Monodisperse solutions

After normalization to the asymptotic baseline, g2(r) decays from two to unity if measured with a perfect instrument. A real instrument always suffers from some loss of coherence, and for a monodisperse solution of ideal, non-interacting solute molecules the intensity autocorrelation function g2(r) takes the form... 

Figure 21.3. Concentration distribution of solute in solution at sedimentation equilibrium. Curve A represents ideal behavior of a monodisperse solute curve B represents nonideality and curve C represents a polydisperse system.

We have outlined how TDFRS not only provides a useful tool for the study of the Ludwig-Soret effect in multicomponent liquids, but can also contribute valuable pieces of information towards solving the puzzles encountered in polymer analysis. Though TDFRS is conceptually simple, real experiments can be rather elaborate because of the relatively low diffraction efficiencies, which require repetitive exposures and a reliable homodyne/heterodyne signal separation. As an optical scattering technique it has much in common with PCS, and the diffusion coefficients obtained in the hydrodynamic limit (q —> 0) for monodisperse solutions are indeed identical. 

Monodisperse Suspensions. The absolute accuracy of the instruments were tested with a set of monodisperse solutions. Table I shows the results of this comparative study. 

Many of the structural proteins are components of the protein capsids/ cores that surround the nucleic acid, or are components of Upid envelopes. Each has its own challenges for structure determination. Membrane proteins are generally difficult to crystallize or to study by NMR in monodisperse solutions. Domains that are not integral to the membrane can of course be cleaved off for structural studies. For the most part, the capsids and core proteins have been studied as assemblies isolated from natural sources or reconstituted/expressed to mimic those of the infectious virus. By the standards of diffraction methods, these are large complexes, the smallest of which contain >1 MDa of protein. 

The term in brackets on the left-hand side of Eq. (3-37) is called the intrinsic viscosity or limiting viscosity number. It reflects the contribution of the polymeric solute to the difference between the viscosity of the mixture and that of the solvent. The effects of solvent viscosity and polymer concentration have been removed, as outlined earlier. It now remains to be seen how the term on the right-hand side of Eq. (3-37) can be related to an average molecular weight of a real polymer molecule. To do this we first have to express the volume V of the equivalent hydro-dynamic sphere as a function of the molecular weight A/ of a monodisperse solute. Later we substitute an average molecular weight of a polydisperse polymer for M in the monodisperse case. 

Breadth of Phase Transition. The abruptness or width of the coexistence region of the order-disorder transition has also been estimated theoretically. For a monodisperse solution of a relatively high molecular weight polymer, an intrinsic coexistence region, ATc, exists because of standard flnite-size fluctuation effects. Numerical calculations yield the result (22)... 

As a first application we calculate the second virial coefficient A2 of the osmotic pressure for a monodisperse solution According to Eq. (Id)... 

Based on the thermodynamics of ideal solutions, it can be shown that for monodispersed solutes ... 

Hence, the time autocorrelation function provides a direct means to determine the diffusion coefficient D in dilute monodisperse solutions ... 

Constraint release has a limited effect on the diffusion coefficient it is important only for the diffusion of very long chains in a matrix of much shorter chains and can be neglected in monodisperse solutions and melts. The effect of constraint release on stress relaxation is much more important than on the diffusion and cannot be neglected even for monodisperse systems. Constraint release can be described by Rouse motion of the tube. The stress relaxation modulus for the Rouse model decays as the reciprocal square root of time [Eq. (8.47)] ... 

We can also determine the value of interaction b from the second virial coefficient A2, a basic quantity associated with osmotic pressure as well as radiation scattering. Hence, for a monodisperse solute of molecular mass M, the... 

A complementary method consists in determining the exponent v by measuring the second virial coefficient Av This coefficient is related to the average volume Rq occupied by a chain. For a monodisperse solute... 

In Chapter 5, Section 10, we discussed the result of osmotic pressure experiments, made at different concentrations p. In particular, we noted that, if we renormalize the concentration p by the volume Ro associated with the polymer chain in the limit p- 0, we can exactly superpose the results IIM/RTp( — nfi/C) obtained for two different molecular masses Mn of the solute and two different solvent qualities (see Fig. 5.10, Chapter 5). More generally, the osmotic pressure of a monodisperse solution is written in the following form (good solvent)... 

For a monodisperse solute, the structure function 1/(5) is directly related to the osmotic compressibility... 

However, there are other types of experimental tests of the theory of polymers in solution these are experiments in which the chemically homogeneous solute is a mixture of two (or more) samples whose polymer chains have very different average sizes. However, we cannot test the theories of strongly polydisperse solutions as we did in the weakly polydisperse case i.e. by referring to a monodisperse solute of molecular mass equal to the average molecular mass of the sample. Polydispersion here becomes an essential parameter. 

Newman et al. (1974) have recently studied the concentration-dependence of the diffusion coefficient and the sedimentation coefficient in a highly monodisperse solution of the single-stranded circular DNA from the fd Bacteriophage. Their results are shown in Fig. 13.5.3. From these data it is possible to determine the coefficients in the expression... 

Degradation during flow of entangled monodisperse solutions is quite different in character and degree from degradation of either isolated or entangled polydisperse molecules. Polydispersity may play an important role in the development of transient networks and their subsequent extensibility. 

The Debye equation is based on the following physical description of the sample. This is a monodisperse solution of identical particles, which are in random orientations relative to the incident primary beam, and act as independent entities (i.e. there are no interparticle spatial correlations). The above derivation has presumed also that the particles are in vacuo. If they are in solution, they are required to form a two-phase system of solute and solvent. In biology, this corresponds to dilute solutions of pure proteins or glycoproteins in a low-salt buffer. Complications arise in the case of polyionic macromolecules in low-salt buffers, such as nucleic acids. Here, interparticle correlation effects can readily occur and the macromolecule is surrounded by an ion-cloud of opposite charge (i.e. a three-phase system). Other complications can arise in the cases of polydisperse distributions of macromolecules, oligomerization or dissociation phenomena, and conformational changes. Different formuhsms have to be derived for the analyses of these systems. 

Figure 6-6. Reduced chemical potential Am/RT of the solvent (degree of polymerization Xi = 1) as a function of the volume fraction 02 of monomeric units of a monodisperse solute with degree of polymerization X2 = 100 and with different interaction parameters xo. Calculations according to Equation (6-32) with o = 0.

The phase-separation behavior of quasibinary systems is different from that of binary systems. This phenomenon can be most simply described in terms of the cloud-point curves of ternary systems consisting of a solvent and two monodisperse solutes. The cloud-point curve corresponds to the special case of phase separation where the volume of one of the phases tends toward zero. 

The crucial assumption of the Doi-Edwards theory is that the primitive chain reptates in a tube fixed in space. However, in monodisperse solutions, all chains are wriggling simultaneously, so that the tube around each chain is never fixed but successively renewed by different chains. Hence, the Doi-Edwards theory is not self-consistent. This fact has given rise to recent measurements of the tracer diffusion coefficient as a function of the molecular weight and concentration of the matrix component. 

Flow Cytometer Lab-on-a-Chip Devices, Fig. 5 Microfabricated Coulter counter (a) scanning electron microscope image of the Coulter counter. The 3.5 pm-deep reservoirs and the inner Ti/Pt electrodes, which control the voltage applied to the pore but pass no current, are only partially shown. The inset shows a magnified view of this device s pore, which has dimensions 5.1 X 1.5 X 1.0 pm, and (b) relative changes in baseline current versus time for a monodisperse solution of 87 nm diameter latex colloids and a polydisperse solution of latex colloids with diameters 460, 500, 560, and 640 nm [5]... 

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