Debye-Hiickel potential, polyelectrolyte

Since neutral block copolymer micelles are a convenient model system for the investigation of the steric stabilization potential, polyelectrolyte block copolymer micelles may serve to study electro-steric interaction. Similar to the case of neutral block copolymer micelles, the interaction potential u(r) can be probed by measuring the shear modulus of micellar gels as a function of distance. Assuming a simple Debye-Hiickel potential yields for the shear modulus by taking the second derivative... 

This simple Debye-Hiickel or Yukawa potential captures the essence of electrostatic interactions in many polyelectrolyte systems, and as the outcome of a linear analysis, provides discrete interaction terms that can be directly summed to obtain the net electrostatic energy associated with the multiple charges of a polyelectrolyte system. When rigor beyond the Debye-Hiickel approximation is mandatory, electrostatic energies must be evaluated through the full... 

Fig. 3. Schematic plots of the decay of the electrostatic potential if(r) near a polyelectrolyte chain of unspecified chain radius. The potential decay predicted by a nonlinear Poisson-Boltzmann analysis can be superimposed onto one predicted by a linearized Debye-Hiickel approximation in the far field region if the surface potential of the Debye-Hiickel is appropriately adjusted. However, in this case there remains strong deviation between the two approaches in the region nearer the chain.

Almost all theoretical works considering polyelectrolytes [7,26,27,29, 141-145] have treated counterions and salt ions at the Debye-Hiickel (DH) level by preaveraging the electrostatic interactions between charged monomers over the small ions degrees of freedom. In this approximation the presence of counterions and salt ions leads to an effective screening of the electrostatic interaction between charged monomers, which interact via a screened electrostatic (Yukawa) potential. Of course, such an... 

To investigate the effect of the Debye-Huckel approximation on the solution properties, Stevens and Kremer [152] performed molecular dynamics simulations of salt-free solutions of bead-spring polyelectrolyte chains in which the presence of counterions was treated via a screened Coulomb potential, and compared the results with their simulations with explicit counterions [146,148]. To elucidate the effect of the Debye-Hiickel approximation, the dependence of the mean square end-to-end distance, R ), osmotic pressure, and chain structure factor on polymer concentration was examined. Stevens and Kremer found that (i ) tends to be larger at low densities for DH simulations and is smaller at higher densities. However, the difference in (i ) between DH simulations and simulations with explicit counterions is within 10%. This trend seems to be a generic feature for all N in their simulations. The functional form and density dependence of the chain structure factor are very close in both simulations. The most severe Debye-Huckel approximation affects the dependence of the osmotic pressure on polymer concentration. It appears that in the DH simulations not only is the magnitude of the osmotic pressure incorrect, but also the concentration dependence is wrong. 

In the following chapters of this book, we will discuss fluorescence measurements performed mainly on PE solutions in thermodynamically poor solvents. Therefore, the behavior of PEs in poor solvents is our main sphere of interest, but we will first mention the classical Kuhn treatment of polyelectrolytes in )9-solvents [53]. The potential energy of a polyelectrolyte chain in a given conformation (described by a set of r, position vectors of segments) can be written within the framework of the mean-field Debye-Hiickel (DH) theory [54] as a sum of three contributions the energy corresponding to (i) the entropic elasticity of harmonic bonds, Ui, with bond lengths I, which connect the monomers in the polymer chain. This contribution depends on the set of all position vectors, r, ... 

For low-polyelectrolyte concentrations, R 00 and Eqs. [304] and [305] give the standard bulk Debye-Hiickel solution for the potential near an ion of radius a... 

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