Concentrated solution, diffusion behavior

In many process design applications like polymerization and plasticization, specific knowledge of the thermodynamics of polymer systems can be very useful. For example, non-ideal solution behavior strongly governs the diffusion phenomena observed for polymer melts and concentrated solutions. Hence, accurate modeling of... 

Under conditions of nonlimiting interfacial kinetics the normalized steady-state current is governed primarily by the value of K y, which is the relative permeability of the solute in phase 2 compared to phase 1, rather than the actual value of or y. In contrast, the current time characteristics are found to be highly dependent on the individual K. and y values. Figure 16 illustrates the chronoamperometric behavior for K = 10, log(L) = —0.8 and for a fixed value of Kf.y = 2. It can be seen clearly from this plot that whereas the current-time behavior is sensitive to the value of Kg and y, in all cases the curves tend to be the same steady-state current in the long-time limit. This difference between the steady-state and chronoamperometric characteristics could, in principle, be exploited in determining the concentration and diffusion coefficient of a solute in a phase that is not in direct contact with the UME probe. 

As the concentration of MeOH increases, the divergent diffusion behavior between the two membrane types is a reflection of fhe difference in MeOH solubility and its concentration dependence within each membrane. This was verified by solvenf upfake measurements. Upon increasing MeOH concentration, Nafion 117 showed a steady increase in mass, while a sharp drop in total solution uptake was observed for BPSH 40. The lower viscosity of MeOH also affecfs fhe fluidity of the solution within the pores. The constant solvent uptake and the increased fluidity of the more concentrated MeOH solutions accounted for fhe slight increase in diffusion coefficienf of Nafion 117. For BPSH 40, increasing the MeOH concentration resulted in a decrease in MeOH diffusion. The solvent uptake measurements showed very similar behavior, indicating that the membrane excludes the solvent upon exposure to higher MeOH concentrations. 

The dynamic behavior of liquid-crystalline polymers in concentrated solution is strongly affected by the collision of polymer chains. We treat the interchain collision effect by modelling the stiff polymer chain by what we refer to as the fuzzy cylinder [19]. This model allows the translational and rotational (self-)diffusion coefficients as well as the stress of the solution to be formulated without resort to the hypothetical tube model (Sect. 6). The results of formulation are compared with experimental data in Sects. 7-9. 

Fig. 14. Comparison of the convection-diffusion behavior of acetonitrile and hemoglobin within an ATR flow-through cell as calculated by the convection-diffusion model described in the text. The concentrations of the two molecules were periodically varied between zero and a non-zero value with a frequency of 67 mHz the flow rate was 1.5mL/min. Dark areas represent high concentrations of the solute molecules (65).

Equation 4.2 can take various forms, depending upon the behavior of D. The simplest case is when D is constant. However, as discussed below, D may be a function of concentration, particularly in highly concentrated solutions where the interactions between solute atoms are significant. Also, D may be a function of time for example, when the temperature of the diffusing body changes with time. D may also depend upon the direction of the diffusion in anisotropic materials. 

The analytical solutions to Fick s continuity equation represent special cases for which the diflusion coefficient, D, is constant. In practice, this condition is met only when the concentration of diffusing dopants is below a certain level ( 1 x 1019 atoms/cm3). Above this doping density, D may depend on local dopant concentration levels through electric field effects, Fermi-level effects, strain, or the presence of other dopants. For these cases, equation 1 must be integrated with a computer. The form of equation 1 is essentially the same for a wide range of nonlinear diffusion effects. Thus, the research emphasis has been on understanding the complex behavior of the diffusion coefficient, D, which can be accomplished by studying diffusion at the atomic level. 

Additionally in pervaporation separation follows the solution-diffusion mechanism. Therefore the molecular size of the permeating molecules becomes very important to characterize the permeation behavior [43], It is known that acetic acid has larger molecular size (0.40 nm) than water molecules (0.28 nm). As the amount of acetic acid increases in the feed mixture it becomes difficult for acetic acid molecules to diffuse through the less swollen membrane, so separation factor increases at high acid concentrations. 

The proposed model consists of a biphasic mechanical description of the tissue engineered construct. The resulting fluid velocity and displacement fields are used for evaluating solute transport. Solute concentrations determine biosynthetic behavior. A finite deformation biphasic displacement-velocity-pressure (u-v-p) formulation is implemented [12, 7], Compared to the more standard u-p element the mixed treatment of the Darcy problem enables an increased accuracy for the fluid velocity field which is of primary interest here. The system to be solved increases however considerably and for multidimensional flow the use of either stabilized methods or Raviart-Thomas type elements is required [15, 10]. To model solute transport the input features of a standard convection-diffusion element for compressible flows are employed [20], For flexibility (non-linear) solute uptake is included using Strang operator splitting, decoupling the transport equations [9],... 

A variety of RO membrane models exist that describe the transport properties of the skin layer. The solution-diffusion model( ) is widely accepted in desalination where the feed solution is relatively dilute on a mole-fraction basis. However, models based on irreversible thermodynamics usually describe membrane behavior more accurately where concentrated solutions are involved.( ) Since high concentrations will be encountered in ethanol enrichment, our present treatment adopts the irreversible thermodynamics model introduced by Kedem and Katchalsky.(7.)... 

Reverse osmosis is simply the application of pressure on a solution in excess of the osmotic pressure to create a driving force that reverses the direction of osmotic transfer of the solvent, usually water. The transport behavior can be analyzed elegantly by using general theories of irreversible thermodynamics however, a simplified solution-diffusion model accounts quite well for the actual details and mechanism in most reverse osmosis systems. Most successful membranes for this purpose sorb approximately 5 to 15% water at equilibrium. A thermodynamic analysis shows that the application of a pressure difference, Ap, to the water on the two sides of the membrane induces a differential concentration of water within the membrane at its two faces in accordance with the following (31) ... 

Many surfactant solutions show dynamic surface tension behavior. That is, some time is required to establish the equilibrium surface tension. If the surface area of the solution is suddenly increased or decreased (locally), then the adsorbed surfactant layer at the interface would require some time to restore its equilibrium surface concentration by diffusion of surfactant from or to the bulk liquid. In the meantime, the original adsorbed surfactant layer is either expanded or contracted because surface tension gradients are now in effect, Gibbs—Marangoni forces arise and act in opposition to the initial disturbance. The dissipation of surface tension gradients to achieve equilibrium embodies the interface with a finite elasticity. This fact explains why some substances that lower surface tension do not stabilize foams (6) They do not have the required rate of approach to equilibrium after a surface expansion or contraction. In other words, they do not have the requisite surface elasticity. 

The behavior of the diffusion coefficient in supersaturated solutions can be explained in two different ways, one based on thermodynamics, and the second based on metastable solution structure and nucleation theory. If we think of this thermodynamically, it is useful to look at equations used to predict concentration-dependent diffusion coefficients. Two examples are listed below... 

Capone [219] has summarized more recent analysis of the diffusion behavior, and an example is the work by Baojin et al. [249]. The rate of diffusion is modeled from cylindrical coordinates again based on Pick s law. The composition of actual filaments from the spin bath was analyzed, and the coagulant was a DMP water system. Correlations are presented for diffusion coefficients and flux ratios as functions of jet stretch, polymer solution concentration, and coagulation temperature. The flux ratios, they reported, are similar to those reported in Paul s data, 20 years earlier. The diffusion coefficients are in the same range of 4-10 X lO cm /s that Paul found for DMAC-H2O systems. 

---