Introduction

Lyocell is the generic name of cellulosic fiber produced from dissolved wood pulp in an organic solvent using dry-jet wet spinning [1,2,3]. Commercially, lyocell fibers are cellulose fiber spun from the polymer solution of cellulose-NMMO-water. The process used for the production of lyocell fibers is environmentally acceptable and occupies a much smaller footprint compared to the conventional rayon process. In the lyocell production process, dissolving grade pulp is first suspended in an aqueous solution of NMMO, which forms a slurry. Subsequently, the water content of the slurry is gradually reduced to around 11–14 wt% under vacuum at a higher temperature at around 100 °C, resulting into the complete dissolution of cellulose to form a homogeneous cellulose solution. This solution is then extruded via an air-gap into a coagulation bath at a higher stretch ratio (> 6.0) where the precipitation of cellulose takes place and forms the wet fibers, which are then dried to produce lyocell fibers [2, 4,5,6]. In the commercial lyocell process, uniform and homogeneous slurry formation is an important prerequisite to produce a consistent cellulose solution for spinning. The behavior of the slurry is mainly decided by the swelling phenomenon of the pulp in an aqueous NMMO solution [7,8,9,10]. Aqueous NMMO readily disintegrates the pulp layer during this swelling and penetrates the inter-molecular crystalline area of cellulosic fiber in the pulp to swell the structure of the cellulose completely. The benefit of this method is better penetration of NMMO deep into the cellulose crystalline area to make the dissolution process faster [11].

The mechanism of cellulose swelling in pulp starts with penetration of solvent molecules into inter-crystalline “amorphous” regions and then enters into the interstices between the fibrillary structural units of cellulose [12,13,14]. During swelling, as time progresses, the swelling solvent penetrates via the interlinking regions from both ends into the elementary crystallites and attains the equilibrium. In an aqueous NMMO solution, initially, water acts as a solvent to some extent and then the NMMO solvent molecules penetrate into the cellulose network and break the hydrogen bond of the inter-and intra-crystalline interstices region [8, 15, 16]. These, in turn, cause structural changes in the cellulose pulp and alter the open space present between the fibers [8]. This swelling phenomenon is dictated to a great extent by temperature and NMMO concentration [9]. Hence, to understand the swelling kinetics phenomena of cellulose in an aqueous NMMO solution, it requires experimental data and a theoretical model which covers the wide range of NMMO concentration and temperature.

As explained above, the swelling mechanism of cellulosic pulp in aqueous NMMO solution is described in the literature based on its structural changes such as crystallinity and open space between fibers (pores) [8, 15]. There is no theoretical correlation reported on the swelling kinetics of cellulose in an aqueous NMMO solution. However, there are few empirical correlations reported related to the quantification of pulp swelling kinetics in water. H. Schott assumed second-order swelling kinetics considering the initial stage limited swelling which is retarded by cellulose network and asymptotically attains the equilibrium in water [17, 18], however, this proposed kinetic equation does not validated with experimental data. Botková et al. investigated the swelling kinetics of handmade sheets formed from the disintegrated pulp in the water system at 30 °C and fitted experimental data in first-order kinetics [19]. However actual pulp sheets will behave differently than handmade sheets due to processing. Geffert et al. have also investigated the swelling of cellulosic porous materials in water systems and developed an empirical correlation explaining swelling kinetics with mathematical description [20, 21]. They showed the effect of multiple recycled pulps produced by disintegration and drying on swelling kinetics at higher temperatures > 80 °C. In the commercial lyocell process, cellulose pulp undergoes simultaneous swelling and disintegration in the slurry preparation step in the solution of the NMMO-water system. Recently, Jablonsky et al. used unbleached and oxygen delignified kraft pulp for the study of swelling kinetics in the water at 23 °C without disintegration and subsequent drying and modeled based on first-order kinetics [22]. The state of pulp used by them is equivalent to the one used in this study. They observed that there is a decrease in the swelling ratio with the degree of delignification [22]. The above literature study shows that practically no study has been made till date to quantify swelling kinetics of porous cellulosic pulp in the NMMO-water system.

Hence, an attempt has been made to develop a mathematical model to predict swelling kinetics of cellulose pulp corrected with pore contribution under different temperatures and NMMO concentrations. In view of this in the present study, the range of NMMO concentration is covered from 0 to 55 wt% and that of temperature is from 308.15 to 338.15 K. The concentrations of NMMO were selected in the homogeneous swelling region below 60 wt% NMMO [15]. The concentration and temperature-based correlations were used to express the rate constant and equilibrium swelling quantity in a second-order equation. Also, the resulted mathematical model was validated with the reported swelling ratio of oxygen delignified kraft pulp in pure water at 23 °C.

In order to develop a reliable correlation, a suitable method for quantification of swelling is essential. Few methods are reported for the same, e.g. Botková et al. used a continuous monitoring device to measure the change in the dimension of pulp and paper sheet in pure water [19]. This method requires a special type of equipment to measure the change in sheet thickness accurately and also provides information only on the relative change in the dimensions. Ritcher and Glidden developed an offline measurement technique to quantify swelling of the pulp sheet [23]. The changes in dimensions and mass at equilibrium conditions were measured and based on these values swelling ratio was determined. This offline swelling quantification method is more reliable and provides direct information on the change in dimension and mass [7, 8]. Hence this method is implemented in the present study.

Initially, the experimental procedure used for the measurement of swelling kinetics is described below. Then, the second-order swelling kinetics model is presented. Next, the methodology is described to estimate the model parameters. This is followed by the result analysis to obtain the model parameters. Finally, the best-fitted correlations of swelling kinetics are discussed in detail.

Experimental section

Materials

Commercial bleached kraft hardwood pulp obtained from Grasim Industries Ltd, Nagda, India was used. The aqueous NMMO was obtained from Amines and Plasticizers Ltd, Mumbai, India.

Experimental measurement of pulp swelling

The methods of the sample preparations were similar as reported by Richter et al. and Sayyed et al. [7, 23] The square pieces of ̴ 2.5 cm × 2.5 cm of pulp sheet were cut, and care was taken to maintain the uniformity of size. For the equalization of moisture (averaged 5–6% calculated on a bone-dry basis), these sheets were then stored in a closed Mason jar. The weight (M0) and volume (V0) of each pulp piece were accurately measured.

The swelling was performed at different concentration of NMMO/water solution in the range of 0 wt% to 55 wt% at different temperatures of 308.15 K, 318.15 K, 328.15 K, and 338.15 K. From the 76 wt% NMMO solution, six different concentrations of NMMO between 0 to 55 wt% in an aqueous solution of NMMO was prepared by adding water. In the weighing bottle (250 ml capacity, 50 mm × 55 mm size), the excess quantity of NMMO/Water of 200 ml was taken and it was sealed tightly around the lid using Teflon tape. Then the bottle is immersed in a constant hot water bath, to achieve the desired temperature of the NMMO solution. The temperature of NMMO was then checked by inserting a thermometer. Pulp piece was inserted in the bottle and was then allowed to swell for 0.5,1, 2, 3, 4, 5, 7, 10, 15, and 20 min. By tilting the pulp sheet for 2 min, the residual NMMO was drained. The complete removal of the solvent from the surface was confirmed by weighing the pulp after completion of swelling at different times (0, 0.25, 0.5, 1, 1.5, and 2 min) and observed no change in the weight after 1 min in all swelling experiments. The swelled pulp sheet was kept in a tilted position for 2 min to assure the surface solvent gets completely removed. After draining, the sheet was transferred to a flat glass plate and their swelled weights (M) and volumes (V) were measured and recorded by taking an average of 3 measurements.

The swelling ratio (W) was calculated using Eq. 1.

$$W=\frac{M-{M}_{0}}{{M}_{0}}$$
(1)

Here M and M0 represent the weight of the cellulose pulp sheet at the sampling time and initial dried pulp sheet, respectively.

Swelling of cellulose pulp in aqueous NMMO solution

The typical swelling kinetic plots of cellulose pulp in an aqueous NMMO solution of 0, 20, 40, and 55 wt% NMMO at different temperatures are shown in Fig. 1.

Fig. 1
figure 1

Swelling kinetics of cellulose pulp in aqueous NMMO solution a 0 wt% NMMO, b 20 wt% NMMO, c 40 wt% NMMO, d 55 wt% NMMO. Filled symbols represent overall swelling and open symbols represent corrected swelling ratio with trapped swelling liquid in the pore structure of pulp. Reaction conditions: temperature 298.15 K to 338.15 K

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It has been reported that the liquid penetrates into the pulp through a porous structure and then interacts with the cell wall [24]. Some part of the liquid gets bounded with the cell wall and remaining occupies porous structure present in the pulp and has no interaction with cellulose, which are represented in Fig. 2. Hence it is important to consider bounded liquid only for the study of swelling kinetics. The cell walls of pulp fiber behave like a hydrogel. There are numerous studies reported related to the determination of hydrogel porosity using the swelling measurement method with a different type of liquid system [25,26,27]. They assumed isotropic swelling for the determination of porosity from weight swelling ratio (qw) and volume swelling ratio values (qv). The Similar approach was used to determine the extent of liquid occupied in the porous structure. Based on this, liquid in the bound form was estimated.

Fig. 2
figure 2

Schematic representation of swelling behavior of cellulosic pulp: SEM image of swelled pulp in aqueous NMMO solution

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The fraction of liquid present (P) in the pores was calculated from weight (qw) and volume (qv) swelling ratios using the following equation:

$$P=1-\frac{{q}_{v}}{1+\left({q}_{w}-1\right){\rho }_{c}/{\rho }_{l}}$$
(2)

Here ρc is the density of pure cellulose as 1.50 g/cm3 and ρl is the density of aqueous NMMO solution (Table S1, Supporting Information).

The following equation is used to estimate total pore volume from measured volume (V) and weight (M) of the swelled pulp sheet.

$${V}_{p}=\frac{PV}{M}$$
(3)

Since swelling experiments were performed in the excess quantity of aqueous NMMO solution it is assumed that the properties of aqueous NMMO solution present in the pores are equivalent to that of bulk liquid. Thus, the quantity of aqueous NMMO solution trapped in the pores (Mp) using Eq. 4:

$${M}_{p}={\rho }_{l}{V}_{p}M={\rho }_{l}PV$$
(4)

The bounded form of aqueous NMMO solution (Mb) is obtained by subtracting trapped liquid in the pores from the total weight of swelled pulp

$${M}_{b}=M-{M}_{p}$$
(5)

The swelling ratio is corrected in Eq. 6 by accounting for aqueous NMMO solution trapped in the pores as

$${W}_{b}=\frac{{M}_{b}-{M}_{0}}{{M}_{0}}$$
(6)

The values of overall and corrected swelling weight ratio were plotted in Fig. 1 which shows that variation in corrected swelling ratio with time follows a similar trend as that of overall swelling ratio however the extent of difference diminishes as temperature increases at any particular time. The trend remains the same because, liquid first occupies the pores and then interacts with cellulose [24], which results in the constant quantity of trapped liquid in the pore structure over the measured time period and the difference diminishes due to the expansion of the liquid in the pore. Thus, the behavior of swelling kinetics solely depends on the bounded form of liquid. Henceforth this corrected pulp swelling ratio will be used for pulp swelling study.

It was observed that pulp swelling has two distinct phases. An initial phase is fast and short with a high swelling rate and a second, slow, and long phase where a low swelling rate is observed which approaches an equilibrium condition. It is seen from Fig. 1 that these phases shift with the change in concentration of NMMO and temperature. The swelling phenomenon of cellulose in an aqueous NMMO solution is mainly decided by solvent molar volume and solvent dipolarity [28, 29]. Solvent molar volume plays a vital role in the initial phase, whereas solvent dipolarity in the second phase. Solvent molecules are diffused through capillaries, pores, and cavities present in the cellulose pulp sheet and then interact with cellulose-based on its di-polarity and eventually break inter-chain hydrogen bonds [30, 31]. In the initial phase of swelling, water diffuses faster and interacts with the amorphous region of cellulose and breaks the hydrogen bond due to its lower molar volume compared to the NMMO molecule [30]. At higher water content in an aqueous NMMO solution, the effective molar volume of the solvent is less. This facilitates the fast water dominating diffusion rate of solvent in cellulose. Hence at a lower concentration of NMMO, the rate of swelling is faster in the initial stage. As NMMO concentration increases further, this rate reduces. However, in the second phase higher molar mass NMMO dominates. Where due to high di-polarity more extent of swelling occurs due to the breaking of hydrogen bonds. Hence at high NMMO concentrations, a high degree of equilibrium swelling is observed. In order to express both of these phases, the second-order swelling kinetics model as proposed by H. Schott for cellulose in water was extended for cellulose in an aqueous NMMO solution [17].

Swelling kinetics model

The swelling behavior of cellulose in water closely followed second-order kinetics as reported by H. Schott [17]. The same kinetic model is extended to the swelling mechanism of cellulose in an aqueous NMMO solution. According to the Schott model, the second-order rate equation is defined as

$$\frac{d{W}_{b}}{dt}={k\left({W}_{b\infty }-{W}_{b}\right)}^{2}$$
(7)

Here k is the swelling rate constant, Wb is bounded swelling ratio at any time t and Wb∞ is the maximum bounded swelling ratio when cellulose reaches equilibrium.

Integrating Eq. 7 with initial conditions Wb = 0 at t = 0, results in

$${W}_{b}=\frac{k{W}_{b\infty }^{2}t}{1+k{W}_{b\infty }t}$$
(8)

The following expression obtained after rearrangement of Eq. 8

$$\frac{t}{{W}_{b}}=\frac{1}{k{W}_{b\infty }^{2}}+\frac{t}{{W}_{b\infty }}$$
(9)

At the very beginning of swelling (t → 0), \(1/k{W}_{b\infty }^{2}\gg 1/{W}_{b\infty }\). In the limit, Eq. 8 results:

$$\underset{t\to 0}{\mathrm{lim}}\left(\frac{d{W}_{b}}{dt}\right)=k{W}_{b\infty }^{2}$$
(10)

Equation 10 represents the initial swelling rate which corresponds to the initial phase where the solvent penetrates into the cellulose and starts interacting with the amorphous region but before the strain on the cellulose network begins to retard swelling.

Subsequently, at a longer time, \(1/{W}_{b\infty }\gg 1/k{W}_{b\infty }^{2}\), the swelling of cellulose attains the maximum or equilibrium swelling and the following relation is obtained from Eq. 8:

$${W}_{b,t\to \infty }={W}_{b\infty }$$
(11)

Regression of model parameters

The equilibrium bounded swelling ratio (Wb∞) and rate constant (k) are the parameters of the second-order rate equation, which are assumed to be the function of temperature and NMMO concentration.

Temperature dependence of Wb∞ and k are correlated in terms of Arrhenius type relationship, whereas NMMO concentration dependence is expressed in polynomial form. The following are corresponding correlations.

$${W}_{b\infty }={e}^{-\frac{{a}_{T}}{RT}}\sum_{i=0}^{n}{a}_{i}{w}_{NMMO}^{i}$$
(12)
$$k={e}^{-\frac{{k}_{T}}{RT}}\sum_{i=0}^{n}{k}_{i}{w}_{NMMO}^{i}$$
(13)

Here aT and kT are Arrhenius constant, \({w}_{NMMO}\) the initial weight fraction of NMMO in aqueous NMMO solution, ai, and ki is ith (i = 0, 1, 2, 3) correlation constant used for NMMO concentration.

These constants of Eqs. 12 and 13 for different polynomial forms (n = 1, 2, 3) are regressed using the measured swelling ratio for different initial NMMO concentrations and temperature. The nonlinear least-square method of Levenberg–Marquardt is used for the regression analysis [32]. Root mean square error (RMSE) and parity plot are used for judging the quality of the regression. RMSE is defined as

$$RMSE=\sqrt{\frac{\sum_{i=1}^{{n}_{data}}{\left({W}_{b,exp,i}-{W}_{b, model,i}\right)}^{2}}{{n}_{data}}}$$
(14)

Results and discussion

Regression analysis of swelling data for different temperatures (308.15 K to 338.15 K) and NMMO concentrations (0 to 55 wt%) was performed to determine correlation coefficients defined in Eqs. 12 and 13. In Table 1, the regression parameters of constants aT, kT, ai and ki (i = 0, 1, …, n) for n = 1, 2, 3 with RMSE for the best fit values of the constants are listed. It is seen that the value of RMSE for n = 1 is significantly larger than those for n = 2 and n = 3. Hence, the linear form of correlation (n = 1) is not used for further analysis. RMSE values of the quadratic and cubic form are not significantly different from each other, but n = 2 involves less number of constants. Hence, the correlation in the quadratic form (n = 2) is accepted for NMMO concentration-dependent constants.

Table 1 The least-square estimates of constants aT, kT, ai and ki, obtained from swelling data
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Based on the above analysis, the final form of temperature and NMMO concentration-dependent constants is given below

$${W}_{b\infty }={e}^{-\frac{6094}{RT}}\left(4.0233-2.2391{w}_{NMMO}+28.610{w}_{NMMO}^{2}\right)$$
(15)
$$k={e}^{-\frac{1621.6}{RT}}\left(22.579-45.205{w}_{NMMO}+21.328{w}_{NMMO}^{2}\right)$$
(16)

The accuracy of correlation is shown through the parity plots of swelling of cellulose in aqueous NMMO solution for different concentrations and temperatures. The abscissa represents experimental values and the ordinate represents the values predicted using the constants. It is seen from Fig. 3 that the experimental data lie close to the parity line. Predicted values obtained from Eq. 8 using correlation presented in Eqs. 15 and 16. Experimental data for all temperatures and NMMO concentrations are from the present study. The experimental data lie close to the parity line.

Fig. 3
figure 3

Parity plot of swelling of cellulose pulp in aqueous NMMO solution for different concentrations of NMMO and temperature. The X-axis represents experimental values and the Y-axis represents the values predicted using the constants. Reaction conditions: temperature 298.15 K to 338.15 K, solvent concentration (NMMO) 0, 10, 20, 30, 40, 50, 55 wt%

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Further to validate the model, the swelling ratio was predicted and compared with reported values (converted in terms of weight ratio) by Jablonsky et al. [22] for oxygen delignified cellulose pulp without disintegration in the pure water at 23 °C and shown in Fig. 4. A close match with reported data except at a lower time range can be seen. The slight deviation at a lower time range is probably due to the very narrow time span used by Jablonsky et al. [22] than the one used in developing this model.

Fig. 4
figure 4

Comparison of swelling data extracted from Jablonsky et al. [22] for oxygen delignified kraft pulp in water at 23 °C with prediction obtained from the present study

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The predicted equilibrium swelling ratios are plotted with NMMO concentrations for different temperatures in Fig. 5. It is observed that at lower NMMO concentrations, below ~ 20 wt%, the swelling ratio is low and then increases exponentially with NMMO concentration for all temperatures. It is known that the swelling ratio increases due to the formation of hydrogen bonds between cellulose and NMMO molecules [7, 31, 33]. Water content decreases the overall power of the solution to form hydrogen bonds and simultaneously competes in forming hydrogen bonds [30, 33, 34]. This together decreases the hydrogen bond-forming capability of aqueous NMMO solution to a greater extent especially when the water content is high (> 80 wt%). However, when the NMMO concentration is high enough where NMMO bond-forming capacity dominates water then the extent of swelling increases drastically as shown in Fig. 5.

Fig. 5
figure 5

Predicted equilibrium swelling ratios with different NMMO concentrations for different temperatures. Reaction conditions: temperature 298.15 K to 338.15 K, solvent concentration (NMMO) 0, 10, 20, 30, 40, 50, 55 wt%

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It has also been observed that the equilibrium swelling ratio increases marginally with the increase in temperature. NMMO exists in the form of crystalline hydrates in an aqueous NMMO solution [35,36,37]. Hence the availability of NMMO for swelling is lower at low temperatures.

Effect of temperature on predicted equilibrium swelling ratio is used to calculate the enthalpy of mixing between dry cellulose pulp and solvent. This effect was analyzed by using the Gibbs–Helmholtz equation.

$$\frac{{\mathrm{d}(\mathrm{ln}W}_{\infty })}{\mathrm{d}(1/T)}=\frac{-{\Delta H}_{m}}{R}$$
(17)

Here ΔHm is the enthalpy of mixing between the dry cellulose pulp and an infinite amount of the swelling liquid. Rearrange and differentiate Eq. 12 with respect to temperature provide the following relation

$${a}_{T}={\Delta H}_{m}$$
(18)

The fitted value of aT indicated in Eq. 15 which represents ΔHm is 6.094 kJ/mol. The positive value of ΔHm indicates the swelling behavior of cellulose in aqueous NMMO solution is an endothermic process. The swelling process of pulp is primarily a process of absorbing aqueous NMMO solution in the range of 0–55 wt% NMMO concentration, which can be described by adsorption theory [38]. In the adsorption process, the reported value of ΔHm from 2.1 to 20.9 kJ/mol represents physical adsorption and that from 20.9 to 418.4 kJ/mol indicates chemical adsorption [39]. The value of ΔHm (6.094 kJ/mol) in the present work proves that physical adsorption plays a vital role in the pulp swelling process.

The effect of NMMO concentration on swelling rate constant at different temperatures is shown in Fig. 6. It is observed that the swelling rate constant decreases with an increase in NMMO concentration. This is because as the concentration of NMMO increases the diffusion rate decreases due to the higher molar volume of NMMO [30, 40]. Mobility of aqueous NMMO solution increases with temperature largely in the lower NMMO concentration regime, which enhances the diffusion, and hence swelling rate constant increases with an increase in temperature as seen in Fig. 6.

Fig. 6
figure 6

Effect of NMMO concentration on swelling rate constant at different temperatures. Reaction conditions: temperature 298.15 K to 338.15 K, solvent concentration (NMMO) 0, 10, 20, 30, 40, 50, 55 wt%

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The variation of the predicted initial rate of swelling with NMMO concentration is plotted in Fig. 7 for different temperatures. It can be seen that the initial rate of swelling with pure water is more than that with 10 and 20 wt% NMMO. In the case of pure water, water diffuses very fast due to its smaller molecular size and hence initial rate is higher. While in the case of 10 and 20 wt% NMMO concentration, NMMO molecules reduce the mobility of water molecules hence initial rate is lower than that of pure water. Later region of NMMO concentrations (> 20 wt% NMMO solution), a large number of NMMO molecules are available for interaction with cellulose chain which results increase in the initial rate of swelling with NMMO concentration.

Fig. 7
figure 7

Variation of the initial rate of swelling with NMMO concentration at different temperatures. Reaction conditions: temperature 298.15 K to 338.15 K, solvent concentration (NMMO) 0, 10, 20, 30, 40, 50, 55 wt%

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Conclusions

A mathematical model was successfully developed for predicting the extent of swelling at different NMMO concentrations and temperatures for cellulose and aqueous NMMO system. The offline mass-based swelling measurement method with corrected pore contribution in the cellulosic pulp is used for the measurement of swelling ratio. Swelling ratio increases with an increase in NMMO concentration and temperature. Two distinct phases are observed during the swelling process. In the initial phase, the swelling rate is high which eventually slows down, and equilibrium swelling is attained. The swelling kinetics can be well fitted by the second-order rate equation. The relation of the rate constant and equilibrium swelling, with temperature, follows Arrhenius form and with NMMO concentration follows quadratic form. The swelling of cellulose in an aqueous NMMO solution is endothermic in nature. Thus, this swelling kinetic study would help for a better understanding of the Lyocell process at the slurry stage and would also be helpful in a new area of research in this context.