Diffusion constant, solvent

Tlere, y Is the friction coefficien t of the solven t. In units of ps, and Rj is th e random force im parted to th e solute atom s by the solvent. The friction coefficien t is related to the diffusion constant D oflh e solven l by Em stem T relation y = k jT/m D. Th e ran doin force is calculated as a ratulom number, taken from a Gaussian distribn-... 

Here, y is the friction coefficient of the solvent, in units of ps and Rj is the random force imparted to the solute atoms by the solvent. The friction coefficient is related to the diffusion constant D of the solvent by Einstein s relation y = kgT/mD. The random force is calculated as a random number, taken from a Gaussian distribu-... 

The proportionality constant in this equation, D, is called the diffusion constant. A typical solute has D 1 X 10 9 m2 s . Values of D do not vary greatly with the solute but are inversely proportional to the viscosity of the solvent. 

In fact, the diffusion constant in solutions has the form of an Einstein diffusion of hard spheres with radius Re. For a diffusing chain the solvent within the coil is apparently also set in motion and does not contribute to the friction. Thus, the long-range hydrodynamic interactions lead, in comparison to the Rouse model, to qualitatively different results for both the center-of-mass diffusion—which is not proportional to the number of monomers exerting friction - as well as for the segment diffusion - which is considerably accelerated and follows a modified time law t2/3 instead of t1/2. 

The equations of motion (75) can also be solved for polymers in good solvents. Averaging the Oseen tensor over the equilibrium segment distribution then gives = l/ n — m Y t 1 = p3v/rz and Dz kBT/r sNY are obtained for the relaxation times and the diffusion constant. The same relations as (80) and (82) follow as a function of the end-to-end distance with slightly altered numerical factors. In the same way, a solution of equations of motion (75), without any orientational averaging of the hydrodynamic field, merely leads to slightly modified numerical factors [35], In conclusion, Table 4 summarizes the essential assertions for the Zimm and Rouse model and compares them. 

One of the most popular applications of molecular rotors is the quantitative determination of solvent viscosity (for some examples, see references [18, 23-27] and Sect. 5). Viscosity refers to a bulk property, but molecular rotors change their behavior under the influence of the solvent on the molecular scale. Most commonly, the diffusivity of a fluorophore is related to bulk viscosity through the Debye-Stokes-Einstein relationship where the diffusion constant D is inversely proportional to bulk viscosity rj. Established techniques such as fluorescent recovery after photobleaching (FRAP) and fluorescence anisotropy build on the diffusivity of a fluorophore. However, the relationship between diffusivity on a molecular scale and bulk viscosity is always an approximation, because it does not consider molecular-scale effects such as size differences between fluorophore and solvent, electrostatic interactions, hydrogen bond formation, or a possible anisotropy of the environment. Nonetheless, approaches exist to resolve this conflict between bulk viscosity and apparent microviscosity at the molecular scale. Forster and Hoffmann examined some triphenylamine dyes with TICT characteristics. These dyes are characterized by radiationless relaxation from the TICT state. Forster and Hoffmann found a power-law relationship between quantum yield and solvent viscosity both analytically and experimentally [28]. For a quantitative derivation of the power-law relationship, Forster and Hoffmann define the solvent s microfriction k by applying the Debye-Stokes-Einstein diffusion model (2)... 

Loutfy and coworkers [29, 30] assumed a different mechanism of interaction between the molecular rotor molecule and the surrounding solvent. The basic assumption was a proportionality of the diffusion constant D of the rotor in a solvent and the rotational reorientation rate kOI. Deviations from the Debye-Stokes-Einstein hydrodynamic model were observed, and Loutfy and Arnold [29] found that the reorientation rate followed a behavior analogous to the Gierer-Wirtz model [31]. The Gierer-Wirtz model considers molecular free volume and leads to a power-law relationship between the reorientation rate and viscosity. The molecular free volume can be envisioned as the void space between the packed solvent molecules, and Doolittle found an empirical relationship between free volume and viscosity [32] (6),... 

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 growth rate, characterized by the change of the radius with time, is proportional to the driving force for the phase separation, given by the differences between 2 > the chemical composition of the second phase in the continuous phase at any time, and, its equihbrium composition given by the binodal line. The proportionahty factor, given by the quotient of the diffusion constant, D, and the radius, r, is called mass transfer coefficient. Furthermore the difference between the initial amount of solvent, (])o, and c]) must be considered. The growth rate is mathematically expressed by [101]... 

From diffusion measurements it is known that the translatory mobility of solvent molecules in polymers changes abruptly at the glass temperature. Below T0 the diffusion constants are of the order of 10-12 to 10 14sq. cm. /sec., above Tg of the order of 10 4 to 10 8sq. cm./sec. (13). We have found that even below the glass temperature, plasticizer molecules may perform rather rapid motions, preferably rotatory motion. If the plasticizer has long aliphatic side chains, as with phthalates, rotating motions in the side chains should also result in a narrow... 

In various applications the following model has been used, which is of more general interest. Consider a molecule having a number of internal states or levels i. From each i it can jump to any other level j with a fixed transition probability yjti per unit time. Moreover the molecule is embedded in a solvent in which it diffuses with a diffusion constant depending on its state i. The probability at time t for finding it in level i at the position r with margin d3r is P,(i% t) d3r. While the molecule resides in i the probability obeys... 

Neutral NIPA gel is the most extensively studied among known gels from the standpoint of phase transition, and thus, various physical properties around the transition have been reported. These include the shear and bulk modulus [20, 24], the diffusion constant of the network [25], spinodal decomposition [26], specific heat [21], critical properties of gels in mixed solvents [8] and the effect of uniaxial [27] and hydrostatic [28] pressures on the transition, and so... 

The diffusion constant D is a function of both molecular weight and shape. It can be measured by observing the spread of an initially sharp boundary between the protein solution and a solvent as the protein diffuses into the solvent layer. Once we know the value of the diffusion constant, we can combine the information with the sedimentation data and calculate the molecular weight of the protein. 

Although detailed microscopic calculations of the problem mentioned above are not available, there exist several computer simulation studies [102, 117], which also find the anomalous enhanced diffusion, even for simple model potentials such as the Lennard-Jones. The physical origin of the enhanced diffusion is not clear from the simulations. The enhancement can be as large as 50% over the hydrodynamic value. What is even more surprising is that the simulated diffusion constant becomes smaller than the hydro-dynamic prediction for very small solutes, with sizes less than one-fifteenth of the solvent molecules. These results have defied a microscopic explanation. 

Figure 5. Segmental diffusion constant depending on temperature and solvent viscosity data are calculated according to Equation 1 and the graph according to Equation 2...

The rates of bimolecular reaction between two extremely reactive species in solution may be limited by the rate at which the two species diffuse together such reactions are referred to as diffusion-controlled. Solvent will affect the rate of these reactions by hindering the mobility of the reactants and, consequently, the rate constant is a function of solvent viscosity. Typically, radical-radical (termination) reactions are diffusion-controlled and thus their rates are governed by solvent viscosity. 

Before closing, let us add a few comments about the diffusion constant D in the equations above. For diffusion of solutes like A and B in a solvent, it is customary to introduce diffusion constants Da and Db such that the associated fluxes J A and Jg are given relative to the flux of the solvent molecules. Since they are present in an overwhelming quantity as compared to the solutes, this will be equivalent to a center-of-mass reference frame, and if the system is stationary, to a laboratory reference... 

The diffusion constant Di of a particle in a solvent is related to the viscosity of the solvent by the Stokes-Einstein relation known from hydrodynamics ... 

It is obvious that if V is large at certain places I will get contributions practically only from these places. If the solution is ideal and no field of force is applied to the column, we may say that

different point of view. The dissolved molecules diffuse between the solvent molecules, which cause them to move in a force field of very variable intensity. If this were not so we should expect the diffusion constant to be the same everywhere. Denoting by D this constant of diffusion, we get... 

CR, cryoscopic method DV, diffusion constant and intrinsic viscosity EB, ebullioscopic method EG, end-group titration IV, intrinsic viscosity-molecular weight relationship in other solvents LS, light scattering MV, melt viscosity-molecular weight relationship OS, osmotic pressure PR, analysis of polymerization rate SD, sedimentation and diffusion constants SE, sedimentation equilibrium (Archibald s method) SV, sedimentation constant and intrinsic viscosity [see Eq. (72)]. 

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