You have to check the bulk density in both phases!! Fox example at constant temperature, how is the behaviour of bulk densities when pressure increase? Remember that the interfacial tension is proportional to the density difference (parachor approach)

Regards

Dear Shahin, let me try to answer your question (but I am cautious in my interpretation :-). I think that the answer is in understanding whether the interface layer becomes compressed or expanded, as compared with bulk phases.

To clarify my point, I will consider two immiscible liquids, X and Y, separated by a Gibbs surface (at a given constant temperature). In this case, as usually, we may define the excess Gibbs free energy at the interface layer and the Gibbs isotherm:

-Ad(gamma)=n(ex,X)d(chem. potential of liquid X)+n(ex,Y)*d(chem. potential of liquid Y),

where n(ex, X or Y) represents the interface excess of component X or Y as obtained by extrapolating the bulk phases to the separating plane; A is the surface area of this plane, and "gamma" is the interface tension; "d"indicates the differential.

When you change the pressure, the chemical potentials of both components in their bulk phases change, too, with the derivatives equal to the molar volumes of component X or Y (v(X) and v(Y), respectively). Therefore, the derivative of interface tension by pressure will be

 –(n(ex,X)*v(X)+n(ex,Y)*v(Y)).

Lets' assume that the interface layer becomes strongly compacted, due to mutual interactions between components X and Y. In this case, the excess amounts n(ex,X or Y) should be positive, thus indicating this "concentrating" at the interface. Therefore, the overall derivative is negative, and the increasing pressure will reduce  the interface tension. But, on the other hand, we would expect that such strong mutual interactions resulting to compaction of the interface layer would lead to miscibility of liquids X and Y, isn't? This case woul be probably correct for a solid-liquid contact, with a high affinity.

However, water and octane dislike each other, and may want to keep a distance, at least, at the molcular level. We may easily imagine that the interface layer becomes "expanded", somehow more "gas-like", with a reduced density as compared with densities of separated  bulk phases. In this case, the excess amounts of lqiuids X and Y will become negative. The overall derivative will be, in opposite, positive! Pressure will increase the interface tension!

The same conclusion I could obtain if I would use considering the thin interface layer characterized by its volume instead of considering the Gibbs dividing plane. In this case

-Ad(gamma)=n(X)d(chem. potential of liquid X)+n(Y)*d(chem. potential of liquid Y)+V(interface layer)dP,

where n(X) and n(Y) denote the amounts of components X and Y in this thin interface layer (not the excesses!), and V is its volume; the derivative of the interface tension will be given as

 –(n(X)*v(X)+n(Y)*v(Y))+V(interface layer)..

The first two terms calculate the volume of the interface layer, based on the amounts of components X and Y present in this layer, where the mol amounts are multiplied by the molar volumes of the bulk liquids X and Y! The third term is just the actual volume of the interface layer. If the overall sum is positive, I,.e., the interface layer is expanded, as compared with a hypothetical layer having the same composition as a real one but characterized with molar volumes of components coinciding with those in the bulk phases, then, increasing pressure will increase the interface tension. That is exactly the case of two immiscible liquids tending to repulse each other. What would you say? Your interesting question stimulated me to recall some 'strange" phenomena in the transport through the alkane-water interface. Now I begin to think that the explanation is in this expanded ("gas-like") interface. Well, more than this! It is due to freezing water at the interface with a saturated hydrocarbon! The "old, classic" interpretation of hydrophobic effect   of non-polar solutes in aqueous environments involved some claims that water is "frozen" around non-polar solutes. Therefore, water may be also  "frozen" at the water-alkane interface where certainly a hydrophobic effect takes place. . Ice has smaller density as compared with bulk water; here we are – that is the reason for expanding the interface layer, and for an apparent anomaly - increasing the interface tension with increasing pressure. You have a method to quantify the volumetric changes in the interface layer :-).

Thank you professor Borisover for your detailed and useful answer.