Polymer scaling laws

SAL Salabat, A. and Nasirzadeh, K., Measurement and prediction of water activity in PEG + (NH4)2S04 + H2O systems using polymer scaling laws, J. Mol. Liq., 103-104,... 

Classical rubber elasticity theory predicts that the plateau modulus,, for a diluted polymer, is proportional to the square of the volume fraction of the polymer. Scaling law interpretations for dilute solutions by P.G. 

A further difference with polymers arises from the fact that the chain length f of the worm-like micelles depends on the surfactant concentration because aggregation is an equilibrium process. The polymer scaling laws which deal both with polymer chain length and concentration can be tested here only for surfactant concentration c. Moreover, the relation between f and c is not well known according to different models, f scales with c or c2. In the Cates theory... 

L. Schafer, T. A. Witten. Renormalization field theory of polymer solutions. I. Scaling laws. J Chem Phys 66 2121-2130, 1977 A. Knoll, L. Schafer, T. A. Witten. The thermodynamic scaling function of polymer solution. J Physique 42 161-m, 1981. 

VII. POLYMER CHAINS IN RANDOM POROUS MEDIA A. Scaling Laws in Equilibrium... 

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

Applying MD to systems of biochemical interest, such as proteins or DNA in solution, one has to deal with several thousands of atoms. Models for systems with long spatial correlations, such as liquid crystals, micelles, or any system near a phase transition or critical point, also must involve a large number of atoms. Some of these systems, including synthetic polymers, obey certain scaling laws that allow the estimation of the behaviour of a large system by extrapolation. Unfortunately, proteins are very precise structures that evade such simplifications. So let us take 10,000 atoms as a reasonable size for a realistic complex system. 

These recent results for dense polybead systems are very encouraging. One must wait for tests on realistic polymers with complicated chemical structures and side groups, however, before definitive conclusions can be drawn. The scaling law for the embedding algorithm has to be explored in more detail for the most cumbersome polymer structures. 

Since the prediction of the solute diffusion coefficient in a swollen matrix is complex and no quantitative theory is yet possible, Lustig and Peppas [74] made use of the scaling concept, arriving at a functional dependence of the solute diffusion coefficient on structural characteristics of the network. The resulting scaling law thus avoids a detailed description of the polymer structure and yet provides a dependence on the parameters involved. The final form of the scaling law for description of the solute diffusion in gels is... 

SR Lustig, NA Peppas. Solute diffusion in swollen membranes. IX. Scaling laws for solute diffusion in gels. J Appl Polym Sci 36 735-747, 1988. 

Special theoretical insight into the internal relaxation behavior of polymers can also be provided on the basis of dynamic scaling laws [4,5]. The predictions are, however, limited since only general functional relations without the corresponding numerical prefactors are obtained. 

A simple scaling law has been postulated to define the relationship between polymer length and Rg under various solvent conditions (Flory, 1953) ... 

Many polymer properties can be expressed as power laws of the molar mass. Some examples for such scaling laws that have already been discussed are the scaling law of the diffusion coefficient (Equation (57)) and the Mark-Houwink-Sakurada equation for the intrinsic viscosity (Equation (36)). Under certain circumstances scaling laws can be employed advantageously for the determination of molar mass distributions, as shown by the following two examples. 

The more time-consuming task is the establishment of the scaling law, which requires a series of polymer samples of narrow molar mass distribution and known molar mass. Their sedimentation coefficients have to be measured as a function of concentration and extrapolated back to c — 0 in order to obtain So(M) (Figure 18). 

The same authors then discuss the determination of the entire molar mass distribution from sedimentation velocity runs via scaling laws for the polymer polystyrene in cyclohexane, where the scaling law is also known [78] ... 

We hope that this chapter on the molecular weight determination of synthetic polymers has illustrated that in the case of a complex polymer it is preferable to use several experimental methods for the molecular weight determination to obtain a full picture. Owing to the different sensitivity of the various methods some are blind for low molar masses while others are blind at low concentrations. As exemplified, often scaling laws can be utilized to compare results of different methods and different sensitivities. 

Kilbride BE, Coleman JN, Foumet P, Cadek A, Hutzler S, Roth S, Blau WJ (2002). Experimental observation of scaling laws for alternating current and direct current conductivity in polymer-carbon nanotube composite thin films. J. Appl. Phys. 92 4024—4030. 

A successful theoretical description of polymer brushes has now been established, explaining the morphology and most of the brush behavior, based on scaling laws as developed by Alexander [180] and de Gennes [181]. More sophisticated theoretical models (self-consistent field methods [182], statistical mechanical models [183], numerical simulations [184] and recently developed approaches [185]) refined the view of brush-type systems and broadened the application of the theoretical models to more complex systems, although basically confirming the original predictions [186]. A comprehensive overview of theoretical models and experimental evidence of polymer bmshes was recently compiled by Zhao and Brittain [187] and a more detailed survey by Netz and Adehnann [188]. 

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