Critical line force

This report is concerned with contact angle hysteresis and with a closely related quantity referred to as "critical line force (CLF)." More particularly, it is concerned with the relationship between contact angle hysteresis and the magnitude of the contact angle itself. Two sets of liquid-solid-vapor systems have been investigated to provide the experimental data. One set consists of Teflon [poly(tetrafluoroethylene), Du Pont] and a series of liquids forming various contact angles at the Teflon-air interface. The second set consists of polyethylene and a similar series of liquids. In neither case was the ratio of air to test liquid vapor at the boundary line controlled, but it can be assumed that the ambient vapor phase operative in all the systems was close to an equilibrium mixture. 

The exact form of functional relationship between the resistance force and the contact angle hysteresis depends, of course, on the parameters chosen. Rosano has presented an equation based on the plate method [14] and very recently Furmidge[10] has shown the essential relationship between Rosano s equation and the widely used equation for a droplet moving on an inclined plane. Since the total resistance force that must be overcome before the boimdary line will move is directly proportional to the length of the line, we can tentatively define a force just necessary to start a unit length of boundary line moving as the critical line force. 

In studying this critical line force we have chosen to use the device first described by Jamin as shown in Figure 1 [13]. This consists of a horizontally placed cylindrical tube containing a series or "chaplet of individual liquid indexes separated by air spaces. An air pressure differential is applied across the chaplet, and the velocity of the last index on the low pressure side is noted as a function of increasing pressure differential. 

When the minimum pressure necessary to move the boundary line Pi - P2 had been established, the critical line force was calculated as follows ... 

Figure 7. Apparatus for measuring critical line force...

The relationship between contact angle hysteresis and critical line force in this cylindrical tube system is easily derived from the geometry of the system and the Laplace equation for capillary pressure. It is... 

Figure 8, Critical line force vs. cosine of equilibrium contact angle for liquids on poly ethylene (top) and Teflon (bottom)...

Calculation of Advancing and Receding Contact Angles from Experimentally Determined Relationships among Surface Tension, Equilibrium Contact Angle, and Critical Line Force... 

Figure 10. Relation of critical line force to contact angle hysteresis 0 a cos Q R - 6 A (bottom) for test liquids...

These fluctuations, which are referred to as order-parsmeter fluctuations in studies of critical phenomena (3). comprise the driving forces for transport in the system. For liquid mixtures near a critical mixing point, the order parameter is concentration, and for pure gases near the vapor-liquid critical point, the order parameter is density. For gas mixtures such as supercritical solutions near the critical line, the order parameter is again density, which is a function of composition and temperature compared to a pure gas where density is a function of only temperature at constant pressure. 

Section A-B in Fig. 2 shows that the solubility falls as the contaminant is diluted by the fluid. The rapid rise in solubility in Sec. B-C occurs at pressures quite higher than the critical pressure because of the rapid rise in density, and therefore solvating power, of the SCF at around this pressure. This r on has been defined by King as the threshold pressure which is the pressure at which the solute begins to dissolve in the SCF. l Obviously, this pressure is technique dependent and varies with the analytical method sensitivity used to measure the solute concentration in the SCF. A decrease in solubility, as shown in r on C-A may occur at higher pressures due to r ulsive forces that may squeeze the solute out of solution. For moderately volatile solutes, a rise in solubility, as shown in section i>- , can occur if there is a critical line in the mixture phase diagram at higher pressures. 

If we take account of the relatively large molar volumes in the critical region, it does not seem worthwhile to try to improve the calculations by taking account of differences in sizes of cells as in Ch. X. We must also keep in mind that the applicability of the ideas developed in Ch. IX-X to critical conditions is somewhat uncertain because the number of first neighbours around each molecule has already decreased to about one half of its value in the solid state. For this reason fluctuations in the intermolecular forces will play a more important role (cf. Ch. VII, 6). We shall however obtain some interesting results about the influence of intermolecular forces on thc critical line which seem at least in qualitative agreement with the few published experimental data. 

Using (12.6.6)-(12.6.7) we have now the final explicit form for the critical line in terms of intermolecular forces... 

First indirect experimental observations of the critical Casimir force were made by Chan and Garcia [152]. They measured the thickness of He films on a copper substrate and detected a thinning of the films close to the critical point of transition to superfluidity, indicating an attractive critical Casimir force. For a He/ He mixture close to the tricritical point, the same authors found a repulsive critical Casimir force, which caused film thickening on the copper substrate [153] (for a later, refined theoretical analysis, see Ref [154]). The tricritical point is the point in the phase diagram where the superfluidity transition line terminates at the top coexistence line of He/He. 

Certain boilers employ forced circulation, whereby a pump helps impart the circulation through the downcomer lines to the waterwaH header, particularly to improve or control circulation at low loads. Forced-circulation pumps are also required in high pressure and supercritical pressure boilers, because once the pressure within a boiler approaches the critical pressure, 22.1 MPa (3208 psia), the densities of the water and steam become similar, limiting or eliminating the potential for natural circulation. 

Flexible rotors are designed to operate at speeds above those corresponding to their first natural frequencies of transverse vibrations. The phase relation of the maximum amplitude of vibration experiences a significant shift as the rotor operates above a different critical speed. Hence, the unbalance in a flexible rotor cannot simply be considered in terms of a force and moment when the response of the vibration system is in-line (or in-phase) with the generating force (the unbalance). Consequently, the two-plane dynamic balancing usually applied to a rigid rotor is inadequate to assure the rotor is balanced in its flexible mode. 

The turbulent regime for Cq is characterized by the section of line almost parallel to the x-axis (at the Re" > 500). In this case, the exponent a is equal to zero. Consequently, viscosity vanishes from equation 46. This indicates that the friction forces are negligible in comparison to inertia forces. Recall that the resistance coefficient is nearly constant at a value of 0.44. Substituting for the critical Reynolds number, Re > 500, into equations 65 and 68, the second critical values of the sedimentation numbers are obtained ... 

The shear forces are mainly in the range of 1 to lONm. This exposure causes cell death between 20 and 80% depending on the exposure duration which is between a few seconds and several hours. Studies performed in a bioreactor have an exposure duration of several days. The results are partly contradictory. Tramper et al. [30] found a critical stress level of 1.5 Nm" for insect cells, whereas Oh et al. [31] could not show an influence on hybridoma cells even at high stirrer speed. This shows that each cell line reacts different and that there is a necessity for defined stress systems if the results is to be comparable. 

FIG. 10 Critical force needed to rupture all the bonds as a function of (3, the angle at which the force is exerted. Simulation results are given by solid circles. Tangential critical force TCF/cos P is also plotted for comparison (solid line). (Reprinted with permission from Ref. 35. Copyright 1996 American Chemical Society.)... 

To continuously separate FT wax products from ultrafine iron catalyst particles in an SBCR employed for FTS, a modified cross-flow filtration technique can be developed using the cross-flow filter element placed in a down-comer slurry recirculation line of the SBCR. Counter to the traditional cross-flow filtration technique described earlier, this system would use a bulk slurry flow rate below the critical velocity, thereby forcing a filter cake of solids to form between the filter media and the bulk slurry flow, as depicted in Figure 15.2b. In this mode, multiple layers of catalyst particles that deposit upon the filter medium would act as a prefilter layer.10 Both the inertial and filter cake mechanisms can be effective however, the latter can be unstable if the filter cake depth is allowed to grow indefinitely. In the context of the SBCR operation, the filter cake could potentially occlude the slurry recirculation flow path if allowed to grow uncontrollably. 

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