Effect of counterion size on polyelectrolyte conformations and thermodynamics.

We present a theoretical model to study the effect of counterion size on the effective charge, size, and thermodynamic behavior of a single, isolated, and flexible polyelectrolyte (PE) chain. We analyze how altering counterion size modifies the energy and entropy contributions to the system, including the ion-pair free energy, excluded volume interactions, entropy of free and condensed ions, and dipolar attraction among monomer-counterion pairs, which result in competing effects challenging intuitive predictions. The PE self-energy is calculated using the Edwards-Muthukumar Hamiltonian, considering a Gaussian monomer distribution for the PE. The condensed ions are assumed to be confined within a cylindrical volume around the PE backbone. The dipolar and excluded volume interactions are described by the second and third virial coefficients. The assumption of freely rotating dipoles results in a first-order coil-globule transition of the PE chain. A more realistic, weaker dipolar attraction, parameterized in our theory, shifts it to a second-order continuous transition. We calculate the size scaling-exponent of the PE and find exponents according to the relative dominance of the electrostatic, excluded volume, or dipolar effects. We further identify the entropy- and energy-driven regimes of the effective charge and conformation of the PE, highlighting the interplay of free ion entropy and ion-pair energy with varying electrostatic strengths. The crossover strength, dependent on the counterion size, indicates that diminishing sizes favor counterion condensation at the expense of free ion entropy. The predictions of the model are consistent with trends in simulations and generalize the findings of the point-like counterion theories.

Driving forces of the complex formation between highly charged disordered proteins.

Highly disordered complexes between oppositely charged intrinsically disordered proteins present a new paradigm of biomolecular interactions. Here, we investigate the driving forces of such interactions for the example of the highly positively charged linker histone H1 and its highly negatively charged chaperone, prothymosin α (ProTα). Temperature-dependent single-molecule Förster resonance energy transfer (FRET) experiments and isothermal titration calorimetry reveal ProTα-H1 binding to be enthalpically unfavorable, and salt-dependent affinity measurements suggest counterion release entropy to be an important thermodynamic driving force. Using single-molecule FRET, we also identify ternary complexes between ProTα and H1 in addition to the heterodimer at equilibrium and show how they contribute to the thermodynamics observed in ensemble experiments. Finally, we explain the observed thermodynamics quantitatively with a mean-field polyelectrolyte theory that treats counterion release explicitly. ProTα-H1 complex formation resembles the interactions between synthetic polyelectrolytes, and the underlying principles are likely to be of broad relevance for interactions between charged biomolecules in general.

Polymer complexation: Partially ionizable asymmetric polyelectrolytes.

Theories of bulk coacervation of oppositely charged polyelectrolytes (PE) obscure single molecule level thermodynamic details, considered significant for coacervate equilibrium, whereas simulations account for only pairwise Coulomb interaction. Also, studies of effects of asymmetry on PE complexation are rare compared to symmetric PEs. We develop a theoretical model, accounting for all entropic and enthalpic contributions at the molecular level, and the mutual segmental screened Coulomb and excluded volume interactions between two asymmetric PEs, by constructing a Hamiltonian following Edwards and Muthukumar. Assuming maximal ion-pairing in the complex, the system free energy comprising configurational entropy of the polyions and free-ion entropy of the small ions is minimized. The effective charge and size of the complex, larger than sub-Gaussian globules as for symmetric chains, increase with asymmetry in polyion length and charge density. The thermodynamic drive for complexation is found to increase with ionizability of symmetric polyions and with a decrease in asymmetry in length for equally ionizable polyions. The crossover Coulomb strength demarcating the ion-pair enthalpy-driven (low strength) and counterion release entropy-driven (high strength) is marginally dependent on the charge density, because so is the degree of counterion condensation, and strongly dependent on the dielectric environment and salt. The key results match the trends in simulations. The framework may provide a direct way to calculate thermodynamic dependencies of complexation on experimental parameters such as electrostatic strength and salt, thus to better analyze and predict observed phenomena for different sets of polymer pairs.

Effect of free energy barrier on pattern transition in 2D diffusion limited aggregation morphology of electrodeposited copper.

Fractal like morphology is a very interesting feature during electrodeposition of metals and shows pattern transition with changes in deposition conditions. In this article, we have explained the thermal effects in the two dimensional DLA morphology on the basis of thermal free energy and another free energy barrier resulting from the electric field. The results obtained from free energy hypothesis are consistent with experiments showing the transition voltage for electrodeposition of copper ions to be around 6 V.