Paper Structure

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. • This can be seen in Figure 2. • This 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. • 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

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. • 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.

15. 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.

16. 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.

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

18. 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

19. 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.

20. 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

21. 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.

22. 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.

23. 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.

24. 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.

25. 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.

26. 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.

27. 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.

28. 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.

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. • 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.

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. • 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

31. 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

32. 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.

33. 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) ~ 2.2-2.6, while the in plane value ~ 1.1-1.6.

34. 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.

35. 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.

36. 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.

37. 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.

38. 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.

39. 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.

40. 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.

41. 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.

42. 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.

43. 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.

44. 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.

45. 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.

46. 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.

47. 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.

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 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

49. 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.

50. 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.