**2. Experimental**

#### **2.1 Materials**

Pulp 1 was purchased from Sappi Saiccor (Durban, South Africa). Pulp 2 and 3 were supplied by Weyerhaeuser (Covington, Washington, USA). N-methlymorpholine-N-oxide (NMMO) aqueous solution from BASF (Ludwigshafen, Germany) had an initial water content of 50% (*wt*). N, N-dimethlylacetamide (DMAc) was obtained from Fluka (Lausanne, Switzerland). LiCl was obtained from Yili Finer Chemical Ltd. (Shanghai, China). Polystyrene standard was purchased from Waters (Milford, Massachusetts, USA), with nominal *MWs* of 1.3×104, 3.03×104, 6.55×104, 18.5×104, 66.8×104, and 101×104.

#### **2.1.1 Rheological measurement method**

**Preparation of cellulose/NMMO·H2O solution**. A mixture of cellulose pulp in 87% NMMO·H2O (wt) was placed in a dissolving tank maintained at 100°C and stirred continually. The mixture gradually turned into a brown and clear homogeneous liquid. The solution with 12%, 11%, and 9% cellulose in NMMO·H2O solution (wt) was obtained, respectively.

**Rheological measurements**. Rheological measurements were recorded on the RS1 rheometer [13] (Thermo Haake, Karlsruhe, Germany) in a frequency scanning mode of 0.2 to 620 (*rad·s-1*) with a cone plate (*Ti*, 35/10). Dynamic rheological properties (storage modulus *G'* and loss modulus *G''*) were obtained in the linear viscoelastic region at the temperature of 75°C, 90°C, and 105°C, respectively. The data were analyzed with RheoSoft software (Thermo Haake).

**Relative differential** *MWD* **curve and method of calculating** *PDI*. Tuminello [6-8, 14] has a method of computing the relative differential *MWD* of linear polymer melts which is based on an analogy with polymer solutions. For the latter, it is well known that

$$\mathbf{G}\_{N, \text{solution}}^{0} \approx \text{ } \mathbf{q} \text{ } \mathbf{G}\_{N, \text{melt}}^{0} \tag{1}$$

Where *φ* is the volume fraction of the polymer. Tuminello assumes that the unrelaxed chains of a melt at any frequency (*ωi*), are "diluted" by, but not entangled with, the relaxed (lower *Mi*) chains. He also assumes that each monodisperse fraction (*Mi*), has a single relaxation frequency (*ωi*), below which it makes no contribution to the modulus – thus behaving as a step function. Hence, in Eq. (1), if one associates <sup>0</sup> *GN solution* , with the storage modulus *G'*(*ωi*), <sup>0</sup> *GN melt* , with the rubbery plateau modulus <sup>0</sup> *GN* , and *φ* with the mass fraction of unrelaxed chains of *M* ≥ *M i*, then

$$\mathbf{C}(M) = \mathbf{1} - \left[ G'(\alpha \flat) / \, G\_N^0 \right] \,^{0.5} \tag{2}$$

where *C*(*M*) is the mass fraction of relaxed chains, and

$$\text{MWD} = \frac{\text{d}(\text{C}(M))}{\text{d}(\log M)} \tag{3}$$

where *M* is the molecular weight.

It is well know for the relations

314 Materials Science and Technology

investigated. Furthermore, the calculation of the *PDI* of cellulose was developed. In addition, it also realized the conversion of the reciprocal of the frequency to the actual *MW*

Pulp 1 was purchased from Sappi Saiccor (Durban, South Africa). Pulp 2 and 3 were supplied by Weyerhaeuser (Covington, Washington, USA). N-methlymorpholine-N-oxide (NMMO) aqueous solution from BASF (Ludwigshafen, Germany) had an initial water content of 50% (*wt*). N, N-dimethlylacetamide (DMAc) was obtained from Fluka (Lausanne, Switzerland). LiCl was obtained from Yili Finer Chemical Ltd. (Shanghai, China). Polystyrene standard was purchased from Waters (Milford, Massachusetts, USA), with

**Preparation of cellulose/NMMO·H2O solution**. A mixture of cellulose pulp in 87% NMMO·H2O (wt) was placed in a dissolving tank maintained at 100°C and stirred continually. The mixture gradually turned into a brown and clear homogeneous liquid. The solution with 12%, 11%, and 9% cellulose in NMMO·H2O solution (wt) was obtained,

**Rheological measurements**. Rheological measurements were recorded on the RS1 rheometer [13] (Thermo Haake, Karlsruhe, Germany) in a frequency scanning mode of 0.2 to 620 (*rad·s-1*) with a cone plate (*Ti*, 35/10). Dynamic rheological properties (storage modulus *G'* and loss modulus *G''*) were obtained in the linear viscoelastic region at the temperature of 75°C, 90°C, and 105°C, respectively. The data were analyzed with RheoSoft software

**Relative differential** *MWD* **curve and method of calculating** *PDI*. Tuminello [6-8, 14] has a method of computing the relative differential *MWD* of linear polymer melts which is based

Where *φ* is the volume fraction of the polymer. Tuminello assumes that the unrelaxed chains of a melt at any frequency (*ωi*), are "diluted" by, but not entangled with, the relaxed (lower *Mi*) chains. He also assumes that each monodisperse fraction (*Mi*), has a single relaxation frequency (*ωi*), below which it makes no contribution to the modulus – thus behaving as a step function. Hence, in Eq. (1), if one associates <sup>0</sup> *GN solution* , with the storage modulus *G'*(*ωi*), <sup>0</sup> *GN melt* , with the rubbery plateau modulus <sup>0</sup> *GN* , and *φ* with the mass fraction of unrelaxed

 *C*(*M*) = 1– [*G'*(*ωi*)/ <sup>0</sup> *GN* ] 0.5 (2)

<sup>0</sup> *GN solution* , ≈ *φ*<sup>2</sup> <sup>0</sup> *GN melt* , (1)

on an analogy with polymer solutions. For the latter, it is well known that

nominal *MWs* of 1.3×104, 3.03×104, 6.55×104, 18.5×104, 66.8×104, and 101×104.

scale, obtaining *MWD* scale curve of cellulose [11,12].

**2.1.1 Rheological measurement method** 

**2. Experimental** 

**2.1 Materials** 

respectively.

(Thermo Haake).

chains of *M* ≥ *M i*, then

where *C*(*M*) is the mass fraction of relaxed chains, and

$$1/\omega \ll \eta\_0 \ll M^{3.4} \tag{4}$$

where *ω* is the frequency of dynamic rheology data, *η0* is the zero-shear viscosity of dynamic rheology data, therefore, log (1/*ω*) is proportional to log *M*, then

$$\text{MND} = \frac{\text{d}(\text{C}(M))}{\text{d}(\log(1/o))} \tag{5}$$

The above equations were employed in the calculation of the relative differential *MWD* curve of the cellulose pulps.

The relative differential *MWD* curve is a Wesslan function which is the logarithm of the normal distribution function and is especially applied to measure the *MWD* of polymers. The Wessllan function is given by [15, 16]:

$$Y = \frac{1}{\sigma \sqrt{2\pi}} \quad \exp\left(-\frac{1}{2\sigma^2} (M - M\_p)^2\right) \tag{6}$$

where *Y* is the relative content percent of *MW*, *M* is the molecular weight, *σ* is the standard deviation, and *Mp* is the peak *MW* on the *MWD* curve.

There is a correlation between the peak value of the ordinate (*Yextre*) and the *σ* with respect to the logarithm normal distribution, as given by *Eq*.(6) [15, 16]:

$$Y\_{extr} = \frac{1}{\sqrt{2\pi}\sigma} \tag{7}$$

Then the PDI value is calculated by *Eq*.(7) [15, 16]:

$$\text{PDI} \equiv \exp(\sigma^2) \tag{8}$$

#### **2.1.2 GPC measurements**

**Dissolution of cellulose in LiCl/DMAc***.* A 10mg sample of each of the three pulps was placed in a 10mL centrifuge tube, respectively. Then 5mL of distilled water was added to each tube. The mixtures were stirred for 5min and left overnight to pre-activate the cellulose. The samples were centrifuged at 4000rpm for 15min. The supernatant fluid was decanted and 5mL of DMAc was added, respectively. After stirring for 15min, the centrifugation and the decantation steps were repeated. The whole solvent exchange procedure was repeated five times. Finally, 1.25mL of 8% LiCl/DMAc (wt/vol) was added, stirred for 60s, and left for approximately one week to dissolve completely, with occasional gentle stirring. The dissolved cellulose solutions were diluted to 20ml with DMAc to give a

Rheological Method for Determining Molecular Weight and Molecular Weight Distribution 317

Fig. 1. Master curves of the pulp1 at different cellulose concentrations in NMMO·H2O

solutions: (a) 9%, (b) 11%, and (c) 12%.

final cellulose concentration of 0.5mg/mL in 0.5% LiCl/DMAc (wt/vol). Then the solution was filtered through a 0.45μm membrane filter.

**GPC analysis**. The *MWDs* for the three cellulose pulps were determined by *GPC* in a liquid chromatography (Waters1525) with a refractive index detector (Waters 2410).The mobile phase of 0.5% LiCl/DMAc (wt/vol) was pumped into the system at a ow rate of 1ml/min. Columns were Waters styragel HR 3, 4, and 5 (300mm × 7.8mm) preceded by a guard column. The system was operated at 50°C controlled by a column heater (Waters column temperature system). Injection volume was 200μL. Run time was 45min. A linear calibration curve was constructed with polystyrene standards directly dissolved in 0.5 % LiCl/DMAc (wt/vol). Data acquisition and *MWD* calculations were performed using Breeze software (Waters, Milford, MA, USA). Furthermore, the *GPC* data were calibrated using the Eawkins and Maddock calibration equation, as follows [1, 2]:

$$\log M\_{\rm I} = \log M\_{\rm 2} + \frac{2}{3} \log \frac{(\kappa\_{\theta})\_2}{(\kappa\_{\theta})\_1} \tag{9}$$

where subscript 1 and 2 represent the unknown sample and the standard sample, respectively. Here, they are the cellulose and the polystyrene. *Kθ* values [1] of cellulose and polystyrene are 0.528 and 0.081 in the 0.5% LiCl/DMAc (wt/vol) system, respectively.

### **3. Results and discussion**

#### **3.1 Effect of cellulose concentration in NMMO·H2O solution on the rheology-based results**

Fig.1a, 1b, and 1c show *G'* and *G''* master curves of the pulp 1 at various cellulose concentrations in NMMO·H2O solutions according to the time-temperature superposition theory [13]. In terms of the Tuminello diluted assumption theory [6-8, 14], these curves are converted to relative differential *MWD* curves of the pulp 1 at different cellulose concentrations in NMMO·H2O solutions shown in Fig.2a. Using the same procedure, relative differential *MWD* curves of the pulp 2 and 3 are obtained at various cellulose concentrations in NMMO·H2O solutions and presented in Fig.2b and 2c, respectively. Meanwhile, the calculated log (1/*ωp*), *σ* and *PDI* values of the three pulps are given in Table 1.

Fig.2a, 2b, and 2c show that the relative differential *MWD* curves almost overlap with each other when the cellulose concentrations are 12% and 11%, respectively. However, when the cellulose concentration is 9%, the relative differential *MWD* curves shift slightly to the left.

The data in Table 1 clearly indicate that log (1/*ωp*) are higher and *PDI* decrease with increasing concentration. However, the relative deviation of the cellulose concentration between 9% and 11% is up to 16%, and the relative deviation of the cellulose concentration between 11% and 12% is less than 0.9%. Obviously, the calculated log (1/*ωp*) and *PDI* are approximately equal when the cellulose concentrations are 12% and 11%.

This phenomenon originates from the relaxation time (*τ*) of molecular chains and the steadystate recoverable compliance (*Je <sup>0</sup>*) in a polymer solution. With increasing concentration, the polymer chains are more strongly entangled with each other and get more and more

final cellulose concentration of 0.5mg/mL in 0.5% LiCl/DMAc (wt/vol). Then the solution

**GPC analysis**. The *MWDs* for the three cellulose pulps were determined by *GPC* in a liquid chromatography (Waters1525) with a refractive index detector (Waters 2410).The mobile phase of 0.5% LiCl/DMAc (wt/vol) was pumped into the system at a ow rate of 1ml/min. Columns were Waters styragel HR 3, 4, and 5 (300mm × 7.8mm) preceded by a guard column. The system was operated at 50°C controlled by a column heater (Waters column temperature system). Injection volume was 200μL. Run time was 45min. A linear calibration curve was constructed with polystyrene standards directly dissolved in 0.5 % LiCl/DMAc (wt/vol). Data acquisition and *MWD* calculations were performed using Breeze software (Waters, Milford, MA, USA). Furthermore, the *GPC* data were calibrated using the Eawkins

2

where subscript 1 and 2 represent the unknown sample and the standard sample, respectively. Here, they are the cellulose and the polystyrene. *Kθ* values [1] of cellulose and polystyrene are 0.528 and 0.081 in the 0.5% LiCl/DMAc (wt/vol) system, respectively.

**3.1 Effect of cellulose concentration in NMMO·H2O solution on the rheology-based** 

Fig.1a, 1b, and 1c show *G'* and *G''* master curves of the pulp 1 at various cellulose concentrations in NMMO·H2O solutions according to the time-temperature superposition theory [13]. In terms of the Tuminello diluted assumption theory [6-8, 14], these curves are converted to relative differential *MWD* curves of the pulp 1 at different cellulose concentrations in NMMO·H2O solutions shown in Fig.2a. Using the same procedure, relative differential *MWD* curves of the pulp 2 and 3 are obtained at various cellulose concentrations in NMMO·H2O solutions and presented in Fig.2b and 2c, respectively. Meanwhile, the calculated log (1/*ωp*), *σ* and *PDI* values of the three pulps are given in

Fig.2a, 2b, and 2c show that the relative differential *MWD* curves almost overlap with each other when the cellulose concentrations are 12% and 11%, respectively. However, when the cellulose concentration is 9%, the relative differential *MWD* curves shift slightly to the left. The data in Table 1 clearly indicate that log (1/*ωp*) are higher and *PDI* decrease with increasing concentration. However, the relative deviation of the cellulose concentration between 9% and 11% is up to 16%, and the relative deviation of the cellulose concentration between 11% and 12% is less than 0.9%. Obviously, the calculated log (1/*ωp*) and *PDI* are

This phenomenon originates from the relaxation time (*τ*) of molecular chains and the steady-

polymer chains are more strongly entangled with each other and get more and more

*<sup>0</sup>*) in a polymer solution. With increasing concentration, the

approximately equal when the cellulose concentrations are 12% and 11%.

<sup>3</sup> log <sup>2</sup>

( ) ( ) 

1

(9)

was filtered through a 0.45μm membrane filter.

and Maddock calibration equation, as follows [1, 2]:

log *M1* = log *M2* +

**3. Results and discussion** 

state recoverable compliance (*Je*

**results** 

Table 1.

Fig. 1. Master curves of the pulp1 at different cellulose concentrations in NMMO·H2O solutions: (a) 9%, (b) 11%, and (c) 12%.

Rheological Method for Determining Molecular Weight and Molecular Weight Distribution 319

Table 1. Logarithm of relative molecular weight log ((1*/ωp*)/(*s·rad-1*)), standard deviation *σ*, and polydispersity index *PDI* of the three pulps calculated by the rheology-based method

difficult to relax the entanglement network, leading to a longer *τ*. Meanwhile, with

solution will tend towards equilibrium because the gyration radius, the end-to-end distance, and the "degree" of mutual entanglement of the chains will reach critical points,

with *PDI* of the polymer [17, 18]. Therefore, *MW* and *PDI* of the polymer move towards stabilization with the polymer concentration reaches a critical point. Accordingly, in the rheology-based method, a high enough concentration of cellulose in NMMO·H2O solution

**3.2 Prediction of MW scale and MWD of cellulose using the rheology-based method Methodology.** Definition of the Rouse terminal relaxation time is well known as follows

where *ρ* is the density, *η0* the zero-shear viscosity, *R* the universal gas constant, *M* the molecular weight, *T* the temperature. *ωchar* is the 79th percentile point of the zero shear normalized flow curve and normally symbolized as *ω.79*. The *ω.79* is the gradually-changed frequency from Newtonian to non-Newtonian behavior for a liquid of polymer solution, with which the apparent viscosity of the polymer solution decreases and the untangling

According to the Rouse terminal relaxation time theory [14], a polymer solution always has a corresponding characteristic relaxation frequency (*ωchar*). Here, the *ωchar* is the *ω.79* corresponding to the point of maximum curvature of the flow curve. Therefore the calculated *M* is believed to be the peak *MW* (*Mp*). *Mp* indicates the maximum probability of

increasing concentration, the polymer solution shows a decreasing *Je*

has to be used in order to obtain reliable and stable data.

polymers do not fully and freely stretch with rearranging configurationally. However, when the polymer concentration reaches a certain value, τ and *Je*

It is well known that the *τ* is a function of *MW* of the polymer [17, 18], and *Je*

*τ<sup>1</sup>* = <sup>0</sup> 2 6 *M RT* 

effect is stronger than the entangling effect among molecular chains.

 

log (1/*ωp*) -1.5024 -1.5296 -1.4888 *σ* 1.1091 1.0956 1.1036 *PDI* 3.4217 3.3211 3.3850

log (1/*ωp*) -1.5099 -1.5437 -1.4982 *σ* 1.1118 1.0986 1.1069 *PDI* 3.4424 3.3429 3.4046

log (1/*ωp*) -1.7982 -1.8398 -1.7635 *σ* 1.1853 1.1410 1.1650 *PDI* 4.0753 3.6759 3.8854

*<sup>0</sup>* because the entangled

= 1/*ωchar* (10)

*<sup>0</sup>* of the polymer

*<sup>0</sup>* is correlated

Samples Pulp1 Pulp 2 Pulp 3

Concentration (12 %)

Concentration (11 %)

Concentration (9 %)

respectively [17].

[14].

with different concentrations.

Fig. 2. Relative differential *MWD* curves of the three pulps at different cellulose concentrations in NMMO·H2O solutions: (a) pulp 1, (b) pulp 2, and (c) pulp3.


Fig. 2. Relative differential *MWD* curves of the three pulps at different cellulose concentrations in NMMO·H2O solutions: (a) pulp 1, (b) pulp 2, and (c) pulp3.


Table 1. Logarithm of relative molecular weight log ((1*/ωp*)/(*s·rad-1*)), standard deviation *σ*, and polydispersity index *PDI* of the three pulps calculated by the rheology-based method with different concentrations.

difficult to relax the entanglement network, leading to a longer *τ*. Meanwhile, with increasing concentration, the polymer solution shows a decreasing *Je <sup>0</sup>* because the entangled polymers do not fully and freely stretch with rearranging configurationally.

However, when the polymer concentration reaches a certain value, τ and *Je <sup>0</sup>* of the polymer solution will tend towards equilibrium because the gyration radius, the end-to-end distance, and the "degree" of mutual entanglement of the chains will reach critical points, respectively [17].

It is well known that the *τ* is a function of *MW* of the polymer [17, 18], and *Je <sup>0</sup>* is correlated with *PDI* of the polymer [17, 18]. Therefore, *MW* and *PDI* of the polymer move towards stabilization with the polymer concentration reaches a critical point. Accordingly, in the rheology-based method, a high enough concentration of cellulose in NMMO·H2O solution has to be used in order to obtain reliable and stable data.

#### **3.2 Prediction of MW scale and MWD of cellulose using the rheology-based method**

**Methodology.** Definition of the Rouse terminal relaxation time is well known as follows [14].

$$
\pi\_1 = \frac{6\eta\_0 M}{\pi^2 \rho RT} = 1/\alpha\_{\text{char}} \tag{10}
$$

where *ρ* is the density, *η0* the zero-shear viscosity, *R* the universal gas constant, *M* the molecular weight, *T* the temperature. *ωchar* is the 79th percentile point of the zero shear normalized flow curve and normally symbolized as *ω.79*. The *ω.79* is the gradually-changed frequency from Newtonian to non-Newtonian behavior for a liquid of polymer solution, with which the apparent viscosity of the polymer solution decreases and the untangling effect is stronger than the entangling effect among molecular chains.

According to the Rouse terminal relaxation time theory [14], a polymer solution always has a corresponding characteristic relaxation frequency (*ωchar*). Here, the *ωchar* is the *ω.79* corresponding to the point of maximum curvature of the flow curve. Therefore the calculated *M* is believed to be the peak *MW* (*Mp*). *Mp* indicates the maximum probability of

Rheological Method for Determining Molecular Weight and Molecular Weight Distribution 321

Samples Pulp 1 Pulp 2 Pulp 3

log *Mp* 5.12 4.96 5.14

*σ* 1.109 1.096 1.104

*PDI* 3.42 3.32 3.38

Table 2. Logarithm of molecular weight scale log*Mp*, standard deviation *σ*, and

with 12% cellulose concentration in NMMO·H2O solution.

in Fig.4. Meanwhile, the calibrated *GPC* data are listed in Table 3.

converted to the *MW* scale in the rheology-based method.

based method.

polydispersity index *PDI* values of the three pulps calculated by the rheology-based method

**3.3 Comparison of the results from the rheology-based method and the GPC method**  Because of the lack of commercial cellulose standards with a narrow distribution, the narrow distribution polystyrenes standards are employed to measure *MW* and *MWD* of cellulose. The *MWD* curves of the three pulps measured by the *GPC* method are illustrated

Comparing the data of Table 2 with those of Table 3, one can observe that log *Mp* calculated by the rheology-based method is nearly equal to log *Mp'* determined by the *GPC* method. Therefore, it is feasible and reasonable that the calculated *M* with Eq. (10) is regarded as the peak *MW* (*Mp*) on the *MWD* curve. Consequently, the reciprocal of the frequency is

The results of Table 3 show that *PDI'* of the three pulps is pulp 1 > pulp 3 > pulp 2, which are consistent with the results from the rheology-based method. Moreover, more information can also be obtained from Fig.4 by the GPC method than that from Fig.3 by the rheology-based method. For example, pulp 1 shows a symmetrical distribution, and moderate *MW* components are dominating. Pulp 3 shows a slightly asymmetrical distribution, moderate *MW* components are the major composition, and it has a little lower *MW*. For pulp 2, it is asymmetrical and slightly protuberant in the lower MW region, which indicates the presence of a higher low *MW* content. However, such useful information can not be reflected from the *MWD* scale curves obtained by the rheology-based method directly. It shows that further modification is still needed for the application of the rheology-

molecular weight on the curve of *MWD*. In addition, it can be believed that the maximum probability of molecular weight begins to untangle, which would lead to a decreasing apparent viscosity of the polymer solution.

Accordingly, the peak *MW* scale is obtained by *Mp* = *M* = *π*2*ρRT*/(6*η0ω.79*) and the *MWD* scale curve is obtained with shifting the abscissa (log *Mp* - log (1/*ωp*)) units. Here, the *Mp* and (1/*ωp*) indicate the peak *MW* on the *MWD* scale curve and the relative *MWD* curve, respectively.

For the 79th percentile point of the normalized flow curve, the Vinogradov extrapolation leads to [14]:

$$
\omega\_{.79} = 0.3365 m\_0^2 / m\_1 \tag{11}
$$

where *m0* and *m1* are respectively the intercept and the slope on the curve of stress versus viscosity at the low frequency of the dynamic data. In the current paper, the *m0* and *m1* are obtained in the frequency range from 0.2 to 2 (*rad·s-1*).

**Results.** Fig.2a, 2b, and 2c respectively represent relative *MWD* curves of the three pulps at various cellulose concentrations in NMMO·H2O solutions. Choosing a concentration of 12%, these relative *MWD* curves are converted to the *MWD* scale curves of the three pulps shown in Fig.3, using the above-mentioned method. Furthermore, in terms of the *MWD* scale curve, log *Mp*, *σ*, and *PDI* of the three pulps are calculated and given in Table 2. From Fig.3 and Table 2, it can be found that the relation of peak *MW* is pulp 3 > pulp 1 > pulp 2. For the *MWD* of the three pulps, pulp 1 appears the broadest, next is pulp 3, and then pulp 2.

Fig. 3. *MW* scale curves of the three pulps by the rheology-based method with 12% cellulose concentration in NMMO·H2O solution.

molecular weight on the curve of *MWD*. In addition, it can be believed that the maximum probability of molecular weight begins to untangle, which would lead to a decreasing

Accordingly, the peak *MW* scale is obtained by *Mp* = *M* = *π*2*ρRT*/(6*η0ω.79*) and the *MWD* scale curve is obtained with shifting the abscissa (log *Mp* - log (1/*ωp*)) units. Here, the *Mp* and (1/*ωp*) indicate the peak *MW* on the *MWD* scale curve and the relative *MWD* curve,

For the 79th percentile point of the normalized flow curve, the Vinogradov extrapolation

where *m0* and *m1* are respectively the intercept and the slope on the curve of stress versus viscosity at the low frequency of the dynamic data. In the current paper, the *m0* and *m1* are

**Results.** Fig.2a, 2b, and 2c respectively represent relative *MWD* curves of the three pulps at various cellulose concentrations in NMMO·H2O solutions. Choosing a concentration of 12%, these relative *MWD* curves are converted to the *MWD* scale curves of the three pulps shown in Fig.3, using the above-mentioned method. Furthermore, in terms of the *MWD* scale curve, log *Mp*, *σ*, and *PDI* of the three pulps are calculated and given in Table 2. From Fig.3 and Table 2, it can be found that the relation of peak *MW* is pulp 3 > pulp 1 > pulp 2. For the *MWD* of the three pulps, pulp 1 appears the broadest, next is pulp 3, and then pulp 2.

Fig. 3. *MW* scale curves of the three pulps by the rheology-based method with 12% cellulose

*ω.79* = 0.3365*m0*2/*m1* (11)

apparent viscosity of the polymer solution.

obtained in the frequency range from 0.2 to 2 (*rad·s-1*).

concentration in NMMO·H2O solution.

respectively.

leads to [14]:


Table 2. Logarithm of molecular weight scale log*Mp*, standard deviation *σ*, and polydispersity index *PDI* values of the three pulps calculated by the rheology-based method with 12% cellulose concentration in NMMO·H2O solution.

#### **3.3 Comparison of the results from the rheology-based method and the GPC method**

Because of the lack of commercial cellulose standards with a narrow distribution, the narrow distribution polystyrenes standards are employed to measure *MW* and *MWD* of cellulose. The *MWD* curves of the three pulps measured by the *GPC* method are illustrated in Fig.4. Meanwhile, the calibrated *GPC* data are listed in Table 3.

Comparing the data of Table 2 with those of Table 3, one can observe that log *Mp* calculated by the rheology-based method is nearly equal to log *Mp'* determined by the *GPC* method. Therefore, it is feasible and reasonable that the calculated *M* with Eq. (10) is regarded as the peak *MW* (*Mp*) on the *MWD* curve. Consequently, the reciprocal of the frequency is converted to the *MW* scale in the rheology-based method.

The results of Table 3 show that *PDI'* of the three pulps is pulp 1 > pulp 3 > pulp 2, which are consistent with the results from the rheology-based method. Moreover, more information can also be obtained from Fig.4 by the GPC method than that from Fig.3 by the rheology-based method. For example, pulp 1 shows a symmetrical distribution, and moderate *MW* components are dominating. Pulp 3 shows a slightly asymmetrical distribution, moderate *MW* components are the major composition, and it has a little lower *MW*. For pulp 2, it is asymmetrical and slightly protuberant in the lower MW region, which indicates the presence of a higher low *MW* content. However, such useful information can not be reflected from the *MWD* scale curves obtained by the rheology-based method directly. It shows that further modification is still needed for the application of the rheologybased method.

Rheological Method for Determining Molecular Weight and Molecular Weight Distribution 323

In the present work, the data obtained by the *GPC* method are relative values because of the use of polystyrenes standards, nevertheless the *GPC* method is an effective way for observing the differences of *MW* and *MWD* of cellulose. The relative data can not reflect the real *MW* characteristics of cellulose, so the data from *GPC* can not be used to calibrate the results from the rheology-based method. Even so, the comparison of the results from the two methods shows that it may be feasible to compare the *MW* and *MWD* of cellulose by the

Prediction of *MW* scale and *MWD* of cellulose by means of a rheology-based method was developed. With this method, insignificant effect of cellulose concentration on predicting *MW* and *MWD* of cellulose was found using a rheology-based method when the cellulose concentration in the NMMO·H2O solution is high enough. Furthermore, a method of calculating *PDI* of cellulose was established according to the Wesslan function which is the logarithm of the normal distribution function. For the cellulose/NMMO·H2O solution, the cellulose *MW* values calculated by the Rouse terminal relaxation time can be considered as the peak *MW* on the *MWD* curves of cellulose. Consequently, the reciprocal of the frequency

Meanwhile, the results obtained by the rheology-based method were compared with those measured by the *GPC* method. All obtained results from the two methods are only relative values. The comparison shows that the calculated peak *MW* are approximately equal, the calculated *PDI* have the same trends, but the shapes of the *MWD* curves do not match. *GPC* method is advantageous to depict finer characteristics of the *MWD* of cellulose. In spite of that, the rheology-based method is simple and fast. Therefore it is a useful and easy way to

is converted to the *MW* scale, obtaining *MWD* scale curves of cellulose.

analyze the *MW* scale and *MWD* of cellulose in the fiber industry.

W. H. Tuminello and N. C. Mauroux, *Polym. Eng. Sci.,* 31(10), 1496 (1991). W. H. Tuminello, T. A. Treat, and A. D. English, *Macromolecules,* 21, 2606 (1988).

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rheology-based method.

**4. Conclusions** 

**5. References** 


Table 3. Logarithm of molecular weight log*Mp'*, mass-average molecular weight *Mw* , number-average molecular weight *Mn* , and polydispersity index *PDI'* of the three pulps measured by the *GPC* method.

Fig. 4. *MWD* curves of the three pulps by the *GPC* method.

In the present work, the data obtained by the *GPC* method are relative values because of the use of polystyrenes standards, nevertheless the *GPC* method is an effective way for observing the differences of *MW* and *MWD* of cellulose. The relative data can not reflect the real *MW* characteristics of cellulose, so the data from *GPC* can not be used to calibrate the results from the rheology-based method. Even so, the comparison of the results from the two methods shows that it may be feasible to compare the *MW* and *MWD* of cellulose by the rheology-based method.
