Scaling theories

1 Scaling Theory In the preceding subsection, we employed a blob model to derive power relationships for various thermodynamic quantities that characterize the semidilute solutions. The simple assumption produced useful results that agree with those obtained in experiments. We can derive the same results without assuming a blob. We will use the scaling theory here for that purpose. 

Let us first consider the osmotic pressure O. The virial expansion of II reduced by Oideai needs to be a power series of the dimensiomess concentration of the polymer chains, pR o /N  

In general, the factor in the square bracket of Eq. 4.17 is a universal function of pR o lN defined for pR JN O. By universal, we mean that the functional form is independent of A or ft except through pR Q /N. Therefore, different polymer-solvent combinations share a common function, as along as the solvent is good to the polymer or the polymer chain follows the same statistics. The universal 

Note that a dimensionless quantity NU/(pk T) is equated to another dimensionless quantity/n(A ) with x = pR /N. 

As a — 0, /n(x) approaches unity, that is, the solution is ideal in the dilute-solution limit. The scaling theory assumes that /n(x) for large x asymptotically approaches a power of x with a scaling exponent m yet to be determined. Thus, 

The central idea of the scaling theory, as proposed by Abrahams and coworkers (Abrahams, Anderson, Licciardello and Ramakrishman 1979, Abrahams, Anderson, and Ramakrishman 

The next step in the argument is to consider how G L) changes with the size of the sample. Abrahams et al. (1979) make the scaling hypothesis that when small volumes are combined into a larger one of size bL, G L) is the only quantity needed to calculate the new G(bL), so that. 

It then follows that when the scaling is considered as a continuous function of L 

The form of P(G) is constructed from its asymptotic values. When the conductivity is large, for example, well above the mobility edge, then the normal macroscopic transport theory must apply, for which o is independent of the size of the sample, so that from Eq. (7.57), p is unity. Generalized to d dimensions, this gives 

At the other extreme of low conductivity, the electrons are localized with an exponential decay of the wavefunction, which represents the conductivity. 

It is believed that a length such as must tend to infinity at a second-order transition, just as the relevant lengths do at a critical or Neel point. 

The important paper of Abrahams et a/. (1979) first made clear that the conductivity r( ) of a degenerate electron gas at zero temperature in a disordered environment must tend continuously to zero as the Fermi energy EF tends to Ec. Abrahams et al defined a dimensionless conductivity 

It is then argued that, if we fit together these blocks to form larger blocks, the only relevant quantity determining the new value of G is that for the smaller ones. Expressing this relationship as a differential equation, we find 

The important point is that in three dimensions p has a zero, so that near G0, dlnG/dlnL vanishes therefore near this point G is constant (G0) and thus 

Scaling theory does not give the value of G0, but in previous sections we found it to be approximately 0.03, and we deduce that this value must be universal. 

Except for water-soluble polymers, most synthetic polymers dissolve only in organic solvents, which are usually volatile and thus more difficult to deal with in the ultracentrifuge. Furthermore, unlike proteins whose polydispersity is near unity, polymer (synthetic) solutions are often heterogeneous. For this reason, sedimentation experiments are used less frequently in (synthetic) polymer chemistry. Investigators usually avoid using the ultracentrifuge for the determination of molecular weight. 

However, while the ultracentrifuge is no longer heavily used in biochemistry (because of the development of sodium dodecyl sulfate (SDS)-polyacrylamide gel electrophoresis for the estimation of molecular weight, see Chapter 13), it has become an important tool for the study of dimensions of synthetic polymers in solution. 

In the semidUute range, we may imagine the solution as a continuum formed by entangled macromolecules which can be divided into spheres (or blobs) of radius equal to the screen length If we write the Svedberg equation in its approximate form, 

Equation (11.5) predicts that in the semidilute range of concentration (c c ) the sedimentation coefficient should be independent of the molecular weight of the polymer and the plot of log 5 versus log c should give the slope —0.50. 

This theory has been partially confirmed by sedimentation experiment (Langevin and Rondelez, 1978). The value of the slope so far found was —0.50 0.10. We now have some evidence to believe that in the semidilute range of polymer solution the solvent is forced through in orderly fashion around the blob of radius C but still cannot penetrate the interior of the blob. Note that this theory is reminiscent of the pearl necklace model and the hydrodynamic equivalent sphere. 

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The shape of the (7 and curves is theoretically well established by critical scaling theory. The temperature dependence is given by... 

Modem scaling theory is a quite powerful theoretical tool (appHcable to Hquid crystals, magnets, etc) that has been well estabUshed for several decades and has proven to be particularly useful for multiphase microemulsion systems (46). It describes not just iuterfacial tensions, but virtually any thermodynamic or physical property of a microemulsion system that is reasonably close to a critical poiat. For example, the compositions of a microemulsion and its conjugate phase are described by equations of the foUowiug form ... 

Changing the distance between the critical points requires a new variable (in addition to the three independent fractional concentrations of the four-component system). As illustrated by Figure 5, the addition of a fourth thermodynamic dimension makes it possible for the two critical end points to approach each other, until they occur at the same point. As the distance between the critical end points decreases and the height of the stack of tietriangles becomes smaller and smaller, the tietriangles also shrink. The distance between the critical end points (see Fig. 5) and the size of the tietriangles depend on the distance from the tricritical point. These dependencies also are described scaling theory equations, as are physical properties such as iuterfacial... 

M. E. Fisher, H. J. Nakanishi. Scaling theory for the criticality of fluids between plates. J Chem Phys 75 5857-5863, 1981. 

In the case of a single patch, the size dependence of the system follows directly from the finite size scaling theory [133]. In particular, the critical point temperature scales with the system size as predicted by the equation... 

P. D. Gujrati, Scaling theory of polydispersity, infinite chain and the Flory-Huggins approximation. Phys Rev B 40 5140-5143, 1989. 

D. Stauffer. Scaling theory of percolating clusters. Phys Rep 54 1-74, 1979. 

According to the scaling theory of the adsorption transition [2,35], one expects for e near e. in the limit A oo a power law behavior... 

The main predictions of the scaling theory [40], concerning the dynamics behavior of polymer chains in tubes, deal with a number of characteristic times the smallest time rtube measures the interval of essentially Rouse relaxation before the monomers feel the tube constraints significantly, 1 < Wt < Wrtube = and diffusion of an inner monomer is... 

A simulational test [19] of the power law (39) can be seen in Fig. 15(a). As it is clear from the raw data, presented in the insert to Fig. 15(a) the data for zjD 1 settle down to constant values, and a region where zjD is still small and a power law is seen cannot be easily identified. The scaling theory, however, is expected to hold only for distances z that are large on the microscopic scale [74]. Studying such a regime where z is very large in... 

K. Kremer, K. Binder. Dynamics of polymer chains confined into tubes Scaling theory and Monte Carlo simulations. J Chem Phys 7 6381-6394, 1984. 

The theoretical foundation for describing critical phenomena in confined systems is the finite-size scaling approach [64], by which the dependence of physical quantities on system size is investigated. On the basis of the Ising Hamiltonian and finite-size scaling theory, Fisher and Nakanishi computed the critical temperature of a fluid confined between parallel plates of distance D [66]. The critical temperature refers to, e.g., a liquid/vapor phase transition. Alternatively, the demixing phase transition of an initially miscible Kquid/Kquid mixture could be considered. Fisher and Nakashini foimd that compared with free space, the critical temperature is shifted by an amoimt... 

S. S. Schiffman, M.L. Reynolds and F.W. Young, Introduction to Multi-dimensional Scaling Theory, Methods and Applications. Academic Press, New York, 1981. 

Figure 9. Experimental data for the effective viscosity of the foam bubble regime in Berea sandstone as a function of the foam superficial velocity. The solid line is drawn according to the scaling theory with values of the two sets of parameters e and 6 listed.

Fisher, M. E. Barber, M. N., Scaling theory for finite-size effects in the critical region, Phys. Rev. Lett. 1972, 28, 1516-1519... 

Computer simulation and analytical methods have both been used, based on diffusion equation, partition function and scaling theory approaches. There are a number of parameters which are common to most of these theories some of these are also relevant to theories of polymer solutions, i.e. 

In scaling theories, Rc, Rm> and Z are directly correlated to NA and Mb for the investigated micelles. Two limiting cases have to be distinguished, the starlike or hairy micelles with N < Nb and the crew-cut micelles with Na > Nb (Sect. 2.3). 

Scaling theories are restricted to long polymer chains in good solvents and do not include finite chain effects and polymer-solvent interactions. These models should be complemented by more detailed mean-field calculations and molecular simulations. 

The constant in Equation (5.112) cannot be readily evaluated using scaling theory. Our transformation applies equally well to the radius of gyration or the root mean square end-to-end length, only the numerical constant changes. We would like to be able to apply this idea to the role of concentration in semi-dilute and concentrated polymer regimes. In order to do this we need to define a new parameter s, the number of links or segments per unit volume ... 

So far we have invoked reptation theory to describe the behaviour in the melt state. We can use scaling theory in the form of Equations (5.122) and (5.123) to express the concentration dependence of the modulus and viscosity. By inspection of Equations (5.128) and (5.130) ... 

Non-dilute solutions also allow for theoretical descriptions based on scaling theory [16, 21]. When the number of polymer chains in the solution is high enough, the different chains overlap. At the overlapping concentration c , the long-scale density of polymer beads becomes uniform over the solution. Consequently c can be evaluated as... 

The distribution of the center-to-end distance, F(R(,), in a star can also be predicted from scaling theory. For EV chains, it is expected to be close to Gaussian [26], except for small R. Applying scaling arguments and RG theory, Ohno and Binder [27] obtained a power-law behavior for small R, F(Rj,)=(Rj,/) with the exponent value 0(f)=l/2 for high f. They also considered the case of a star center adsorbed on a planar surface, evaluating the bead density profiles and the distribution of center-to-end distance in the directions perpendicular and parallel to the surface in terms of similar power-laws. 

Fig. 2. Bead density profiles. Solid line Brushes, mean-field and scaling theory (step function) dashed-dotted line generalization of the Milner et al. theory for brushes in the theta state dashed-double dotted line Milner et al. theory for brushes (EV chains) dashed line EV stars dotted line EV combs. Variable r is scaled to give zero bead density for the smooth curves of brushes at r=l. The brush curves are normalized to show equal areas (same number of units). The comb and star densities are arbitrarily normalized to show similar bead density per volume unit as the step function and EV curves for brushes at the value ol r where these curves intercept...

Interestingly, scaling theory predicts for brushes that the density of units is independent from the distance to the surface and that the extension of the chain (or the brush height) is proportional to its contour length as if the chains were completely extended. Thus... 

Freed et al. [42,43], among others [44,45] have performed RG perturbation calculations of conformational properties of star chains. The results are mainly valid for low functionality stars. A general conclusion of these calculations is that the EV dependence of the mean size can be expressed as the contribution of two terms. One of them contains much of the chain length dependence but does not depend on the polymer architecture. The other term changes with different architectures but varies weakly with EV. Kosmas et al. [5] have also performed similar perturbation calculations for combs with branching points of different functionalities (that they denoted as brushes). Ohno and Binder [46] also employed RG calculations to evaluate the form of the bead density and center-to-end distance distribution of stars in the bulk and adsorbed in a surface. These calculations are consistent with their scaling theory [27]. 

The internal distribution of beads in a uniform star predicted by the Daoud and Cotton scaling theory [11] can be tested by computing bead density profiles. These profiles can be compared with the scaling predictions for the density within EV blobs, Eq. (16), that can be explicitly written for EV stars as... 

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