The Interfacial Pressure

The difference between the interfacial tension y of the bare interface and the interfacial tension y of the interface containing the monolayer is called the interfacial 

The interfacial pressure is a two-dimensional (2D) pressure, expressed in force per unit length. For fluid interfaces, rr is readily obtained because both y and y can be directly measured. The interfacial pressure at solid interfaces is defined according to Equation 7.1 as well. However, in this case, n cannot be measured but may be accessible through Equation 3.87. 

Although there is a formal analogy with the three-dimensional (3D) pressure of a gas in a given volume, the origin of the interfacial pressure is different. The pressure a gas exerts on the walls of the container in which it is confined is due to the gas molecules colliding against the wall, whereas the interfacial pressure is the difference in contractile force in a bare interface and a monolayer, respectively. The 3D analogue of the interfacial pressure is the osmotic pressure difference between a solution and a solvent (see Section 3.6). 

Interfacial equations of state, relating the interfacial pressure n to the adsorbed amount T, the available interfacial area A, and the temperature T, can be formulated analogously to the 3D equivalent relating the pressure, number of molecules, volume, and temperature. 

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Fig. 9. Residual stresses owing to thermal expansion mismatch between a particle with radius a and thermal expansion coefficient and a matrix with thermal expansion coefficient The stresses illustrated here are for and P is the interfacial pressure.

In Eq. (5.26), Tt is the interfacial pressure of the aqueous-organic system, equal to (Yo - Y) e to the difference between the interfacial tensions without the extractant (Yo) and the extractant at concentration c (y)], c is the bulk organic concentration of the extractant, and is the number of adsorbed molecules of the extractant at the interface. The shape of a typical n vs. In c curve is shown in Fig. 5.4 rii can be evaluated from the value of the slopes of the curve at each c. However, great care must be exercised when evaluating interfacial concentrations from the slopes of the curves because Eq. (5.26) is only an ideal law, and many systems do not conform to this ideal behavior, even when the solutions are very dilute. Here, the proportionality constant between dHld In c and is different from kT. Nevertheless, Eq. (5.26) can still be used to derive information on the bulk organic concentration necessary to achieve an interface completely saturated with extractant molecules (i.e., a constant interfacial concentration). According to Eq. (5.26), the occurrence of a constant interfacial concentration is indicated by a constant slope in a 11 vs. In c plot. Therefore, the value of c at which the plot n vs. In c becomes rectilinear can be taken as the bulk concentration of the extractant required to fully saturate the interface. 

The interfacial tension of mixed adsorbed films of 1-octadecanol and dodecylammonium chloride has been measured as a function of temperature at various bulk concentrations under atmospheric pressure. The transition interfacial pressure of 1-octadecanol film has been observed to increase with the addition of dodecylammonium chloride and then to disappear. The interfacial pressure vs mean area per adsorbed molecule curves have been illustrated at a constant mole fraction of adsorbed molecules. With the aid of the thermodynamic treatment developed previously, we find that the mutual interaction between 1-octadecanol and dodecylammonium chloride molecules in the expanded state is similar in magnitude to the interaction between the scime kind of film-forming molecules. 

The interfacial pressure II vs the mean area per adsorbed molecule A curves are useful to make clear the film behavior. By making use of Figures 2 and 4, we can obtain the II vs A curves at constant m as shown in Figure 6(a), where II and A are defined, respectively, by... 

Pavlovskaya, G., Semenova, M., Tsapkina, E., Tolstoguzov V. (1993). The influence of dextran on the interfacial pressure of adsorbing layers of 1 IS globulin Viciafaba at the planar w-decane/aqueous solution interface. Food Hydrocolloids, 7, 1-10. 

A similar experiment to that noted above can be performed, but now let the interface be populated by a molecular layer at constant n and known interface electrical potential. A molecule adsorbing at such an interface must do work against the electrical potential barrier, as well as against the interfacial pressure. We get... 

This leads to a rather simple scaling form of the interfacial pressure... 

Considering the interfacial pressure model, the flow rates should be between the higher and lower limits. APpiow can be evaluated by considering the pressure loss in the microchannels. Assuming that the pressure at the opened outlet is atmospheric pressure, Patm, then the pressure P of each phase from the pressure loss AP is expressed as follows ... 

The liquid-liquid interface formed between two immissible liquids is an extremely thin mixed-liquid state with about one nanometer thickness, in which the properties such as cohesive energy density, electrical potential, dielectric constant, and viscosity are drastically changing from those of bulk phases. Solute molecules adsorbed at the interface can behave like a 2D gas, liquid, or solid depending on the interfacial pressure, or interfacial concentration. But microscopically, the interfacial molecules exhibit local inhomogeneity. Therefore, various specific chemical phenomena, which are rarely observed in bulk liquid phases, can be observed at liquid-liquid interfaces [1-3]. However, the nature of the liquid-liquid interface and its chemical function are still less understood. These situations are mainly due to the lack of experimental methods required for the determination of the chemical species adsorbed at the interface and for the measurement of chemical reaction rates at the interface [4,5]. Recently, some new methods were invented in our laboratory [6], which brought a breakthrough in the study of interfacial reactions. 

At very low molecular densities, i.e. at very low Interfacial pressures, the mono-layer exhibits gaseous behaviour. The molecules are far apart, but, unlike in a three-dimensional gas, they are not completely disordered. Because of their amphi-polar nature, the molecules exhibit a preferential orientation relative to the surface-normal. As stated in sec. 3.1, the interfacial pressure exerted by an ideally dilute monolayer is equivalent to the osmotic pressure of an ideal three-dimensional solution. Ideal gaseous monolayer behaviour means obe3dng relation [3.1.1]. 

Equation [3.4.9] may be compcired with [2.2.25]. As with interfacial tensions, the interfacial excess grand potential may play a part in the molecular interpretation of the interfacial pressure. 

A variety of authors have paid attention to the question of how the charging of a monolayer affects the (Helmholtz or Gibbs) energy, and hence the interfacial pressure. (See for instance refs. ) Thermod3mamics can help to answer some of the basic questions that have given rise to unnecessary confusion in the literature. 

The measurements according to the integral method were performed at various values for a, both before and beyond (= 0.39 nm ). Figure 3.93 shows results obtained after 24 h. As in the differential method, the diffusion coefficient at a packing density of 0.38 nm /molecule is =10" m s. At 0.40 nm /molecule the monolayer is still closely packed, but the Interfacial pressure has dropped to nearly zero, and, correspondingly, D has strongly increased. Beyond 0.40 nm / molecule it is almost independent of the cholesterol concentration in the monolayer and equal to = 10" ° m s" ... 

The rate may thus be evaluated from a plot of log A vs t. If measurements of the interfacial pressure, n, are made at a fixed interfacial area (i.e., n — t curves are obtained), the rate of adsorption at a given value of n is... 

Two experimental results indicate that there is an adsorption energy barrier related to the interfacial pressure. First, the presence of an energy barrier becomes evident only after an interfacial pressure of 0.1 mN m-1 is attained (Table II). In the second experiment, different compounds were spread at the air/water interface and the rate of adsorption of pepsin and lysozyme were measured under conditions where charge effects were minimized (MacRitchie and Alexander, 1963a). It was found that the rates of adsorption for these proteins were independent of the nature of the surface film and depended only on the surface pressure. 

When a protein adsorbs from a solution in which the pH is close to its isoelectric point, the rate of adsorption at mobile interfaces is controlled by the rate of diffusion to the interface and the interfacial pressure barrier. However, when the protein molecule takes on a net electrical charge, an additional barrier to adsorption appears, owing to the electrical potential set up at the interface by the adsorbed protein. 

When a protein molecule adsorbs, an area of interface of the order of 100 A has to be cleared for adsorption to occur (Section III,B). It seems reasonable to assume that once an adsorbed molecule has been compressed until its area in the interface, due to pressure displacement of segments, falls below this critical value, it will be unstable in the adsorbed state and will desorb. This transition state for desorption may be reached in two ways (1) at constant interfacial pressure and total area, by fluctuations in energy of the adsorbed molecules about the mean value, resulting in certain molecules achieving the transition state configuration (2) by compression of the film, thus increasing the interfacial pressure and decreasing the molecular area until the latter has been reduced to the critical value. 

When protein solutions are shaken, insoluble protein is often seen to separate out (Bull and Neurath, 1937). The coagulation occurs at the interface and may be observed when protein is allowed to adsorb from solution at a quiescent interface (Cumper and Alexander, 1950) or when spread protein monolayers are compressed (Kaplan and Frazer, 1953). This is an interesting type of phase separation in which a three-dimensional coagulum is formed from the two-dimensional monolayer, once a certain critical value of the interfacial pressure is exceeded. The concentration of protein in the monolayer when the critical pressure is reached may be thought of as the solubility in the interface under those conditions. When this concentration is exceeded, precipitation occurs. A simple model may help to illustrate how free energy considerations govern the coagulation. 

The terms pk)Ai — Pk)vk) oik + < k)Ai otk are referred to as the interfacial pressure difference effect (or the concentration gradient effect) and the combined interfacial shear and volume fraction gradient effect [67] [115], respectively. The interfacial pressure difference effect is normally assumed to be insignificant for the two-fluid model [54, 4, 125, 119]. That is, for two-phase flows one generally assumes that... 

Nevertheless, it is common to neglect the variations in surface tension at the interfaces, and the interfacial pressures are generally related by the Young-Laplace equation [244] [138] [54] ... 

In a similar manner as for the single averaging methods in sects 3.4.1, 3.4.2 and 3.4.3, the stress terms are normally rewritten introducing an interfacial averaged pressure (pk)ei and an interfacial averaged viscous stress (Ofc)ej- In this method the interfacial pressure is normally defined by ... 

Ramshaw and Trapp [172] incorporated surface tension effects, which had traditionally been ignored. Travis et al ]220] considered viscous stresses, La-hey et al [134] considered the added mass force, Prosperetti and Wijngaar-den [167] considered compressibility effects. Trapp [219] considered Reynolds stresses, while Stuhmiller [213], Boure ]22], Pauchon and Banerjee ]161, 162], Sha and Soo [188], Prosperetti and Jones ]168] and Holm and Kupershmidt [109] have put their main focus on the adoption and interpretation of the interfacial pressure forces (see sects 3.4.1 and 3.4.6). The simplest choice for the interfacial pressure distribution pk)A, is to assume that it is equal to the fluid bulk pressure. This implies that (pg)vg = pi)vt = Pq)ai = pi)ai for a two-phase system and as a result the current engineering practice is obtained (e.g., ]168, 125, 119]). However, this approach leads to an equation set involving a single pressure, which has real characteristics only when the two fluid velocities are equal [219]. 

Considering low-pressure bubbly flows a slightly more general assumption is that the interfacial pressure in the gas phase is related to the average pressure of the gas phase by ... 

The appropriate pressure in this case is the capillary pressure given by the Young-Laplace equation. This assumes that there is essentially no effect of the flow on the interfacial pressure change. One of the principal curvatures of the interface is zero hence... 

Clearly, then, we must know the concentration, temperature, and charge distributions at the interface in order to define the surface tension variation required to solve the hydrodynamic problem. However, these distributions are themselves coupled to the equations of conservation of mass, energy, and charge through the appropriate interfacial boundary conditions. The boundary conditions are obtained from the requirement that the forces at the interface must balance. This implies that the tangential shear stress must be continuous across the interface, and the net normal force component must balance the interfacial pressure difference due to surface tension. 

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