Tortuosity, polymer fractionation

HW Osterhoudt. Transport properties of hydrophilic polymer membranes. The influence of volume fraction polymer and tortuosity on permeability. J Phys Chem 78 408-411, 1974. 

In addition to the criticisms from Anderman, a further challenge to the application of SPEs comes from their interfacial contact with the electrode materials, which presents a far more severe problem to the ion transport than the bulk ion conduction does. In liquid electrolytes, the electrodes are well wetted and soaked, so that the electrode/electrolyte interface is well extended into the porosity structure of the electrode hence, the ion path is little affected by the tortuosity of the electrode materials. However, the solid nature of the polymer would make it impossible to fill these voids with SPEs that would have been accessible to the liquid electrolytes, even if the polymer film is cast on the electrode surface from a solution. Hence, the actual area of the interface could be close to the geometric area of the electrode, that is, only a fraction of the actual surface area. The high interfacial impedance frequently encountered in the electrochemical characterization of SPEs should originate at least partially from this reduced surface contact between electrode and electrolyte. Since the porous structure is present in both electrodes in a lithium ion cell, the effect of interfacial impedances associated with SPEs would become more pronounced as compared with the case of lithium cells in which only the cathode material is porous. 

Here, Da is diffusion coefficient in the amorphous phase alone, oc is the volume fraction of crystalline polymer, and t is a scalar quantity that denotes the tortuosity of diffusional path of the solute. The value of Da may be estimated by the Peppas-Reinhart model if the amorphous regions of the polymer are highly swollen. This substitution yields... 

Hence, the addition of inorganic impermeable nanoplatelets improves the barrier properties of polymers. This is attributed mostly to the lengthening of the diffusion path of the permeating gas molecules due to the increase of the tortuosity. Increasing the aspect ratio of the platelets and their volume fraction improves these... 

The characteristics of pore structure in polymers is a key parameter in the study of diffusion in polymers. Pore sizes ranging from 0.1 to 1.0 pm (macroporous) are much larger than the pore sizes of diffusing solute molecules, and thus the diffusant molecules do not face a significant hurdle to diffuse through polymers comprising the solvent-filled pores. Thus, a minor modification of the values determined by the hydrodynamic theory or its empirical equations can be made to take into account the fraction of void volume in polymers (i.e., porosity, e), the crookedness of pores (i.e., tortuosity, x), and the affinity of solutes to polymers (i.e., partition coefficient, K). The effective diffusion coefficient, De, in the solvent-filled polymer pores is expressed by ... 

Above the the rate of polymer chain relaxation is faster than the diffusion of CO2, and hence Fickian diffusion is to be expected. The diffusion of CO2 is believed to occur within the amorphous domains of the polymer matrix, and for this reason the diffusion in semi-crystalline polymers may be more complex than it in the case for glassy polymers. In the case of semi-crystalline polymers, CO2 is not soluble in the crystalline domains, and therefore the degree of crys-taUinity and hence the amorphous fraction available for CO2 molecules may influence the diffusion characteristics. Furthermore, C02-induced crystallization is likely to lead to an increase in the tortuosity factor, and thus the diffusion path length may increase as a function of time. Syndiotactic polystyrene and poly(4-methyl-l-pentene) [45] are semi-crystalline polymers which have crystalline phases (helical in the case of sPS) with lower densities than that of the amorphous phase and are exceptions, as CO2 access is not restricted to the amorphous domains, in fact CO2 diffuses faster in the helical sPS than in the amorphous polymer [46]. 

Nanoscale fillers such as clays are used to improve the barrier properties of plastic films. The platelike fillers, in particular, with the relatively higher tortuosity deliver lower gas permeabilities (Alperstein, 2005). At a volume fraction d> of the nanofiller, the permeability P of a gas through the polymer (Picard et al., 2007), depends on the value of tortuosity t of the nanoscale filler ... 

In addition, the presence of impermeable filler in a polymer forces the diffusant molecule to travel further around the filler particles. This physical blocking effect is known as tortuosity, because the filler forces the diffusant to take a more indirect, or tortuous, path through the material. The degree of tortuosity imposed is dependent upon the anisotropy and orientation of the filler particles with respect to the direction of diffusion. For example, platy particles oriented perpendicularly to the diffusion vector will be particularly effective in retarding diffusion. The permeability of a composite can be calculated using an equation that allows for the reduction in permeant solubility and for the tortuosity (Equation 8.3). Where P and Pp are the permeability of the composite and the unfilled polymer, respectively. The terms w and t refer to the width and thickness of the filler and (pp and (pf represent the volume fraction of polymer and filler. 

FIGURE 2.10 Theoretical predictions based on path tortuosity [eq. (2.9)], as a function of (a) filler aspect ratio a = 1 to 1000 (b) filler aspect ratio and alignment (5 = 1 perfect smectic alignment—dashed lines S = 0 random orientation —solid lines) (c) filler aspect ratio for a constant volume fraction (pv = 5%. (d) Comparison of the same theoretical predictions (parameters as indicated) with experimental values for water vapor permeabilities in various polymer-montmorillonite nanocomposites. (From Refs. 39-41.)... 

The effective diffusivity in the secondary particle. Deg, can be estimated using the conventional expression for effective diffusivity in porous heterogeneous catalysts, Eq. (42), where is the monomer bulk diffusivity in the reaction medium, and and T are the void fraction and tortuosity of the polymer particle, respectively. The fact that both e and r are likely to vary as a function of the degree of fragmentation and expansion of the secondary particle is certainly one of the difficulties in getting a good estimate for D. ... 

A decrease of the solubility is expected in nanostructured polymer blends due to the reduced polymer matrix volume, as well as a decrease in diffusion due to a more tortuous pathway for the diffusing molecules. The reduction of the diffusion coefficient is higher than that of the solubility coefficient. Indeed, the volume fraction of nanoplatelets is low and, thus, the reduction of the matrix volume is small. The major factor, then, is the tortuosity, which is connected directly to the shape and the degree of dispersion of nanoplatelets. Better dispersed clay systems increase the tortuosity path of the diffusing molecules whereas larger aggregates decrease the aspect ratio of the nanoparticles and can act as a low-resistance pathway for the gas transport. 

In gel electrolytes, in which a liquid electrolyte is imbibed into a polymer matrix, calculation of the effeetive diffusivity and ionie eonductivity based on the apparent volume fraction of electrolyte in the polymer may be eomplieated by solvation of the polymer by the solvent, inereased tortuosity presented by the polymer, and possible interactions of the ions with solvating groups on the polymer. One way to handle these effects empirically is to treat the tortuosity... 

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