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Rehbinder’s Effect, Spontaneous Dispergation Processes, and Formation of Nanosystems
Published in Victor M. Starov, Nanoscience, 2010
Alexander V. Pertsov, N. V. Pertsov
Diffusion field, described by Equation 23.19, is in correspondence with the following dependence of the excess chemical potential of the dissolved solid against the coordinate: Δμ(x)=RTc(x)c0=RTln[eΔμ0/RT(1−xδ)+xδ].
Properties of Binary Solvent Mixtures
Published in Yizhak Marcus, Solvent Mixtures, 2002
In a binary mixture A+B, the partial vapor pressure of component A (or its fugacity, if a correction for vapor nonideality is required), pA, is related to its excess chemical potential μAE by () μAE=RTlnpA/xApA∘=RTlnfA
Basic Principles of Thermodynamics of Polymer Solutions
Published in Yuri S. Lipatov, Anatoly E. Nesterov, Thermodynamics of Polymer Blends, 2020
Yuri S. Lipatov, Anatoly E. Nesterov
By the quenching the solution of non-crystallizable polymer, one can reach temperature at which the solution separates into two phases: one enriched with polymer and the second is dilute solution. Further quenching makes enriched phase more concentrated and dilute phase more diluted. That implies that the system has the regions of insolubility that are narrowed with increasing temperature up to the upper critical solution temperature, UCST. At this temperature, the heat and entropy of mixing is positive and excess chemical potential, μ1E, decreases with increasing temperature. In the lattice theory, the exchange contribution, RTχ1φ22, may exceed the negative combinatorial entropy of mixing. However, at the increasing temperature, the entropy of mixing becomes prevailing because χ1 diminishes with temperature. One has to bear in mind that the critical concentration for polymer solutions is low and decreases with the increase in the molecular mass of the polymer, whereas the critical temperature grows (Fig. 1.4). The calculated binodals have been obtained from the experimental data on the critical points, using the Flory-Huggins theory. From the condition for critical point, i.e., (δμ1/δφ2)P,T=0 and (δ2μ1/δφ22)P,T=0 by differentiation of Eq 1.44, we find: [] 1−(1−r−1+2χ1φ2)(1−φ2)=0 [] 1−2χ1(1−φ2)2=0
Phase separation of active Brownian particles in two dimensions: anything for a quiet life
Published in Molecular Physics, 2021
Sophie Hermann, Daniel de las Heras, Matthias Schmidt
Recall that the adiabatic force density contribution is determined via the density functional Equation (14), or equivalently, expressed as a force field, In the considered adiabatic reference system of spherical swimmers, the excess free energy functional is independent of orientation. Thus the adiabatic force field is a gradient expression independent of φ. The force field can be expressed as where denotes the adiabatic excess chemical potential. Since we base our local density approximation on scaled particle theory, is given as [67] where is a rescaled packing fraction to approximately model the soft interparticle interaction potential and the possibility of the spherical swimmers to penetrate each other to some extent. The rescaling with factor is necessary, because the scaled particle theory assumes hard particles and allows to model various repulsive interparticle interactions. Alternatively, can be interpreted as a packing fraction with higher jamming density .
Integral equation theories for fluid with very short-range screened Coulomb plus power series interactions
Published in Molecular Physics, 2023
All other thermodynamic functions can be obtained from the virial pressure , reduced excess chemical potential , and .