Determination of Molecular Weight and Its Distribution of Rigid-Rod Polymers Determined by Phase-Modulated Flow Birefringence Technique

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1 Determination of Molecular Weight and Its Distribution of Rigid-Rod Polymers Determined by Phase-Modulated Flow Birefringence Technique YUM RAK OH, YOUNG SIL LEE, MOO HYUN KWON, O OK PARK Department of Chemical Engineering, Korea Advanced Institute of Science and Technology, Kusong-dong, Yusong-gu, Taejon , Korea Received 19 January 1999; revised 5 November 1999; accepted 12 November 1999 ABSTRACT: The phase-modulated flow birefringence (PMFB) method is widely accepted as one of the most sensitive and accurate techniques suitable for experimental tests on the molecular theory of polymer solutions. The objective of this study is to develop a systematic method to determine molecular weight and distribution of rigid-rod polymers by the PMFB technique. Using molecular theory for rigid polymers, birefringence n and orientation angle have been expressed as a function of molecular weight and distribution. n has been shown to be proportional to c i M 4 i, and cot 2 turned out to have a linear relationship with c i M 7 i / c i M 4 i. From the experimental results for PBLG solutions, birefringence and orientation angle data were in some degree matched with the theory presented John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: , 2000 Keywords: rigid polymer; molecular theory; phase-modulated flow birefringence; birefringence; orientation angle; poly( -benzyl-l-glutamate), molecular weight distribution INTRODUCTION It has been well known that molecular weight (MW) and molecular weight distribution (MWD) of polymeric materials have a considerable effect on the macroscopic properties. Adherence, toughness, tensile strength, brittleness, gas permeability, and environmental stress crack resistance are largely determined by MWD. 1 Polydispersity is particularly important to polymer processing problems such as extrusion and fiber spinning. For automobile tires, the MWD plays an important role in explaining fatigue, which occurs usually by periodically applied stresses. In general, Correspondence to: O Ok Park ( oopark@sorak. kaist.ac.kr) Journal of Polymer Science: Part B: Polymer Physics, Vol. 38, (2000) 2000 John Wiley & Sons, Inc. when MWD is narrow, the fatigue resistance is large, so the prediction of MW and MWD is essential to research over many areas of polymer science and engineering. There are various techniques for measuring number-average molecular weight, such as membrane osmometry and end group analysis, while light scattering, ultracentrifugation, etc., yield the weight-average molecular weight. But the measurement of MWD is somewhat restricted. One of the measurement techniques for MWD is gel permeation chromatography (GPC), which was proposed by Moor and developed by Dow Chemical Co., and the other is dynamic storage modulus which was proposed by Wu to measure MWD of insoluble polymers such as polymer melts. 2,3 There are some problems in applying GPC, one of the most popular techniques for MWD deter- 509

2 510 OH ET AL. mination. First of all, it takes a lot of time, and exchanging the columns to use the different solvents is difficult. Phase-modulated flow birefringence (PMFB) 4 provides simultaneous measurements of shape and size, average orientation, and rotational diffusivity of solute particles. In the previous article, 5 we have already examined the flow birefringence and orientation angle of polymer solutions with bimodal distributions. When the polymer solution is oriented by shear flow, their relaxation time is determined by particle length, i.e., the molecular weight. It has been quite successful for the bimodal distribution so far. But, to analyze the general distribution, i.e., multimodal case, we need a general theoretical model for data analysis. Here, new theoretical equations will be proposed and utilized for the analysis of experimental data to extract the molecular weight distribution. The PMFB method is used to analyze rheo-optical properties of the polymer solution, and a proposed theoretical model will be compared with the experimental results. PMFB method is simple, and is little influenced by solvent. Rheological behaviors of polymers in solution largely depend on their chain structures. In consideration of these respects we are going to propose two distinct molecular models for the rigid rods and flexible chains. Each model predicts the effect of the molecular weight distribution on the rheo-optical properties. The case of the rigid model is presented here, and the flexible model will be discussed later. One of the merits in this experiment is that it does not take too much time to examine and analyze the experimental data, and optical data obtained is reasonably correct and accurate. THEORY The relationship between refractive index and molecular polarizability for isotropic polymer solution can be explained well by 4 c 3 n2 1 n 2 2. (1) Here, n is the refractive index of polymer solution, c the number of molecules per unit volume of polymer solution, and denotes the molecular polarizability, which is an optical property that increases in proportion to the molecular size. Figure 1. A rigid rod-like polymer model having axisymmetric geometry in solution. 1 and 2 is the molecular polarizability in the direction of principal axis and the transversal axis, respectively. Consider an axisymmetric rigid rod-like polymer in solution subjected to the flow field shown in Figure 1. When the incident light is propagating along the x 3 -axis, the polarizability tensor, which has symmetric form, can be written as follows, ij a 0 0 a a E ij (2) where a,, and E ij can be given by a /3, 1 2, and E ij (3) 0 0 The refractive index tensor n ij can be represented in a similar way: n n ij 0 n 2 0 n a ij n 3 n 1 n 2 E ij (4) with Kronecker ij. Here, n a denotes (2n 2 n 1 )/3 and n 1, n 2 represent the refractive indices with respect to axes x 1 and x 2, respectively. To obtain the orientation angle for macromolecules in solution and the birefringence of the polymer solution, we apply the Lorentz Lorenz formula for each axis of the rod, and expand to a Taylor series on.

3 RIGID-ROD POLYMERS AND PMFB 511 n 2 1 8/3 c 1/2 a 1 4/3 c a 2/3 c 1, 1 8/3 c a 1 4/3 c a n 1 1 8/3 c 1/2 a 1 4/3 c a 4/3 c /3 c a 1 4/3 c a (5a) (5b) Then from eq. (5), the difference of refractive index, n 1 n 2, becomes n 1 n 2 2 n a c. (6) 9n a E ij obtained in a polymer axes frame is expressed into the laboratory frame using a rotation tensor. E ij 3u i u j ij (7) where u i is the unit vector defining the orientation of a rigid macromolecule in the laboratory frame. By using eqs. (5) and (7), we can obtain the average refractive index tensor for macromolecules with the distribution function (, ) and constant length. n ij n 11 n 12 n 21 n 22 n a 1 3 n 1 n 2 ij n 1 n 2 u i u j. (8) The angular brackets represent the average over an orientation distribution function (, ), which describes the probability of a particular orientation. The refractive index tensor can be decomposed into components parallel and perpendicular to the plane normal to the ray direction in a given system. Then the birefringence n, the difference in the eigenvalues of the real part of the refractive index tensor, and the orientation angle, defined as the angle between the principal axis of the refractive index and the laboratory frame, are given by eqs. (9) and (10) as follows. tan 2 2 u 1 u 2 u 1 u 1 u 2 u 2, (9) n n 1 n 2 u 1 u 1 u 2 u u 1 u 2 2 1/2. (10) The average refractive index tensor n ij for the polydisperse system in the rectangular coordinates can be obtained from eqs. (6) and (8). n ij 2 n a 27n a c k k ij 2 n a 9n a c k k u i u j k. (11) In the above equations, subscript k represents the k component of the macromolecules. Substitution of eq. (11) into eqs. (9) and (10) yields the expressions for tan2 and n for the polydisperse system. tan 2 2 c i i u 1 u 2 i c i i u 1 u 1 i c i i u 2 u 2 i, (12) n 2 n a 9n a c i i u 1 u 1 i u 2 u 2 i 2 4 c i i u 1 u 2 i 2 1/2. (13) The following assumptions are proposed to solve the above two expressions; (1) dilute solution without hydrodynamic interaction between the molecules, and (2) weak and simple shear flow field with dimensionless 1. Here, 6 and relaxation time is given by L 2 /12kT. and L denote the friction coefficient and length of the macromolecules respectively. k and T are Boltzmann s constant and absolute temperature. Because u l and u 2 in the rectangular coordinates can be replaced by sin, cos and sin sin in the spherical coordinates, the direction vectors u l and u 2 of the macromolecules can be expressed in spherical harmonics by u 1 u 1 u 2 u 2 1/3 P 2 cos 2, u 1 u 2 1/6 P 2 sin 2, (14a) (14b) where P 2 (1/2)(3 cos 2 1) is an associated Legendre polynomial. Using the orthogonality between the orientation distribution function and

4 512 OH ET AL. spherical harmonic function, eq. (14) can be solved into u 1 u 1 u 2 u 2 1/90 2 O 4, u 1 u 2 1/30 O 2. (15a) (15b) Substituting eq. (15) into eqs. (9) and (10) gives a theoretical relationship for the dependence of molecular weight distribution upon birefringence of the dilute polymer solution under the simple shear flow and orientation angle of the macromolecules. 2 c 4 il 1 i, (16) tan 2 12kT c i L i 7 n n a 270kTn a 2 2kT c il i /2 c i L 4 i. (17) EXPERIMENTAL Rheo-optical measurements were done by using the PMFB technique that has been reported in detail in a previous article of Oh and Park. 5 To identify the pertinence of theoretical models on rigid rod-like polymer, poly( -benzyl-l-glutamate) was used as the experiment material for optical measurements at room temperature. All samples, including the monodisperse and bimodal MWD solutions, were prepared by dissolving accurately weighed amounts. The preparation of sample solution for operating PMFB system has some restrictions. First, to minimize the light scattering from polymer particles, their size should be less than the wavelength of light source. Second, sample solution should be clear to neglect any dichroism. Finally, form birefringence can be easily neglected by decreasing the difference of refractive index between solute and solvent. PBLG in m- cresol was chosen as a standard sample of the rigid rod-like model because PBLG shows a rigid helix structure due to intermolecular hydrogen bonding. The m-cresol reveals the very high viscosity 20.8 centipoise at 25 C in comparison with other solvents. A flow cell of coaxial cylinder type was used. PBLG has been proposed to have a helical chain structure with side groups folding down along the backbone, resulting in a rigid conformation. At low concentrations PBLG solutions are isotropic, but with increasing the concentration PBLG rods align in parallel to form a liquid crystalline phase. The aspect ratio between the major and the minor axes can be easily determined, owing to a regular structure of the PBLG molecule, which has unit length of 0.15 nm. The PBLG used in this study was manufactured by Sigma Co. Four different molecular weights of 116,000 (PB4), 188,000 (PB3), 236,000 (PB2), and 300,000 (PB1) were examined, and samples with different volume fractions were prepared from 500 to 2000 ppm. In addition, to obtain the bimodal distribution, we prepared mixtures of samples with different molecular weights i.e., PB1 and PB2, named PBl2, PB1, and PB4 (PBl4), and PB2 and PB4 (PB24). All bimodal mixtures were prepared with equal weight ratios. Especially, in preparation of sample solutions, the errors of concentrations can cause a great error in analyzing the data. Thus, all the sample solutions were carefully weighed and prepared. RESULTS AND DISCUSSION The PMFB technique provides us with two optical properties of a polymer solution in which dissolved macromolecules are partially oriented by an applied shear field. The birefringence n reflects the extent of alignment, and the orientation angle represents the average direction of macromolecules. Our theoretical studies found that these two optical properties closely depend upon molecular weight and its distribution. In addition, theoretical results in this study show that the dependence of these optical properties, n and, on molecular weight or molecular weight distribution are affected by the chain flexibility of macromolecules in solution. In Figure 2(a), birefringence was plotted as a function of shear rate, and distinguished by the weight concentration C of PB1. Birefringence data are well superposed into a master curve, independent of concentration, and increase linearly with shear rate in the low shear rate region. If cl 14 in eq. (17) is negligible compared to cl 8, eq. (17) can be simplified to n n a 270kTn a c i L 4 i. (18)

5 RIGID-ROD POLYMERS AND PMFB 513 n and C is maintained even though n was rationalized by weight concentration instead of number concentration. Over the range of concentrations employed in the experiments we finally obtain n C. (20) Now let us consider the effect of concentration of the polymer solution on the orientation angle. For a dilute solution without hydrodynamic interactions between the macromolecules, the concentration of the solution does not affect the orientation angle of a rigid rod. For the monodisperse polymer system with constant molecular weight, Figure 2(b) shows that cot 2 is superposed into a master curve independent of concentration, and increases linearly with the shear rate. From Figure 2(b), one can qualitatively find the effect of cot 2 expressed as a form of eq. (19a); cot 2 A 2 c 7 c i M 4 (21) i Figure 2. (a) Steady-state flow birefringence, and (b) orientation angle of poly( -benzyl-l-glutamate) with molecular weight of 300,000 in m-cresol solution (PB1). Optical measurements were performed on four different concentrations (F 500 ppm, ƒ 1000 ppm, 1500 ppm, and 2000 ppm) as a function of shear rate up to the strength of 200/s, and measured birefringence data were all reduced by corresponding weight concentration C of the solution. Because PBLG is a rigid macromolecule with linear chains, the chain length can be replaced by the molecular weight, from which the following expression can be obtained; n A 1 c i M i 4 (19) with A 1 denoting a proportionality constant. Because Figure 2 was obtained for PBLG of a single molecular weight, the linear relationship between with A 2 denoting a proportionality constant. From Figure 2, together with eqs. (19) and (21), we can find A 1 and A 2 for PB1, and we calculate c i M i 4 and ( c i M i 7 )/( c i M i 4 ) for subsequent samples using the same values A 1 and A 2. Experimental results compared with calculated ones in Figures 3 and 4 are in good accordance. Thus, we define other average molecular weights as and M 4 M n M 7 c 4 c i M i c 7 1/6 c i M i 1/3, (22) (23) c n 1/ n 1 c i M i. (24) The data of M 7 calculated from Figures 3 and 4 and shown in Figure 5 also are in a good agreement with calculated values, which were directly obtained from the number concentrations and molecular weights for each sample. Thus, we can say that a new polydispersity index (PDI) can be

6 514 OH ET AL. Figure 3. Comparison of experimental data of ( c i M i 4 / c i M i ) 1/3 with calculated values for four different concentrations of PBLG solution with several molecular weights and distributions. Four different molecular weights of 300,000 (PB1), 236,000 (PB2), 188,000 (PB3), and 116,000 (PB4), and three bimodal mixtures of equal weight ratios of PB12, PB14, PB2, and PB24 were experimented. accurately determined by birefringence and orientation angle as PDI M 7 M. (25) 4 Figure 5. Comparison of experimental data of ( c i M i 7 / c i M i ) 1/6 with calculated values for the same PBLG solutions as in Figure 3. Figure 6 shows ratios of experimental PDI to the calculated ones. For monodisperse systems, PDI should have a value of 1, independent of molecular weight. PBLG samples used in the experiments have quite narrow polydispersity less than about As shown in the figure, experimental PDI data have somewhat larger values, up to 7% higher, than the calculated ones, which is mainly due to the polydispersity of PBLG samples. Finally, it is necessary to mention the flexibility of PBLG samples. Ookubo et al. 9 reported that Figure 4. Comparison of experimental data of [( c i M i 7 / c i M i )/ c i M i 4 / c i M i )] 1/3 with calculated values for the same PBLG solutions as in Figure 3. Figure 6. Ratio of experimental PDI data to calculated values for four different concentrations of PBLG solutions.

7 RIGID-ROD POLYMERS AND PMFB 515 PBLG above the molecular weight of 10,000 has a little flexibility at the dynamic frequency mode. Warren et al. 10 reported that intrinsic moduli of PBLG solutions with MW from to agree well with the theory for the rigid rods. Thus, we considered PBLG samples in shear flows up to 200 s 1 to behave as as rigid rod. CONCLUSION The effects of molecular weight and molecular weight distribution on the birefringence and the orientation angle for rigid rod polymers were investigated by theory and experiments. From the experimental results for PBLG in which the molecular chain is rigid enough to form mesophase, it was found that n increased almost linearly with increasing the shear rate and concentrations over the range from 500 to 2000 ppm. Furthermore, taking into account the effect of molecular weight and its distribution on n and cot 2, we obtained n c i M i 4, while cot 2 is roughly proportional to c i M i 7 / c i M i 4. We defined a new PDI(M 7/M 4) to characterize the broadness of the molecular weight distribution. From the experimental results for PBLG solutions, birefringence and orientation angle data were in some degree matched with the theory presented. It may be concluded that the rigid-rod model can be readily applicable for determination of MWD by PMFB data. REFERENCES AND NOTES 1. Billmeyer, R. W.; Martin, J. R.; Johnsons, J. F. Polym Eng Sci 1982, 22, Wu, S. Polym Eng Sci 1985, 25, McGrory, W. J.; Tuminello, W. H. J Rheol 1990, 34, Frattini, P. L.; Fuller, G. G. J Rheol 1984, 28, Oh, Y. R.; Park, O. O. J Chem Eng Jpn 1992, 25, Bird, R. B.; Curtiss, C. F.; Armstrong, R. C.; Hassagar, O. Dynamics of Polymeric Liquids; John Wiley & Sons: New York, Van Krevelen, D. W. Properties of Polymers; Elsevier: Armsterdam, Blumstein, A. Liquid Crystalline Order in Polymers; Academic Press: New York, Ookubo, N.; Komatsubara, M.; Nakajima, H.; Wada, Y. Biopolymers 1976, 15, Warren, T. C.; Schrag, J. L.; Ferry, J. D. Biopolymers 1973, 12, 1905.