Introduction

It is long known that cellulose materials swell and soften when placed in water, a feature which is prominently exploited during papermaking. There, a bond between two pulp fibers can form and achieve a high area in molecular contact because the fibers’ resistance against mechanical deformation is lowered. Viscose fibers are also reported to be formed into all-viscose paper (Weber et al. 2013, 2014) or used as additives to improve properties of standard paper (Yu et al. 1999). Additionally, viscose fibers can be viewed as model systems for paper fibers, exhibiting a simpler composition and surface as well as bulk structure.

In most applications, cellulose based materials are exposed to an atmosphere with non-constant humidity. Obviously, a change in humidity will result in a change in the water content of cellulose fibers which, in turn, affects their mechanical properties. To test the humidity dependent viscoelastic properties of cellulose fibers or cellulose films, dynamic mechanical analysis (DMA) is often applied (Zhou et al. 2001; Yakimets et al. 2007). There, a complex numbered elastic modulus is detected, with the real part named storage modulus and the imaginary part loss modulus. However, when it is of interest to detect the mechanical properties in terms of the more common Young’s modulus and tensile strength, tensile tests are performed (Burgert et al. 2003; Sun et al. 2009).

In particular, nanoindentation (NI) is frequently applied to cellulose fibers, when local properties need to be determined (Gindl et al. 2004, 2006; Lee et al. 2007). To properly use this method, it is essential to prepare smooth sample surfaces. In fact, all NI studies deal with microtome cut longitudinal or radial cross-sections, to provide a smooth and flat sample area.

To obtain mechanical properties of cellulose fibers relevant for bond formation, it would be best to measure them directly at the pristine fiber surface. This is usually not done by NI, since pulp fibers, as well as some viscose fiber types, have rough surfaces. An obvious solution is to first image the surface area and then hand-pick sufficiently flat areas. However, scanning with a nanoindenter can destroy the sample surface, especially in the case of soft samples such as cellulose. To circumvent this problem, atomic force microscopy based NI (AFM-NI) can be applied. There, the surface can be imaged in the gentle tapping mode without damaging it (Zhong et al. 1993). AFM-NI has been commonly applied to polymers (Clifford and Seah 2005; Tranchida et al. 2006; Jee and Lee 2010), and biological matter (Roos and Wuite 2009; Fuhrmann et al. 2011), including pulp fibers (Yan and Li 2013; Ganser et al. 2014a) and viscose fibers (Ganser et al. 2014b).

In this work, we determine the humidity dependent reduced modulus \(E_r\) and the hardness \(H\) of viscose fibers by AFM-NI and compare them to those of pulp fibers. Furthermore, the water content \(w\) of viscose fibers as a function of relative humidity \(\varphi _r\) is detected using a gravimetric sorption system (GSS). This allowed to describe \(H\) and \(E_r\) also as a function of the fibers’ water content.

Methods and materials

Atomic force microscopy based nanoindentation

AFM-NI experiments were performed by employing an Asylum Research MFP-3D AFM (Santa Barbara, CA). The system is equipped with a closed-loop planar sample scanner and has been operated in tapping mode for imaging and in contact mode to collect force-vs.-distance plots. For all experiments, ND-DYIRS full diamond probes (Advanced Diamond Technologies, Romeoville, IL) were used. These probes have a four-sided pyramid as a tip, with an angle of about 75° between two opposite side planes. The radius at the tip apex is less than 50 nm. In the MFP-3D AFM—as with most commercial systems—the probe is tilted by 11°, leading to an asymmetric tip shape.

To study the effects of water uptake on a sample’s mechanical properties, a setup to control relative humidity was employed. The used setup is described by Ganser et al. (2014a) and was modified by adding the possibility of heating the buffer segment. This modification allowed to reach relative humidities \(\varphi _r\) within 0.05 and 0.95. The tolerance of the SHT21 humidity sensors (Sensirion AG, Switzerland) is \(\varDelta \varphi _r = 0.03\) from \(\varphi _r = 0.2\) to \(\varphi _r = 0.8\) and increases linearly to \(\varDelta \varphi _r = 0.05\) beyond this interval. To ensure that at a selected value for \(\varphi _r\) an equilibrium in water uptake has been reached, subsequent AFM scans of the fiber under investigation were performed until no drift was observed in the images. Then, the geometrical swelling of the fiber was assumed to be completed (Ganser et al. 2014a).

The detailed procedure of AFM-NI is given by Ganser et al. (2014a), therefore only a brief outline will be presented here. After the collection of force-versus-distance plots, the cantilever’s deflection was subtracted from the distance to obtain force-versus-indentation depth plots. These plots were then evaluated according to Oliver and Pharr (1992) with applied modifications for viscoelastic materials (Tang and Ngan 2003). From this procedure, hardness

$$H = \frac{P_{max}}{A(x_c)}$$
(1)

and reduced modulus

$$E_r = \frac{\sqrt{\pi }}{2}\,\frac{S}{\sqrt{A(x_c)}}$$
(2)

are obtained. \(P_{max}\) denotes the maximum applied load, \(A(x_c)\) the projected area of the tip at the contact depth \(x_c\), and \(S\) describes the slope at the beginning of the unloading segment \(S = \frac{dP}{dx}|_{x_{max}}\). The quantities \(P_{max}\), \(x_c\), and \(S\) are illustrated in an exemplary force-vs.-indentation depth plot in Fig. 1.

The hardness obtained by Eq. 1 is related to a material’s tensile strength, whereas the reduced modulus (Eq. 2) is linked to the Young’s modulus. In principle, the Young’s modulus can be calculated from \(E_r\), provided that the material is isotropic. However, this is neither the case for spun fibers nor for natural pulp fibers, therefore, no corrections were applied and \(E_r\) is presented as the result.

The load schedule used to collect all force-versus-distance plots had a loading and unloading rate of 10 μN s−1, a holding time at the maximum load of 10s to compensate for viscoelastic effects, and a holding time of 30 s close to the end of the unloading part at 5 % of the maximum load to allow determination of the thermal drift rate and subsequent correction thereof. In Fig. 1, the effect of the holding time at \(P_{max}\) is clearly visible as it leads to an increase in the indentation depth at constant load.

Fig. 1
figure 1

Exemplary force-versus-indentation depth plot to illustrate the necessary quantities needed to calculate hardness and reduces modulus. This particular curve was recorded on a viscose fiber surface at \(\varphi _r \approx 0.05\)

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An advantage of AFM-NI over classical NI is that with the AFM’s gentle tapping mode, an inspection of the surface before indentation is possible. Although modern nanoindenters also provide a scanning stage, the recording of the surface topography is done in contact mode, which is known to damage soft surfaces during scanning.

It should be noted that AFM-NI differs from classical NI with respect to the load application. In classical NI, a transducer is used to apply a load perpendicular to the sample’s surface. During an AFM-NI experiment, however, the AFM’s cantilever bends during load application, thereby slightly tilting the tip with respect to the surface. Therefore, the load is not applied perfectly perpendicular to the surface. Consequently, \(H\) and \(E_r\) obtained by AFM-NI are not directly comparable to those obtained by NI and a calculation of the Young’s modulus from \(E_r\) is not straightforward, even if the sample is isotropic.

All viscose fiber samples were classical viscose fibers provided by Kelheim Fibres (Kelheim, Germany). Pulp fibers were bleached softwood kraft pulp fibers. Since the pulp is a mixture of spruce and pine, it is not possible to determine which exact fiber type was investigated. The fibers were placed on a drop of uncured nail polish which was subsequently cured (Schmied et al. 2012; Fischer et al. 2014). This preparation procedure ensures a fixation of the fibers and inhibits their vertical movement during AFM-NI (Ganser et al. 2014a).

Gravimetric sorption

The GSS that was used to measure sorption isotherms is a home-built semiautomatic gas sorption apparatus based on a symmetric two-pan vacuum ultra-microbalance (Sartorius Instruments, model S3D) described elsewhere (Groß and Findenegg 1997). To record an isotherm, gas or vapor is dosed towards the sample until a certain target pressure is reached. After a default equilibration time, the mass uptake of the sample—placed inside a gold coated glass bowl—is recorded. In order to be able to separate the sample’s mass increase from the one of the sample bowl, a second bowl is fixed the same way at the other side of the balance’s beam. With this, the measured mass increase is no longer influenced by the sample holder. The above described procedure is repeated for selected pressure values of the sorption branch until the saturation pressure is reached. For desorption, the pressure is reduced via a connection with a vacuum pump. In this work, only the sorption is shown, which includes adsorption and absorption. The former denotes the attachment of a substance to an interface, whereas the latter is used to describe that a substance is incorporated into the bulk of another material. Both, adsorption and absorption, occur in the presented GSS data and are not separated from each other.

To measure the sorption isotherm of viscose fibers, a loose bundle of fibers with a dry mass of 20.6 mg was prepared. The bundle was placed inside a gold coated glass bowl of 10 mm diameter and 6 mm height. The gold coating is hydrophobic and ensures, therefore, that the water will attach preferentially to the sample and not to the bowl.

Results and discussion

Hardness and reduced modulus

As expected, a decrease of hardness and reduced modulus with increasing relative humidity \(\varphi _r\) was observed for classical viscose fibers. This fact becomes already evident when comparing the residual indents on viscose fiber surfaces, presented in Fig. 2. The AFM topography images in Fig. 2a show that the residual indents become larger at higher humidities. At \(\varphi _r = 0.10\), the indents have a side length of approximately 150 nm while at \(\varphi _r = 0.94\) the side length is about 400 nm. The line profiles in Fig. 2b also indicate an increase in the persistent indentation depth from about 30 nm at \(\varphi _r = 0.10\) to about 50 nm at \(\varphi _r = 0.94\).

Fig. 2
figure 2

Indents on viscose fibers at different humidities. a 1.0 μm × 0.6 μm AFM topography images. b Line profiles in vertical direction, indicated by the dotted line in (a)

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It is already apparent from Fig. 2 that at low humidities pile-ups occur at the left and right sides of the indents. This is visualized more clearly in Fig. 3. In Fig. 3a, the topography is presented and in Fig. 3b line profiles at the positions indicated in Fig. 3a are plotted. These profiles clearly reveal that a severe pile-up of material at the left and right sides of an indent is present. Since the cantilever is oriented horizontally by performing the indent in Fig. 3a, the pile-up could be caused by the lateral force component due to the bending of the cantilever in AFM-NI. On the other hand, the fiber’s longitudinal axis was oriented perpendicular to the cantilever, i.e., vertically in Fig. 3a. Since viscose fibers are anisotropic, this could also explain a pile-up only on two sides. By rotating the fiber with respect to the cantilever and performing indents, a pile-up on three sides was found. One pile-up was caused by the bending of the cantilever and the other two pile-ups by the anisotropy, as they were similar to those in Fig. 3a. Therefore, we believe that the pile-ups visible in Figs. 2 and 3 stem from the fibers’ anisotropy as well as from the bending of the cantilever during AFM-NI.

The consequence of such a pile-up of material is that it increases the contact area during indentation and hence leads to an overestimation of \(H\) and \(E_r\). Because of this fact, it is necessary to correct the \(H\) and \(E_r\) values obtained by the method of Oliver and Pharr for the pile-up. This correction was performed according to Kese et al. (2005) by measuring the widths \(w_1\) and \(w_2\) and length \(l\) of the pile-ups and approximating the additional area by a semi-ellipse. All indents performed at relative humidities below 0.9 were corrected. Above this value of \(\varphi _r\), no pile-up was observed.

Fig. 3
figure 3

The occurrence of the pile-up effect on a viscose fiber (a) illustrated by a 10 μm × 6 μm AFM topography image and (b) two line profiles to demonstrate its extent. \(w_1\) and \(w_2\) describe the width of the pile-up and \(l\) its length. The dotted line and the dashed line in (a) mark the position of the respective line profile in (b)

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The evolution of the corrected mechanical properties of classical viscose fibers with relative humidity is plotted in Fig. 4. The hardness (Fig. 4a) as well as the reduced modulus (Fig. 4b) exhibit an approximately linear decrease with increasing relative humidity. The reduced modulus, however, seems to feature a kink at \(\varphi _r \approx 0.75\) after which a stronger decrease is observed. A definite identification of a kink is hindered by the large standard deviation in \(E_r\), and a linear decrease is equally likely from the present data. For hardness \(H\), the data points at \(\varphi _r \approx 0.95\) deviate slightly from a linear trend, but not significantly.

Fig. 4
figure 4

The dependence of mechanical properties of viscose fibers on relative humidity. a Hardness and b reduced modulus. The dotted line represents a linear fit

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In the literature, such a kink in the elastic modulus has been reported (Zhou et al. 2001). There, the elastic modulus of regenerated cellulose films was determined by DMA as a function of relative humidity, and the storage modulus as well as the loss modulus were reported to exhibit a kink at \(\varphi _r \approx 0.80\).

As a comparison, the change of a kraft pulp fiber’s mechanical properties with relative humidity is presented in Fig. 5. These values were determined simultaneously as those of the classical viscose fiber number 3 presented in Fig. 4. It was achieved by gluing a pulp fiber next to a viscose fiber and recording force distance curves in an alternating manner. Hardness of a kraft pulp fiber, plotted in Fig. 5a, and reduced modulus (Fig. 5b) exhibit a similar behavior as classical viscose fibers on increasing humidity. It is not possible to decide whether a kink in \(E_r\) is present or not because only seven data points on one pulp fiber were recorded. For classical viscose fibers, on the other hand, 25 data points were obtained on four different fibers. But it is obvious that the measured \(E_r\) of pulp fibers at \(\varphi _r \approx 0.95\) is decreased more than what is predicted by a linear relationship between \(E_r\) and \(\varphi _r\).

Fig. 5
figure 5

The dependence of mechanical properties of kraft pulp fibers on relative humidity. a Hardness and b reduced modulus

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When a fiber is immersed in water and swollen for several hours, \(H\) and \(E_r\) of fully swollen fibers in the wet state are obtained. The results of such experiments are given in Table 1. These values are not plotted in Figs. 4 and 5, because they were obtained by providing liquid water for swelling, which is in contrast to all other values presented in these figures. Also, these wet-state values are so small that in Figs. 4 and 5 they would appear to be 0.

Table 1 Hardness and reduced modulus of fully swollen fibers in water, including values for viscose fibers at \(\varphi _r = 1.0\), predicted by Eq. 8
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With these data, all states of humid cellulose fibers were sampled and a comparison between pulp and viscose is possible. While the hardness \(H\) in humid nitrogen is basically the same for pulp fibers as well as for viscose fibers, the reduced modulus \(E_r\) seems higher for viscose fibers. To investigate this in more detail, the ratios of hardness and reduced modulus between viscose and pulp fibers, i.e.

$$r_H = \frac{H_v}{H_p}$$
(3)

and

$$r_E = \frac{E_{r,v}}{E_{r,p}}$$
(4)

are plotted in Fig. 6 as a function of \(\varphi _r\). The indices \(v\) and \(p\) denote a quantity which is obtained from viscose or pulp fibers, respectively. The error bars were calculated from the individual standard deviations by employing error propagation. Considering only the values obtained in humid nitrogen atmosphere, \(r_H\) is mostly close to unity, whereas \(r_E\) reaches values between 1.2 and 1.5 from \(\varphi _r = 0\) until \(\varphi _r = 0.5\) and is then also approximately constant at 1. Due to the large errors, however, one can assume that \(E_r\) is also the same for both materials.

Fig. 6
figure 6

Comparison of \(H\) and \(E_r\) of viscose and pulp fibers by the ratios \(r_H\) (filled squares) and \(r_E\) (open circles). The relative humidity values are the same for both ratios but the modulus ratios were shifted to the left by \(\varDelta \varphi _r = 0.015\) for clarity’s sake. The dashed line indicates a ratio of 1. The two rightmost data points were obtained from fibers swollen in water

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When pulp and viscose fibers were immersed in water and swollen for several hours, it appears that the reduced modulus is still similar for both materials \((r_E \approx 1)\). However, the hardness in water is higher for viscose fibers than for pulp fibers with \(r_H \approx 2\) in this state, as it is visible in Fig. 6. According to a Welch test (Welch 1947) at a significance level of 0.01, there is a significant difference in hardness between fully swollen pulp and viscose fibers. Their moduli, however, do not differ significantly. Both observations are in accordance to earlier findings (Ganser et al. 2014b).

Creep effects

Due to the fact that the \(P\,-\,x\) plots are recorded with a hold time of 10 s at \(P_{max}\), it is straightforward to extract also a measure for the creep effect. This is the distance which the indenter sinks into the material at constant load, as is evident from Fig. 1. The creep distance \(x_{creep}\) is plotted against the relative humidity in Fig. 7a, b for viscose fibers and pulp fibers, respectively. It is not surprising that \(x_{creep}\) is increasing with increasing humidity. The creep distance for viscose seems to be well described by the relation

$$x_{creep}(\varphi _r) = a_1 + a_2 \varphi _r + a_3 \exp \left( a_4 \varphi _r \right),$$
(5)

which has been found by us empirically. Equation 5 fits best the data obtained on viscose fibers. While \(x_{creep}\) measured on the pulp fiber seems also likely to follow Eq. 5, its validity for this case is less certain due to a lack of data points, compared to viscose fibers. Apparently, until \(\varphi _r \approx 0.7\), the increase in creep distance is mostly linear. Only for \(\varphi _r > 0.7\), the exponential part kicks in and \(x_{creep}\) increases stronger than linear. Note that the creep distance of fully swollen viscose fibers in water lies on the same curve as the ones measured in humid air. This suggests that there is no or little difference in viscoelastic properties between viscose fibers in humid air at \(\varphi _r = 1\) and fully swollen viscose fibers in water. It is unclear if this holds also for pulp fibers. For the pulp fiber, the kink is located at \(\varphi _r \approx 0.9\) and is clearly induced by the the high value of \(x_{creep}\) measured in water.

Fig. 7
figure 7

Increase in indentation depth at maximum load as a function of relative humidity for (a) classical viscose and (b) a pulp fiber

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Water uptake

The reason why viscose fibers become softer with increasing relative humidity is that water from the environment is absorbed by the material’s bulk. There, water will weaken the hydrogen bonds which are the dominant bond type within the material. It is therefore of interest to convert the relative humidity \(\varphi _r\) to the water content, defined as

$$w = \frac{m_{H_2O}}{m_{viscose}}.$$
(6)

\(m_{H_2O}\) denotes the mass of the sorbed water by the viscose and \(m_{viscose}\) the viscose fibers’ dry mass. With the AFM setup outlined in Sect. 2.1, only \(\varphi _r\) can be detected, but not \(w\), therefore a GSS is used which allows the measurement of a material’s mass increase as a function of the water vapor pressure—and hence \(\varphi _r\). This setup has been described in Sect. 2.2. The sorption isotherm of the same type of viscose fibers characterized with AFM-NI is presented in Fig. 8. To allow the transformation of \(\varphi _r\) to \(w\), the isotherm was fitted with the Guggenheim–Anderson–de Boer (GAB) model (Anderson 1946; de Boer 1953; Guggenheim 1966), which is frequently used to describe sorption isotherms of various materials, including cellulose (Delwiche et al. 1991; Belbekhouche et al. 2011; Sun 2008). The GAB model describes \(w(\varphi _r)\) as

$$w(\varphi _r) = \frac{AB \varphi _r}{(1 - B \varphi _r)(1 - B \varphi _r + AB \varphi _r)}.$$
(7)

\(A\) is the sorption constant as is appears in the Brunauer–Emmett–Teller (BET) isotherm (Brunauer et al. 1938) and \(B\) accounts for the fact that the heat of adsorption for the layers two to nine is less than those following these (Anderson 1946; Delwiche et al. 1991).

GSS measurements with our setup required a mass of about 20 mg, which corresponds to a number of approximately 4000 single fibers (5 mm fiber length, 30 μm fiber diameter, 1500 kg/m3 cellulose density) or a total fiber length of almost 20 m. Although effects attributed to the fiber network (such as capillary condensation at fiber-fiber crossings) are present in this approach, we believe them to be negligible. The water content obtained by GSS is, therefore, considered as a good estimate for the water content in a single fiber.

It is obvious from Fig. 8 that the experimentally obtained data points follow the GAB model only for \(\varphi > 0.2\). Thus, the fit with Eq. 7 was performed within the interval \(0.2 \le \varphi _r \le 1.0\). This will affect only the transformation of the \(H\) and \(E_r\) values obtained at the lowest humidities at around \(\varphi _r = 0.05\) to 0.15, which are bunched together at the very left in Fig. 4.

Fig. 8
figure 8

Sorption isotherm of viscose fibers at a temperature of 17 °C. Filled circles GSS experiment, dotted line fit with the GAB model

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The plot of hardness and reduced modulus as a function of the water content \(w\) is shown in Fig. 9a, b, respectively. Apparently, the assumption of a linear dependence on \(w\) is not appropriate, since it will lead to negative values for \(H\) and \(E_r\) at \(w = 0.304\) (which corresponds to \(\varphi _r = 1\)). A description of the Young’s modulus \(E\) as a function of \(w\) for hydrogen bond dominated solids was given by Nissan (1976). There, the decrease of \(E\) was considered to be caused by the dissociation of hydrogen bonds due to water infiltrating the material. This yielded the exponential relation

$$\frac{E(w)}{E_0} = \exp \left( a - bw \right).$$
(8)

\(E_0\) denotes the Young’s modulus at zero water content and is usually determined by fitting data in the interval \(0 \le w \le w_c\) with \(E/E_0 = \exp (-w)\), \(w_c\) is postulated to be equal to the BET monolayer and \(w_c \approx 0.05\). Then, \(a\) and \(b\) depend only on \(w_c\), the hypothetical quantity of water needed to provide one water molecule for each OH group of the cellulose molecule, and the average number of hydrogen bonds that will break cooperatively (Nissan 1976). However, our collected data of \(H\) and \(E_r\) for \(0 \le w \le w_c\) is insufficient to determine \(E_0\) accurately. Therefore, only the plausibility of Eq. 8 for high water content is checked. Least squares fits of \(H(w)\) and \(E_r(w)\) with Eq. 8 are presented in Fig. 9. The reduced modulus follows Eq. 8 as expected, even though a large scattering in \(E_r\) is present (see Fig. 9b). The change in hardness due to the uptake of water from humid air is also well described by Eq. 8 (Fig. 9a), which was derived initially only for the modulus. However, hardness and modulus do have a relationship, as can be seen by comparing Fig. 4a, b, where \(H\) and \(E_r\) decrease both with increasing \(\varphi _r\). In any case, it is obvious that the exponential relation is in both cases more plausible than a linear relation, when the predicted values for \(w = 0.304\) are considered.

Fig. 9
figure 9

Hardness (a) and reduced modulus (b) of viscose fibers as a function of the water uptake. The water content \(w\) was determined from Fig. 8

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It is now interesting to compare values of \(H\) and \(E_r\) predicted for \(w = 0.304\) by Eq. 8 with values measured on fully swollen fibers in water. The model developed by Nissan (1976) does only take into account the uptake of water into cellulose from humid air and cannot be applied when bulk water is present. Using Eq. 8, hardness and reduced modulus at \(\varphi _r = 1\) are predicted as \(H(w)|_{0.304} = 51\) MPa and \(E_r(w)|_{0.304} = 1.1\) GPa. These values are clearly different from the wet-state values presented in Table 1. This means that viscose fibers in humid air will never reach \(H\) and \(E_r\) values as low as those of fibers swollen in water. A possible explanation for this behavior is that at \(\varphi _r = 1\) not all pores or interfibril spaces are filled with water. Only when the viscose fibers are swollen in liquid water, all spaces within the fiber are infiltrated by water and thus the fibers’ structure is weakened as much as possible, leading to the lowest values of hardness and reduced modulus. Compared to humid atmosphere at \(\varphi _r = 1\), the viscose fibers’ hardness is decreased by a factor of almost 4 and the reduced modulus by a factor of about 20 when swelling them for several hours in water. This suggests that viscose fibers still retain a high \(E_r\) and a relatively high \(H\), even when swelling them at humidities close to \(\varphi _r = 1\). Only the exposure to liquid water will reduce the fibers’ resistance against deformation to a minimum. Such a highly weakened state is desirable in, e.g., papermaking where soft swollen fibers will lead to the formation of strong interfiber bonds (Persson et al. 2013).

Conclusions

In this work, we have detected hardness \(H\) and reduced modulus \(E_r\) of viscose fiber surfaces as a function of the surrounding relative humidity \(\varphi _r\) by AFM-NI. The results obtained on viscose fibers were compared to those of pulp fibers, which appear to have very similar mechanical properties at all humidities. A significant difference between the hardness of viscose and pulp was found only for fully swollen fibers in water, which is consistent with earlier results (Ganser et al. 2014b). Additionally, an exponentially increasing creep distance was found for increasing relative humidity. By measuring the viscose fibers’ water uptake as a function of \(\varphi _r\) with a GSS, the decrease in \(H\) and \(E_r\) could be related to the fibers’ water content \(w\). There, a model based on hydrogen bond dissociation by water (Nissan 1976) was applied and allowed for extrapolating \(H\) and \(E_r\) to \(\varphi _r = 1\). Apparently, only a small decrease of \(H\) and \(E_r\) takes place between \(\varphi _r = 0.95\) and \(\varphi _r = 1\) and the materials still retain a relatively high resistance against deformation. In other words, it is substantial that cellulose is swollen in liquid water in order to reach maximum deformability. Such a low resistance against deformation is mandatory in papermaking where it leads to strong interfiber bonds (Persson et al. 2013). A maximum in creep effects, however, might already be reached at \(\varphi _r \approx 1\).