Diffusion coefficients in polymers

The diffnsion coefficients of polymers lie between those of solids and liquids. Diffusion of high-molecnlar-weight assumes importance when the polymer forms a solnte of a dilnte solntion or one component of a polymer-polymer blend. A polymer blend can be miscible or immiscible, compatible or incompatible. When two polymers are mixed to yield a product with improved properties, they are said to be a compatible blend. When two polymers mix at the molecular level, they are said to be a miscible blend. Concentrated systems in which the volume fraction of the polymer solnte is large make up another category where diffnsion has to be treated in a different manner compared with other systems. 

A polymer molecule dissolved in a solvent can be envisioned as a necklace comprising spherical beads connected by a string [23]. The polymer molecules are separated and only interact through the solvent. The Stokes-Einstein equation for the diffnsion coefficient of the polymer can be used for a Flory theta solvent. The root-mean-sqnare 

Dq includes the solute s activation energy. This must be sufficient to overcome any attractive forces that constrain near-neighboring polymer segments. This coefficient can be expected to vary with the free volume of the polymer chains. Only a fraction of the free volume, the hole free volume, will be accessed by the solute  

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JL Duda, YC Ni, JS Vrentas. An equation relating self-diffusion and mutual diffusion coefficients in polymer-solvent systems. Macromolecules 12 459-462, 1979. 

The sorption coefficient (K) in Equation (2.84) is the term linking the concentration of a component in the fluid phase with its concentration in the membrane polymer phase. Because sorption is an equilibrium term, conventional thermodynamics can be used to calculate solubilities of gases in polymers to within a factor of two or three. However, diffusion coefficients (D) are kinetic terms that reflect the effect of the surrounding environment on the molecular motion of permeating components. Calculation of diffusion coefficients in liquids and gases is possible, but calculation of diffusion coefficients in polymers is much more difficult. In the long term, the best hope for accurate predictions of diffusion in polymers is the molecular dynamics calculations described in an earlier section. However, this technique is still under development and is currently limited to calculations of the diffusion of small gas molecules in amorphous polymers the... 

The expressions (12.5)-(12.8) are also convenient ways of measuring diffusion coefficients in polymers. A permeant is contacted with a film of material of known geometry until equilibrium is reached. The film is then removed from the permeant solution, washed free of contaminants, and the rate of release of the permeant is measured. From the release curves, the diffusion coefficient and permeant sorption can be obtained. 

Figure 12.4 The pulse scheme of the three-pulse echo sequence to determine Xe diffusion coefficients in polymers and other porous systems. The shaded areas are magnetic field gradient pulses with amplitude g and length 8. The time between the two gradient pulses A determines the time during which the diffusion path length is...

Brandsch, J. Mercea, P. Piringer, O. Modeling of Additive Diffusion Coefficients in Polymers. New Developments in the Chemistry of Packaging Materials ACS Symposium Series, ACS, Washington, D.C. 1999. 

Some properties of diffusion coefficients in polymers. J. Phys. Chem. 61, 178 (1957). 

In the aq/polymer/org situation, the organic solvent typically penetrates the polymer causing it to swell considerably, and the situation is very similar to that of MMLLE. With a fixed composition of the membrane, the possibilities for chemical tuning (such as application of carriers) of the separation process are greatly reduced compared to SLM extraction or MMLLE. Also, as diffusion coefficients in polymers are lower than in liquids, the mass transfer is slower, leading to slower extractions. On the other hand, as the membrane is virtually insoluble, any combination of aqueous and organic liquids can be used, and the entire system becomes very stable. 

Harland RS and Peppas NA. On the accurate experimental determination of drug diffusion coefficients in polymers. S.T.P. Pharm. Sci. 1993 3(5) 357-361. 

Following the ideas developed by Guillet and his co-workers, a method using Capillary Column Inverse Gas Chromatography (CCIGC) was developed (1.21 to measure diffusion coefficients in polymer-solvent systems at conditions approaching infinite dilution of the volatile component. 

NMR measurement of solvent self-diffusion coefficients in polymer solutions NMR measurement of protein diffusion coefficients in solution and in synthetic membranes... 

C. E. Rogers and D. Machin, "The Concentration Dependence of Diffusion Coefficients in Polymer-Penetrant Systems", CRC Critical Rev, in Macromol. Sci., April 1972, pp. 245-313. 

The disadvantage of an BCD is mainly associated with its comparatively slow response time since the driving mechanism deals with ions which generally have low diffusion coefficients in polymers. 

The effect of free volume on penetrant diffusion coefficients in polymers is often described using concepts from the Cohen and Turnbull model (7. This statistical mechanics model provides a simplistic description of diffusion in a liquid of hard spheres. A hard sphere penetrant is considered to be trapped in a virtual cage created by its neighbors. Free volume is defined as the volume of the cage less the volume of the penetrant. Free volume fluctuations, which occur randomly due to thermally-stimulated Brownian motion of neighboring hard spheres, provide opportunities for the penetrant to execute a diffusion step if the gap (i,e, free volume fluctuation) occurs sufficiently close to the i netrant to be accessible and is of sufficient size to acconunodate it. The diffusion coefficient of a penetrant is given by ... 

DpOL Monomer diffusion coefficient in polymer phase [m s- l... 

For a dilute solution of polymer Aina low molecular weight solvent B, the polymer molecules are modeled as bead-spring chains. Resistance in the motion of beads is characterized by a friction coefficient As the number of beads is proportional to the polymer molecular weight M, we have Dab 1 / Vm. Table 2.8 shows some values of diffusion coefficients in polymers. In a flow of dilute solution of polymers, the diffiisivity tensor is anisotropic and depends on the velocity gradient. The Maxwell-Stefan equation may predict the diffusion in multicomponent mixtures of polymers. 

For food packages, polymeric materials should exhibit an adequate carbon dioxide oxygen ratio (generally lower than 7). The process of permeation involves dissolution of the gas in one side of the membrane, diffusion of the gas through it and release of the gas from the other side of the membrane. A conventional method for determining permeability and diffusion coefficients in polymers involves the measurement of membrane weight gain versus time until the final mass equilibrium mass is reached. 

Q. Chen, K. Schmidt-Rohr, Measurement of the local 1H spin-diffusion coefficient in polymers. Solid State Nucl. Magn. Reson. 29 (2006) 142—152. 

The activation energy of diffusion depends on the polymer chain mobility and the size of the penetrant relative to the intra- and inter-molecular polymer chain gaps. Diffusion coefficients in polymers always increase with increasing temperature, that is, Ed is always positive. In most polymers, gas solubilities decrease with increasing temperature therefore, Hg is generally negative. 

The miscibility of a polymer blend depends on temperature and blend composition, making the investigation of the dynamic behavior of miscible polymer blends very challenging. Associated with the dynamics of miscible polymer blends is the mutual diffusion that, as in determining the self-diffusion coefficient in polymer melts and solutions presented in Chapter 4, can be discussed using molecular theory. Thermodynamic interactions and free-volume effects determine the mutual diffusivity in miscible polymer blends. 

Experimentally translational diflusion coefficients can be determined with an ultracentrifuge (see Section 1.3) in special diffusion cells using Eqs. (A11)-(A14) (800) and with dynamic light scattering (DLS). Since 1970, when the dynamic light scattering became available, nearly all diffusion coefficients have been measured by this method. Based on the theory of Pecora (853,854), the experimentally measured autocorrelation function g 2 (t) is linked with the diffusion coefficient in polymer solutions (Refs. 730,855) ... 

Before we begin, we should list the guidelines that tend to stick in everyone s mind. Diffusion coefficients in gases, which can be estimated theoretically, are about 0.1 cm / sec. Diffusion coefficients in liquids, which cannot be as reliably estimated, cluster around 10 cm /sec. Diffusion coefficients in solids are slower still, 10 cm /sec, and they vary strongly with temperature. Diffusion coefficients in polymers and glasses lie between liquid and solid values, say about 10 cm /sec, and these values can be strong functions of solute concentration. 

Although the diaphragm cell has been used by some researchers to measiue diffusion coefficients in polymer solutions, the results are likely to be influenced by the mechanical stirring of the solutions, which can cause extension and orientation of the polymer molecules. Polymer molecules can also adsorb on the membrane. As a consequence, it is preferable to use methods that do not require the fluid to be subjected to any shear stress. This can be achieved in free-diffusion experiments [15]. 

Duda, J. L., Y. C. Ni, and J. S. Vrentas, An Equation Relating Self-Diffusion and Mutual Diffusion Coefficients in Polymer-Solvent Systems, Macromolecules, 12, 459 62, 1979. 

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