Intrinsic viscosity

Viscosity measurements of the polymer solutions are carried out with an Ubbelodhe viscometer. The viscosities are measured in dilute aqueous solution in neutral pH. The time of flow for solutions is measured at four different concentrations (i.e. 0.1%, 0.05%, 0.025% and 0.0125%). From the time of flow of polymer solutions (t) and that of the solvent (t, for distilled water), relative viscosity(Ti, ), specific viscosity (tj p), reduced viscosity (t ) and inherent viscosity(Tij jj) are obtained by the following relations  

Intrinsic viscosity [t]] of a polymer solution is related to its viscosity average molecular weight by the Mark-Houwink-Sakurada relationship  

Elemental Analysis (C, H, N, O S) helps us understand the actual extent of grafting. From the synthesis details (Table 5.1) we get the percentage grafting. Now from percentage grafting and from the known elemental composition of the polysaccharide and that of the graft polymer, we can calculate the theoretical elemental composition, which we can simply compare with that actually obtained from the elemental analyzer. 

Furthermore, since most polysaccharide doesn t contain nitrogen, and if the grafted chain is polyacrylamide, then we can confirm grafting simply by observing the presence of nitrogen in the grafted product (and its absence in the polysaccharide) as reported in the case of GG-g-PAM (Table 5.2) [49]. 

If the grafted chain is polyacrylic acid or polymethyl methacrylate, then we look for an increase in the percentage of oxygen as proof of grafting. 

The intrinsic viscosity [17] is a quantity characteristic of a polymer. It represents an increase in the solution viscosity when the concentration is raised to a certain level. As expected, a polymer molecule with a greater dimension has a larger [77]. 

Experimentally, it is expressed by Mark-Houwink-Sakurada equation  

The classical method to determine and a of a given polymer is as follows. First, prepare fractions of different molecular weights either by synthesis or by fractionation. Next, make dilute solutions of different concentrations for each fraction. Measure the viscosity of each solution, plot the reduced viscosity as a function of polymer concentration, and estimate [17] for each fraction. Plot [17] as a function of the molecular weight in a double logarithmic scale. This method has been extensively used to characterize polymer samples because the exponent a provides a measure of the chain rigidity. Values of a are listed in Table 3.2 for some typical shapes and conformations of the polymer. The value of a is around 0.7-0.8 for flexible chains in the good solvent and exceeds 1 for rigid chains. In the theta solvent, the flexible chain has a = 0.5. 

Apparently, a is greater for a more extended conformation. It is reasonable because a polymer molecule with a greater dimension for a given contour length will experience a greater friction to move in the solvent. We will obtain the formulas of [77] for linear flexible chains in the theta solvent and the good solvent in the next section. 

It is an experimental fact that the viscosity of a polymer solution is generally much larger than that of the solvent alone even at low polymer concentrations, and it increases with increasing molecular weight at a fixed mass concentration. The measurement of solution viscosity can, therefore, be used to estimate polymer molecular weights. Indeed, a large number of sophisticated viscometers exist for the accurate measurement of solution viscosity and its variation with concentration, shear rate, and temperature. Details of these instruments and methods of data analysis are discussed at length in Chapter 14. For molecular- 

If the radius of the capillary is R, its length is L, and the viscosity of the solution (assumed Newtonian) is t], then according to the well-known Hagen-Poiseuille equation [22], the volumetric flow rate Q through the capillary is given by 

If we consider each polymer molecule in dilute solution to be an isolated random coil of spherical shape and volume v, we may apply Eq. (8.6.4) with given by Eq. (8.6.3) and the volume fraction of polymer by 

Multiplying and dividing the right-hand side of Eq. (8.6.5) by MN, the product of the polymer molecular weight and Avogadro s number, yields 

The viscosity number ought to be independent of polymer concentration. However, the Einstein equation is vahd only for noninteracting spheres this simation prevails as the concentration tends to zero. Consequently, we can extrapolate data to infinite dilution, and the result is known as the intrinsic viscosity or limiting viscosity number [t]]. In the past, this quantity was measured in units of deciliters per gram recent practice has been to use miUihters per gram. Data can generally be represented in terms of the Huggins equation. 

In macromolecular chemistry, the relative viscosity rj, is often measured. The relative viscosity is the ratio of the viscosity of the solution to that of the solvent  

The specific viscosity risp is obtained from the relative viscosity by subtracting one unit from ri,  

The plot of Psp/c versus c or (1/c) In versus c often gives a straight line, the intercept of which is [p]. Huggins (1942) showed that the slope is 

For molecules of high intrinsic viscosity a correction must be made for the effect of the rate of shear strain. For relatively low intrinsic viscosity, the rate of shear strain does not have any appreciable effect 

Each polymer coil in a solution contributes to viscosity. In very dilute solutions, the contribution from different coils is additive and solution viscosity r] increases above the solvent viscosity t/s linearly with polymer concentration c. The effective virial expansion for viscosity at low concentration is of the same form as Eq. (1.76) for osmotic pressure and Eq. (1.96) for light scattering  

The term that is linear in concentration contains the intrinsic viscosity [r]] and the quadratic term includes the Huggins coefficient h, which plays the role of the second virial coefficient for viscosity. Intrinsic viscosity [rj is the initial slope of a plot of relative viscosity r]/r]s against concentration. Since 

Intrinsic viscosity is related to the linear size of the coil R and the molar mass M by the Fox-Flory equation  

This equation will be derived in Chapter 8. Dilute solution viscosity measurements are important characterization tools for polymers because 

K and a are tabulated for nearly all linear polymers in various solvents, 

The quantity of interest is the excess of t] over the viscosity of the pure solvent, T]s- It was Einstein, who demonstrated that this excess can be directly interpreted when dealing with a suspension of spheres. He derived the following power series 

0 denotes the volume fraction occupied by spheres and 7 is a numerical factor (7 == 2.5). The result is remarkably simple, as it implies that the extra viscosity is only dependent on the volume fraction of the spheres, irrespective of whether there are many small spheres or larger spheres in smaller numbers. 

Equation (6.156) can be applied to a solution of polymers since the macromolecules, being hydrodynamically impermeable, behave like hard spheres with volumes as given by the hydrodynamic radius. Detailed theoretical treatments suggest a minor correction because it is found that the hydrodynamic radius to be used in viscosity measurements differs slightly from that applied in the representation of the diffusion coeflicient. While the latter is given by Eq. (6.148), viscosity measurements have to be based on the relation 

In works on polymer solutions a particular quantity has been introduced in order to specify the extra viscosity. It is called intrinsic viscosity , denoted [77], and defined as 

Equation (6164) is known as the Mark-Houwink-Sakurada relation . It generally holds very well, as is also exemplified by the data obtained for two different solutions of poly(isobutylene) presented in Fig. 6.17. 

The oldest, simplest and most widely used method for obtaining information about the molecular weight of a polymer is based on the measurement of the viscosity of dilute solutions. We will see that this quantity is less sensitive to molecular weight than the zero-shear viscosity of the melt. However, the apparatus required is much simpler and can be used in combination with GPC to determine the molecular weight distribution. Furthermore, it is often impossible using a commercial rheometer to determine the zero-shear viscosity of a melt. 

Several quantities are used to describe the low-shear-rate limiting viscosity of a solution tj in terms of the viscosity of the solvent, rj, and the concentration of polymer, c. These are defined as follows. 

The units for c (concentration) in all these definitions are g/cm, and those for [77] are thus cmVg. 

Because it is evaluated in the limit of infinite dilution, the intrinsic viscosity provides information about the average size of molecules in a solution in which there is no interaction between molecules. In practice, for a linear, monodisperse polymer, the relationship used to calculate the molecular weight from the intrinsic viscosity is the one proposed by Mark [40], Houwink [41] and Sakurada [42] and given here as Eq. 2.79. 

Although the viscosity and fluorescence results discussed above were obtained in dilute solution, the polymer concentration was 10 unit moll (i.e about 0.1%), they can also reflect some contribution of coil/coil interactions. Consequently, a more rigorous approach should be the study of these systems at the infinite dilution limit, for example, by measuring the intrinsic viscosity, [ ], instead of the specific one. However, such a study presents two main drawbacks. One is that intrinsic viscosity measurements cannot be given at very low salt concentrations, in our case lower than 0.2% NaCl. The other is that these measurements are very time consuming, at least five measurements must be done for each sample at each salt concentration. 

To check if there is any significant difference between the results obtained by measuring the specific or the intrinsic viscosity, we performed some intrinsic 

A very different behavior was observed with the sample 3-C8. In 10% NaCl solution the intrinsic viscosity of this sample exceeds that of the precursor by 30% (Fig. 4.5 and Table 4.1) in a similar way as the specific viscosity does (Fig. 4.2), and k takes a value typical of nonassociating systems (0.48). Obviously the hydrophobic interactions between octyl groups are not strong enough to induce hydrophobic aggregation. 

Until now the solution properties of the copolymer coil were discussed only in terms of electrostatic repulsions between acrylate units and hydrophobic 

It has been shown that the expansion of a copolymer coil depends largely on the number of heterocontacts, i.e. contacts between unlike monomers [26, 27]. The larger the number of heterocontacts the higher the expansion of the copolymer coil. As a consequence, the intrinsic viscosity of the copolymer is higher than that of the equivalent homopolymers. The importance of this effect increases as the solvent quality decreases [27]. 

Both the colligative and the scattering methods result in absolute molecular weights that is, the molecular weight can be calculated directly from first principles based on theory. Frequently these methods are slow, and sometimes expensive. In order to handle large numbers of samples, especially on a routine basis, rapid, inexpensive methods are required. This need is fulfilled by intrinsic viscosity and by gel permeation chromatography. The latter is discussed in the next section. 

Of course, the relative viscosity is a quantity larger than unity. The specific viscosity is the relative viscosity minus one  

The specific viscosity, divided by the concentration and extrapolated to zero concentration, yields the intrinsic viscosity  

For dilute solutions, where the relative viscosity is just over unity, the following algebraic expansion is useful  

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Anotlier simple way to obtain the molecular weight consists of measuring tire viscosity of a dilute polymer solution. The intrinsic viscosity [q] is defined as tire excess viscosity of tire solution compared to tliat of tire pure solvent at tire vanishing weight concentration of tire polymer [40] ... 

The Mark-Houwink-Sakurada equation relates tire intrinsic viscosity to tire polymer weight ... 

The intrinsic viscosity of a solution of particles shaped like ellipsoids of revolution is given by the expression... 

Simha equation), where a/b is the length/diameter ratio of these cigarshaped particles. Doty et al.t measure the intrinsic viscosity of poly(7-benzyl glutamate) in a chloroform-formamide solution and obtained (approximately) the following results ... 

At first glance, the contents of Chap. 9 read like a catchall for unrelated topics. In it we examine the intrinsic viscosity of polymer solutions, the diffusion coefficient, the sedimentation coefficient, sedimentation equilibrium, and gel permeation chromatography. While all of these techniques can be related in one way or another to the molecular weight of the polymer, the more fundamental unifying principle which connects these topics is their common dependence on the spatial extension of the molecules. The radius of gyration is the parameter of interest in this context, and the intrinsic viscosity in particular can be interpreted to give a value for this important quantity. The experimental techniques discussed in Chap. 9 have been used extensively in the study of biopolymers. 

This chapter contains one of the more diverse assortments of topics of any chapter in the volume. In it we discuss the viscosity of polymer solutions, especially the intrinsic viscosity the diffusion and sedimentation behavior of polymers, including the equilibrium between the two and the analysis of polymers by gel permeation chromatography (GPC). At first glance these seem to be rather unrelated topics, but features they all share are a dependence on the spatial extension of the molecules in solution and applicability to molecular weight determination. 

Both the intrinsic viscosity and GPC behavior of random coils are related to the radius of gyration as the appropriate size parameter. We shall see how the radius of gyration can be determined from solution viscosity data for these... 

The intrinsic viscosity Vred- Equation (9.14) shows that 77 ... 

With this terminology in mind, we can restate the objective of this section as the interpretation of the intrinsic viscosities of solutions of rigid molecules. If the solute molecules are known to be spherical, comparison of Eqs. (9.10) and (9.14) shows that the intrinsic viscosity for such systems is given by... 

The serum albumin molecule is known to have an approximately spherical shape (see Example 8.7) and is foundf to have an intrinsic viscosity in aqueous buffer solutions of 3.7 cm g". Using p = 1-34 g cm as the density of the... 

Based on these ideas, the intrinsic viscosity (in 0 concentration units) has been evaluated for ellipsoids of revolution. Figure 9.3 shows [77] versus a/b for oblate and prolate ellipsoids according to the Simha theory. Note that the intrinsic viscosity of serum albumin from Example 9.1-3.7(1.34) = 4.96 in volume fraction units-is also consistent with, say, a nonsolvated oblate ellipsoid of axial ratio about 5. 

For solutions of rigid particles, then, the intrinsic viscosity exceeds 2.5 as a result of some combination of the following effects ... 

Figure 9.3 Intrinsic viscosity according to the Simha theory in terms of the axial ratio for prolate and oblate ellipsoids of revolution.

It is a frustrating aspect of Eq. (9.20) that the observed intrinsic viscosities contain the effects of ellipticity and solvation such that the two cannot be resolved by viscosity experiments alone. That is, for any value of [77], there is a whole array of solvation-ellipticity values which are consistent with the observed intrinsic viscosity. 

In all of these derivations concerning rigid bodies, no other walls are considered except the particle surfaces. Before we turn to the question of the intrinsic viscosity of flexible polymers, let us consider the relationship between the viscosity of a fluid and the geometry and dimensions of the container in which it is measured. 

Experiments based on the Poiseuille equation make intrinsic viscosity an easily measured parameter to characterize a polymer. In the next section we consider how this property can be related to the molecular weight of a polymer. 

Table 9.3 lists the intrinsic viscosity for a number of poly(caprolactam) samples of different molecular weight. The M values listed are number average figures based on both end group analysis and osmotic pressure experiments. Tlie values of [r ] were measured in w-cresol at 25°C. In the following example we consider the evaluation of the Mark-Houwink coefficients from these data. 

Table 9.3 Intrinsic Viscosity as a Function of Molecular Weight for Samples of Poly(caprolactam) ...

Since viscometer drainage times are typically on the order of a few hundred seconds, intrinsic viscosity experiments provide a rapid method for evaluating the molecular weight of a polymer. A limitation of the method is that the Mark-Houwink coefficients must be established for the particular system under consideration by calibration with samples of known molecular weight. The speed with which intrinsic viscosity determinations can be made offsets the need for prior calibration, especially when a particular polymer is going to be characterized routinely by this method. 

Hven fractionated polymer samples are generally polydisperse, which means that the molecular weight determined from intrinsic viscosity experiments is an average value. The average obtained is the viscosity average as defined by Eqs. (1.20) and (2.40) as seen by the following argument ... 

The experimental (subscript ex) intrinsic viscosity is proportional to some average the nature of which we seek to verify value of M raised to the power a, according to Eq. (9.34),... 

This concludes our discussion of the viscosity of polymer solutions per se, although various aspects of the viscous resistance to particle motion continue to appear in the remainder of the chapter. We began this chapter by discussing the intrinsic viscosity and the friction factor for rigid spheres. Now that we have developed the intrinsic viscosity well beyond that first introduction, we shall do the same (more or less) for the friction factor. We turn to this in the next section, considering the relationship between the friction factor and diffusion. 

All that can be concluded from the data given in the preceding example is that the particle is not an unsolvated sphere. However, when an appropriate display of contours is examined for f/fo (e.g.. Ref. 2), the latter is found to be consistent with an unsolvated particle of axial ratio about 4 1 or with a spherical particle hydrated to the extent of about 0.48 g water (g polymer). Of course, there are a number of combinations of these variables which are also possible, and some additional experimental data—such as the intrinsic viscosity—are needed to select that combination which is consistent with all experimental observations. 

The intrinsic viscosity of poly(7-benzyl-L-glutamate) (Mq = 219) shows such a strong molecular weight dependence in dimethyl formamide that the polymer was suspected to exist as a helix which approximates a prolate ellipsoid of revolution in its hydrodynamic behaviorf ... 

Fox and Floryf used experimental molecular weights, intrinsic viscosities, and rms end-to-end distances from light scattering to evaluate the constant in Eq. (9.55). For polystyrene in the solvents and at the temperatures noted, the following results were assembled ... 

The intrinsic viscosity of polystyrene in benzene at 25°C was measuredf for polymers with the following molecular weights ... 

Mandelkern and Floryt have assembled the sedimentation coefficients and intrinsic viscosities for polymers of various molecular weights. As shown... 

The polymers dissolve in l,l,l,3,3,3-hexafluoro-2-propanol [920-66-1/, hot phenols, and /V, /V- dim ethyl form am i de [68-12-2] near its boiling point. The excellent solvent resistance notwithstanding, solvents suitable for measurement of intrinsic viscosity, useflil for estimation of molecular weight, are known (13,15). 

Solution Polymers. Acryflc solution polymers are usually characterized by their composition, solids content, viscosity, molecular weight, glass-transition temperature, and solvent. The compositions of acryflc polymers are most readily determined by physicochemical methods such as spectroscopy, pyrolytic gas—liquid chromatography, and refractive index measurements (97,158). The solids content of acryflc polymers is determined by dilution followed by solvent evaporation to constant weight. Viscosities are most conveniently determined with a Brookfield viscometer, molecular weight by intrinsic viscosity (158), and glass-transition temperature by calorimetry. 

SAN resins show considerable resistance to solvents and are insoluble in carbon tetrachloride, ethyl alcohol, gasoline, and hydrocarbon solvents. They are swelled by solvents such as ben2ene, ether, and toluene. Polar solvents such as acetone, chloroform, dioxane, methyl ethyl ketone, and pyridine will dissolve SAN (14). The interactions of various solvents and SAN copolymers containing up to 52% acrylonitrile have been studied along with their thermodynamic parameters, ie, the second virial coefficient, free-energy parameter, expansion factor, and intrinsic viscosity (15). 

The molecular weight of SAN can be easily determined by either intrinsic viscosity or size-exclusion chromatography (sec). Relationships for both multipoint and single point viscosity methods are available (18,19). Two intrinsic viscosity and molecular weight relationships for azeotropic copolymers have been given (20,21) ... 

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