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Additionally, the zero-span tensile strength of the MFC at 50 dry wt% mineral was measured, and for most fibre species was over 2.5 times the long span tensile strength of the MFC, when normalised by fibre content. This MFC zero-span tensile strength was also found to vary strongly with mineral content, implying that bonding failure was not completely suppressed (likely because smaller MFC particles are more prone of slipping out of the gap between the clamps). The notion that MFC fails mainly by network failure even in a zero-span tensile test is supported by electron microscopy of breakage sites by Varanasi et al. (2012). This makes the measured MFC zero-span strength a gross underestimate of the true value, and so fibril failure is responsible for much less than 40% of the breakage. Therefore, the fibre weakness term will be assumed to be zero as the MFC zero-span tensile strength is believed to be much higher than the bonding strength. The Page Equation is therefore reduced to: 12 A q s B PL RB A ð Þ 2 Þ Hemicellulose on microfibril surfaces would be expected to increase relative bonded area since extended hemicellulose chains allows for more exten- sive and intimate contact between microfibrils. Addi- tionally, hemicellulose appears to result in finer microfibrils when MFC is produced. Finer diameter microfibrils could be expected to be more flexible, and so be more capable of deforming for a more intimate contact with other particles. It is also expected that finer microfibrils would be more susceptible to capillary forces drawing them in contact with other surfaces during drying. These effects would imply that a high hemicellulose content would lead to a high relative bonded area. It is conceivable that these same factors could lead to a higher specific bond strength s B within a given region of bonded area (i.e. more hemicellulose chains likely means a higher fraction of the bonded area in mo l ec ul ar contact). 1 T ¼ ð The perimeter to cross-sectional area ratio, P/ A , would also correlate with increased bonding; since this increases as fibril diameter decreases, this would also correlate with hemicellulose content. Though P/ A , RB A , and s B are all believed to be influenced by hemicellulose content, their relative influences cannot be distinguished. If hemicellulose correlated linearly with all three, then it would be expected that, all other
things being equal, tensile index would be propor- tional to the cube of the hemicellulose content; however, Fig. 1 shows that this correlation is linear, and the maximum R 2 value of the fit between hemicellulose and MFC tensile strength was obtained with a hemicellulose exponent value close to unity. Therefore, rather than distinguishing between the three, the RB A term, the P/ A term, and the s B term in the denominator of the Page Equation are replaced with the hemicellulose content H in Eq. (3) below. Since the larger MFC particles are the primary load- bearing particles, the corresponding MFC length term L MFC is expected to be analogous to the fibre length term L in the Page Equation, and so this substitution is alsomade: 3 Þ where k is a proportionality constant. Density q is constant for all MFC, so can be combined with k under a single coefficient B 1 : 1 T ¼ k 12 q L MFC H ð
1 T ¼
1 B 1 L MFC H
ð
4 Þ
which can be inverted to obtain the MFC tensile index: T ¼ B 1 L MFC H ð 5 Þ The product of L MFC and H was calculated for all fibre sources and plotted against MFC tensile index in Fig. 3. As can be seen, the fit to a general straight line is much better when MFC length is included in this analysis compared to just hemicellulose content alone, with the R 2 improving from 0.63 to 0.87. However, this fit requires an intercept that is not accounted for in Eq. (5). This intercept is at least in part because fibrillation can take place in the absence of hemicel- lulose, as the cotton linter and jeans MFC demonstrate. Other contributions to this intercept may be due to some of the assumptions being inaccurate, for example if the contribution of fibre weakness to the strength was significant, or if the tensile strength at 80% mineral loading was not directly proportional to mineral-free tensile strength for all samples. Despite this, the fit is good, and so an empirical residual strength term, r 0 , is added to Eq. (5), representing at least in part the MFC tensile strength when hemicel- lulose is absent.
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