Generalities
$J_i$	Molar flux of component $i$ (gaseous species diffusing across the solid membrane)
${\mathcal{P}}_i$	Permeability (coefficient) of component $i$ (gaseous species diffusing across the solid membrane)
$p_i$	Partial pressure of component $i$ in the gas mixture
$f_i$	Fugacity of component $i$ in the gas mixture
$l$	Membrane thickness
$c_i$	Penetrant molar concentration in the membrane phase
$c_{u,i}$ $, c_{d,i}$	Molar concentration (moles of gas/membrane volume) in the membrane phase on the upstream and downstream sides, respectively
$D_i$	Local diffusivity or diffusion coefficient of species $i$ in the membrane
z	Coordinate indicating the position along the membrane thickness
$L_i$	Mobility (or self-diffusion diffusivity) of species $i$ in the membrane
${\mu }_i$	Chemical potential of species $i$
$S_i$	Solubility coefficient of species $i$ in the membrane $S_i=\frac{∆c_i}{∆p_i}$
${\overline{D}}_i$	Concentration-averaged diffusivity of species $i$ in the membrane
${\omega }_i$	Mass fraction of the fluid in the membrane
${\alpha }_{i,j}$	Selectivity of component $i$ versus component j in the membrane
${\alpha }_{i,j}^S$	Solubility-Selectivity of component $i$ versus component $j$ in the membrane
${\alpha }_{i,j}^D$	Diffusivity-Selectivity of component $i$ versus component $j$ in the membrane
$S_0$	Pre-exponential factor for solubility coefficient
$\Delta H_s$	Heat of sorption
$D_0$	Pre-exponential factor for diffusion coefficient
$E_D$	Activation energy of diffusion
$P_0$	Pre-exponential factor for permeability coefficient
$E_P$	Activation energy of permeation
$\Delta H_c$	Heat of condensation of the fluid species
$\Delta H_m$	Heat of mixing of the fluid species in the membrane
Robeson’s upper bound
${\beta }_{i,j}$	Gas couple-dependent parameter (position of the upper bound)
${\lambda }_{i,j}$	Gas couple-dependent parameter (slope of the upper bound)
$d_{k,j}$	Kinetic diameter of the larger molecule
$d_{k,i}$	Kinetic diameter of the smaller molecule
$a$	Parameter used in the correlation for ${\beta }_{i,j}$ and ${\lambda }_{i,j}$
$𝓫$	Parameter used in the correlation for ${\beta }_{i,j}$ and ${\lambda }_{i,j}$
$𝓯$	Parameter used in the correlation for ${\beta }_{i,j}$ and ${\lambda }_{i,j}$
Activity coefficient models for solubility
${\gamma }_i$	Activity coefficient (of the fluid in the membrane)
$n_i$	Number of moles of component $i$ in a mixture
$G^{ex}$	Excess Gibbs free energy of a mixture
${\overline{G}}_i^{ex}$	Partial molar excess Gibbs free energy for a component $i$ in a mixture
LF EoS
$G$	Gibbs free energy
$\mathcal{N}$	Total number of molecules
$k_B$	Boltzmann’s constant
$n_c$	Number of components (gases + polymer)
$M_i$	Molar mass of component $i$
${\rho }_i$	Density of component $i$
$v_i^{*}$	Molar volume of a lattice cell of component $i$
$r_i^0$	Number of lattice cells occupied by a molecule of pure component $i$
${\epsilon }_i$	Non-bonded interaction energy between two lattice cells occupied by component $i$
$T_i^{*}$	Characteristic temperature of component $i$ $T_i^{*}=\frac{{\epsilon }_i}{k_b}$
$p_i^{*}$	Characteristic pressure of component $i$ $p_i^{*}=\frac{{\epsilon }_i}{v_i^{*}}$
${\rho }_i^{*}$	Characteristic density of component $i$ ${\rho }_i^{*}=\frac{M_i}{r_i{v}_i^{*}}$
${\tilde{T}}_i$	Reduced temperature of component $i$ ${\tilde{T}}_i=\frac{T}{T_i^{*}}$
${\tilde{p}}_i$	Reduced pressure of component $i$ ${\tilde{p}}_i=\frac{p}{p_i^{*}}$
${\tilde{\rho }}_i$	Reduced density of component $i$ ${\tilde{\rho }}_i=\frac{{\rho }_i}{{\rho }_i^{*}}$
$\rho$	Density of the mixture
$\tilde{T}$	Reduced temperature $\tilde{T}=T/T^{*}$
$\tilde{p}$	Reduced pressure $\tilde{p}=p/p^{*}$
$\tilde{\rho }$	Reduced density $\tilde{\rho }=\rho /{\rho }^{*}$
$k_{ij}$	Binary interaction parameter between $i$ and $j$
${\omega }_i$	Mass fraction of component $i$
${\varphi }_i$	Volume fraction of component $i$ in close-packed conditions ${\varphi }_i=\frac{{\omega }_i/{\rho }_i^{*}}{{{\displaystyle \sum }}_i^N{\omega }_i/{\rho }_i^{*}}$
${\rho }^{*}$	Characteristic density of the mixture $\frac{1}{{\rho }^{*}}={\displaystyle \sum }_i^{n_c}\frac{{\omega }_i}{{\rho }_i^{*}}$
$p^{*}$	Characteristic pressure of the mixture $p^{*}={\displaystyle \sum }_i^{n_c}{\varphi }_i{p}_i^{*}-{\displaystyle \sum }_i^{n_c-1}{\displaystyle \sum }_{j>i}^{n_c}{\varphi }_i{\varphi }_j\Delta p_{ij}^{*}$ $∆p_{ij}^{*}=p_i^{*}+p_j^{*}-2\left(1-k_{ij}\right)\sqrt{p_i^{*}·p_j^{*}}$
$T^{*}$	Characteristic temperature of the mixture $T^{*}=\frac{p^{*}}{{{\displaystyle \sum }}_i^N\frac{p_i^{*}{\varphi }_i}{T_i^{*}}}$
$v^{*}$	Average close-packed molar volume in the mixture $v^{*}=\frac{T^{*}R}{p^{*}}$
$r_i$	Number of lattice cells occupied by a molecule in mixture $r_i=\frac{r_i^0{v}_i^{*}}{v^{*}}$
NRHB EoS
${\epsilon }_i^{*}$	Characteristic energy in the LF model and in the NRHB model
${\epsilon }_{i,h}^{*}$	Enthalpic contribution to the characteristic energy
${\epsilon }_{i,s}^{*}$	Entropic contribution to the characteristic energy
$E_{\alpha \beta }^0$	Association energy between group α and a functional group $\beta$
$S_{\alpha \beta }^0$	Association entropy between a functional group α and a functional group $\beta$
SAFT EoS
$A^{res}$	Residual Helmholtz free energy (at fixed temperature and volume)
$A^{hs}$	Hard-sphere term of residual Helmholtz free energy
$A^{disp}$	Dispersion term of residual Helmholtz free energy
$A^{chain}$	Chain term of residual Helmholtz free energy
$A^{assoc}$	Association term of residual Helmholtz free energy
${\mu }_i^{IG}$	Chemical potential of species i in the ideal gas state
NET-GP model for solubility
${\mu }_i^{NE}$	Non-equilibrium chemical potential of species $i$
${\mu }_i^{Eq}$	Equilibrium chemical potential of species $i$
ρpol	Non-equilibrium density of the glassy polymer
${\mu }_i^{NE\left(pol\right)}$	Non-equilibrium chemical potential of species $i$ in the polymer phase
${\mu }_i^{Eq\left(gas\right)}$	Equilibrium chemical potential of species $i$ in the gas phase
$\overline{\omega }$	Composition vector in the polymer phase
$\overline{y}$	Composition vector in the gas phase
${\rho }_{pol}^0$	Non-equilibrium density of the dry glassy polymer
$k_{sw,i}$	Swelling coefficient correlating the glassy polymer density to the gas $i$ partial pressure
DMS model for solubility
$k_{D,i}$	Henry’s law constant
$C_{H,i}^{\prime }$	Langmuir capacity constant
$b_i$	Langmuir affinity constant
$k_{D0}$	Pre-exponential factor for temperature-dependence of $k_D$
$b_0$	Pre-exponential factor for temperature-dependence of $b$
$∆H_D$	Enthalpy of sorption for Henry’s mode of sorption
$∆H_b$	Enthalpy of sorption for Langmuir’s mode of sorption
GAB model for solubility
$v$	Penetrant adsorbed mass ratio
$p$	Adsorbate pressure
$p^{*}$	Reference pressure value associated with the adsorbate
$v_m$	Capacity of the 1st adsorption monolayer
$h$	Dimensionless binding parameter
$r_{ij}$	Parameter related to sorbate–sorbate interactions in multicomponent GAB
$h_{ij}$	Parameter related to sorbate–sorbate interactions in multicomponent GAB
Fractal model for solubility coefficient
$F_g^{ef}$	Effective cross-sectional area of the sorbed gas molecules
$D_f$	Global fractal dimension parameter
$S_0$	Minimum solubility of a gas molecule
${\phi }_{cl}$	Relative fraction of the closely packed segments in clusters
$T_g$	Polymer glass transition temperature
$X_{cr}$	Degree of crystallinity
$d_f$	Fractal dimension of the polymer structure
$A_{cr}$	Cross-sectional area of a macromolecule
$C_S$	Characteristic ratio representing the index of chain flexibility
Free-volume theory for diffusion coefficient
$D_{1,self}$	Self-diffusion coefficient of fluid 1 in polymer 2
$D_1^0$	Pre-exponential factor = diffusion in a fluid with infinite free volume
$E_D^0$	Energy required for a jump into an adjacent free-volume void
$\gamma$	Coefficient accounting for overlaps of free volume available to adjacent molecules
$V_i^{*}$	Occupied volume
$V_F$	Average free volume per jumping unit
$\xi \equiv \frac{V_1^{*}}{V_2^{*}}$	Ratio of occupied volumes
$K_{1i},K_{2i}$	Parameters for component i related to pure component viscosity
${\alpha }_1$	Thermodynamic factor of mutual diffusivity
$FFV$	Fractional free volume
$A,B$	Adjustable parameters for correlation between diffusivity and FFV
$V_{pol}$	Polymer-specific volume
$V_{pol}^{*}$	Polymer-occupied volume
Fractal model for diffusion coefficient
$D_0$	Universal constant of the model, equal to 3.8 × 10−7 cm2/s
$f_g$	Relative free volume
$d_h$	Diameter of a microvoid
$d_m$	Diameter of the penetrant gas molecule
$d_s$	Polymer chain spectral dimension
Maxwell–Stefan model for diffusion coefficient
$Đ$	Maxwell–Stefan diffusivity
$v_A-v_B$	Velocity of species A relative to species B
$\mathsf{\Gamma }$	Thermodynamic correction factor
$N_i$	Total molar flux of species $i$ (with respect to a fixed reference frame)
${𝓃}_t$	Total molar concentration of the fluid mixture
$x_i$	Mole fraction of component $i$ in the mixture
Partial Immobilization Dual-Mobility Model for Permeability
$N_i$	Total molar flux of component $i$
$C_{D,i}$	Penetrant concentrations in the Henry’s region
$C_{H,i}$	Penetrant concentration in the Langmuir’s region
$D_{D,i}$	Diffusivity in the Henry region
$D_{H,i}$	Diffusivity in the Langmuir region
$F_i$	Ratio between diffusivity in Henry’s region and Langmuir’s regions
$K_i$	Ratio between Dual-Sorption Mode Model parameters
$p_{u,i}$	Upstream partial pressure of component $i$
$p_{d,i}$	Downstream partial pressure of component $i$
Standard Transport Model for Permeability
$L_i$	Mobility coefficient of component $i$
$L_{i,0}$	Infinite dilution mobility coefficient of component $i$
$\beta$	Plasticization factor
$M_i$	Penetrant molecular weight
$Z_i$	Penetrant compressibility factor
${\rho }_{pol}$	Polymer density
$\varrho$	Size selectivity of the polymer
$\tau$	Polymer-dependent parameter in the empirical correlation between penetrant mobility and critical volume
${\varrho }_0$	Pre-exponential factor for parameter $\eta$
$V_c$	Critical volume of penetrant
Widom insertion method
${\mu }_i^{ex}$	Excess chemical potential of penetrant $i$ inside the polymer
$\Delta U_{\mathrm{test}}^{\mathrm{inter}}$	Potential energy of interaction between the test molecule and the other molecules
Simulation of diffusivity
$⟨(r_i\left(t\right)-r_i\left(0\right){)}^2⟩$	Mean squared displacement (MSD) averaged over all molecules
t	Time
$D_{i,self}^{MD}$	Self-diffusion coefficient computed in the MD simulations
$\eta$	Shear viscosity computed in MD simulations
$\Xi$	Dimensionless constant equal to 2.837298 for periodic (cubic) lattices
$\Lambda$	Simulation box length