Concentration between plates

The first virial coefficient term The first term (/,) in equation (10.16) has an osmotic character since its contribution to the force between the plates, obtained from — dJJdh), is 2 RT r%/h = RTc2, where C2 = mean polymer concentration between plates. Dolan and Edwards (1974 1975) obtained a similar term but, due to an algebraic error, they presented the term with the opposite sign, implying not a repulsive interaction but rather an attraction. 

Plate and frame coolers using HasteUoy C-276 plates have been used successfuUy. Anodically protected plate coolers are available as weU as plate coolers with plates welded together to minimize gasketing. Another promising development is the introduction of plate coolers made of HasteUoy D205 (105). This aUoy has considerably better corrosion resistance to concentrated sulfuric acid at higher temperatures than does C-276. Because of the close clearance between plates, cooling water for plate coolers must be relatively clean. 

T] h = distance between plates. Use log mean concentration difference. 

We have been able to derive an approximate analytical expression for A as a function of which describes the curves shown in Figure 3 quite accurately over nearly the whole concentration range, from dilute solutions up to 0.6. The derivation is based upon the greatest eigenvalue of the matrix which, in the case of nonadsorbing long chains at low concentrations between the plates, is easily computed (19). We will report the derivation elsewhere and give here only the result, which in the present context may be considered as an empirical expression ... 

This model assumes that the air/water interface from the blade to the Wilhelmy plate can be divided into a number of equal small cells. We apply a simple argument that the rate of mass transfer by diffusion is proportional to the difference in concentration between the neighboring cells, while the concentration and the surface pressure within each cell are assumed homogeneous. 

We perform a series of experiments. We purchase a stock of identical plates and prepare several solutions of beetroot juice of varying concentration. One plate is soaked per solution. The chemisorptive bond between the dye and plate is strong, and equilibrium is reached after only a few seconds. The excess juice is decanted off for analysis, e.g. by means of optical spectroscopy and the Beer-Lambert law, provided we know the extinction coefficient s for the juice. 

Dissolve the antigen (pure protein or hapten-carrier conjugate) with a concentration between 0.1 and 50 pg/ml in Soln. A. Pipet 0.1-0.2 ml of the antigen solution into the wells of a 96-well microtiter plate (high binding capacity) and incubate on a shaker at RT for 1 h or at 4 °C in a humid chamber overnight. 

Prepare a solution of affinity-purified anti-RIP antibody at a concentration between 2 and 4 pg antibody/mL in carbonate/bicarbonate buffer Add 100 pL to each well of a microtiter plate This step requires at least 10 mL of the solution per plate (see Note 17). 

The orifice-type flow measurement has been modified, and new, special-purpose devices have been introduced to meet particular process requirements. One such unique design is the annular orifice used to measure the hot and dirty gases in the steel industry. Here, the process flow passes through an annular opening between the pipe and a disk-shaped, concentrically located plate the pressure difference is detected between the upstream and downstream faces of that disk. (Refer to Table 3.81 for recommendations on selecting the right orifice plate design for various applications.)... 

Often, the total capacitance is defined to be Q/V, which in the present example has the same value as the differential capacitance. This is true even for geometries other than planar parallel plates, as for example, in cases of capacitors having concentric cylindrical plates or concentric spherical plates. It is only in the realm of space charge where we must be concerned with differences between the total capacitance and the differential capacitance. The space charge situation will be discussed later. 

Osmotic pressure arises because the total concentration of all ions [irrespective of their charge, i.e., c+ + c = 2n cosh (e%jkT) is always higher near the surface than in the reservoir (where c, + c = 2n). Thus the liquid in the reservoir tends to flow into the space between plates to dilute this concentration and thereby force the plates apart. So the osmotic pressure contribution is always repulsive. Of course, the sign of the net force will depend on whether the attractive Maxwell stress or the repulsive osmotic pressure is larger. 

Since the Hamaker constant for proteins in a physiological solution is of the order of 10 20 J, Eq. (18) provides a concentration of approximately 2 M even for the large value of /(=31 X10 50 Jm3 corresponding to Cl- ions [14]. This corresponds to an ideal gas pressure of approximately 50 atm. Consequently, the long-range repulsion or attraction between plates due to the van der Waals interactions between a gas of neutral particles and the plates is expected to be negligible in most cases. 

At very large separations (d /), the attraction induced by the van der Waals interactions of ions, which decays only with the second power of the distance (Eq. (16)), becomes much larger than the traditional van der Waals attraction between plates, and its magnitude is proportional to the electrolyte concentration. 

The hydration force is important for distances between plates less than a few nanometers. Since the DLVO potential barrier between spherical particles or droplets is located at separation distances of the order of the Debye length, it is clear that at least at high electrolyte concentrations the hydration force becomes relevant. The qualitative effect of hydration was earlier recognized, regarding the stability of silica colloids at high electrolyte concentrations1 or the stability of amphoteric latex particles at high concentrations of some electrolytes.12... 

If the polymers are charged, the segment concentration and the force between plates can be calculated numerically only. 

Fig. 4. The segment concentration distribution between plates for various salt concentrations. Only half space from one plate to the midway is plotted. >=12 A, and the other parameter values used in calculations are as for Fig. 3. (1) c =0.001 M (2) c=0.01 M (3) c=0.1 M (4) c=l M.

The effect of salt concentration on the force is presented in Fig. 3 for a Hamaker constant of 0.5 x 10 20 J. As the salt concentration increases, the force between plates at constant surface potential becomes more attractive at large distances. 

The concentration of the segments is plotted in Fig. 7 for D=20 A. As expected, the polymer segment concentration at the middle is larger for higher bulk concentrations. A higher segment concentration can lead to an increased attraction between plates because it increases the number of bridges. 

However, it also leads to a higher concentration of counterions dissociated from the polyelectrolyte that generate repulsion [the fourth term on the right-hand side of Ecj. (17)]. This higher concentration can generate a higher repulsion between plates for the same electrical potential at the middle. 

The stiffness of the chains, represented by the term Hi) in Eq. (2), together with the electrical and van der Waals interactions determine the conformation of the polyelectrolyte chains and hence their charge and segment concentration distributions in solution. The chains become more extended with increasing stiffness. As a result, at short distances, the force between plates is repulsive for... 

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