Theta solvent, polymers Functions

Staudinger showed that the intrinsic viscosity or LVN of a solution ([tj]) is related to the molecular weight of the polymer. The present form of this relationship was developed by Mark-Houwink (and is known as the Mark Houwink equation), in which the proportionality constant K is characteristic of the polymer and solvent, and the exponential a is a function of the shape of the polymer in a solution. For theta solvents, the value of a is 0.5. This value, which is actually a measure of the interaction of the solvent and polymer, increases as the coil expands, and the value is between 1.8 and 2.0 for rigid polymer chains extended to their full contour length and zero for spheres. When a is 1.0, the Mark Houwink equation (3.26) becomes the Staudinger viscosity equation. 

If c and dc/dx are known as a function of x and the measurement is carried out in a theta solvent, the molecular weight M of monodisperse polymers can now be calculated precisely. If the solvent is not a theta solvent, the obtained value of M is an apparent molecular weight from which the true value can be calculated upon plotting 1/M vs. c and extrapolation to c —> 0. For polydisperse samples, one has to insert the average of dc/dx in the above equation, and the thus calculated molecular weight represents a weight-average,... 

Leiva et al. [65] have reported for poly(itaconates) monolayers the surface behavior at the air - water interface at different surface concentrations. They have found that for these type of polymers, the air - water interface at 298 K, is a bad solvent, very close to the theta solvent. At the semidilute region concentration, the surface pressure variation was expressed in terms of the scaling laws as a power function of the surface concentration. According to equation (3.3), the log it vs log T plot shows a linear variation with slope 2 v/(2 u-1). 

According to the statistical-mechanical theory of rubber elasticity, it is possible to obtain the temperature coefficient of the unperturbed dimensions, d InsjdT, from measurements of elastic moduli as a function of temperature for lightly cross-linked amorphous networks [Volken-stein and Ptitsyn (258 ) Flory, Hoeve and Ciferri (103a)]. This possibility, which rests on the reasonable assumption that the chains in undiluted amorphous polymer have essentially their unperturbed mean dimensions [see Flory (5)j, has been realized experimentally for polyethylene, polyisobutylene, natural rubber and poly(dimethylsiloxane) [Ciferri, Hoeve and Flory (66") and Ciferri (66 )] and the results have been confirmed by observations of intrinsic viscosities in athermal (but not theta ) solvents for polyethylene and poly(dimethylsiloxane). In all these cases, the derivative d In sjdT is no greater than about 10-3 per degree, and is actually positive for natural rubber and for the siloxane polymer. 

Since solvent power is a function of temperature, there is for any solvent a temperature, 0, at which ideal behavior should be observed—the solvent is then referred to as the theta-solvent. As the solvent power is increased, an isotropic expansion of the polymer molecule occurs, and the root-mean-square displacement length is then given by... 

As z increases from 0 (theta conditions) to 2 (good solvent), the function also increases from 0 to about 0.2. For simplicity, dilute solutions are used to avoid the need for determination of higher-order virial coefficients. In this case, A2 provides a direct measure of the intermolecular interactions in polymer solutions and can be directly related to the respective Flory-Huggins parameter (x, vide infra, Sect. 2.6.1.1). However, since A3 oc A2M , Eq. 2.29 can be written as... 

Fig. 6.7. Intrinsic viscosity [q] as a function of the molar mass for different polymer-solvent systems. In addition to the experimental data, theoretical possible slopes of the [rj]-/M-relationships are shown for a better visualization. Data for poly(glutamic acid benzyl ester) (PGE) in trichloromethane at 25 C, poly(acrylamide) (PAAm) in H2O at 25 and poly(styrene) (PS) in c/s-decaline at 25 °C are taken from [77], poly(styrene) (PS) in the theta-solvent-cyclohexane at 34.5 X from [64], and hydroxyethyl starch (HES) in H2O at 25 C from [103]...

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. 

Problem 4.8 Compare the osmotic compressibility of a given polymer in the good solvent and in the theta solvent by drawing a sketch of n/Ifijeai as a function of b p in a double logarithmic scale. 

It should be emphasized that Figure 10 was calculated for an idealized freely jointed chain melt in which the site diameter and bond length are the same. In order to make quantitative contact with experiments, it is necessary to more faithfully represent the monomer architecture through the intramolecular functions Cl y(k). A model that captures more of the local chemical structure of real polymer chains is the well-known rotational isomeric state model. In order to mimic a chain in a theta solvent or a melt, intramolecular repulsions are included between sites separated by less than or equal to four bonds (the pentane effect ). ... 

The SANS experiments on PDMS in COj were designed to compare the chain dimensions with those in organic solvents and to test the prediction [7] that they will adopt ideal configurations, unperturbed by excluded volume effects, at a critical theta pressure XPo) as they do in polymer solutions at the theta temperature. The effect of temperature on the thermodynamic state of PDMS - COj solution is illustrated in Fig. 4. The concentration fluctuations diverge as the temperature approaches the phase boundaries, which are themselves a function of pressure. When the correlation length decreases to the theoretical value (0) Rg(0)/3 s 41 A, the condition is reached and this demonstrates that, unlike PS-AC, the solvent quality may be changed from the "poor to theta solvent by varying the temperature, as in PS-... 

Figure 19-11. Reduced viscosities as a function of the reduced shear stress of colloidal silica suspensions (diameter of 100 nm) in the presence of addedpolymer (polystyrene). The solvent used is decalin which is a near theta solvent for polystyrene. The size ratio of the polymer radius of gyration to the colloid radius (Rg/R) is 0.02S. The colloid volume fraction ((f>) is kept fixed at 0.4. In the absence of added polymer (Cp/c = 0), the particles behave as hard spheres and as more polymer is added to the system, the particles begin to feel an attraction. The colloid-polymer suspensions at (p of 0.4 shear thin between a zero rate viscosity of r o and a high shear rate plateau viscosity r]x,. The shear thinning behavior (in the absence and presence of polymer) is well captured by equation (19-10) with n = 1.4. Note rjo, rjao and cTc are functions of volume fraction and strengths of attraction but weakly dependent on range of attraction (Shah, 2003c Rueb, 1997).

As well as controlling chain dimensions, solvent quality affects the thermodynamics of dilute polymer solutions. This is because interactions between polymer chains are modified by the presence of solvent molecules. In particular, solvent molecules will change the excluded volume for a polymer coil, i.e. how much volume it takes up and prevents neighbouring chains from occupying. In a theta solvent, the excluded volume is zero (this holds for the excluded volume for a polymer segment or the whole coil). The solution is said to he ideal if the excluded volume vanishes. Deviations from ideality for polymer solutions are described in terms of a virial equation, just as deviations from ideal gas behaviour are. The virial equation for a polymer solution in terms of polymer concentration is given by Eq. (2.9). The second virial coefficient depends on interactions between pairs of molecules in particular it is proportional to the excluded volume. Therefore, in a theta solvent, = 0. If the solvent is good then Ai > 0, but if it is poor Ai < 0. If the solvent quality varies as a function of temperature and theta (0) conditions are attained, this occurs at the theta temperature. 

Figure 12.11 Viscosity of 925 kDa polybutadiene in 25°C (O) phenyloctane (good solvent) and ( ) dioctylphthalate (Theta solvent) as a function of polymer volume fraction, and fits to stretched exponentials and power laws in

Fig. 2. Chain stiffhess, a, as a function of polymer chain area, nm. ( ) theta solvent conditions (0) non-theta solvent conditions (A) deviant points. The solid line is a least-squares fit to alt but the deviant points. From ref 5.

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