and and basis weight of fibers, respectively and b is the basis weight of the paper.

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**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. It gives a precise measure of the effective number of fiber layers in the sheet. For papermaking fibers, bf = 5-10 g/m2, so that printing papers have c = 5-20 layers of fibers.

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

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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' is P(c') = e(c'-c)/c'!, for c'³0 = 0, otherwise. 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.

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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 fiber centers within a unit area. The validity of the Poisson distribution is demonstrated in Figure 3.

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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/bf). The probability of an area being covered by at least one fiber is 1-e-c. This is dependent on the assumption of randomness.

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**At high average coverages, the Poisson distribution is similar to a Gaussian distribution (e-c2).**

This is shown in Figure 4. The important difference is that the Gaussian distribution contains negative values, but the Poisson distribution is meaningless for negative values.

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

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

Fiber segments are defined as the the sections between crossings. We define the segment length as the distance between the centroids. Consider a test fiber in a large network. Divide its length into square sections. The number of sections is lf/wf. If the fiber width is small, we expect the Poisson distribution to be valid.

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**P(lfree) = e-lfreec/wf**

The 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, lfree = kwf is P(lfree) = e-lfreec/wf 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.

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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 corresponding probability is given by Pc(f)=lf2sinf/A The average over all crossing angles is pc = 2lf2/pwf

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**P(ls)=(2c/pwf)e-2cls/pwf**

For an area containing N fibers, the average number of crossings per fiber is given by nc=2Nlf2/pA=2clf/pwf The crossings occur at random locations, so that the average distance, ls, between them is ls=lf/nc=pwf/2c This assumes that the fibers are very long. Corte and Kallmes obtained the corrected (relative to Poisson) probability P(ls)=(2c/pwf)e-2cls/pwf

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Network Connectivity 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. 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.

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**For coverage c, the total surface area of fibers is**

2Nlfwf = 2cA since each fiber has two sides, top and bottom. Likewise, the paper has two sides, except at points of zero coverage. Thus, 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

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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 are given by B1=(1-e-c)/c-e-c B2=(1-2/c)+(1+2/c)e-c These are all shown in Figure 6.

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**The preceding treatment is approximate.**

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. 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 2bf=10-20 g/m2.

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Percolation 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. 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.

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

cc~5.7wf/lf Thus, with coverage below cc, the network is not connected. At cc, 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. For most papermaking fibers, cc<.1.

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

It is possible to prepare a thin paper of basis weight 2.5g/m2, 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 at ncc~11.4/p=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.

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**The corresponding critical RBA is**

RBAc = cc/2~2.8wf/lf Clearly RBAc <<1, for ordinary papermaking fibers. The remaining issue is reinforcement fibers. Sometimes, the primary fibers may screen the reinforcement fibers from bonding to one another (or vice/versa). See Figure 7.

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**Three-Dimensional Network**

The 3d pore structure of paper controls the density and optical properties directly. 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. We will call on results from computer simulations of particle packing for interpretation.

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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. The 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.

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**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. Some sheet area may be completely empty, with no coverage. Recall that the frequency,n, of these vacancies is given by n = e-c This is appreciable when c<2.

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

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. At 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 of pore space in a handsheet. to slide 77

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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 results of a simulation are shown in Figure 9. The simulation supports the linear relationship p=p¥(c-c0) for c>c0

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In this relationship, p¥ and c0 are constants, which depend on the stiffness of the fibers, as well as on process variables such as wet pressing and calendering. c0 gives the minimum number of fiber layers necessary for pore formation. For real papermaking fibers, the coverage threshold is c0~2-10, depending on flexibility and thickness to width ratio. Figure 10 shows qualitatively how the cross-section dimensions affect pore number.

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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 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. The out of plane eccentricity (MD/ZD) ~ , while the in plane value ~

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**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. Beating is a common method of increasing the flexibility of chemical pulp fibers. The increasing SR number indicates the accumulation of the beating action.

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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. The free span lengths can also characterize pore sizes. The free span length is equivalent to the free segment length lfree. Recall that lfree obeys an exponential distribution.

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**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 is not the same as the size distribution of entire pores. The local height of a pore doesn't determine the shape of the three dimensional pore. Measurements of the local pore height distribution are shown in Figure 12.

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

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Relative Bonded 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. 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.

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**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 to RBA=1-(1+p)(1-n)/c where p is the number of pores and n=e-c. This equation still ignores the z-directional projection of fiber surfaces. The maximum fiber surface available for bonding is still 2lfwf.

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

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Measurement of RBA The data in Figure 13 came directly form cross-sectional images. Preparation and measurement of these cross-sections are tedious. 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). This gives the bonding degree irrespective of orientation.

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One method of determining the RBA uses the Kubelka-Munk light scattering coefficient, S[m2/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 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.

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This doesn't guarantee that the two fibers are bonded chemically, since the bonding distance is shorter. When 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 to represent changes in the bonded area of paper.

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The remaining problem for calculation RBA is to determine the light scattering coefficient for completely unbonded fibers, S0. Then the RBA is given by RBA=1-S/S0 In beating trials, one frequently uses for S0, the value at which tensile strength versus S extrapolates to zero. However, this value is unreliable, because; The surface area may change in beating, as is the case for mechanical pulps and high yield chemical pulps.

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

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 S0. This is generally not possible, and thus, RBA cannot be reliably determined using light scattering from paper.

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**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: RBA=(r-r0)/r¥ where r0 and r¥ are positive constants. However, there is no verification of the validity of this equation. The constants are not known.

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**Layered and Felted Sheet Structure**

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. 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. The fibers form an ordered sequence in the vertical direction. In a felted structure there is no clear sequence.

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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. 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: h=1/lf 0lfòSdl

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**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. The probability distribution of h for all fibers characterize the degree of layering in the sheet. In 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.

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

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

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

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**The basis weight variation depends on:**

The randomness of single fiber distribution. Fiber interactions. Flocculation - increases the variability of basis weight. And hydrodynamic forces in the web forming process. Turbulence can decrease basis weight by breaking flocs. “Hydrodynamic smoothing” improves sheet uniformity.

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

Formation effects: Print unevenness - resulting from local porosity. Tensile strength. 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.

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

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Measurements 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”. 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. Functional properties depend only on the latter.

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

The measured values must be calibrated to basis weight. A good method is to use b-radiation for which the transmitted intensity decays exponentially with basis weight and the absorption coefficient is independent of furnish, for b-sources that emit no g-rays. C14, Pr147 or Kr85 are pure b-sources. X-rays can be used, but attenuation is greater for fillers than fibers.

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

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**Papers of different basis weight may be ranked differently if ranked by sb, fN or COV(b).**

In addition to amplitude of variation, formation has spatial characteristics. The most important of these is length scale. The microscopic and specific perimeter (Figure 17) characterize the length scale. The specific perimeter is defined by the boundary line between areas where basis weight is above or below the mean value.

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

Microscale is the average separation in a given direction between the borderlines. The power spectrum (Figure 18) theoretically contains all of the information about the spatial variation of basis weight. The spectrum shows how the variance, sb2, distributes to different wavelengths.

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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 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.

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**Formation of Random Fiber Network**

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. This implies the Poisson statistics. For a given basis weight, the number of fibers in an A is N=bA/lfwf where wf is the fiber coarseness (m/l).

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**If we set mf=lfwf, then we have N=bA/mf **

For a Poisson distribution, the variance is N and the standard deviation of basis weight is sb=ÖNmf/A Thus, the specific formation becomes fN=Ö(mf/A) where A is interpreted as the square of the spatial resolution of the measurement. To slide 66

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**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. This discussion has assumed that all fibers have the same mass and length and that the sheet is isotropic.

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

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

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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. This is the mechanism of hydrodynamic smoothing. As a result, handsheets have better formation than completely random networks.

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**Nf=(sbmeasured/sbrandom)2,**

Fiber flocculation is the main factor in base sheet formation. Consider clumps, or flocs, of several fibers as the basic constituents of paper. The standard deviation of basis weight increases roughly as the Önf, where nf is the average number of fibers in the floc. Dotson showed that an estimate for nf is Nf=(sbmeasured/sbrandom)2, when using a measurement window of A=lf2.

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

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. They considered the crowding number ncrowd=pClf2/6wf where wf is the fiber coarseness (mass/length).

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The crowding number gives the average number of fibers in a sphere of diameter equal to the fiber length. High values give poor formation. Formation has been shown to increase exponentially with ncrowd for a single machine. Kerekes and Schell found a threshold for flocculation for ncrowd~60. With 2 mm fibers having coarseness of .2 mg/m, the threshold occurs at C=5kg/m3 (5% in mass concentration).

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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. On the wire, adjusting dewatering elements controls turbulence.

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

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

The speed difference creates a shear field in the suspension that breaks flocs and causes turbulence.

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

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**Paper Properties versus Formation**

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. Later we will discuss the effect on tensile strength and cockling. For these, formation couples with local fiber orientation and dried in strains.

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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. 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.

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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. 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.

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**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 increases with increasing calendering pressure. The correlation becomes stronger at high ink coverage, for the highest calendering impression.

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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 same machine. The correlation is evident for both, but more pronounced for Figure 26.

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

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

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

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Fiber Orientation For machine made papers, the fibers tend to align with the machine direction. We refer to this anisotropy as fiber 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.

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

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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. 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.

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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. The difference between the surfaces (two sidedness) causes curl. The fibers also have a small component in the z direction. This is completely different from the in plane orientation.

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**Laminar Shear on the Wire**

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. Other significant effects are: The acceleration and deceleration of the suspension flow in the headbox and wire section, respectively. The random disturbances from turbulence.

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**The shear field rotates the fibers toward the machine direction. **

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

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

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**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. The rate of change of the fiber orientation is given by df/dt=-Gvssinf where vs is the velocity difference in the machine direction and G is a constant proportional to the viscosity.

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**The solution to this differential equation is tan(f/2)=tan(f0/2)e-Gvst **

where f0 is the initial value of the angle. This solution applies while the fiber rotates freely before it contacts the mat. If f0 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(f)=1/p(1-q2)/(1+q2-2qcos(2f)) where q=tanh2(bvs), b being a constant.

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

An alternate derivation leads to an elliptical distribution, but with a different q given by q=2tan2[tan-1(2bvs)/2] Other distributions are often used to characterize measurements. The most common alternative is the von Mises distribution.

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**f(f)=1/p[1+Sancos(2nf)] **

Any fiber orientation distribution that is symmetric about the machine direction can be expressed as a Fourier series f(f)=1/p[1+Sancos(2nf)] The elliptical distribution has simple Fourier coefficients, an=2qn. These are important, since the MD/CD ratio of elastic modulus of tensile strength is given by R=EMD/ECD=(6+4a1+a2)/(6-4a1+a2) This ratio, R, serves as an orientation index that measures the degree of orientation anisotropy. to slide 127

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**Figure 31 shows how R changes with the suspension-to-wire **

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-wire speed difference vs using the elliptical distribution. Comparison with Figure 28 indicates that this model overestimates R at large vs. to slide 99 to slide 100

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**Effect of Fiber Properties**

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 dvs/dz is constant. 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.

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

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

Curly fibers give less anisotropy than straight fibers. Fiber interactions and flocculation resist the rotation of fibers. Thus, fibers may bend during drainage. The curl of fibers as a function of orientation can be described in terms of hindered rotation.

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**Other Hydrodynamic Effects**

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. This is caused by the converging slice channel that accelerates the suspension. Without turbulence, the elliptical orientation distribution has q=qjet=(k-1)/(k+1) where k=(Ain/Aout)1/2, with Ain/out being the areas of in and out channels, respectively.

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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. Curve II in Figure 31 shows how the distribution changes as a result of the jet anisotropy. Another factor omitted form the laminar shear analysis is turbulence. Turbulence destroys jet anisotropy from the suspension.

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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. The effect of turbulence on the orientation distribution can be qualitatively described as an average of elliptical distributions, f(f+Df,vs+Dvs) over Df and Dvs. Figure 31 shows that turbulence during drainage limits R for large velocity differences.

101
**Pulsating drainage generates turbulence in the headbox and on the wire.**

Disintegration of flocs and other viscous forces rapidly dampen small scale turbulence. Small eddies merge and coalesce into larger vortices. 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.

102
**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. 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.

103
The only feature missing from the fiber orientation mechanism is the asymmetry between “rush” (vs>0) and “drag” (vs<0) in Figure 28. 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. This increases the effective speed difference in drag and decreases it in rush. This is consistent with the asymmetry seen in Figure 28.

104
**In gap formers, the deceleration of suspension speed is easy to understand from the nip pressure.**

The Bernoulli equation 1/2rDv2=-Dp implies that a pressure pulse changes the absolute velocity of the suspension. On 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.**

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.

106
Orientation Angle If the suspension flows on the wire exactly in the machine direction then the orientation must be symmetric, f(f)=f(-f), relative to the machine direction. Thus, the average orientation direction <f>=o. 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-<f>)=f(<f>-f). <f> 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. It depends on the direction of suspension flow relative to the wire. We assume that the suspension velocity has a transverse component, as shown in Figure 33. This may come from a small misalignment in the suspension velocity.

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**<f>=tan-1(vCD/vs)**

Since the speed of the suspension, vs, is close to the speed of the wire, vw, the misalignment angle can be very large. In the absence of fluctuations, the misalignment angle is given by <f>=tan-1(vCD/vs) where vCD is the transverse component of the suspension velocity. If the magnitude of vs 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 figure indicates that the orientation angle goes through zero at vs=0, indicating that vCD goes to zero faster than vs for this machine. The excursions to large positive or negative values reflect the sensitivity to small changes in vs or vCD near vs=0.

110
CD Variations 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. 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.

111
Large scale variations in <f> 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. 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.

112
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 edge areas shrink more than the the central areas. 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 orientation angle profile results. This is shown in Figure 35. Local 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. Fiber orientation in the plane of the sheet can vary through the thickness of the sheet. Fibers may z-directional orientation. These two aspects are totally independent and effect the paper properties in different ways. 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.

115
**The z direction orientation angle, q, relative to the plane is limited by**

q£tan-1(d/lf)»d/lf where d is the paper thickness. Even though q is small, this is still described as z directional fiber orientation. This 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.**

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. 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.

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

This results from an asymmetry in the z-directional orientation angle in the MD-CD plane. The cause of this asymmetry is unknown.

118
Two-Sidedness 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. The prevailing flow conditions when a particular layer drains determines the layer structure. 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.

119
**The layered fiber orientation structures are sensitive to former type.**

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 figures show “drag”, vs < 0 and “rush”, vs>0 conditions. For the Fourdriner, the fiber orientation index changes systematically from top to bottom. 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.

121
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. 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 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.**

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

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

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**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. This is illustrated in Figure 40. Image 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. This is shown in Figure 41. Image analysis can be used for this case as well. Full sheet properties are obtained by averaging over all of the layers of the original sheet.

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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. 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.

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

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

The result is sensitive to moisture content of the paper. 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. These ellipsoids can be averaged over a larger area to obtain representative results.

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BENDING MOMENTS AND SHEARING FORCES IN BEAMS

BENDING MOMENTS AND SHEARING FORCES IN BEAMS

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