The Hydrodynamic Volume

By definition, M[t ], canceling to centimeters cubed per mole, is the hydro-dynamic volume (Barth, 1986). M d (Glasstone and Lewis, I960) has identical dimensions  

If log[iq]M for each eluted fraction of a standard polysaccharide is plotted against its corresponding vd, a universal graph circumscribed by log [ q]M standard vs wel standard may be drawn that fits unknown polysaccharide fractions with identical vel. An unknown M may be computed by substitution in the equation 

The procedure is as follows a standard graph of [r)]AT standard vs veA is constructed, [Tq]uriknown is measured, and [T]]M, .tandard corresponding to the unknown vei is read from the graph. Inserting the data in Eq. (4.49), 

By another definition, the hydrodynamic volume is the volume of a single particle in solution (Tanford, 1961) or by invoking the unsolvated, spherical molecular model, (4/3)pi(Vr )3 cm3. The total volume of particles in 1 g of a dispersed solute is (4/3)pi(Vr )3 cm3(N/M). According to 

Seymour and Carraher (1981), (Vr )3 cm3 is the effective hydrodynamic volume. In terms of the specific volume, 

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An interesting outgrowth of these considerations is the idea that In r versus K or Vj should describe a universal calibration curve in a particular column for random coil polymers. This conclusion is justified by examining Eq. (9.55), in which the product [i ]M is seen to be proportional to (rg ), with r = a(rg 0 ) - This suggests that In rg in the theoretical calibration curve can be replaced by ln[r ]M. The product [r ]M is called the hydrodynamic volume, and Fig. 9.17 shows that the calibration curves for a variety of polymer types merge into a single curve when the product [r ]M, rather than M alone, is used as the basis for the cafibration. 

The presence of inorganic salts in solutions of poly(ethylene oxide) also can reduce the hydrodynamic volume of the polymer, with attendant reduction in intrinsic viscosity this effect is shown in Figure 7. 

Calibration curves for PS and PMMA are shown in Figs. 15.3-15.5. The slight differences in courses of calibration curves for PS in THF, chloroform, and toluene, as well as the curve for PMMA in THF (Fig. 15.3), can be explained by the flow rate variations for different pumping systems and by the hydrodynamic volume effects, respectively. The calibration curves for PMMA in mixed eluents THF/toluene are shown in Fig. 15.4. Three percent of THF in toluene assured a reasonable SEC elution of PMMA. However, more chloroform was needed to obtain a good SEC elution of PMMA in mixed eluent chloroform/toluene... 

Solvent can affect separation in two different ways. Because water is a better solvent for these four columns than water/methanol, based on the swelling or void volume of the columns in Table 17.9, the separation should be better in water than in water/methanol. The relative viscosity of a 0.5% PEO standard from Aldrich (Lot No. 0021kz, MW 100,000) in water and in water/methanol with 0.1 M lithium nitrate is 1.645 and 1.713, respectively. This indicates that the hydrodynamic volume of PEO in water is smaller than in water/methanol. The difference in hydrodynamic volume between two PEO standards should also be larger in water/methanol than in water. Hence, the separation for PEO should be better in water/methanol than in water. The results in Table 17.8 indicate that separation efficiency is better in water than in water/methanol... 

Several factors can contribute to the difference in retention times for PEO in different mobile phases the viscosity of a mobile phase, the hydrodynamic volume of a PEO, and the swelling or void volume of a column. Shodex and TSK columns should swell more in water than in water/methanol, and PEO should therefore come out later in water than in water/methanol. PEO should also elute later in water than in water/methanol because water/methanol is a better solvent for PEO than water. The viscosity of the 50 50 water/methanol mobile phase is higher than the viscosity of water. PEO should therefore elute later in water/methanol than in water due to the difference in viscosity. The results in Table 17.9 indicate that the difference in retention time for PEO in water and in water/methanol depends more on the swelling of columns and the hydrodynamic volumes of PEO than the viscosities of mobile phases. 

Chain degradation in turbulent flow has been frequently reported in conjunction with drag reduction and in simple shear flow at high Reynolds numbers [187], Using poly(decyl methacrylate) under conditions of turbulent flow in a capillary tube, Muller and Klein observed that the hydrodynamic volume, [r ] M, is the determining factor for the degradation rate in various solvents and at various polymer concentrations [188], The initial MWD of the polymers used in their experiments are, however, too broad (Mw/Iiln = 5 ) to allow for a precise... 

Interactions with xanthan were investigated for some GAX fractions of wheat bran [109]. Whereas, for lowly substituted GaMs a synergy in viscosity was observed at low total polymer concentrations, yielding a maximum of the relative viscosity at nearly equal proportions of both polysaccharides [124], the xanthan/xylan mixtures at the same experimental conditions showed no synergy. The observed decrease in the relative viscosity values upon addition of the xylan indicates that a certain interaction with xanthan takes place, but that it leads to a contraction in the hydrodynamic volume. The authors suggested that structural and conformational differences between GaM and GAX might be the reason for this observation. 

Yet as long ago as 1966 the problem of calibration in GPC was solved. In that year, Benoit and his co-workers recognised that GPC separates on the basis of the hydrodynamic volume of the polymer molecules in solution. The intrinsic viscosity [rj] is related to the hydrodynamic volume, V, by the equation ... 

Advanced computational models are also developed to understand the formation of polymer microstructure and polymer morphology. Nonuniform compositional distribution in olefin copolymers can affect the chain solubility of highly crystalline polymers. When such compositional nonuniformity is present, hydrodynamic volume distribution measured by size exclusion chromatography does not match the exact copolymer molecular weight distribution. Therefore, it is necessary to calculate the hydrodynamic volume distribution from a copolymer kinetic model and to relate it to the copolymer molecular weight distribution. The finite molecular weight moment techniques that were developed for free radical homo- and co-polymerization processes can be used for such calculations [1,14,15]. 

Hyperbranched poly(ethyl methacrylate)s prepared by the photo-initiated radical polymerization of the inimer 13 were characterized by GPC with a lightscattering detector [51]. The hydrodynamic volume and radius of gyration (i g) of the resulting hyperbranched polymers were determined by DLS and SAXS, respectively. The ratios of Rg/R are in the range of 0.75-0.84, which are comparable to the value of hard spheres (0.775) and significantly lower than that of the linear unperturbed polymer coils (1.25-1.37). The compact nature of the hyperbranched poly(ethyl methacrylate)s is demonstrated by solution properties which are different from those of the linear analogs. 

As a result, the branches in the G TPPH2 family are more flexible, thus causing a decrease in the hydrodynamic volume compared to the theoretical fully extended structure in the gas phase. 

The hydrodynamic volume separation mechanism of SEC, along with the different molecular size/weight relationships of branched and linear polymers of identical chemical composition, can be exploited with the SEC/LALLS method to gain information about polymer branching. In the studies described in this paper both conventional SEC and SEC/LALLS are used to obtain data about branching in samples of poly(vinyl acetate) (PVA) and polychloro-prene (PCP). 

With use of a THF/n-Heptane mixture for example, in GPC 2 the hydrodynamic volume of polymer mola ules rich in one monomer component can be much more affected than those rich in the other. Steric exclusion paration can then distinguiA compsitional differences. 

Figure 19 shows the Universal Calibration Curve obtained for the coupled GPCs in terms of the hydrodynamic volume in THF (Ref Table II). The addition of the n-Heptane caused a dramatic shift downstream of the polystyrene standards. 

In the absence of adsorption, inclusion, or exclusion, a polymer is fractionated on a GPC column according to the hydrodynamic volume.40138 The hydrodynamic volume is a function of monomer identity, as well as polymer molecular weight, branching, and cross-linking. The polymer chains in any given chromatographic fraction have roughly the same hydrodynamic volume. 

The need for a viscosimeter capable of online measurement of the intrinsic viscosity of a polymer sample arose when it was shown that polymer molecules elute in order of decreasing hydrodynamic volume.81 The hydrodynamic volume can be related to the product of intrinsic viscosity and molecular weight. 

Electrostatic repulsion of the anionic carboxylate groups elongates the polymer chain of partially hydrolyzed polyacrylamides increasing the hydrodynamic volume and solution viscosity. The extensional viscosity is responsible for increased resistance to flow at rapid flow rates in high permeability zones (313). The screen factor is primarily a measure of the extensional (elonga-tional) viscosity (314). The solution properties of polyacrylamides have been studied as a function of NaCl concentra-tion and the parameters of the Mark-Houwink-Sakaruda equation calculated... 

The second new program allows the user to compare the shapes of molecular weight distributions. For example, if we have the cumulative distribution of hydrodynamic volume for two polymers we can plot the hydrodynamic volume corresponding to the 10th percentile of the distribution for polymer A against the similarly defined hydrodynamic volume for polymer B. Such a plot, made for the entire distribution of both polymers, is called a "quantile... 

For some polymers, like polystyrene or poly(methyl methacrylate), narrow standards of known molar mass and small polydispersity are commercially available, which can be used for calibration. Unfortunately, such standards are not available for all polymers and then the obtained true molar masses of a specific polymer might differ by a factor of two from the value obtained by calibration with, e.g., polystyrene [30] (see Section 9.1). This problem can be resolved by the so-called universal calibration, which is based on the finding that the retention volume of a polymer is a single-valued function of the hydrodynamic volume of the polymer, irrespective of its chemical nature and... 

Returning to Staudinger s derivation, it must be revised on two grounds. First the kinematics of motion is three rather than two-dimensional and the hydrodynamic volume spherical rather than cylindrical, i.e. 3. The detailed calculation for thin ellipsoidal particles (13) shows an approximate proportionality of the intrinsic viscosity with M1 7, a considerable difference from eq. (1) for large M. 

Branched polymers are often analyzed by means of size exclusion chromatography (SEC). It should always be kept in mind that fractionation in SEC is on the basis of the hydrodynamic volume, i.e. [41], Analysis based on a mass... 

Scheme 1 summarizes four different approaches used to characterize dendrimer structures by photophysical and photochemical probes 1. Non-covalent, inter-molecularly bound interior probes - to study the internal cavities and the encapsulation abilities of dendrimers. 2. Non-covalent, intermolecularly bound surface probes - to study surface characteristics of dendrimers. 3. Covalently linked probes on dendrimer surfaces - to study the molecular dynamics of dendrimers. 4. Covalently linked probes at the dendrimer central core - to study the site isolation of the core moiety and define the hydrodynamic volume of dendrimers by the concentric dendrimer shells. Critical literature in these four categories will be described using representative examples. 

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