Nanocellulose is an emerging material that needs to be well characterized to control its performance during industrial applications. Gel point (Øg) is a convenient parameter commonly used to estimate the aspect ratio (AR) of cellulose nano/microfibers (CNFs/CMFs), providing critical information on the nanofiber network. However, its estimation requires many sedimentation experiments, tedious and time consuming. In this study, a simpler and faster technique is presented to estimate Øg, based on one or two sedimentation experiments, reducing the experiments by a factor of at least 2.5. Here, this new methodology is successfully validated by using the Øg of different CNF/CMF hydrogels calculated with the traditional methodology, showing an error lower than 7%. The error in the estimation of the AR is lower than 3% in all cases. Furthermore, the two mathematical models currently used to estimate Øg, the smoothing spline and the quadratic fit, are compared and the mathematical assumptions improved.
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Introduction
Gel point (Øg) is defined as the volume concentration of a suspension at the boundary between the semi-dilute and dilute regions. This concentration is considered the lowest at which all flocs are interconnected, forming a self-supporting network (Martinez et al. 2001; Nasser and James 2006; Raj et al. 2016a; Zhang et al. 2012).
In the last few years, Øg has been used to estimate the aspect ratio (AR) of different cellulose materials such as cellulose fibers, fines, microfibers (CMFs) and nanofibers (CNFs) (Martinez et al. 2001; Sanchez-Salvador et al. 2020a; Varanasi et al. 2013) using two methods: the Effective Medium Theory (EMT) and the Crowding Number (CN) theory (Celzard et al. 2009; Kerekes and Schell 1992).
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Performing the Øg methodology requires at least 5 sedimentation experiments, each with a different initial fiber concentration (Co), which is time consuming and a drawback, especially in industry. Another difficulty is that sedimentation time increases from 2 to 10 days with the fibrillation degree of the fiber sample (Sanchez-Salvador et al. 2020a).
Two mathematical methods have been used to calculate gel point (Øg). In the first, the Øg was estimated from a quadratic fit using Eq. 1 according to Varanasi et al. (2013). In this case, Øg results from the derivative at the origin of the curve Co vs Hs/Ho as indicated by Eq. 2 (Martinez et al. 2001).
where Hs/Ho is the sediment height (Hs) normalized by the initial height (Ho) as measured from fiber sedimentation directly in the graduated cylinder.
The second method was developed using the curve fitting tool (CSAPS) in MATLAB. Øg is obtained from the first derivative of each curve in the y-intercept (Raj et al. 2016b). In this case, a smoothing spline is fitted to the data with a smoothing parameter (p) oscillating in the interval [0, 1]. For p = 0, the equation solution is the least-squares straight line fit to the data, while, on the other extreme, for p = 1, the solution is the variational or natural cubic spline interpolant. As p moves from 0 to 1, the smoothing spline changes as the parameter approaches p = 1 the equation is closer to the data than the straight line. Traditionally, a constant “p” parameter is used but it must be optimized for each of the different types of nanofibers as explained in Sect. 2.1.
Here, we compare the use of both mathematical methods and their optimization, since in many cases they are used regardless of its mathematical limitations. In addition, we present a simpler and faster technique with mathematical assumptions to characterize Øg and subsequently, the aspect ratio of nano/microfiber suspensions. Besides, these simplifications may be applied to other kinds of suspensions. The main novelty of this method is the reduction in the number of sedimentation experiments to estimate Øg using only one sedimentation experiment, at an optimal Co (Co,opt), based on a classical approximation of derivatives by a small finite difference close to zero. Thus, the number of experiments and time labour are reduced by at least a factor of 2.5. It is important to adequately select the optimal Co (Co,opt), in order to obtain a sedimentation deposit neither too small to have a high experimental error nor too big to have a disproportionate increment. If this is not achieved, it will be necessary to do a second experiment with a new Co closer to Co,opt. The optimal concentration will depend on the type of fiber source, the pretreatment or the fibrillation degree.
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Theory
Quadratic fit and smoothing spline function
After carrying out the sedimentation experiments with suspensions at different CNF/CMF concentrations, Hs/Ho is measured in the graduated cylinders (See in Supplementary 1, Fig. S1), following the traditional methodology. Then, Øg is estimated using the quadratic fit or the CSAPS method.
When the quadratic fit is used, the conditions to obtain are based on Eqs. 1 and 2. In Eq. 1, the quadratic fit is forced to have an independent term equal to zero. This fact makes the parameter “b” in Eq. 1 slightly lower than both the quadratic fit with the independent term and that calculated with the smoothing spline.
Using the CSAPS smoothing spline function requires to select the value of the “p” parameter based on two criteria:
The selection of the minimum smoothing parameter (p) to represent the data close to the origin accurately, with the y-intercept very close to zero.
The first derivative of the function must provide a smooth and gradual increase with no local fluctuations.
The selection of the smoothing parameter (p) in CSAPS smoothing spline function varies for each sample, not being able to use the same “p” value for all CNFs/CMFs. Figure 1 shows the differences between CSAPS smoothing spline functions using different values of “p” parameter for the results obtained in this study with the same CNF/CMF sample. In Fig. 1a, the “p” parameter is too low, and the curve shows a y-intercept far from zero and a linear trend. In Fig. 1c, the “p” parameter is too high, and the first derivative shows local fluctuations because the curve tries to connect the experimental values without obtaining a curve. However, the y-interception is almost zero. Therefore, the optimal parameter is an intermediate “p” value” as shown in Fig. 1b. As consequence, the Øg calculated with different “p” values that does not adequately represent the gel point curve may have variations even higher than 30%. Further, the “p” parameter cannot be a fixed value for all samples as in some instances this value would show local fluctuations and an y-intercept very close to zero (similar to Fig. 1c); the same “p” value in other samples would show a line far from the origin in the y-interception as in Fig. 1a. This is the reason why each sample should be selected with an appropriate p-value that adequately represents the Øg curve. For this, it will be necessary to find a mathematical model that defines a minimum p-value as shown in results, where the smoothing spline curve approaches the curve of the gel point, with a low error in the y-intercept and no local fluctuations of the curve.
Fig. 1
Selection of the optimum p parameter of CSAPS smoothing spline function. Experimental data are fitted to the traditional quadratic fit; a “p” value of 0.99; b “p” value of 0.9999; c “p” value of 0.99999999
×
Simplifying assumptions
The assumptions to simplify the Øg calculation are shown in Eq. 3. The derivative is replaced by a classical assumption, an increment between a determined concentration Co(i) and a theoretical concentration of zero, Co(0). Therefore, the derivative is approximated as the quotient between the difference of concentrations and the difference in the relation of heights. Both Co (0) and (Hs/Ho (0)) are considered zero.
Øg is the experimental gel point, Øg (est) is the estimated gel point for a specific Hs/Ho(i) and Co(i) the selected initial concentration.
Methods
Determination of the optimal concentration
The selection of the best initial concentration (Co,opt) requires a compromise solution. On the one hand, to obtain the minimum difference between Øg(est) and the experimental Øg, the closer the initial concentration to zero, the better. However, on the other hand, when the Hs/Ho ratio is measured in graduated cylinders, the experimental error is significant at low Hs/Ho (low initial concentration) and decreases at higher values.
Therefore, the determination of a relation between Co,opt, to calculate Øg(est), and Øg should be analyzed. For that, 25 gel point curves were studied from multiple CNFs/CMFs, most of them from published studies (Ang et al. 2020; Raj et al. 2016a; Sanchez-Salvador et al. 2020a) and others not published, and used to evaluate their relation and the best correlation as a function of Øg. From them, the points of the curves Co vs Hs/Ho were used to obtain Hs for each point, the quadratic fit and also the smoothing spline fit calculated without local fluctuations nor y-intercept far from zero, as Fig. 1 indicates.
An example to determine the Co,opt, from a gel point curve previously graphed, is shown in Fig. 2. The orange line shows Øg(est) of a sample curve for each Hs/Ho (Eq. 4), assuming Eq. 1 to calculate Co. On the other hand, Øg (est) obtained from experimental points are shown with the orange bullets, observing, as expected, a greater deviation at low Hs/Ho.
Estimated gel point (Øg (est)) vs. Initial concentration (Co). Obtention of the optimal initial concentration to simplify gel point method
×
Øg(est) increases with Co and moves away from the Øg calculated by the quadratic fit (grey line, Fig. 2). As previously mentioned, the Øg quadratic fit is forced to have an independent term equal to zero. Therefore, the “b” parameter is slightly lower than the quadratic fit with independent term and the smoothing spline fit. However, the quadratic fit is useful to describe the Øg curve as in Eq. 4. On the other side, Øg calculated with the smoothing spline of MATLAB (green line), this line crosses the Øg(est). The exact interception depends on the “p” parameter selected that gives a different Øg value, as mentioned in Sect. 2.1.
To consider the experimental error due to the visualization of the Hs using the common 250 mL graduated cylinders, we assumed an estimated error of Hs ± 1 mL. Hs/Ho and Øg(est) were recalculated for each sample using Eq. 4, substituting Hs/Ho by ((Hs ± 1) /Ho). The experimental error effect is shown in Fig. 2 by the dashed blue lines indicating the maximum lower and upper error.
Comparing Øg(est) with the possible experimental error, the upper blue line shows a minimum error value at the concentration indicated by the red line. Reading the initial concentration in the red line it is possible to obtain the Co,opt and Øg(est) with the minimum deviation. At this point, the difference between Øg(est) and Øg calculated with the smoothing spline and a correct estimation of the “p” parameter is minimum.
Experimental validation
To validate the simplifications of the new methodology, sedimentation experiments of 8 different types of CNFs and CMFs were carried out preparing suspensions at different initial concentrations. Three cellulose sources (cotton, eucalyptus and recycled paper) were tested to validate the simplifications with a wide range of morphologies and chemical compositions in order to try the method for all kinds of CNFs/CMFs. All samples were treated in a PANDA PLUS 2000 laboratory homogenizer (GEA Niro Soavi, Parma, Italy) at different passes and pre-treated with mechanical or chemical processes or not pre-treated. The characterization of the different samples, according to Sanchez-Salvador et al. (2020b), is summarized in Table 1.
Table 1
Characterization of CNF/CMF hydrogels used to validate the methodology
Source
1
2
3
4
5
6
7
8
Cotton
Recycled paper
Eucalyptus
Pretreatment
Commercial cellulose powder from cotton linters
Refining 5000 revolutions
TEMPO-mediated oxidation, 10 mmol/g pulp of NaClO
Refining 5000 revolutions
TEMPO-mediated oxidation, 5 mmol/g pulp of NaClO
No pretreated
Treatment
Homogenization 2 steps, 300 bars
Homogenization
2 steps, 300 bars
2 steps, 600 bars
2 steps, 900 bars
Homogenization
6 steps, 600 bars
Homogenization
4 steps, 600 bars
Homogenization
2 steps, 300 bars
Homogenization
6 steps, 600 bars
Preparation Øg sample
Low agitation, Magnetic stirring
Low agitation, Magnetic stirring
Low agitation, Magnetic stirring
High agitation, Overhead stirrer
Low agitation, Magnetic stirring
High agitation, Overhead stirrer
Low agitation, Magnetic stirring
Low agitation, Magnetic stirring
CNF/CMF Characterization
Nanofibrillation yield (%)
< 5
< 5
39
78
5
>95
Transmittance 400 nm
7.1
2.1
1.8
15.4
17.5
83.5
Transmittance 800 nm
11.7
9.2
8.7
35.7
29.4
94.8
Polymerization degree (Monomeric units)
232
229
703
201
930
440
Carboxylic groups (mmol/g)
0.06
0.06
0.07
0.81
0.13
0.59
The CNF/CMF suspensions at different Co were prepared using deionized water and stirred. 200 µL of crystal violet 0.1 wt. % was added to dye the fibers. 250 mL of each suspension were settled into graduated cylinders until the sediment reached a steady value to obtain the complete deposition of fibrils. Hs/Ho values were used to build the gel point curves.
Results
Comparison of the two mathematical methods to select the optimal initial concentration
Co,opt vs. Øg was calculated from 25 gel point curves from CNFs/CMFs with different raw materials and treatments. The quadratic fit and the smoothing spline method were used to calculate Øg and to compare both methods in the estimation of Co,opt. In a first approximation to compare both fits, the “p” parameter of the smoothing spline function was selected to obtain a y-intercept less than 0.01. A MATLAB script to determine a suitable p value for each sample is shown in Supplementary 2, which inputs are the experimental points of the curve Co vs. Hs/Ho, and the outputs are the Øg, the p value calculated and the value in the y-interception. Co,opt was calculated as shown in Fig. 2; it does not depend on the Øg methodology, just to calculate the error regarding them. For further details, Supplementary 3 presents an example calculating Co,opt.
The relationship between Co,opt and Øg is observed in Fig. 3, which represents the errors between the Øg,(est) calculated from Eq. 3 with Co,opt as initial concentration and the experimental Øg for each methodology. Figure 3 shows that in both cases, the relationship between the Co,opt and the experimental gel point is almost linear without significant differences in the residual sum of squares.
Fig. 3
Relation between the optimal initial concentration (Co,opt) and the experimental gel point (Øg)
×
However, as Fig. 4 indicates, the error between the experimental Øg and Øg,(est) is different when using the smoothing spline or the quadratic fit methods. When the quadratic fit method is used, the average error for all samples shows that Øg,(est) is higher than Øg in 12.2% ± 8.1, whereas when using the smoothing spline fit, the average error is inferior with a value of − 7.8% ± 6.4. In both cases, the error is biased and depends on the value of Øg.
Fig. 4
Errors between the estimated gel point and the experimental gel point
×
As Fig. 4 indicates, the quadratic fit is not the best tool to estimate adequately the gel point due to the large deviations between Øg,(est) and Øg. However, the smoothing spline option with a “p” value according to a fix y-interception is not adequate either because, for low gel point values, the errors are higher than for high gel point values. Therefore, an alternative is developed to calculate the gel point using the smoothing spline tool but optimizing the “p” parameter for each case.
Optimization of “p” parameter in smoothing spline as a function of gel point to select the optimal initial concentration
To avoid a biased error that depends on the value of Øg, the y-interception error was set as a function of Øg using 7 conditions: (a) < 0.0005·Øg; (b) < 0.001·Øg; (c) < 0.0015·Øg; (d) < 0.0017·Øg; (e) < 0.002·Øg; (f) < 0.0025·Øg; (g) < 0.005·Øg. A new MATLAB script (Supplementary 4) that replace the condition of a y-interception under 0.01 by the conditions a–g, in function of the Øg. Figure 5 relates the Co,opt and the experimental gel point with different y-intercept as function of Øg. The curves can be linearly adjusted in the same way as in Fig. 3 without significant differences. Figure 6 shows the error between the experimental and estimated Øg from these gel point estimations. The best results are obtained with a y-intercept error under 0.0015·Øg although the lower sum of squared residuals is obtained with a y-intercept error < 0.002· Øg. When the y-intercept error is under 0.001· Øg, some samples have local fluctuations in the MATLAB curve. A high y-intercept error (under 0.005 Øg) also shows a bad estimation with a high sum of squared residuals.
Fig. 5
Relation between the optimal initial concentration and the experimental gel point using the difference with the y-intercept as function of Øg
Fig. 6
Errors and statistics between the estimated gel point and the experimental gel point using the difference with the y-intercept as function of Øg
×
×
Validation of the assumptions to simplify gel point method
In order to validate the simplified gel point methodology, the results of full sedimentation experiments from 8 different CNF/CMF samples (presented in Table 1) are considered (Fig. 7).
Fig. 7
Initial concentration of CNFs/CMFs vs. Hs/Ho
×
From Fig. 7, the experimental gel point, using both the quadratic fit and the best smoothing spline fit (y-intercept under 0.0015·Øg), were calculated (Table 2). In most of the cases a higher Øg using the smoothing spline fit is observed. From the Øg calculated by this last method, the Co,opt for a single experiment is calculated with Eq. 5 (obtained from Fig. 5). Calculation of Co,opt is not possible for the sample 1 as the Øg range in Eq. 5 is up to around 30, since the nanocellulose is more in the form of nanocrystals instead of nanofibers. Therefore, Øg reaches a value of 317 which indicates a very low AR of around 10, usually corresponding to nanocrystals. For sample 2, the proximity of Øg with the range limit allows the calculation of Co,opt. The differences between both mathematical methods are due to the forced fit without the independent term of the quadratic fit. Therefore, while the quadratic fit is a good tool to estimate a value in the curve Co vs. Hs/Ho, it is not the best method to obtain Øg.
Estimation AR at different initial concentrations (kg/m3)
Aspect ratio using the Co,opt
~ 0.15
0.2–0.25
0.5–0.75
~ 1
~ 1.5
~ 3
Co,opt
Øg (est)
Error with smoothing spline
1
10.6
–
15.1
–
11.0
10.7
10.8
Out of range
2
33.9
–
40.7
35.7
34.1
33.7
1.77
34.3
1.4%
3
124
124
120
105
102
–
–
0.121
123
1.0%
4
153
145
135
116
110
–
–
0.074
149
2.6%
5
112
113
109
101
98
–
–
0.154
111
1.2%
6
95
95
93
82
79
–
–
0.217
92
2.6%
7
133
133
123
119
112
–
–
0.101
129
3.0%
8
37.4
36.6
37.4
37.8
37.5
37.3
–
1.45
37.3
0.3%
Table 3 shows the Øg (est) for several experimental points (red zone), calculated for the different types of CNFs/CMFs and Co from Fig. 7 and Eq. 3 In addition, Øg (est) was calculated for Co,opt (green zone). In this case, Hs/Ho was obtained from the resolution of the quadratic fit of the data in Fig. 7 at Co,opt and using Eq. 3.
The results show that comparing Øg (est) at the Co,opt with the smoothing spline fit Øg the errors are under 7%. Øg (est) in the experimental points at Co similar to the Co,opt are also closer to the Øg obtained with the sedimentation curve and the smoothing spline fit. Concretely, bold marked results in Table 3 show the results with an error under 7% respect to the smoothing spline fit. This fact indicates a certain versatility in the selection of Co which would allow the use of a Co in the same order of magnitude of Co,opt with a tolerable error without affecting the result to a greater extent. Some clarifications could be done in sample 4, whose Co,opt is 0.074 kg/m3, half of minimum Co studied (0.15 kg/m3), and would show an error slightly higher, around 12% at this concentration. On the other hand, sample 8 shows a low error during a wide interval of Co. This fact is associated to a more linear trend in the curve Co vs. Hs/Ho in this sample, compared to a more pronounced curvature in the others.
Since Øg is used as a tool to estimate AR of CNF/CMF samples, AR was calculated in Table 3 according to the Eq. 6, employing the Crowding number theory (CN) and assuming a cellulose density of 1500 kg/m3 (Varanasi et al. 2013).
As Eq. 6 indicates, AR is inversely proportional to the square root of Øg. Therefore, the error between the AR using smoothing spline and AR with the Co,opt decrease with respect to the Øg error, reaching always values under 3%.
Considerations to apply the simplify gel point methodology
The application of this simplification to measure Øg in CNFs and CMFs does not require the previous preparation of the Øg curve with more than 5 graduated cylinders. However, the selection of the Co is a key factor to obtain an adequate Øg (est), so a second additional experiment may be required in case of not obtaining a favourable result in the first one, using the Eq. 5 to recalculate a new Co according to the Øg (est) obtained in the first sample.
To select Co in the first experiment, the recommendation would be to obtain a sedimentation height around 4–12% of the total height, since a lower value would be difficult to measure with precision, and a higher Hs would cause a great deviation in the substitution of the derivative for an increment (Eq. 3), moving away from the limit Hs/Ho close to zero. The selection of Co depends on several factors as the raw material, the pretreatments or the intensity of the mechanical treatment.
A first approximation to select the value of Co for the first graduated cylinder could be carried our according to the pretreatment process:
Refining: from 0.05–0.5 kg/m3, using lower Co values as refining intensity increases.
Enzymatic pretreatments: around 0.2–0.5 kg/m3 due to the ratio length/diameter is very similar after pretreatment and mechanical treatments.
Samples in a powder state: The decrease in length produces a lower AR and higher Øg as commercial cellulose powder from cotton linters, so the Co should be higher, > 1 kg/m3.
TEMPO-mediated oxidation: from 0.5 to 3 kg/m3, using higher Co values as the oxidant dose increases.
In addition, the intensity of the mechanical treatment, as high-pressure homogenization, also produces a variation in the selection of Co. In general terms, a high intensity in the mechanical treatment requires a low Co within the established ranges due to an increase of AR except in TEMPO-mediated oxidation, in which the intense mechanical treatment promotes not only the separation of the fibers, but also helps in the shortening of the fibers affected by the previous TEMPO-mediated pretreatment, increasing probably the Øg and, therefore, Co.
After completing the first experiment Øg(est) is calculated and the Co,opt is obtained from Eq. 5. If the values of Co and Co,opt are too different, a second experiment should be carried out using a new Co close to Co,opt. The results of the CNF/CMF samples studied in Table 3 show that at least when Co is in the interval (0.5·Co,opt, 1.5·Co,opt), the Øg(est) error is under 7% in all cases. This approach cannot be used for micro or nanocrystals, since the AR must be higher than 30.
Conclusion
Nanocellulose is arguably the emerging nanomaterial of the last two decades. However, its full industrial deployment depends on the ability to conveniently characterize and control its quality during manufacturing. The gel point methodology is commonly used to calculate its aspect ratio (L/D) and could also be used to optimize its dispersion degree. However, sedimentation experiments are most tedious to perform which difficult its implementation in industrial environments and are certainly incompatible with industry environment and practice.
This study shows that the gel point can be estimated using one or two sedimentation experiments and selecting an optimal initial concentration (Co,opt), based on the minimal experimental error and on a simplification of the gel point equations. The main simplification is that the derivative is replaced in a classical interpretation by an increment between a determined concentration and a theoretical concentration of zero. Therefore, the derivative is approximated as the quotient between Co and Hs/Ho.
The two usual methods to calculate Øg, the smoothing spline and the quadratic fit, were compared. We conclude that the smoothing spline is generally the most reliable method, provided that the smooth spline fitting parameter is chosen correctly. Here, for the first time, an objective criterion for determining the appropriate smoothing spline parameter is presented.
The results obtained with the fitting procedures were validated with independent full studies of eight CNF/CMF samples. It was possible to estimate the gel point from a single sedimentation experiment at the Co,opt. The difference between the experimental Øg obtained by the Øg curve and the Øg(est) from the simplification was under 7% for all samples and under 3% in the case of the aspect ratio estimation. This was achieved with the smooth spline fitting parameter calculated using a MATLAB script in which y-interception in the curves Co vs. Hs/Ho was < 0.0015 Øg. The main drawback of this simplification is the a priori selection of the Co,opt, so in some occasions if the selected Co is too far from the Co,opt, the selected Co is not adequate and a new sedimentation experiment is required using the equation that relates Øg and Co,opt.
This new method shows that the sedimentation experiments can be reduced by a factor of 2.5–5 and is a promising tool to easily control nanocellulose quality and effectiveness during the different applications, compatible with industrial environments.
Acknowledgments
The authors wish to thank the Economy and Competitiveness Ministry of Spain for the support of the project with reference CTQ2017-85654-C2-2-R, The Community of Madrid for the support to the Program Retoprosost2-S2018/EMT-4459 as well as the support of Universidad Complutense de Madrid and Banco de Santander for the grant of J.L. Sanchez-Salvador (CT17/17). Warren Batchelor and Gil Garnier wish to thank the Australian Research Council, Australian Paper, Carter Holt Harvey, Circa, Norske Skog and Visy for their support through the Industry Transformation Research Hub grant IH130100016.
Declarations
Conflict of interest
The authors declare that they have no conflict of interest.
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