Excess interfacial free energy

In contrast to this, the scaling theory of the PE star collapse developed in [27] suggested that, instead of the formation of a collapsed core, a decrease in the solvent strength may provoke the formation of bundles by the sticking of individual branches to each other. The bundle formation reduces the excess interfacial free energy of the collapsed domains, without a significant penalty in terms of the intramolecular Coulomb repulsion. More recently, the formation of bundles was theoretically predicted in colloidal PE brushes [42]. 

The latter determines the excess interfacial free energy per chain as ... 

In this equation, the first and the second terms describe the respective conformational entropies of stretched core and coronal blocks, whereas the third term, 7 ( core), accounts for excess interfacial free energy [here, s Rcore) is specified by (38)]. The last term in (69) accounts for repulsive interactions in the corona. 

The equilibrium structure of the micelle is now determined by a balance between the osmotic pressure of counterions in the corona and excess interfacial free energy of the core-corona boundary. Here, the micellar corona is equivalent to a quasi-planar PE brush in the annealing osmotic regime [31],... 

For two pure, mutually immiscible liquids having a common flat interface we can define the terms interfacial tension and excess interfacial free energy, based on the same concepts used for the liquid-vapor systems. However, because unlike atoms or molecules at a liquid-liquid interface experience mutual attractions from units in the adjacent phase, those interactions become important in determining the properties of the system. The specific excess interfacial free energy will be dimensionally equivalent to and numerically equal to the interfacial tension. 

In curve C, a large excess V /V = lO ) of IB-piin droplets of Z, is assumed to be present. The higher rate of swelling obtained in the case A as compared to B and C is due to the interfacial free energy term which, because of the lower value of in case A, gives a higher value of AC,fc. i.e. 

Over the last two decades the exploration of microscopic processes at interfaces has advanced at a rapid pace. With the active use of computer simulations and density functional theory the theory of liquid/vapor, liquid/liquid and vacuum/crystal interfaces has progressed from a simple phenomenological treatment to sophisticated ah initio calculations of their electronic, structural and dynamic properties [1], However, for the case of liquid/crystal interfaces progress has been achieved only in understanding the simplest density profiles, while the mechanism of formation of solid/liquid interfaces, emergence of interfacial excess stress and the anisotropy of interfacial free energy are not yet completely established [2],... 

The origin of the excess stress on liquid/vapor interfaces follows from the tendency of the liquid surface to contract. As a molecule inside a mass of liquid is under the effect of the forces of the surrounding molecules, while a molecule on the surface is only partly surrounded by other molecules, some work is necessary to bring molecules from the inside to the surface. This indicates that the force must be applied along the surface in order to increase the area of the surface. This force on the surface appears as excess stress (a difference between normal and transverse components of pressure tensor in the region of the interface) and defines the surface tension of the liquid. Excess stress a, for liquid/vapor interfaces, is always a positive quantity and is equal to the interfacial free energy. 

This study is consistent with the idea that crystal surfaces at temperatures close to melting have some kind of disordered layer or layers, often called liquid-like . Due to the different equilibrium volumes of the liquid and solid phases, this region makes the surface either contract (as in the case of the ice surface) or expand (as it is for Lennard-Jones systems). The positive interfacial excess stress of the ice/water interface therefore makes it similar to liq-uid/vapor interfaces, and the water/vapor interface in particular, for which the excess stress is equal to the interfacial free energy (surface tension). 

It is also clear from Equation (5.2) that surface or interfacial tension - that is, the force per unit length tangential to the surface, measured in units of miUinewtons per metre - is dimensionally equivalent to an energy per unit area measured in millijoules per square metre. Eor this reason, it has been stated that the excess surface free energy is identical to the surface tension, but this is tme only for a single-component system - that is, a pure liquid (where the total adsorption is zero). 

The interfacial free energy is then defined as the excess contribution of a system such as that considered in fig. 35a, containing one interface, and a homogeneous system where 4> z) = 4>a everywhere. Denoting the surface area of the interface by A, we thus obtain the interfacial tension fml [eq, (4)] as... 

On the other hand, the Gibbs free energy function is defined as G = U+ PV - TS from Equations (104) and (124), which may be expressed as, dG = dU + PdV + VdP - TdS -SdT, and the excess Gibbs free energy, Gs, of the interfacial region in a reversible process, for a completely plane interface can be expressed as... 

One of the main objectives in surface science is the prediction of the amount of substance that is adsorbed at an interface. Adsorption and interfacial free energy are related through the Gibbs Adsorption Law. If we define Jj as the excess moles of the component, i, adsorbed at the interface per unit area of an interphase,... 

Surface Tension. The presence of an interface between two phases goes along with an excess free energy that is proportional to the interfacial area. For a clean fluid interface the specific interfacial free energy (in J m 2) equals the surface or interfacial tension (in N-m-1). This is a two-dimensional tension acting in the direction of the interface, which tries to minimize the interfacial area. The surface tension of a solid cannot be measured. 

The excess of free energy per unit interfacial area is therefore... 

The excess (per unit area) of internal energy, e, and entropy, rj, within the interfacial layer can be introduced by analogy with the excess of free energy [6]. These quantities are also dependent on the position of the dividing surface. One can verily that the equations relating o, 8, and r are very similar to those derived in conventional three-dimensional thermodynamics, i.e. ... 

As indicated in the discussion following Equation 1.29, the interfacial tension Y is equal to the smface excess Helmholtz free energy per unit area (F IA) when the reference surface is chosen to make the surface excess mass T vanish. But... 

We identify the measurable change in interfacial tension, dy, with the excess in free energy per unit area due to the adsorption at the interface. This definition is assumed to hold both at equilibrium and out of equilibrium. The free energy excess can be written as a functional of the volume fraction profile of the surfactant, (p x, t), x being the distance from the interface and t the time,... 

Surfactant molecules adsorb from a solution on hydrophobic solid-liquid (SL) and liquid-vapor (LV) interfaces, modifying the interfacial tensions (or excess surface free energies) of the SL and LV interfaces and contact angle. 

In a powder compact, excess volume free energy is present primarily in the form of excess pore surface or interfacial energy (i.e., liquid—vapor and solid-vapor interfaces) consequently, the driving force for sintering can be approximated by... 

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