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in larger Poisson ’ s ratio values at break (Table 1a). As expected, heavy re fi ning (R þ ) followed this trend and suppressed the auxetic response even further (solid squares in Figure 4). Re fi ning generates new fi brils on the surface of fi bers, increas- ing connectivity but simultaneously also causing fl attening, splitting, or breaking of fi bers. We believe that this invites mech- anistic factors (2), (3), and (4) into play (see Section 3.1 and Figure 5d), each of which can diminish or potentially eliminate an auxetic response. Re fi ning is also known to enhance the elas- tic properties of paper. [46] This was indeed observed to be true in our case; density and modulus for unre fi ned handsheets were lower than re fi ned handsheets, which in turn were lower than heavily re fi ned handsheets (density and elastic modulus for all samples are shown in Figure6 ). Furthermore, even though re fi n- ing increases connectivity in the network (thereby increasing the density and modulus), these are likely distributed in all directions randomly (especially in a handsheet when compared to machine- made paper) and not just along the thickness direction. Thus, even though the increase in density and modulus is signi fi cant due tore fi ning, the increase in connectivity between auxetic units along the thickness direction may not be signi fi cant or enough to overturn the trend. This explanation is not to suggest that re fi ning will invariably suppress the auxetic response. In theory, as per our mechanism, thickness could increase if re fi ning could produce connectivities and a packing arrangement ideal for the ef fi cient transfer of local auxetic responses.
3.2.3. Effect of Grammage/Thickness
Figure 6 shows that the density and modulus of unre fi nedhand- sheets for both kinds of pulp and for all thicknesses were about the same. Thus, unsurprisingly, no discernable trend in auxetic response with thickness was observed for unre fi ned handsheets (see open squares in Figure 4). Even after re fi ning, density and modulus remained approxi- mately the same for medium and heavy handsheets within the same re fi nement level. Light re fi ned handsheets (i.e., HW1R, SW1R, HW1R þ , and SW1R þ ), however, were found to have a lower density and modulus than the medium/heavy handsheets. In fact, their density and modulus were close to their unre fi ned counterparts. Why was the density/modulus of thin re fi ned handsheets lower? Since a cellulose fi ber diameter is about 10 – 25 μ m, [13] we hypothesize that 4 – 10 fi bersor 2 – 5aux- etic units could fi t through thickness in a 90 μ m thick sheet. Perhaps, there were an insuf fi cient number of fi ber layers that could bene fi t substantially from increased connectivity and packing possibilities resulting from re fi ning. It should be remembered that the handsheets are formed via roughly a layer-by-layer assembly under hydrostatic pressure. More wet layers would be a cause of higher pressure and better packing. This inef fi cient connectivity in thin sheets could also explain why thin re fi ned handsheets generally exhibited a smaller auxetic response (see smallest squares in Figure 4 and values in Table 1). Although we spot mechanistic differences in the deformation behavior with respect to the thickness of the paper, we believe that even our thinnest handsheets do not approximate to exam- ples of auxetic single-layer thin fi lms such as graphene that can show tunable negative Poisson ’ s ratio due to wrinkling and assemblies of graphene layers into macro fi lms via formation/ assembly conditions. [47,48] We believe that in the case of our handsheets there is suf fi cient thickness to render surface effects negligible and that we can consider them as bulk materials. Furthermore, thick re fi ned handsheets were found to be more auxetic than medium handsheets (see Table 1) even though their density and modulus were similar. This indicates that re fi ning might have been responsible for increasing through-thickness connectivity in thick handsheets compared to medium hand- sheets (Figure 6).
0 2 4 6 Modulus (GPa)
Density (g/cc)
0.4 0.6 0.8
HW5 HW3 HW1 SW5 SW3 SW1
0.57 0.58
2.32
2.55
0.55
2.37
0.57
2.24
0.58 0.58
2.63 2.58
HW5R HW3R HW1R SW5R SW3R SW1R
0.74
3.84
0.73
4.35
0.63
2.57
0.77
4.66
3.2.4. The Case of Nonauxetic Behavior
0.73
4.32
Speci fi cally, only HW1R þ was truly not auxetic, showing a con- sistent decrease in thickness throughout the entire strain range. While the overall cumulative Poisson ’ s ratio for HW3R þ was positive, it exhibited a dip and a subsequent thickness increase like many other samples. The dip (as in HW3R þ ), or even the consistent decrease (as in HW1R þ ) in thickness, could arise from two factors — 1) a conventional nonauxetic elastic/plastic deformation of a near isotropic porous solid and/or 2) presence of fi ber segments that are not fully extended initially as shown in Figure 5(d6). We believe that a combination of heavy re fi ning (R þ ), shorter fi bers (characteristic for HW) or segments, poor connectivity (due to shorter fi bers), or isotropic connectivity (due to heavy re fi ning), both of which deviate from the ideal auxetic model, contributes to the nonauxetic behavior.
0.60
2.84
HW5R+ HW3R+ HW1R+ SW5R+ SW3R+ SW1R+
0.78 0.77
5.25
5.42
0.69
3.39
0.83
5.81
0.79
5.29
0.65
4.31
Figure 6. Density and Young ’ s modulus of handsheets. Density was calculated for all prepared handsheets using their mass, area, and thicknesses. Modulus was calculated for each of the fi ve specimens using load at 0.1% strain, width = 2 cm, and specimen ’ s thickness. Standard deviations for samples are shown in red.
2400589 (8 of 10)
Phys. Status Solidi B 2025 , 2400589
© 2025 The Author(s). physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH
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