The procedure to evaluate DP_{v} of cellulose from rheometry experiments and glass capillary viscometry experiments is performed on 4 hygrothermally aged samples of cellulose paper and one unaged counterpart. First, the paper samples are dissolved to attain a dilute solution of molecularly dispersed polymeric cellulose chains. The dynamic (or kinematic) viscosity of this dilute cellulose solution is measured, and subsequently converted to the intrinsic viscosity of the cellulose polymers, which is eventually used to calculate DP_{v} of the paper samples by applying the well-known Mark-Houwink-Sakurada equation. The individual experimental steps and the underlying equations are successively described below.

### Sample preparation

The measurements were carried out on a set of Whatman No. 1 filter paper sheets. Whatman No. 1 filter paper is 100% cotton paper, made of cotton linters without sizing and fillers. Its chemical composition is fully based on cellulose, i.e., it does not contain hemicellulose or lignin. The use of Whatman No. 1 filter paper is common in paper conservation studies since its simple composition enables an accurate assessment of degradation mechanisms. Moreover, it is widely commercially available. The considered samples were hygrothermally aged at \(90~^{\circ }\)C for 6, 12, 20, and 35 days; further, a reference, unaged paper sheet was used. The accelerated ageing process was performed in 2014 using Lab-Line hybridization tubes (144 ml), according to the standard TAPPI T573 pm-09 [26] that is commonly applied in paper conservation studies. The samples were part of the same pool of Whatman No. 1 samples as those studied in [7, 11]. The relative humidity (RH) in the tube is buffered by the paper and typically stabilizes around 50% during the ageing process [27]. The samples were subsequently kept in a dark room under ambient conditions (\(21.5\pm 1.5~^{\circ }\)C and \(52\pm 2\%\) RH), until they were prepared for the viscosity measurements presented in this paper.

With the above paper specimens, the sample preparation for glass capillary viscometry was performed based on the standard ASTM D1795-13 [19]. The rheometry samples were prepared according to the standard IEC 60450 [28]. The two standards are merely different regarding the specific temperature prescribed during the viscosity measurement, i.e., the glass viscometry and rheometry were performed at \(25~^{\circ }\)C (ASTM) and \(20~^{\circ }\)C (IEC), respectively. Further, for both types of experiments, three samples were prepared per ageing time.

Different dilute cellulose solutions were prepared using cupriethylenediamine (CED) as a solvent. Specifically, paper samples of 14 to 47 mg were dissolved in 10 ml of 50% water-diluted CED. The use of different concentrations is required to guarantee that the viscosity of all samples falls within a range that is suitable for the performance of accurate viscosity measurements with the viscometer. In fact, samples characterized by a longer ageing time are expected to have a lower DP_{v}, which at a constant cellulose concentration results in a lower viscosity. Since a higher concentration results in higher viscosity, a higher mass of paper was used for the preparation of samples with a longer ageing time. The preparation procedure was based on cutting each piece of paper roughly into 2 mm\(^2\) pieces, followed by soaking these pieces for 30 min in 5 ml of demineralized water, after which 5 ml of CED was added. Each sample was then magnetically stirred for 30 min to ensure an adequate and complete dissolution process, immediately after which the viscosity measurement (rheometry or glass capillary viscometry) was performed.

In order to obtain the intrinsic viscosity of each sample, it is necessary to accurately measure the concentration of cellulose in the solution. To this purpose, before preparing the solution, the hygrothermally aged, pre-cut paper pieces were weighed with a precision of 0.01 mg. Subsequently, the moisture content of the paper specimens was measured and the mass of water was subtracted from the cellulose mass to attain an accurate determination of the polymer concentration. The equilibrium moisture content of the paper samples was determined from the isotherms measured at \(21~^\circ\)C using a TGA Q5000 SA sorption analyzer (TA Instruments). Separate samples were used for evaluating the moisture content and the degree of polymerization.

More details of the measurement procedure can be found in the supplementary material of [11].

### Rheometry

Rheometry is an experimental method that enables to measure the viscosity of a liquid by inducing a controlled shear flow. This is done by applying a force *F* on a boundary plate that is in contact with the surface of the fluid, which results in a gradient of deformation from that surface to the opposite, stationary surface. This gradient of deformation corresponds to a (simple) shear strain \(\gamma\), while the ratio between the applied force *F* and the area of the boundary plate equals the corresponding shear stress \(\tau\). Accordingly, the dynamic viscosity \(\eta\) is defined by

$$\begin{aligned} \eta = \frac{\tau }{\dot{\gamma }} \ , \end{aligned}$$

(1)

with \(\dot{\gamma }\) the shear (strain) rate. Note that in the current study rotational rheometry is employed, whereby the shear flow is applied in a rotational manner. In particular, a thin film of liquid is enclosed in between two circular discs or concentric cylinders, and the rotation of one of the discs or cylinders induces a shear flow in the liquid. The torque needed for the application of a constant rotational speed is measured, from which the applied shear stress \(\tau\), and subsequently, through Eq. (1), the dynamic viscosity \(\eta\) of the liquid at the specific shear rate \(\dot{\gamma }\) are calculated.

In the case of Newtonian fluids, the viscosity is invariant under an increasing shear rate. Fluids that deviate from Newtonian behaviour, i.e., non-Newtonian fluids, often exhibit a behaviour referred to as shear thinning, which is characterized by a decrease of the viscosity when exceeding a (sufficiently high) critical shear rate. This effect may be ascribed to small structural changes within the fluid, such as the disentanglement of polymer chains, which facilitate shearing and thereby reduce the viscosity [23]. Dilute solutions of cellulose in CED, however, are expected to exhibit Newtonian behaviour with a constant dynamic viscosity over a considerable range of shear rates.

In the present work, the rheometry measurements were performed with an Anton Paar MCR 501 rotational rheometer equipped with a double-wall Couette geometry (DG26.7-SS). This geometry provides an extra surface area in comparison to other geometries, such as a disk or a cone, which makes it suitable for accurately measuring relatively low viscosities. For each measurement, approximately 3.5 ml of the cellulose-CED solution was inserted in the rheometer’s geometry. Note that this amount is about one third of the 10 ml CED solution needed for the glass capillary viscometry test, which thus effectively reduces the paper mass needed for sample preparation by \((1-3.5/10)\times 100 \% = 65\%\), to between 5 and 17 mg (with the specific mass depending on the ageing time). The use of a relatively low amount of paper material makes rheometry attractive for degradation studies with limited sample availability. After insertion of the CED solution in the rheometer, measurements were carried out at shear rates of 10 to 2000 s\(^{-1}\). The shear rate \(\dot{\gamma }\) was increased in an exponential fashion, using 25 steps to reach the predefined, maximum value. The duration of each step was 30 s, during which the torque was measured and the corresponding shear stress \(\tau\) was computed, leading via Eq. (1) to the dynamic viscosity \(\eta\). It was verified that a period of 30 s was sufficient for reaching steady-state flow conditions. By applying a stepwise increase of \(\dot{\gamma }\), a single experiment provides different values of the dynamic viscosity \(\eta\); in fact, the dynamic viscosity obtained at each individual step can be regarded as an independent measurement value. Accordingly, a large number of (statistically representative) measurement data is obtained from an individual test.

### Glass capillary viscometry

Glass capillary viscometry is an experimental method for evaluating the viscosity of a fluid, and is based on the measurement of the flow time, called the *efflux time*, of a defined liquid volume flowing through a capillary tube. The method makes use of the Hagen-Poiseuille equation, which provides the pressure drop \(\Delta p\) for an incompressible Newtonian fluid that flows through a tube with a constant cross-section as [29]:

$$\begin{aligned} \Delta p =\frac{ 8 \eta V L }{ \pi \, t \, R^4 } \, , \end{aligned}$$

(2)

where \(\eta\) is the dynamic viscosity of the fluid, *V* is the flow volume, *R* and *L* are the respective radius and length of the tube, and *t* is the efflux time. In glass capillary viscometers, the driving force for flow is the hydrostatic pressure of a liquid column. Accordingly, the pressure difference can be written as \(\Delta p =\rho g h_m\), where \(\rho\) is the density of the liquid, *g* is the gravitational acceleration and \(h_m\) is the height of the liquid head above the capillary entrance of the viscometer. Inserting the above relation for \(\Delta p\) in Eq. (2), and reformulating the resulting expression, allows to write the kinematic viscosity \(\nu =\eta /\rho\) as

$$\begin{aligned} \nu = C_0 t \qquad \text{ with } \qquad C_0 = \frac{\pi R^4 g h_m}{8LV} \, , \end{aligned}$$

(3)

whereby the specific value of the factor \(C_0\) essentially depends on the characteristics of the viscometer used and thus can be determined by calibration. In the current study, a Cannon-Fenske Routine 100 viscometer was used, with a calibration factor of \(C_0 = 0.01468~\text {mm}^2 \, \text {s}^{-2}\) at a temperature of \(40~^\circ\)C. For each specimen, the efflux time was measured using a stopwatch. The measurements were repeated until three consecutive readings corresponded within 0.1 s, which were then averaged. The kinematic viscosity was subsequently determined from the average efflux time via Eq. (3). The measurements were performed in the viscosity regime where the Hagenbach-Couette correction factor, which accounts for pressure losses at the capillary ends, can be neglected. In specific, the range of kinematic viscosities \(\nu\) of dilute cellulose solutions in CED are bounded by a maximum value of approximately \(3.0~\text {mm}^2\, \text {s}^{-1}\) [24, 25]. In accordance with the technical specifications of the experimental setup used for performing the glass viscometry experiments, this maximum viscosity value relates to a maximum efflux time *t* of 200 s and a minimum shear rate \(\dot{\gamma }\) of 612 \(\text {s}^{-1}\). Depending on the state of degradation of each sample, the concentration *c* of the dilute cellulose solution was carefully selected to ensure that the experimental results fall within the above range of values.

### Intrinsic viscosity

In order to determine DP_{v} from the rheometry and glass capillary viscometry experiments, the measured viscosity values first needed to be converted to the so-called *intrinsic viscosity*. The intrinsic viscosity \([\eta ]\) of a dilute solution is a measure of the contribution of the solute, i.e., the cellulose polymeric chains, to the viscosity of the solution. It is defined by the ratio between the specific viscosity \(\eta _s\) and the concentration *c* of the cellulose solution, evaluated in the limit of an infinitely dilute solution \(c\rightarrow 0\), i.e.,

$$\begin{aligned} {[}\eta ] = \lim _{c\rightarrow 0}\frac{\eta _s}{c} \, , \end{aligned}$$

(4)

where the specific viscosity \(\eta _s\) follows from

$$\begin{aligned} \eta _s = \frac{\eta - \eta _{solv}}{\eta _{solv}} \, , \end{aligned}$$

(5)

with \(\eta\) and \(\eta _{solv}\) the dynamic viscosities of the solution and the solvent, respectively. From its definition given by Eq. (4), the intrinsic viscosity can be determined by performing dynamic viscosity measurements at different concentrations close to zero, and next extrapolating the result to zero concentration [30,31,32]. However, it is common practice to obtain \([\eta ]\) based on a *single* concentration measurement. For this purpose, Martin’s formula is used [22, 33]:

$$\begin{aligned} \eta _s = [\eta ] \, c \, 10^{k[\eta ]c}, \end{aligned}$$

(6)

where *k* is Martin’s constant, which is equal to 0.13 and 0.14 for the cases of glass capillary viscometry and rotational rheometry, respectively. This slight difference in the value of *k* is due to the fact that the glass viscometry measurements are performed based on the ASTM D1795-13 standard [19] and the rheometry measurements are based on the IEC 60450 standard [28].

In the rheometry experiment, the specific viscosity \(\eta _s\) appearing in the left-hand side of Eq. (6) is computed via Eq. (5), by substituting the dynamic viscosities \(\eta\) and \(\eta _{solv}\) as obtained via Eq. (1). Conversely, in the glass capillary viscometry experiment, the specific viscosity is determined as follows. When the difference in the densities of the dilute solution and that of the solvent is small and negligible (which is the case for the cellulose solutions used in the present study), from the definition of the kinematic viscosity \(\nu =\eta /\rho\), the specific viscosity in Eq. (5) can be alternatively determined as \(\eta _s=(\nu - \nu _{solv})/\nu _{solv}\). Combining this relation with Eq. (3) allows to write the specific viscosity as:

$$\begin{aligned} \eta _s = \frac{t - t_{solv}}{t_{solv}}, \end{aligned}$$

(7)

where *t* and \(t_{solv}\) are the efflux times of the cellulose solution and the solvent, respectively. The specific viscosities \(\eta _s\) following from the rheometry and glass viscometry experiments and the concentrations *c* of the corresponding samples are next inserted in the (nonlinear) Martin’s equation, Eq. (6), which is solved in an iterative fashion to obtain the intrinsic viscosity \([\eta ]\). It is noted that for the glass viscometry experiment the value of \([\eta ]\) can also be determined from \(\eta _s\) by using the tabular data provided in the standard ASTM D1795-13, which leads to the same result.

### Viscosity-average degree of polymerisation (DP_{v})

From the intrinsic viscosity \([\eta ]\) of the specimen, the corresponding value of the viscosity-average degree of polymerisation DP_{v} is calculated from the inverse of the Mark-Houwink-Sakurada equation:

$$\begin{aligned} {[}\eta ] = K \, (\hbox {DP}_{\textrm{v}})^{\alpha }, \end{aligned}$$

(8)

whereby the two empirical constants are taken as \(K=0.91\) and \(\alpha =0.85\) [20].