Semi-concentrated solution

This concentration is denoted as the critical concentration c The critical concentration marks the transition from a dilute to a semi-concentrated solution. This transition is accompanied by great changes in the flow properties of a polymer solution. At concentrations above c the flow behavior is dominated by the inter-molecular interactions of the polymer coils whereas below c mainly the polymer-solvent interactions determine the flow properties. Nearly all technical applications of polymer solutions require concentrations equal to or above c. For example, the... 

Fedors has proposed a relationship between ( /l and c, which is valid over a wide concentration range and can be used to measure > 1 from the determination of the relative viscosity at a single polymer concentration. Nishio and Wada have observed the transition from intramolecular motion within a polystyrene molecule to a co-operative diffusion process on passing from dilute to concentrated solutions, where D also increased despite a corresponding increase in viscosity. The radius of gyration of polymers in semi-concentrated solutions has been investigated by Aharoni and Walsh. ... 

Under quiescent conditions, polymer solutions are divided into four categories depending on the average distance separating the centers of mass of the molecular coils the dilute, the semi-dilute (or semi-concentrated), the concentrated and the entangled state. 

Taking into account the relevance of the range of semi-dilute solutions (in which intermolecular interactions and entanglements are of increasing importance) for industrial applications, a more detailed picture of the interrelationships between the solution structure and the rheological properties of these solutions was needed. The nature of entanglements at concentrations above the critical value c leads to the viscoelastic properties observable in shear flow experiments. The viscous part of the flow behaviour of a polymer in solution is usually represented by the zero-shear viscosity, rj0, which depends on the con-... 

For semi-dilute solutions, two regimes with different slopes are similarly obtained the powers of M, however, can be lower than 1.0 and 3.4. Furthermore, the transition region from the lower to the higher slope is broadened. The critical molar mass, Mc, for polymer solutions is found to be dependent on concentration (decreasing as c increases), although in some cases the variation appears to be very small [60,63]. 

In semi-dilute solutions, the Rouse theory fails to predict the relaxation time behaviour of the polymeric fluids. This fact is shown in Fig. 11 where the reduced viscosity is plotted against the product (y-AR). For correctly calculated values of A0 a satisfactory standardisation should be obtained independently of the molar mass and concentration of the sample. 

For concentrated solutions of polystyrene in n-butylbenzene, Graessley [40] has shown that the reduced viscosity r red Cnred=(r ( y)- rls)/(rlo rls)) can be represented on a master curve if it is plotted versus the reduced shear rate (3 ((3= y/ ycnt= y-A0). For semi-dilute solutions a perfect master curve is obtained if (3 is plotted versus a slope corrected for reduced viscosity, T corp as shown in Fig. 16. 

Viscoelastic properties have been discussed in relation to molar mass, concentration, solvent quality and shear rate. Considering the molecular models presented here, it is possible to describe the flow characteristics of dilute and semi-dilute solutions, as well as in simple shear flow, independent of the molar mass, concentration and thermodynamic quality of the solvent. The derivations can be extended to finite shear, i.e. it is possible to evaluate T) as a function of the shear rate. Furthermore it is now possible to approximate the critical conditions (critical shear rate, critical rate of elongation) at which the onset of mechanical degradation occurs. With these findings it is therefore possible to tune the flow features of a polymeric solution so that it exhibits the desired behaviour under the respective deposit conditions. 

Based on the analogy between polymer solutions and magnetic systems [4,101], static scaling considerations were also applied to develop a phase diagram, where the reduced temperature x = (T — 0)/0 (0 0-temperature) and the monomer concentration c enter as variables [102,103]. This phase diagram covers 0- and good solvent conditions for dilute and semi-dilute solutions. The latter will be treated in detail below. 

Under good solvent conditions the dynamics of semi-dilute solutions was investigated by NSE using a PDMS/d-benzene system at T = 343 K and various concentrations 0.02 c < 0.25. The critical concentration c as defined by (112) is 0.055. 

In view of the fundamental importance of the Gibbs-Thomson formula, and the magnitude of the discrepancies between the figures calculated from it and the experimental results, it is of obvious interest to inquire to What causes the deviations may be due. The first point to be noticed is that the complex substances which exhibit them most markedly form, at least at higher concentrations, colloidal and not true solutions. It is, therefore, very probable that they may form gelatinous or semi-solid skins on the adsorbent surface, in which the concentration may be very great. There is a considerable amount of evidence to support this view. Thus Lewis finds that, if the thickness of the surface layer be taken as equal to the radius of molecular attraction, say 2 X io 7 cms., and the concentration calculated from the observed adsorption, it is found, for instance, for methyl orange, to be about 39%, whereas the solubility of the substance is only about 078%. The surface layer, therefore, cannot possibly consist of a more concentrated solution of the dye, which is the only case that can be dealt with theoretically, but must be formed of a semi-solid deposit. 

The second position assumes that in semi-diluted solutions the polymeric chains are as much strong intertwined that the all thermodynamic values, in particular the osmotic pressure, achieve the limit (at N —>oc) depending only on the concentration of monomeric links, but not on the chain length. 

However, let note, that the assumption about independence of the osmotic pressure of semi-diluted solutions on the length of a chain is not physically definitely well-founded per se it is equivalent to position that the system of strongly intertwined chains is thermodynamically equivalent to the system of gaped monomeric links of the same concentration. Therefore, both Flory-Huggins method and Scaling method do not take into account the conformation constituent of free energy of polymeric chains. 

Accordingly to (19) the osmotic compressibility dlt / dc into diluted solutions does not depend on the concentration of macromolecules (dft / dc = RT) on the contrary, in semi-diluted solutions it becomes (as it follows from (25)) as linear function of relative concentration ... 

In that way, the thermodynamic approach with the use of conformational term of chemical potential of macromolecules permitted to obtain the expressions for osmotic pressure of semi-diluted and concentrated solutions in more general form than proposed ones in the methods of self-consistent field and scaling. It was shown, that only the osmotic pressure of semi-diluted solutions does not depend on free energy of the macromolecules conformation whereas the contribution of the last one into the osmotic pressure of semi-diluted and concentrated solutions is prelevant. 

That fact the scaling method and presented thermodynamic approach from seeming opposite positions lead to practically the same result in the form (8) and (35) can be named as mysterious incident if it were not two circumstances. First is exactly free energy of the conformation makes the main contribution into the osmotic pressure of the semi-diluted and concentrated solutions. The second is the peculiarity of the point c = c. ... 

Semi-dilute solution p2 - As the concentration rises the rods will begin to interact and their diffusion will become restricted. However, we have not allowed for the excluded volume of the rod and have treated it as a line with no thickness. The excluded volume is of the order of bL2 and until the concentration of rods is such that the particles overlap into this excluded volume the spatial distribution of rods is relatively unaffected ... 

A semi-dilute solution has an entangled aspect similar to a network. An individual chain can be envisioned as constituted by a series of blobs of size equal to the transient network mesh size [16], which obviously decreases with increasing concentration. For c=c , is similar to the chain mean size. For c c, however, the mesh size is independent on the chain length. In a good solvent, according to Eqs. (5) and (6), these conditions are satisfied by ... 

Nj,=N/f is the number of beads per branch or arm). For larger chains, however, the solvent can penetrate in outer regions of the star and the situation within these regions is more Hke a concentrated solution or a semi-dilute solution. These portions of the arms constitute a series of blobs, whose sizes increase in the direction of the arm end. The surface of a sphere of radius r from the star center is occupied by f blobs. Then the blob size is proportional to rf. Most internal blobs are placed in conditions similar to concentrated solutions and, consequently, their squared size is proportional to the number of polymer units inside them as in an ideal chain. This permits one to obtain the density of units inside the blob, as a function of r ... 

This result is appHcable to semi-dilute and concentrated solutions [21], and is also useful to check many simulations that do not include HI. For non-draining chains, introducing Gaussian statistics in Eq. (35), and transforming the summations over a large number of units in Eq. (34) into integrals, the translational friction coefficient can finally be written as [ 15]... 

Most important, however, was the discovery by Simha et al. [152, 153], de Gennes [4] and des Cloizeaux [154] that the overlap concentration is a suitable parameter for the formulation of universal laws by which semi-dilute solutions can be described. Semi-dilute solutions have already many similarities to polymers in the melt. Their understanding has to be considered as the first essential step for an interpretation of materials properties in terms of molecular parameters. Here now the necessity of the dilute solution properties becomes evident. These molecular solution parameters are not universal, but they allow a definition of the overlap concentration, and with this a universal picture of behavior can be designed. This approach was very successful in the field of linear macromolecules. The following outline will demonstrate the utility of this approach also for branched polymers in the semi-dilute regime. 

So far, the effects of semi-dilute solutions are qualitatively clear. Ambiguity comes in, however, when the overlap concentration has to be defined quantitatively. This ambiguity arises from the fact that the volume of a macromolecule cannot be uniquely defined. Because of the segment mobility the shape of a macromolecule varies in time such that only a statistical description can be made. As... 

In ultrafiltration and reverse osmosis, in which solutions are concentrated by allowing the solvent to permeate a semi-permeable membrane, the permeate flux (i.e. the flow of permeate or solvent per unit time, per unit membrane area) declines continuously during operation, although not at a constant rate. Probably the most important contribution to flux decline is the formation of a concentration polarisation layer. As solvent passes through the membrane, the solute molecules which are unable to pass through become concentrated next to the membrane surface. Consequently, the efficiency of separafion decreases as fhis layer of concentrated solution accumulates. The layer is established within the first few seconds of operation and is an inevitable consequence of the separation of solvent and solute. 

Orai concentrate soiutions Roxicodone intensoi, OxyFAST, and Oxydose 20 mg/mL solution are highly concentrated solutions. Take care in prescribing and dispensing this solution strength. Fill dropper to the level of the prescribed dose (1 mL = 20 mg 0.75 mL = 15 mg 0.5 mL = 10 mg 0.25 mL = 5 mg). For ease of administration, add dose to approximately 30 mL (1 fluid oz) or more of juice or other liquid. May also be added to applesauce, pudding, or other semi-solid foods. The drug-food mixture should be used immediately and not stored for future use. 

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