Rheological Method for Determining Molecular Weight and Molecular Weight Distribution

Gel permeation chromatography (GPC) method is a widely used and accepted method for measuring the MW and MWD for polymers. However, the method has its limitations. The key of this method is to find the suitable solvents to dissolve the polymer well. But cellulose can not be dissolved in most of the organic solvents because of the interand introhydrogen bonding of the cellulose chains. Lithium chloride/N, N-dimethlylacetamide (LiCl/DMAc) can dissolve cellulose, but the dissolution process is very complicated, which includes pre-activation, solvent exchange, swelling and dissolution. Furthermore, more attention must be paid to each step, and cellulose has to be dissolved for 5-10 days according to the type of cellulose pulp [1-4]. Therefore, it is necessary to develop a simple and fast method which can get the MW and MWD of cellulose.


Introduction
Gel permeation chromatography (GPC) method is a widely used and accepted method for measuring the MW and MWD for polymers. However, the method has its limitations. The key of this method is to find the suitable solvents to dissolve the polymer well. But cellulose can not be dissolved in most of the organic solvents because of the inter-and introhydrogen bonding of the cellulose chains. Lithium chloride/N, N-dimethlylacetamide (LiCl/DMAc) can dissolve cellulose, but the dissolution process is very complicated, which includes pre-activation, solvent exchange, swelling and dissolution. Furthermore, more attention must be paid to each step, and cellulose has to be dissolved for 5-10 days according to the type of cellulose pulp [1][2][3][4]. Therefore, it is necessary to develop a simple and fast method which can get the MW and MWD of cellulose. Wu [5] got the MWD from storage modulus G' and stress relaxation modulus G(t) using approximations derived from the Doi-Edwards description of chain dynamics. Wu's method accurately predicted the MWD of polymers with narrow distribution. However, often, it led to a distorted shape of the MWD for the sample with bimodal distributions. Therefore,  developed a theory based on a diluted assumption in 1986. His method rigorously applies only to linear polymers. Especially, it works better for linear polymers with PI < 3.5. According to his theory, the relative differential MWD of polymer can be determined well from dynamic modulus master curve. Gu [9] applied the diluted assumption theory to the concentrated cellulose in Nmethlymorpholine-N-oxide monohydrate (NMMO·H 2 O) solution. He got relative differential MWD curves of three kinds of cellulose pulps from the dynamic data of cellulose/NMMO·H 2 O solution in 2000. But the results were not compared with the results reported by GPC. In 2004, the relative differential MWD curves of four kinds of cellulose pulps were calculated on the basis of that method and the calculated results were compared with the non-calibrated GPC results by Zhang [10]. In their rheology experiments, the cellulose concentration in NMMO·H 2 O solution was fixed (9%, wt), and the polydispersity index (PDI) of cellulose was not calculated.
In the present work, the effect of cellulose concentration in NMMO·H 2 O solution on prediction of the MW and MWD of cellulose using the rheology-based method was www.intechopen.com 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 scale, obtaining MWD scale curve of cellulose [11,12].

Preparation of cellulose/NMMO·H 2 O solution.
A mixture of cellulose pulp in 87% NMMO·H 2 O (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·H 2 O 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, 35/1 0 ). 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. 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 Where φ is the volume fraction of the polymer. Tuminello assumes that the unrelaxed chains of a melt at any frequency (ω i ), are "diluted" by, but not entangled with, the relaxed (lower M i ) chains. He also assumes that each monodisperse fraction (M i ), has a single relaxation frequency (ω i ), below which it makes no contribution to the modulus -thus behaving as a where M is the molecular weight.
It is well know for the relations where ω is the frequency of dynamic rheology data, 0 is the zero-shear viscosity of dynamic rheology data, therefore, log (1/ω) is proportional to log M, then 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]: where Y is the relative content percent of MW, M is the molecular weight, 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 (Y extre ) and the with respect to the logarithm normal distribution, as given by Eq.(6) [15,16]: Then the PDI value is calculated by Eq.(7) [15,16]:

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 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 flow 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]: where subscript 1 and 2 represent the unknown sample and the standard sample, respectively. Here, they are the cellulose and the polystyrene. K values [1] of cellulose and polystyrene are 0.528 and 0.081 in the 0.5% LiCl/DMAc (wt/vol) system, respectively. Fig.1a, 1b, and 1c show G' and G'' master curves of the pulp 1 at various cellulose concentrations in NMMO·H 2 O solutions according to the time-temperature superposition theory [13]. In terms of the Tuminello diluted assumption theory [6][7][8]14], these curves are converted to relative differential MWD curves of the pulp 1 at different cellulose concentrations in NMMO·H 2 O 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·H 2 O solutions and presented in Fig.2b and 2c, respectively. Meanwhile, the calculated log (1/ω p ), 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.

Effect of cellulose concentration in NMMO·H 2 O solution on the rheology-based results
The data in Table 1 clearly indicate that log (1/ω p ) 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/ω p ) and PDI are approximately equal when the cellulose concentrations are 12% and 11%.
This phenomenon originates from the relaxation time ( ) of molecular chains and the steadystate recoverable compliance (J e 0 ) in a polymer solution. With increasing concentration, the polymer chains are more strongly entangled with each other and get more and more www.intechopen.com   However, when the polymer concentration reaches a certain value, τ and J e 0 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 is a function of MW of the polymer [17,18], and J e 0 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·H 2 O solution has to be used in order to obtain reliable and stable data.

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].
where is the density, 0 the zero-shear viscosity, R the universal gas constant, M the molecular weight, T the temperature. ω char 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 (ω char ). Here, the ω char 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 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 = 2 RT/(6 0 ω .79 ) and the MWD scale curve is obtained with shifting the abscissa (log Mp -log (1/ω p )) units. Here, the Mp and (1/ω p ) 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]: where m 0 and m 1 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 m 0 and m 1 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·H 2 O 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, , 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.

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. 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.

Conclusions
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·H 2 O 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·H 2 O 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 is converted to the MW scale, obtaining MWD scale curves of cellulose.
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 analyze the MW scale and MWD of cellulose in the fiber industry. Materials are important to mankind because of the benefits that can be derived from the manipulation of their properties, for example electrical conductivity, dielectric constant, magnetization, optical transmittance, strength and toughness. Materials science is a broad field and can be considered to be an interdisciplinary area. Included within it are the studies of the structure and properties of any material, the creation of new types of materials, and the manipulation of a material's properties to suit the needs of a specific application. The contributors of the chapters in this book have various areas of expertise. therefore this book is interdisciplinary and is written for readers with backgrounds in physical science. The book consists of fourteen chapters that have been divided into four sections. Section one includes five chapters on advanced materials and processing. Section two includes two chapters on bio-materials which deal with the preparation and modification of new types of bio-materials. Section three consists of three chapters on nanomaterials, specifically the study of carbon nanotubes, nano-machining, and nanoparticles. Section four includes four chapters on optical materials.