Dilute Solution Polymers

The term Ico is composed of thermodynamic and frictional parameters for the polymer in the particular solvent conditions investigated. 

Polymers are not often monodisperse, and each different relaxation time will make a contribution to the observed average f. A popular method of obtaining the diffusion coefficient is to use the cumulants approach outlined by Koppel [27] and the algorithm of Pusey et al. [28] 

Generally only the first two cumulants can be extracted from the correlation functions with any confidence, and the z-average diffusion coeffi- 

Fignre 12.11 Dynamic Zimm plot for polystyrene in toluene. Reproduced with permission of the American Chemi Society from ref. [29] 

We noted earlier that each relaxation time will contribute to T and hence influence the shape of the correlation function. Consequently, all the information on polymer polydispersity is contained within the intensity correlation function because D is proportional to (molecular weight) The extraction of the molecu- 

Comparison with experimental data demonstrates that the bead-spring model allows one to describe correctly linear viscoelastic behaviour of dilute polymer solutions in wide range of frequencies (see Section 6.2.2), if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. The validity of the theory for non-linear region is restricted by the terms of the second power with respect to velocity gradient for non-steady-state flow and by the terms of the third order for steady-state flow due to approximations taken in Chapter 2, when relaxation modes of macromolecule were being determined. 

In Chap. 9 the following definition of the limiting viscosity number or intrinsic viscosity was given 

1) implies that the relationship between rjsp and c can be approximated by a linear proportionality as c approaches zero. At finite concentrations, however, the relationship between r]sp and c is certainly not linear. Therefore some extrapolation method is required to calculate [77] from viscosity measurements at a number of concentrations. 

The most popular extrapolation method was introduced by Huggins (1942)  

Another well-known extrapolation formula was proposed by Kraemer (1938)  

In fact, several authors used both extrapolation methods for the calculation of [77]. In several cases this led to identical values of [77]. Moreover, it was found that 

Einstein derived the following equation for the viscosity of suspensions of rigid, uncharged, spherical particles which do not interact with the suspension medium  

It may be assumed that under the action of a shear stress the polymer coil with the solvent enclosed by it behaves like an Einsteinian sphere, and hence, the viscosity of polymer solution should obey Eq. (3.172) for noninteracting spherical particles. Noninteraction of the polymer coils requires infinite dilution and this is achieved mathematically by defining a quantity called the intrinsic viscosity, [77], according to equation 

If V/j is the hydrodynamic volume of each polymer molecule (it is assumed that all polymer molecules are of the same molecular weight) then 

A more satisfactory form of this equation to satisfy the condition of noninteraction of polymer coils is given in terms of intrinsic viscosity, [77], defined by Eq. (3.173). Thus, 

This equation can be rearranged to give an equation for the hydrodynamic volume of an impermeable (nondraining) polymer molecule in infinitely dilute solution  

It follows from equation (100) that the coexistence curve is obtained as 

The critical values of / and 0 are obtained from d LFJkj T) d = 0 and combining with equation (135). The result is 

As discussed above, the incorporation of density fluctuations leads to corrections to the Flory-Huggins free energy. For example, when three-body interactions are significant, as in the case for temperatures below 0 and where phase separation occurs, it has been shown that and the coexistent polymer volume fraction 0 along the spinodal curve near the critical point obeys /) —These exponents of 1/3 and 1/9 are to be compared with the Flory-Huggins exponents of 1/2 and 1/4 respectively. The experimental data support the new exponents. 

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Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

The inverse scattering function of dilute polymer solutions for small scattering wavenumber qR- 1) obeys tire so-called Zimm relation [22] ... 

In dilute polymer solutions, hydrodynamic interactions lead to a concerted motion of tire whole polymer chain and tire surrounding solvent. The folded chains can essentially be considered as impenneable objects whose hydrodynamic radius is / / is tire gyration radius defined as... 

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] ... 

Use approximations developed for dilute polymer solutions. Note that the values for m + satisfy the requirement for dilution at 10" m but not at 10" m. 

Dilute Polymer Solutions. The measurement of dilute solution viscosities of polymers is widely used for polymer characterization. Very low concentrations reduce intermolecular interactions and allow measurement of polymer—solvent interactions. These measurements ate usually made in capillary viscometers, some of which have provisions for direct dilution of the polymer solution. The key viscosity parameter for polymer characterization is the limiting viscosity number or intrinsic viscosity, [Tj]. It is calculated by extrapolation of the viscosity number (reduced viscosity) or the logarithmic viscosity number (inherent viscosity) to zero concentration. 

Capillary viscometers are useful for measuring precise viscosities of a large number of fluids, ranging from dilute polymer solutions to polymer melts. Shear rates vary widely and depend on the instmments and the Hquid being studied. The shear rate at the capillary wall for a Newtonian fluid may be calculated from equation 18, where Q is the volumetric flow rate and r the radius of the capillary the shear stress at the wall is = r Ap/2L. 

Chisso-Asahi uses a spouted bed process for the production of their coated materials (12). A 12,000 t/yr faciHty is located in Japan. The semicontinuous process consists of two batch fluid-bed coaters. A dilute polymer solution is prepared by dissolving 5% polymer and release controlling agent into a chlorinated hydrocarbon solvent such as trichloroethylene. The solution is metered into the spouted bed where it is appHed to the fertilizer core. Hot air, used to fluidize the granules, evaporates the solvent which is recovered and reintroduced into the process. Mineral talc, when used, is either slurried into the polymer solution or introduced into the fluidizing air. 

Pressures substantially lower than true impact pressures are obtained with pitot tubes in turbulent flow of dilute polymer solutions [see Halliwell and Lewkowicz, Phys. Fluids, IS, 1617-1625 (1975)]. 

Osaki, K. Viscoelastic Properties of Dilute Polymer Solutions. Vol. 12, pp. 1 —64. 

Usually, dilute polymer solutions are isotropic systems, i.e. macromolecular chains can exist in these solutions independently of each other with a random distribution of orientations of the long axes of coils. The solutions of flexible-chain polymers remain isotropic when the solution concentration increases whereas in concentrated solutions of macromolecules of limited flexibility the chains can no longer be oriented arbitrarily and some direction of preferential orientations of macromolecular axes appears, i.e. the mutual orientations of the axes of neighboring molecules are correlated. This means that... 

The following qualitative picture emerges from these considerations in weak flow where the molecular coils are essentially undeformed, the polymer solution should behave approximately as a Newtonian fluid. In strong flow of a highly dilute polymer solution where the macroscopic velocity field can still be approximated by the Navier-Stokes equation, it should be expected, nevertheless, that in the immediate proximity of a chain, the fluid will be slowed down because of the energy intake to stretch the molecular coil thus, the local velocity field may deviate from the macroscopic description. In the general case of polymer flow,... 

The effect of polymer additives on turbulent flow is at the origin of the important phenomenon of drag reduction and has found other industrial applications such as oil recovery and antimisting action. Drag reduction in dilute polymer solutions... 

Wiest JM, Bird RB, Wedgewood LE (1988) On coil-stretch transitions in dilute polymer solutions, RRC Report no. 116, University of Wisconsin-Madison, May... 

Leal LG (1986) In Rabin Y (ed) Studies of flow-induced conformation changes in dilute polymer solutions, Proceedings of the 1985 La Jolla Institute Workshop, Academic International Press, New York, p 5... 

The number average molecular weight is required. This is obtained directly from measurements of a colligative property, such as the osmotic pressure, of dilute polymer solutions (see Chap. VII). It is often more convenient to establish an empirical correlation between the osmotic molecular weight and the dilute solution viscosity, i.e., the so-called intrinsic viscosity, and then to estimate molecular weights from measurements of the latter quantity on the products of polymerization. 

According to the theory of dilute polymer solutions, to be discussed in Chapter XII, the osmotic pressure may be expressed as a function of the concentration by an equation of the form ... 

Hence the theoretical configurational entropy of mixing AaSm cannot be compared in an unambiguous manner with the experimentally accessible quantity ASm- It should be noted that the various difficulties encountered, aside from those precipitated by the character of dilute polymer solutions, are not peculiar to polymer solutions but are about equally significant in the theory of solutions of simple molecules as well. 

The analogy with the virial expansion of PF for a real gas in powers of 1/F, where the excluded volume occupies an equivalent role, is obvious. If the gas molecules can be regarded as point particles which exert no forces on one another, u = 0, the second and higher virial coefficients (42, Azy etc.) vanish, and the gas behaves ideally. Similarly in the dilute polymer solutions when w = 0, i.e., at 1 = , Eqs. (70), (71), and (72) reduce to vanT Hoff s law... 

That the turbidities of dilute polymer solutions agree with the corresponding theoretical relationship, given by Eq. (VII-37), was shown in Figs. 47 and 48. 

The thermodynamic behavior of the dilute polymer solution depends on three factors (1) the molecular weight, (2) the thermodynamic interaction parameters and ki, or ipi and 0, which characterize the segment-solvent interaction, and (3) the configuration, or size, of the... 

In the dilute polymer solution approximation, which may not necessarily be appropriate if the concentration of electrolyte in the external solution is relatively large... 

Parameter occurring in thermodynamic relations for dilute polymer solutions (Chaps. XII and XIV). 

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