Pore potential

The concept of a pore potential is generally accepted in gas adsorption theory to account for capillary condensation at pressures well below the expected values. Gregg and Sing ° described the intensification of the attractive forces acting on adsorbate molecules by overlapping fields from the pore wall. Adamson has pointed out that evidence exists for changes induced in liquids by capillary walls over distances in the order of a micron. The Polanyi potential theory postulates that molecules can fall into the potential field at the surface of a solid, a phenomenon which would be greatly enhanced in a narrow pore. 

In mercury porosimetry it was proposed that the pore potential prevents extrusion of mercury from a pore until a pressure less than the extrusion pressure is reached. Similarly, the pore potential, when applied to gas adsorption, is used to explain desorption at lower relative pressures than adsorption for a given quantity of condensed gas. 

In mercury porosimetry the pore potential can be derived as follows. The force, F, required for intrusion into a cylindrical pore is given by 

If the pore potential, U, is the difference between the interaction of the mercury along the total length of all pores with radius r when the pores 

Expressing force as pressure tim area for pores in a radius interval with mean radius rand mean length /, equation (12.21) can be rewritten as 

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The fluid-pore potential has been chosen as a sum of interactions of the fluid with two commensurate parallel crystalline pore walls. The potential of interaction between fluid particles and the pore walls is given by... 

Thus, the expression for the pore potential is identical to the pressure-volume work differences between intrusion and extrusion and can be expressed as the hysteresis energy, i.e. 

Equation (12.19) predicts that changes in U, the pore potential, will effect the quantity of entrapped mercury and/or the difference in contact angle between intrusion and extrusion. Hence, changes in the pore potential will alter the size of the hysteresis loop. 

Evidence for the pore potential was experimentally obtained from mercury intrusion-extrusion data by impregnating various samples with... 

The effect of pore impregnation with nonpolar material was studied by treating samples with dichlorodimethylsilane (DCDMS). In each case a decrease in hysteresis area, compared to the untreated material, was observed after coating samples with DCDMS. The increases in the extrusion contact angle, with DCDMS compared to untreated sample, resulted in decreases in W q. In some cases impregnation with DCDMS led to greater mercury retention or an increase in IF, over the untreated material. However, this was always accompanied by a larger decrease in and thus a decrease in the pore potential. 

The double-layer model of the membrane consists of many particles (assume a diameter of 0.1 gm) that are impenetrable for solution and carry a surface charge. Electrical double layers exist around each particle, and because the dimensions of the membrane pores are of the same order as the double layers around the particles, double layers exist throughout the membrane pores. The potential measured by the underlying ISFET, with respect to the bulk potential, is on one hand determined by the mean pore potential, which is the net result of the contribution of all surface potentials of the charged particles, and on the other hand by the pH at the membrane-1 SFET interface. The measured ISFET response in equilibrium is therefore the same as that of an ISFET without a membrane, because the distribution of the protons between membrane and solution results in a pH difference, which compensates the mean membrane potential (this is the same mechanism as in the Donnan model). The relation between the surface charge on the particles a (C/cm2) and the surface potential jx of each particle is given by... 

First, assume that the surface charge on the membrane particles does not interact with the mobile protons (no proton release or uptake). An ion step will result in an increase in the double-layer capacitances of the particles and consequently in a decrease of the surface potentials fr, because the charge densities remain constant. The ISFET will measure a transient change in the mean pore potential. As a result of the potential changes, an ion redistribution will take place and the equilibrium situation is re-established. The theoretical maximum ion step response is the change in the mean pore potential. This is comparable with the Donnan model where the theoretical maximum is determined by the change in the Donnan potential at the membrane solution interface. 

The excess potential /l is defined as the difference between the pore potential ore and that for a fictitious pore whose wall was made up of the liquid s solid state liquid- The underlying assumption includes The perturbation in the structure of the solid phase is negligible in the slit pore thus the temperature dependence of the chemical potential of solids in slit pore can be expressed with the entropy of the bulk solid Sj. Further details on the model are available in Ref 3. 

The above dependence of elevation against pore size would be consistent with our perspective of the importance of the pore potential because stronger attraction occurs in a smaller pore. The idea is tested more directly, employing additional two kinds of walls with weaker attraction. In such pores the extent of the elevation is thought to decrease, and a depression in freezing temperature may arise. One is a smooth pore wall made up of U-methane molecules expressed again by 10-4-3... 

More intuitively the elevating effect of the attractive pore potential can be understood as follows. In a pore with strongly attractive potential, a liquid-like state can hold even with lower vapor pressure than the saturated one. When this system is equilibrated with pure liquid or saturated vapor, the excess potential must be balanced with the increase in density... 

The wall curvature of the cylinder pore lowers the pore potential in relation to the slit-like pore of the same width. Consequently the molecular DFT approach calculates that the fluid fills the cylinder pore up at lower pressures as the slit-like pore of the same width. This differences grow in direction of smaller pores. They vanish in the limit H oo oi the bulk phase. 

Under these conditions, overlapping pore potentials compress the adsoihate molecules into a smaller volume than they would otherwise adsorbate molecules into a smaller volume than they would otherwise occupy. The concept of surface area becomes meaningless and the limiting amount adsorbed is a measure of micropore volume rather than monolayer surface. The determined volumes will be higher than the true pore volumes, since the adsorbate molecules will be in a condensed liquid state which may approach the volume they would occupy in the solid state. Type 1 isotherms may also occur for adsorption on high energy level surfaces [S]. 

Pore potential, hi gas adsorption, it is estaUished that overiapping... 

During pressurization the mercury column is under compression as it intrudes, whilst under depressurization the column is under tension, due to pore potential, and can break if the pore potential is high and the pore radius very small. 

In the case of hysteresis, it is necessary that the pore potential causes mercury to extrude from a pore at a lower pressure than it intrudes. Any shape of pore can result in hysteresis and cause mercury entrapment in the first intrusion. Surface roughness. This can cause the mercury to slip-stick so that the mercury thread is broken. 

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