Diffusion coefficient method

Abstract. A recently developed new method to monitor reaction kinetics of intermolecular interaction is reviewed. This method is based on the measurement of the time-dependent diffusion coefficient using the pulsed-laser-induced transient grating technique. Using this method, conformation change, transient association, and transient dissociation on reactions are successfully detected. The principle and some applications to studies on changes in the intermolecular interactions of photosensor proteins (e.g., photoactive yellow protein, phototropins, AppA) in the time domain are described. In particular, unique features of this time-dependent diffusion coefficient method are discussed. 

The self-diffusion coefficient method was developed originally by Lindman and co-workers [42], This method is by far the most powerful and successful application of NMR in the study of surfactant systems. The self-diffusion method permits a full characterization of the system under investigation, since it can yield the values of the free monomeric surfactant concentration, the degree of counterion binding, and the amount of water bound (hydration). Moreover, these data can help one deduce the information on micelle size, shape, and composition from the monomer surfactant concentration in a surfactant mixture ... 

Diffusion coefficient. In Pick s law, the diffusion coefficient, D, is a parameter that relates the flux of a penetrant in a medium to its concentration gradient. A diffusion coefficient value is always given for a particular diffusing molecule/polymer pair. For solid polymers, the diffusion coefficient values of a large number low molecular mass substances range from 1 x 10 to 1 x 10 cw /s. The diffusion theory states that diffusion is an activated phenomenon that follows Arrhenius law. Penetrant concentration and the presence of plasticizers also affect the value of the diffusion coefficient. Methods for the determination of D are discussed in Sec. 11.3. 

The value of coefficient depends on the composition. As the mole fraction of component A approaches 0, approaches ZJ g the diffusion coefficient of component A in the solvent B at infinite dilution. The coefficient Z g can be estimated by the Wilke and Chang (1955) method ... 

For a binary mixture of two components A and B in the gas phase, the mutual diffusion coefficient such as defined in 4.3.2.3, does not depend on composition. It can be calculated by the Fuller (1966) method ... 

This shows that Schlieren optics provide a means for directly monitoring concentration gradients. The value of the diffusion coefficient which is consistent with the variation of dn/dx with x and t can be determined from the normal distribution function. Methods that avoid the difficulty associated with locating the inflection point have been developed, and it can be shown that the area under a Schlieren peak divided by its maximum height equals (47rDt). Since there are no unknown proportionality factors in this expression, D can be determined from Schlieren spectra measured at known times. 

Note that this method of standardizing D values makes no allowance for the possibility that a molecule may change size, shape, or solvation with changes in temperature. In the next section we shall survey the behavior of polymeric materials in an ultracentrifuge. We shall see that diffusion coefficients can be... 

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

Adsorption Kinetics. In zeoHte adsorption processes the adsorbates migrate into the zeoHte crystals. First, transport must occur between crystals contained in a compact or peUet, and second, diffusion must occur within the crystals. Diffusion coefficients are measured by various methods, including the measurement of adsorption rates and the deterniination of jump times as derived from nmr results. Factors affecting kinetics and diffusion include channel geometry and dimensions molecular size, shape, and polarity zeoHte cation distribution and charge temperature adsorbate concentration impurity molecules and crystal-surface defects. 

Water Transport. Two methods of measuring water-vapor transmission rates (WVTR) ate commonly used. The newer method uses a Permatran-W (Modem Controls, Inc.). In this method a film sample is clamped over a saturated salt solution, which generates the desired humidity. Dry air sweeps past the other side of the film and past an infrared detector, which measures the water concentration in the gas. For a caUbrated flow rate of air, the rate of water addition can be calculated from the observed concentration in the sweep gas. From the steady-state rate, the WVTR can be calculated. In principle, the diffusion coefficient could be deterrnined by the method outlined in the previous section. However, only the steady-state region of the response is serviceable. Many different salt solutions can be used to make measurements at selected humidity differences however, in practice,... 

Carbon Dioxide Transport. Measuring the permeation of carbon dioxide occurs far less often than measuring the permeation of oxygen or water. A variety of methods ate used however, the simplest method uses the Permatran-C instmment (Modem Controls, Inc.). In this method, air is circulated past a test film in a loop that includes an infrared detector. Carbon dioxide is appHed to the other side of the film. AH the carbon dioxide that permeates through the film is captured in the loop. As the experiment progresses, the carbon dioxide concentration increases. First, there is a transient period before the steady-state rate is achieved. The steady-state rate is achieved when the concentration of carbon dioxide increases at a constant rate. This rate is used to calculate the permeabiUty. Figure 18 shows how the diffusion coefficient can be deterrnined in this type of experiment. The time lag is substituted into equation 21. The solubiUty coefficient can be calculated with equation 2. 

Flavor and Aroma Transport. Many methods ate used to characterize the transport of flavor, aroma, and solvent molecules in polymers. Each has some value, and no one method is suitable for all situations. Any experiment should obtain the permeabiUty, the diffusion coefficient, and the solubihty coefficient. Furthermore, experimental variables might include the temperature, the humidity, the flavor concentration, and the effect of competing flavors. 

An overview of some basic mathematical techniques for data correlation is to be found herein together with background on several types of physical property correlating techniques and a road map for the use of selected methods. Methods are presented for the correlation of observed experimental data to physical properties such as critical properties, normal boiling point, molar volume, vapor pressure, heats of vaporization and fusion, heat capacity, surface tension, viscosity, thermal conductivity, acentric factor, flammability limits, enthalpy of formation, Gibbs energy, entropy, activity coefficients, Henry s constant, octanol—water partition coefficients, diffusion coefficients, virial coefficients, chemical reactivity, and toxicological parameters. 

Diffiusion Coefficient. The method of Reference 237 has been recommended for many low pressure binary gases (238). Other methods use solvent and solute parachors to calculate diffusion coefficients of dissolved organic gases in Hquid solvents (239,240). Molar volume and viscosity are also required and may be estimated by the methods previously discussed. Caution should be exercised because errors are multiphcative by these methods. 

Experiment diffusion coefficients are scarce and not highly accurate, especially in the liquid phase, leading to prediction methods with marginal accuracy. However, use of the v ues predicted are generally suit le for engineering calculations. At concentrations above about 10 mole percent, predicted values should be used with caution. Diffu-sivities in liquids are lO -lO times lower than those in gases. 

For prediction of gas phase diffusion coefficients in multicomponent hydi ocarbon/nonKydi ocai bon gas systems, the method of Wilke shown in Eq. (2-154) is used. 

In the late 1800s, the development of the kinetic theory of gases led to a method for calculating mmticomponent gas diffusion (e.g., the flux of each species in a mixture). The methods were developed simnlta-neonsly by Stefan and Maxwell. The problem is to determine the diffusion coefficient D, . The Stefan-Maxwell equations are simpler in principle since they employ binary diffnsivities ... 

Multicomponent Mixtures No simple, practical estimation methods have been developed for predicting multicomponent hquid-diffusion coefficients. Several theories have been developed, but the necessity for extensive activity data, pure component and mixture volumes, mixture viscosity data, and tracer and binaiy diffusion coefficients have significantly limited the utihty of the theories (see Reid et al.). 

Penetration theoiy often is used in analyzing absorption with chemical reaction because it makes no assumption about the depths of penetration of the various reacting species, and it gives a more accurate result when the diffusion coefficients of the reacting species are not equal. When the reaction process is veiy complex, however, penetration theoiy is more difficult to use than film theory, and the latter method normally is preferred. 

Ruthven (gen. refs.) summarizes methods for the measurement of effective pore diffusivities that can be used to obtain tortuosity factors by comparison with the estimated pore diffusion coefficient of the adsorbate. Molecular diffusivities can be estimated with the methods in Sec. 6. 

Interdiffusion of bilayered thin films also can be measured with XRD. The diffraction pattern initially consists of two peaks from the pure layers and after annealing, the diffracted intensity between these peaks grows because of interdiffusion of the layers. An analysis of this intensity yields the concentration profile, which enables a calculation of diffusion coefficients, and diffusion coefficients cm /s are readily measured. With the use of multilayered specimens, extremely small diffusion coefficients (-10 cm /s) can be measured with XRD. Alternative methods of measuring concentration profiles and diffusion coefficients include depth profiling (which suffers from artifacts), RBS (which can not resolve adjacent elements in the periodic table), and radiotracer methods (which are difficult). For XRD (except for multilayered specimens), there must be a unique relationship between composition and the d-spacings in the initial films and any solid solutions or compounds that form this permits calculation of the compo-... 

The Gaussian diffusion equation is known as the Pasquill and Gifford model, and is used to develop methods for estimating the required diffusion coefficients. The basic equation, already presented in a slightly different form, is restated below ... 

The diffusion coefficients of the constituent ions in ionic liquids have most commonly been measured either by electrochemical or by NMR methods. These two methods in fact measure slightly different diffusional properties. The electrochemical methods measure the diffusion coefficient of an ion in the presence of a concentration gradient (Pick diffusion) [59], while the NMR methods measure the diffusion coefficient of an ion in the absence of any concentration gradients (self-diffusion) [60]. Fortunately, under most circumstances these two types of diffusion coefficients are roughly equivalent. 

There are a number of NMR methods available for evaluation of self-diffusion coefficients, all of which use the same basic measurement principle [60]. Namely, they are all based on the application of the spin-echo technique under conditions of either a static or a pulsed magnetic field gradient. Essentially, a spin-echo pulse sequence is applied to a nucleus in the ion of interest while at the same time a constant or pulsed field gradient is applied to the nucleus. The spin echo of this nucleus is then measured and its attenuation due to the diffusion of the nucleus in the field gradient is used to determine its self-diffusion coefficient. The self-diffusion coefficient data for a variety of ionic liquids are given in Table 3.6-6. 

The measurement of transport numbers by the above electrochemical methods entails a significant amount of experimental effort to generate high-quality data. In addition, the methods do not appear applicable to many of the newer non-haloalu-minate ionic liquid systems. An interesting alternative to the above method utilizes the NMR-generated self-diffusion coefficient data discussed above. If both the cation (Dr+) and anion (Dx ) self-diffusion coefficients are measured, then both the cation (tR+) and anion (tx ) transport numbers can be determined by using the following Equations (3.6-6) and (3.6-7) [41, 44] ... 

With electrochemical methods such as chronoamperometry, cyclovoltammetry (CV), or conductivity measurements, the diffusion coefficients of charged chemical species can be estimated in highly dilute solutions [16, 17]. 

Self-diffusion coefficients were measured with the NMR spin-echo method and mutual diffusion coefficients by digital image holography. As can be seen from Figure 4.4-3, the diffusion coefficients show the whole bandwidth of diffusion coeffi-... 

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