Sorption state

In this situation, transport equations similar to those discussed previously can be applied. For example, by assuming sorption to be essentially instantaneous, the advective-dispersion equation with a reaction term (Saiers and Hornberger 1996) can be considered. Alternatively, CTRW transport equations with a single ti/Ci, t) can be applied or two different time spectra (for the dispersive transport and for the distribution of transfer times between mobile and immobile—diffusion, sorption— states can be treated Berkowitz et al. 2008). 

The impedance measurements of this system have been combined with oscillometric-gravimetric measurements leading to both the volume of the swelling polymer and the mass of CO2 sorbed at a given gas pressure, cp. Chap. 5. These combined measurements show that there are nearly linear relations between the dielectric permittivity and the gas pressure as well as the volume of the polymer in the sorption state and the gas pressure, cp. data and Figures given in Sect. 3.2. 

Several approximations to Eq. (16) can then be made. The simplest approximation is to assume a constant prefactor and to use the potential energy difference between the transition state and the reactant sorption state. 

The important point is using such criteria to recognize the three-dimensional regions in a polymer structure that most closely resemble the necks between (presumably) larger sorption state regions. 

The relationship between individual sorption states and a fully connected jump network will be discussed in Section IX. Methods for calculating a diffusion coefficient from the structure and rate constants are discussed in Section X. 

Compared to the rigid polymer method, the average fluctuating polymer method improves the treatment of how polymer chains move during the penetrant diffusion process. Rather than remaining fixed in place, polymer chains execute harmonic vibrations about their equilibrium positions. Penetrant jumps are then coupled to elastic fluctuations of the polymer matrix and are independent of structural relaxation of the polymer chains [24,97]. After a penetrant jump completes, chains near the final sorption state will likely show slight elastic deviations as they swell to accommodate the penetrant molecule. Since no chain conformation relaxations are allowed, other polymer chains will essentially retain their initial conformation. The penetrant jump rate then depends only on the local, quasiharmonic fluctuations in the sorption state and the transition state [24,97]. 

FIG. 3 Different chain conformations of atactic polypropylene as a methane penetrant (sphere labeled p) jumps between two sorption states. Text labels indicate the extent along the reaction coordinate. Differences between the positions of dashed and solid triangles indicate the motion of chain segments to create a more open channel for diffusion. Conformations are adapted from Movie 1 in [90]. 

Another direct effect is that high penetrant concentrations can increase polymer chain mobility (plasticization), thereby increasing the diffusion coefficient [1], An indirect effect concerns relative sorption among different sorption states. From microscopic reversibility [22], the ratio of equilibrium solubilities within different sorption states equals the ratio of jump rate constants between them. 

Sorption predictions rely on calculating the chemical potential of the penetrant molecule in each sorption state. Details are described in Chapters 9 and 10 of this book. In the low-concentration limit, the required solubility coefficients for use in Eq. (47) and in diffusion calculations can be calculated by [154,155]... 

The ratio Npe ) J Vpoiym) equals the number of penetrant molecules dissolved in a particular sorption state A per total volume of polymer, and P equals the total pressure. The ensemble average brackets ( ) imply averaging over all penetrant positions within a sorption state A while the polymer conformation fluctuates within the canonical ensemble according to ypofym the total potential energy in the absence of a penetrant molecule. 

For conducting a kinetic Monte Carlo simulation, the most straightforward choice for a network of sorption states and rate constants is that of the original molecular structure. Its key advantage is its one-to-one correspondence with the detailed polymer configuration. However, the small size of a typical simulation box is a disadvantage. For example, in [97] it was observed that anomalous diffusion continued until the root-mean-squared displacement equaled the box size. From this match in length scales, it is not... 

The effect of periodic boundary conditions on the network structure of an amorphous polymer is shown in Fig. 5. The nine images depict a three-dimensional, 23 A edge-length simulation cell of atactic polypropylene, with period replication shown explicitly in two dimensions. Different colors indicate different sorption states of accessible volume [87]. Within a single cell, no positional order is apparent. Within the set of periodically replicated cells, however, an imposed crystallinity is apparent there is one disordered box per unit cell, and one s eye quickly recognizes the periodic array of... 

FIG. 5 Nine periodic replications of geometric sorption states in a particular configuration of atactic polypropylene. Penetrant diffusion occurs via jumps among these states. The states within an individual simulation cell appear disordered, but the crystalline nature of a periodically replicated set is more visually noticeable. 

The final step in obtaining a diffusion coefficient is to simulate the dynamics of a penetrant molecule on the network of sorption states and rate constants. Analogous to the frozen positions of voids and channels in a glassy polymer, the relative sorption probabilities and jump rate constants typically remain constant throughout the diffusion simulation. For uniform rate constants on an ordered lattice, it is possible to solve the dynamics analytically. For the disordered network found for voids through a polymer matrix, a numerical solution is required. 

Because a penetrant is Kkely to explore a sorption state for a long time before jumping, it is reasonable at low concentrations to assume that individual penetrant jumps are uncoupled from one another and that the sequential visiting of states is a Markov process [103]. Since each jump occurs independently, the probabihty density p ii) that a time t elapses before the next jump occurs (the waiting time) is distributed according to a continuous-time Poisson process [103],... 

The amount of time that elapses between penetrant jumps thus depends explicitly on the instantaneous sorption state population and on the distribution of jump rate constants out of occupied states. 

The sorption state populations evolve according to a master equation. For a system in which each sorption state can be represented by a single local minimum,... 

Networks generated using methods of Section IX fall in this category. For some systems, there is a separation of time scales and jumps can be separated into two categories fast jumps among different local minima in the same sorption state and slow jumps between connected sorption states. For this case, the master equation can be written... 

A and B represent different overall sorption states and i and j are individual regions of states A and B, respectively. The ratio ptjpA equals the fraction of sorption in state A that occurs within region i. After some time, the sorption state probabilities stabilize at their equilibrium values p ox p that satisfy microscopic reversibility. 

The master equations can be solved numerically using kinetic Monte Carlo simulation [85,86,155]. To begin, a number of independent, noninteracting ghost penetrants are placed in the sorption states of each network, according to the equilibrium distribution p j = Sa/ Sa-... 

These initial positions are stored in a vector to- The distribution can be sampled using the same method as described above when assigning the sorption coefficients to each sorption state ... 

The number of penetrants Np should be much larger than the number of sorption states Ns in order to sample the probabilities correctly. The number of penetrants in a state i at a time t is denoted below by Ni t). 

The kinetic Monte Carlo simulation should begin by conducting many steps in order to equilibrate the penetrant sorption state positions, after which To is reset to zero. A production run can then begin. In the production run, the net result of many steps (10 -10 per particle) is a list of penetrant sorption state occupancies as a function of time. The mean-squared displacement can be calculated (during or after the simulation) at each time as... 

Characterization of the sorption state of polyatomics is a much more difficult task. Here only some selected examples will be quoted. 

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