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Cellulose (2016) 23:2249–2272
D G elast . elastic free energy of the gel; D G mix free energy of mixing of the gel components and the swelling medium; D G electr. electrostatic free energy In equilibrium the total free energy is 0 and so the following equation is valid: ð 2 Þ D G elast counteracts the osmotic forces described by D G mix and D G electr. . In the case of papermaking fibers it is determined by the constituents of the fiber wall and the arrangement of the different fiber wall layers. D G mix is defined by the molecules in the network of the gel, molecular cellulose fibrils, and the solvent water. These mobile molecules form a mix with the polymer and the solvent. D G electr. is specified by the charges within the gel that gives rise to the osmotic pressure. The status D G = 0 might be valid before sheet forming and is therefore strongly depending of process water quality and used additives. Dewatering and the corresponding approach of the fibers and fibrils is responsible for interfering the fiber water gel and for the intermediate fiber–fiber bonds (Kibble- white 1973; Wa˚gberg and Annergren 1997). Total free energy in a gel: D G D G ¼ G elast þ G mix þ G electr ¼ 0
respective dryness levels. The web dryness is influ- enced by the type of raw material, its composition, and additive usage during sheet formation, even though all mechanical settings for sheet formation and press operation are kept constant. For this reason, the evaluation of IWWS should be done at constant sheet dryness. The effect of changing sheet dryness should be evaluated separately.
Nanometer level (molecular scale)
The nanometer level describes the bonding forces such as electrostatic forces e.g. van der Waals forces. At this level, the approximation of the contacts and bonding forces between fibers and fibrils is the decisive factor. The following paragraphs present the forces and conditions affecting this bonding type according to their mechanistic priority. If the distances between the solid particles are sufficiently small, electrostatic and van der Waals forces can develop (Israelachvili 2006b; Pelton 1993; Wa˚gberg et al. 1987). Second, the theory of molecular fibrillation and partial solubility has been described (Campbell 1930, 1933; Casey 1960; Clark 1978a). To achieve the most accurate possible fiber and fibril approach, a high degree of fiber flexibility is required, which is significantly influenced by internal hydrogen bonds (Hubbe 2006; McKenzie 1984). External hydrogen bonds between fibers will only form during drying (Forgacs et al. 1957; Lobben 1976; Robertson 1959; Williams 1983). For this reason, external hydrogen bonds are not studied in detail in this paper.
Van der Waals forces
If fibers are in sufficient close proximity, van der Waals forces will occur between fibers and fibrils (Eriksson 2006; Hubbe 2006; McKenzie 1984; Pelton 2004; Wa˚gberg and Annergren 1997; Williams 1983) . Figure 3 describes different forces on the surface of cellulose I, II and amorphous cellulose that may act also between the fiber and fibril surfaces. However, this may not be applicable for initially wet paper due to the high water content and the greater distances between the single fibers (Linhart 2005). For interactions to occur, the distances between the fibers and/or fibrils must be very small. The distances described in the literature are between 0.15 and 0.35 nm (Gardner et al. 2008; Linhart 2005). Remark- ably, these distances are considerably smaller than the fiber roughness, which ranges between 10 and 10,000 nm (Heinemann et al. 2011). Figure 4 shows an example of an uneven fiber surface of a common never dried softwood fiber.
Fiber water gel on the fiber surface
In 1963, Voyutskii proposed the formation of a hydro- gel on macromolecules in ‘‘Autohesion and Adhesion of High Polymers’’ (Voyutskij 1963b). In additional papers, the bonds in wet sheets were explained based on a gel-like surface of the fiber (de Oliveira et al. 2008; Lindqvist et al. 2013; McKenzie 1984; Myllytie 2009). In principle, the swelling of a gel can be determined by the energies summarized in Eq. 2 (Flory 1953; Katchalsky 1954; Yin et al. 1992):
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