Real Molecules in Dilute Solution

Two segments of a given polymer molecule cannot occupy the same space and, indeed, experience increasing repulsion as they move closer together. Hence the polymer has around it a region into which its segments cannot move or move only reluctantly, this being known as the excluded volume. The actual size of the excluded volume is not fixed but varies with solvent and temperature. 

Typically in solution, a polymer molecule adopts a conformation in which segments are located away from the centre of the molecule in an approximately Gaussian distribution. It is perfectly possible for any given polymer molecule to adopt a very non-Gaussian conformation, for example an all-trans extended zig-zag. It is, however, not very likely. The Gaussian set of arrangements are known as random coil conformations. 

Solvents for a particular polymer may be classified on the basis of their 9 temperatures for that polymer. Solvents are described as good if 01ies well below room temperature they are described as poor if 0is above room temperature. 

In good solvents, a polymer becomes well solvated by solvent molecules and the conformation of its molecules expands. By contrast, in poor solvents a polymer is not well solvated, and hence adopts a relatively contracted conformation. Eventually of course, if the polymer is sufficiently poor the conformation becomes completely contracted, there are no polymer-solvent interactions, and the polymer precipitates out of solution. In other words, the ultimate poor solvent is a non-solvent. 

One factor which affects the extent of polymer-solvent interactions is relative molar mass of the solute. Therefore the point at which a molecule just ceases to be soluble varies with relative molar mass, which means that careful variation of the quality of the solvent can be used to fractionate a polymer into 

In good solvents, a polymer becomes well solvated by solvent molecules and the conformation of its molecules expands. By contrast, in poor solvents a polymer is not well solvated, and hence adopts a relatively contracted conformation. Eventually of 

One factor which affects the extent of polymer-solvent interactions is relative molar mass of the solute. Therefore the point at which a molecule just ceases to be soluble varies with relative molar mass, which means that careful variation of the quality of the solvent can be used to fractionate a polymer into fairly narrow bands of polymer molar masses. Typically, to carry out fractionation, the quality of the solvent is reduced by adding non-solvent to a dilute solution of polymer until very slight turbidity develops. The precipitated phase is allowed to settle before removing the supernatant, after which a further small amount of non-solvent is added to the polymer solution. Turbidity develops once again, and again the precipitated phase is allowed to settle before removal of the supernatant. Using the technique polymers can be separated, albeit slowly, into fractions of fairly narrow relative molar mass. 

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Figure 3-13. Bead-and-spring representation of a real polymer molecule in dilute solution.

In order to discuss the behavior of single chains in solution, an appropriate geometric description must be chosen. Because only global dimensions are being considered, it is convenient to adopt the model of a chain molecule that consists of m subchains that are themselves long enough to exhibit the asymptotic behavior of a random coil. Each subchain is characterized by an end-to-end vector, f, and the total end-to-end vector, R, is the vector sum of the subchain vectors (Equation 2.26). It is also necessary to specify the location of they enter of mass of each subchain, R. The intersubunit vectors, R i = Ric Ri, are important quantities in the description of the global conformational state of a real-chain molecule in dilute solution. 

In a real situation, the motion of the segments of a chain relative to the molecules of the solvent environment will exert a force in the liquid, and as a consequence the velocity distribution of the liquid medium in the vicinity of the moving segments will be altered. This effect, in turn, will affect the motion of the segments of the chain. To simplify the problem, the so-called free-draining approximation is often used. This approximation assumes that hydrodynamic interactions are negligible so that the velocity of the liquid medium is unaffected by the moving polymer molecules. This assumption was used in the model developed by Rouse (5) to describe the dynamics of polymers in dilute solutions. 

With respect to the above it is noteworthy that Kent et al. [107] performed their study using narrow polydispersity probe and matrix polymers. The insensitivity of Rg versus polymer concentration below C only occurs if the molar mass of the probe and background polymer are similar. If the matrix polymer is of much lower molar mass, it can freely penetrate the probe polystyrene molecules and act as a poor viscous solvent inside the probe coils. In that case a decrease in Rg can also be observed at polymer concentration below C [107], In real free-radical polymerizations, polymer molecules with a wide variety of molar masses will be present simultaneously and it can thus be expected that all macroradicals will experience coil contraction to some extent in dilute solutions (except for the very smallest macroradicals). The magnitude of this effect will thus be dependent upon the molecular weight distribution (MWD) of the polymer and thus also upon the systems polymerization history. 

The analogy with the virial expansion of PF for a real gas in powers of 1/F, where the excluded volume occupies an equivalent role, is obvious. If the gas molecules can be regarded as point particles which exert no forces on one another, u = 0, the second and higher virial coefficients (42, Azy etc.) vanish, and the gas behaves ideally. Similarly in the dilute polymer solutions when w = 0, i.e., at 1 = , Eqs. (70), (71), and (72) reduce to vanT Hoff s law... 

We have performed also a reaction field DFT/Molecular Dynamics simulation of this system. We found that after an initial time, when the complex oscillates within the cage at R(N-H) 2.0 a.u. and R(N-C1) 6.0 a.u., a small temperature variation is enough for allowing the complex to overcome the small energetic barrier and, with time, the distance between Cl" and the NH4 fragments starts to increase. Extrapolating to a real solution environment, the two fragments will be completely surrounded by water molecules, i.e. in a solution at infinite dilution the two ions are fully solvated. 

It is possible however to analyze mathematically well defined models which we hope will give a correct approximation to real physical systems. In this section, we shall be concerned with the simplest case the zeroth-order conductance of electrolytes in an infinitely dilute solution. We shall describe this situation by assuming that the ions—which are so far from each other that their mutual interaction may be completely neglected—have a very large mass with respect to the solvent molecules we are then confronted with a typical Brownian motion problem. 

Equation 6-24 and the equations that follow from it apply to molal activities. However, the concentration can be substituted for activity in very dilute solution where the behavior of the dissolved molecules approximates that of the hypothetical ideal solution for which the standard state is defined. For any real solution, the activity can be expressed as the product of an activity coefficient and the concentration (Eq. 6-26). 

For future theoretical developments in the field of transport properties of binary and higher-order mixtures, the simplest case seems to study the influence of the variation in mass of one of the species on the transport properties. Without a full understanding of this pure mass effect on the transport properties, it is not possible to analyze the effect of the translational-rotational coupling in real molecules. Toward this goal the simplest system that can be considered is a binary mixture at infinite solute dilution where the effect of the solute-solvent mass ratio on the solute diffusion can be studied. 

Computer simulations, such as molecular dynamics (MD) simulations, are helpful tools for investigating the growth mechanism of gas hydrates at the molecular scale. So far, MD simulations of the growth of a CH4 hydrate from a concentrated aqueous CH4 solution were carried out at a temperature much lower than 0 °C. However, in real systems, gas hydrates are grown from a two-phase coexistence of liquid water and a gas at temperatures above 0 and for most gas species, the thermodynamically stable concentration of gas molecules in liquid water is much lower than that in a gas hydrate. Therefore, simulations for understanding of the growth mechanism of gas hydrates in real systems should involve dilute aqueous gas solutions at temperatures above 0 °C. 

In these equations addends J rdnic, and i T-lny. characterize deviation of the solutions from ideal and the work, which is necessary to expend in order to squeeze 1 mole of component i of the ideal solution into real solution. Activity coefficients can be greater or smaller than 1. When pressure of a gas solution or concentration of dissolved substances tends to 0, fugacity coefficients or activities coefficients approach 1. Even in diluted real solutions charged ions and dipole molecules experience electrostatic interaction, which shows up in a decrease of activities coefficient. Only in very diluted solutions this interaction becomes minuscule, and fugacity and activities values tend to values of partial pressure and concentration, respectively. Table 1.3 summarizes calculation formulae for activities values of groxmd water components under ideal and real conditions. 

We want to use the Henryan standard state, which is a state in which the solute exhibits dilute solution behavior. That is, no matter what the actual concentration, the solute behaves as if there is absolutely no solute-solute interactions - each solute molecule thinks it is alone in the solvent. It is a state which obey s Henry s law, which at real concentrations is obviously a hypothetical state, and it lies anywhere on the Henry s law tangent. In 8.3.4 we saw that we could choose a point on this tangent having X = 1, or we could choose any other point. What the other points on this slope mean depends on what concentration scale we are using - if we use a weight percent scale we can choose a weight percent standard state, and if we use a molality scale we can choose a molal standard state. 

Given this failure of the continuum model, it is evidently necessary to treat the solvent as an assembly of molecules. A hard-sphere model is the first approximation. Kinetic theory of diffusion in dilute gases, where the mean free path is much greater than the collision diameter, is well established it can be extended with some success to dense gases, where the two quantities are more nearly equal, and (more speculatively) to hard-sphere models of liquids, where they are comparable. For these highly mathematical theories the reader may consult more specialised works [14]. Analytical solutions are not always to be expected numerical solutions may be required. Computer-simulation calculations have had considerable success, and with the advent of fast computers have become a major source of understanding of real systems (cf., e.g.. Section 7.3.4.5). 

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