Nanomaterials 2022 , 12 , 790
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In order to adapt the Ø g methodology to CNFs, it was necessary to use a dye to visualize the sedimentation line of nanofibers. Cristal violet was selected since it did not influence the sedimentation of the samples. Results showed that highly fibrillated CNFs presented sedimentation curves different from the conventional ones, requiring longer sedimentation times to obtain stable deposits than other cellulosic materials, as previously published [32]. To determine Ø g values, at least five sedimentation experiments with different C o are required. The curve that relates the C o against the H s /H o is plotted. Then, the derivative of the curve close to zero allows the determination of Ø g as Equation (1) shows, in which Ø g andC o have the same units [31,33–35].). Two mathematical methods have been traditionally used to calculate Equation (1). The first one is the fitting to a quadratic equation without an independent term in which Ø g is the first order coefficient [32,36,37], and the other is using the fitting tool CSAPS in MATLAB [28,38]. Both are tedious and time-consuming. Recently, a simplification of the Ø g methodology (Equation (2)) allows to reduce the experiments labor by at least 60%, using only one measured cylinder or two if the C o selected is not suitable, showing an error lower than 7% in Ø g values and 3% in the calculation of the estimated AR [39].
⎞ ⎠
⎛ ⎝
dC o d Hs
∅ g = lim Hs Ho → 0
(1)
Ho
C o ( i ) − C o ( 0 ) ( H s / H o ( i )) − ( H s / H o ( 0 ))
C o ( i ) − 0 ( H s / H o ( i )) − 0
C o ( i ) ( Hs / Ho ( i ))
(2)
( est ) =
=
=
∅ g
To estimate AR from Ø g , the Effective Medium Theory (EMT) and the Crowding Number (CN) theory [36,37] are two possible alternatives. EMT was firstly developed by Celzard et al. [40] to describe the conductivity of a material in which the particles are dispersed, for example, for spheroid graphene particles dispersed in air. AR and Ø g are related by the Equations (3) and (4), according to Celzard et al. and Varanasi et al. [37,40]. ∅ g vol.% vol.% = 9L c ( 1 − L C ) 2 + L c ( 15 − 9L c ) (3)
3 ln
− 2 1 − AR − 2
1 + 1 − AR − 2 1 − 1 − AR − 2
AR − 2 2 1 − AR − 2
(4)
L c =
whereL c is the depolarization factor of the particles. Kerekes and Schell [41] developed the CN theory that relates the Ø g and the AR of the cellulose fibers. Then, Ø g was estimated by Martinez et al. [35] with a CN value of 16 ± 4, based on the analysis of positron emission tomography (PET) measurements of dilute fiber sedimentation experiments. This fact establishes a relationship between Ø g and AR in Equation (5): ∅ g vol.% vol.% = 24/AR 2 (5) However, the volume fraction is more difficult to measure than the solid fraction, Ø g (kg fiber/ kg suspension) as Varanasi et al. [37] have demonstrated. Therefore, the relation betweenØ g (vol./vol.) and Ø g (wt.%/wt.%) may be expressed by Equation (6). ∅ g vol.% vol.% = ∅ g wt.% wt.% · ρ L wt.% vol.% ρ F wt.% vol.% + ∅ g wt.% wt.% · ρ L wt.% vol.% − ∅ g wt.% wt.% · ρ F wt.% vol.% (6) where ρ F is the CMF/CNF density [37] assumed as 1500 kg/m 3 , and ρ L the density of the suspension. If the CMF/CNF dose is very low, under 1 wt.%, the density of the suspension could be approximated to the water density.
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