3D numerical simulations of paper have previously been used to study network properties. For example, a study simulating the wet pressing operation with the ability to calculate the shape of fibers at the contact sites was presented in (Lavrykov et al. 2012). In this study, the deformation of fibers at the bonds naturally results in a specific thickness of the network that can be used to calculate the density. In a similar study, the same simulation techniques were used to study the contribution of fiber properties to the network properties with a focus on the bending stiffness (Lavrykov et al. 2011). In these studies, the fibers were modeled using 3D solid elements. Using such 3D elements requires a fine resolution of the fiber cross-section and is computationally expensive for considering large sizes and grammages. On the other hand, it has the advantage of capturing the cross-sectional deformations of fibers, specifically at the bonds. In this work, the compliance of the bonds is captured through the penalty-based beam-to-beam constraint (Motamedian 2018; Motamedian and Kulachenko 2018).
Since the pulp characterization tools were unable to capture any significant morphological changes induced by refining, a considerable difference in mechanical response is not expected to be made due to pulp variabilities. However, for the sake of consistency, we investigated the impact of these small changes by creating networks using the characterization data corresponding to different levels of refining while keeping other properties such as density and through-thickness profile constant. Later, we also investigated the impact of the factors that could not be detected by the pulp characterization tools. This included the growth in the number of bonds due to increased density, the improved bond strength, and the increased fibrillar content with characteristic sizes outside the detectable range.
In this analysis, we will use network-level computational tools (Kulachenko and Uesaka 2012), which rely on an accurate 3D representation of the network. The method used for network generation and analysis is described below.
We use the pulp characterization data to create the geometry of the network. However, the characterization data are extracted from fibers in the saturated state, in which they are swollen. In the sheet-making process, the fibers are pressed and then they shrink during drying. We use the data from micro-tomography to account for drying shrinkage and the change of the width-to-height ratio of the fibers (shown in Fig. 1) due to pressing. We use a deposition technique from (Kulachenko and Uesaka 2012) following the model of (Niskanen and Alava 1994) to generate the geometry of the networks. The major steps of this procedure are as follows:
We choose a random set of fiber geometry information from the corrected pulp characterization data and create a single fiber. This information includes the fiber length, width, width-to-height ratio, shape factor, and wall-thickness.
This fiber is then placed at a random location and with a random orientation on an imaginary 2D plane (for isotropic handsheets).
The intersections of this newly place fiber with the previously deposited fibers (from top view) are found.
The intersection points are raised vertically with appropriate values to avoid penetration of fibers. However, the fibers can undergo larger pressing at the bond sites and consequently, the distance between their centerlines becomes smaller. The normalized variation of the distance of cross-sectional centers before and after being pressed is demonstrated schematically in Fig. 4 as the press ratio.
The deposited fiber is then smoothed along its length. The smoothing procedure uses a given value for the maximum interface angle and it only permits the fiber segments to move upwards during the smoothing procedure to avoid penetration. The interface angle is schematically shown in Fig. 5.
This procedure is repeated until the desired network grammage is reached.
Schematic representation of the parameter press ratio used in numerical generation of networks
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Fig. 5Schematic representation of the parameter interface angle (Borodulina et al. 2016) used in numerical generation of networks
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This algorithm uses the actual corrected characterization data for fiber dimensions. This means that shortening of fibers and its effect on formation is automatically accounted for in the simulations. A sample of a generated network is shown in Fig. 6.
Fig. 6A sample of a numerically generated network, a in-plane view, and b thickness view
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We used FEM (finite element method) in an implicit solution scheme to analyze the networks. To do so, the generated network was discretized using nonlinear 3-node Timoshenko/Reissner beam elements (Ibrahimbegović 1995) with either solid or hollow rectangular cross-sections. The inter-fiber bonds were simulated as bonded (adhesive) beam-to-beam contact elements (Motamedian 2016, 2018) with a separation criteria. A more detailed description and demonstration of application of the formulations can be found in (Borodulina et al. 2016; Motamedian and Kulachenko 2018, 2019). The geometric, mechanical, and discretization parameters that were used in the simulations are listed in Table 10.
One of the known effects of the fibrillar fines is forming bridges between the neighboring crossing fibers and reinforcing the bonds as shown in Fig. 7a. In some cases, this can result in the formation of webs of fines, as shown in Fig. 7b. Note that the SEM images of Fig. 7 have only captured the fines on the surface while the inner structure is more intricate. To mimic the behavior of fines in the simulations, we introduced a large number of truss elements, which are shown as blue components in Fig. 8. These truss elements have two nodes with three translational degrees of freedom at each node and they solely transmit load in axial tension. The nodes of these elements are connected to the existing nodes of the beams representing the fibers. Owing to their small dimensions, the fibrillary particles tend to converge to the bonds according to literature. To represent this placement, the deposition of the fibrils is made by first selecting an arbitrary node and then finding a second node within a sphere of a designated radius. The sphere radius equals the maximum fine length and it is centered on the first node, which is randomly chosen. This distribution approach yields a fibrillary deposition in which most of the fines gather near the bonds between fibers which is consistent with what is reported in literature. Using this method, no additional degrees of freedom are added to the system. For clarity, this approach is visualized in Fig. 9. The properties of the elements used to simulate the fines can be found in Table 10.
Fig. 7SEM image of paper surface showing, a some fibrillar bridges and b a web structure formed by fines
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Fig. 8Representation of fibrillar bridges with (blue) link elements. (Color figure online)
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Fig. 9Deposition of fines as fibrillar bridges
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Table 10 Parameters and settings used in numerical simulationsFull size table
To show that the captured variability between pulps with different levels of refining do not have a large effect on the properties of the handsheets, we created five different networks using the characterization data corresponding to each level of refining (15 networks in total). We considered similar other properties for these networks. The resulting averaged stress–strain curves are shown in Fig. 10. These results show that the pulp variations captured through characterization do not have a considerable effect on the properties of the handsheets. Given that no considerable differences were revealed, in the generation of the subsequent networks we used a set of pulp data consisting of the combined data captured for all three pulps.
Fig. 10The effect of pulp variations captured through pulp characterization on the simulated stress–strain curves in comparison with the experimental results
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It is suggested that the improvements caused by refining can mainly be attributed to the increased density (Sehaqui et al. 2013). According to the measurements shown in Table 8, the density of handsheets is greatly affected by refining. The change in the density is mainly due to the difference in the thickness of the handsheets. According to our network generation algorithm, there are three parameters that can greatly affect the thickness of the handsheets: the width-to-height ratio of the fiber cross-sections, the press ratio at the bonded sites, and the interface angle between two bonded fibers. The data extracted from the micro-tomography images showed that neither the width-to-height ratio of free fiber segments nor the interface angle distribution are considerably affected by different refining levels. Consequently, while using the micro-tomography extracted data on width-to-height ratio and interface angle in generating networks, we focused on the press ratio at bond sites as the main reason affecting the thickness of the networks. Unfortunately, it was not possible to extract data on the press ratio parameter from tomography images due to difficulties in precisely tracking fibers at the bonded sites caused by the limited resolution of the images. To ensure the correctness of the network representation, we extracted the through thickness density profiles of the handsheets from the tomography data. In the network generation process, we used the press ratio parameter to match these density profiles and also the measured thicknesses of the handsheets. The density profiles of the unrefined and mostly refined samples, and examples of density the profiles of corresponding numerically generated networks are shown in Fig. 11.
Fig. 11Density profiles of unrefined and mostly refined handsheets from tomography data and numerical network generations
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For each level of refining five different networks are generated. The averaged resulting stress–strain curves from the analysis of the generated networks are shown in Fig. 12 together with the experimental stress–strain curves, for comparison. As can be seen in the results, although the increase in the density of the handsheets affects the strength and stiffness of the networks through the increased number of bonds, it is not sufficient to explain the total improvements.
Fig. 12Effect of density on the simulated stress–strain curves in comparison with the experimental results. The bars show the maximum and minimum of the strength results from 10 experimental or 5 numerical samples for each density and the curves represent the averages
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Refining is often described as a process that leads to increased bond area and strength. As we use a pointwise beam-to-beam contact model to represent the inter-fiber bonds, we cannot study the effect of the increased bonded area directly but rather through an indirect change of bonding properties. The increase in the bond strength can be related to the increased bonded area between fibers, which may likely happen upon densification or the use of strength additives. Consequently, we studied the effect of stronger bonds combined with the previously considered effect of density. The stress–strain curves from our analyses of the same network (only one of the networks, keeping the geometry, meshing, etc. constant), with different strengths for inter-fiber bonds, are shown in Fig. 13. As expected, the increase in bond strength improves the strength of the network; however, the stiffness of the sample remains unchanged. This result is consistent with the experimental results on the handsheets from unbeaten pulp in response to the addition of bond strength additives. Those experimental results had shown that using strength additives in unbeaten pulp does not noticeably affect the density and stiffness, but it does improve the strength of paper.
Fig. 13Effect of bond strength on simulated stress–strain curves
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Previous experimental study has shown that both refining and the addition of fibrillary fines have the same effect on the mechanical properties and the dewatering characteristics of paper products (Afra et al. 2013). This suggests that the creation of fines can be a major reason behind the modifications achieved by refining. A microscopic photo of a fiber and fines of both fibrillar and chunk type can be seen in Fig. 14. To get a better understanding of the characteristics of the fines in the handsheets, microscopic images of the pulps were taken. The goal was to qualitatively compare the shape and amount of fines in the differently refined pulps. The microscopy images of the pulps were taken using an Olympus BX50 System Microscope. Representative microscopic images of the pulps can be seen in Fig. 15.
Fig. 14Microscopic images of pulp showing a fiber and some fines
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Fig. 15Microscopic images of differently refined pulps
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Although we can see from the microscopic images that refining has resulted in an increased fine percentage, the amount and size of fibrillar fines in our pulps are not known due to the limited resolution of measuring devices. Consequently, we conducted a parameter study to investigate the effect of fine size, the mechanical properties, and the percentage on the stiffness and strength of the networks. We have chosen the reference set of values for all parameters as presented in Table 11, and investigated the effect of varying each of those parameters while keeping the others as the reference value. The range of studied variations are given in Table 11. We have added the fines to a network corresponding to the unrefined sample, unless stated otherwise.
Table 11 Initial parameter values and the range of variations in the fine parameter studyFull size table
The pulp characterization tools have limited resolution and the fines below certain size will not be detected. This fact was previously reported in the literature (Brodin and Eriksen 2014). In this study, we used a constant mass fraction of the fines equal to 3% and varied the diameter of the fines. We generated a base network and assumed a specific diameter size for fine cross-sections. We started adding link elements to the network until the target mass fraction was reached. We then repeated the same procedure for different diameters of fines. The resulting stress–strain curves from the analysis of the mentioned networks are shown in Fig. 16. As seen in the figure, fines with large diameters of 16 or \(10 \, \upmu \hbox {m}\), which are on the verge of what can be detected by the characterization tools, have almost no effect on the properties of the networks and fines with smaller cross-sections, which are in fact not detected by the characterization devices, have the largest impact on both the strength and the stiffness of the networks. As we have considered a fixed mass percentage of fines, decreasing the fine diameter results in a large increase in the number of fines. This leads to a more effective distribution of fines, and better collective stiffening and strengthening of the network, especially close to the bonded sites where most of the fines are located.
Fig. 16Effect of fine diameter on simulated stress–strain curves
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For this part of the study, fines with the diameter of \(1 \, \upmu \hbox {m}\) were added to a base network (with a thickness corresponding to the experimental handsheet from the unrefined pulp, R0) and a denser network (corresponding to the handsheet from the mostly refined pulp, R2000) at different mass fractions. The results in response to the addition of fines are shown in Fig. 17. The baselines indicate the experimental results for both stiffness and strength in case of the mostly refined and densified sheets. Unless the densification is not taken into account, the experimentally measured stiffness cannot be matched within the reasonable range of fines fraction. This indicates that accounting for densification is important. Obviously, there is a direct relationship between the fines content and the amount of refining, which is consistent with previous studies (Kibblewhite 1972) but was not detected by the tools used for the pulp characterizations. The degree of the change in response to adding fines for both the stiffness and strength is comparable with the findings in which controlled amounts of fibrillar fines were added to the pulp (without refining) to improve the strength and stiffness (Boufi et al. 2016; Johnson et al. 2016; Luukko and Paulapuro 1999).
Fig. 17Effect of fine mass percentage on a stiffness and b strength of networks
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We can now shed the light on the exact mechanisms brought by fines through observing how the strain energy is partitioned between different forms of deformation. We separated the energy into elongation and bending (stored in the fibers), inter-fiber bonding and fines. The bonding energy is collected from the penalty-based contact elements. The other forms of deformations, including torsion and shear, are small and presented combined. The computed partitioning is shown in Fig. 18, where the strain energy output from the simulations of a network without and with 3% fines are compared. In both cases, most of the energy is stored in the longitudinal deformation. Given that fines act as bridges connecting fibers together, it is not surprising that the greatest effect they have is on the strain energy stored in the bonds. They also help in immobilizing the fibers, and therefore, also reduce the bending energy.
Fig. 18Strain energy stored in different components of the network in a absence and b presence of fines
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Other than carrying part of the load applied to the network, an increased amount of fines helps to smooth the strain field, which in turn can lead to the higher load capacity of the network. Figure 19 shows the strain localization from simulations of networks with different fine content.
Scattering coefficient is related to the total specific surface area and has traditionally been used to assess the degree of bonding. It was shown that fines are able to reduce the scattering coefficient at constant density (Lehtonen 2004). This was linked to the higher number of smaller pores resulted by the addition of fines at constant pore volume. Using experimental methods, separating out fiber fractions and doing hot wet-pressing, Lehtonen also concluded that fines ensure better stress transfer between bonded fibers and this effect is activated during wet-pressing. It collaborates well with the finding of this study.
Fig. 19Effect of fine percentage on stress localization (from numerical simulations)
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Previous study has shown that the length of fines increases with the amount of energy used in refining (Retulainen et al. 1993). In this section, we again consider a fixed mass fraction of fines equal to 3%; however, we study the effect of maximum length of fines. Increasing the maximum length combined with the assumed mechanism of fine distribution will yield more fibrillar fines connecting distant regions of the fibers. The results from this study are shown in Fig. 20. It can be seen that the increase in the length of fines does not give a consistent trend. Increasing the maximum length has a positive effect on the strength of the networks up to a certain value between 50 and \(100 \, \upmu \hbox {m}\) and afterwards the effect becomes negative.
Fig. 20Effect of fine length on simulated stress–strain curves
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Increasing the length of fines results in fewer fines with the given fines fraction; however, this reduction is not as drastic as the in case of increasing the diameter. This reduction in the number of fines tends to reduce the improvement in the strength and stiffness. Meanwhile, longer fines can connect fibers at longer distances, which can contribute to better stiffness or strength improvement. The results show that the positive effect of increased length is more dominant to a certain value, after which the negative effect takes over. These results also suggest that the main effect from the fines comes through the collective reinforcement of the bond surroundings.
In this part, we study the effect of fine strength and stiffness on the strength and stiffness of the networks, separately. The stiffness of the fines depends on their type, while the strength is dependent on the bond between fines and fibers. The results shown in Fig. 21 correspond to varying the elastic moduli in the range of 50–120 GPa for the fines, while keeping all of the other properties of the networks intact. We have used maximum strain as the failure criteria for fines. Upon reaching this strain, the fines are behaving ideally plastic, that is, without an incremental addition of stress upon increased strain. The effect of varying this failure criteria from a maximum allowable strain of 0.5% to 8% is shown in Fig. 22.
In case of increased stiffness with a given strain-to-failure of 3%, both the stiffness and the strength increased. The increase of the strength can be explained by the fact that fines can store more energy prior to failure. A similar effect is achieved by increasing the strain to failure. At the same time, increasing the strength of fines while keeping their stiffness unchanged does not affect the stiffness of the network because there are very few fines broken in the initial part of the loading with the property range used in the study.
Fig. 21Effect of elastic modulus of fines on a stiffness and b strength of networks
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Fig. 22Effect of strain to failure of fines on a stiffness and b strength of networks
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The strength of paper pulp depends on the strength, the distribution and arrangement of fibers in paper pulp. The most important is the strength of the fibers, which determines the mechanical properties of the pulp and paper.
Paper pulp strength is measured by a number of indicators, including pretreatment PH value, Capa value, tightness, tensile index, tear index, breaking index etc. Paper pulp strength is closely related to the beating and making process in the paper pulping process. The effect of paper pulp beating on fiber, especially the degree of fiber broom, is the main factor affecting fiber binding. Because of that the fiber surface area and hydrogen bond number are related to beating. At the same time, the size of the fiber binding force is related to the chemical composition, physical properties and arrangement of the fibers. In other words, fiber lignin, hemicellulose, cellulose content, etc, will affect the fiber binding to some extent.
Therefore, pulp strength is affected by many factors. Mainly include: pulp types, hemicellulose, cellulose, lignin, paper-making additives, papermaking operation etc.
Different kinds of paper pulp are different in physical structure and chemical composition. Generally speaking, chemical wood pulp fiber adhesion is the largest, cotton pulp is the second, mechanical wood pulp is the worst. The first step to improve paper pulp strength is to select better paper pulp types.
The pulp with high hemicellulose content is easy to absorb water and swell during beating, which increases the specific surface area and binding area of the fiber, and improves the binding force of the fiber. Hemicellulose has shorter molecular chains than cellulose, and is more hydrophilic. When it is beating by the paper pulp beater, it is easy to absorb water and expand and fine fibrosis. Therefore, the pulp containing hemicellulose is easier to beat and the binding force between fibers is larger.
Of course, the high content of hemicellulose is not good for beating and papermaking. The main reason is that the hemicellulose absorbs water and expands too fast, the beating is too high, so the paper is transparent and brittle, and the strength is low. On the other hand, with more hemicellulose and more short fibers, the strength of the fibers will decrease and the adhesion of the fibers will be reduced.
In general, cellulose molecular chain is long, it is high degree of polymerization, the fiber itself is strong, when beating by the paper pulp beater, is not easy to cut off. Therefore, when it has already been cut to an appropriate length, the fiber has been fully broomized, and the adhesion between the fibers is greater.
There was a negative linear correlation between lignin content on pulp fiber surface and tensile index and breaking resistance index. With the increase of lignin content, the tensile index and breaking resistance index of the paper decreased. The blending ratio has no obvious effect on tensile index and breaking index of finished paper, but has certain effect on tear index of pulp tensioning.
Lignin mainly distributes in the primary wall and the secondary wall of the fiber. Because of the very low hydrophilicity of lignin, the existence of lignin affects the swelling and fine fibrosis of the fiber, so the pulp containing more lignin is not easy to beat and the binding force of the fiber is poor.
Paper pulp beating must be strictly controlled. Therefore, starting from beating, the beating degree, wet weight and water retention value were measured every certain time, and the changes of fiber morphology were observed. At the same time , the paper should be sampled and the physical strength of pulp should be measured to study the change of pulp performance during paper pulping process.
The addition of hydrophilic substances, such as starch, protein, vegetable gum, etc. Adding things like these to the pulp increases the interfiberal adhesion, because these substances themselves have the same polar hydroxyl groups as cellulose, because the hydroxyl groups bind to hydrogen bonds.The combination of fiber is stronger than that of pure fiber. The addition of hydrophobic substances, such as inorganic fillers and alum, can reduce the adhesion between fibers, which is because the addition of these substances will separate the fibers from the fibers and reduce the interface between the fibers, thus affecting the adhesion of the fibers.
This is the technology work in the paper pulping process, such as beating, such as drying. Because of that paper strength is not only related to fiber itself, but also to paper fiber binding strength. The strength of the bond is related to paper making. In general, the bonding strength between the pulp fibers is proportional to the ability of the fiber to form hydrogen bonds between the fibers.
For the same kind of paper pulp, the fiber's specific surface area and contact area are constant. If lignin is adsorbed on the fiber surface, the ability of forming hydrogen bond between fibers will be affected, which will lead to the decrease of interfiber bonding strength.
Paper pulp beating machine is an important equipment in papermaking process.
It plays an important role in pulp, as well as in the strength of paper products, and needs careful selection. CNBM can introduce to customers about pulp strength, and can provide relevant equipment for beating machines. Please contact us for details of the type of paper pulper machine. Welcome to consult!
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