A qualitative approach to adsorption mechanism identification on microporous carbonaceous surfaces

Let us consider a movement of an individual particle of a volatile phase (adsorptive molecule) near an adsorbent solid wall, in time intervals t _int between collisions with other adsorptive molecules. The averaged, in time t, molecule movement is limited by an equipotential boundary surface U _bdry (t). On this surface a kinetic energy of two degrees of freedom k _B τ(t) is compensated by a corresponding decrease in an average solid–fluid potential U _avge (t). Thus, the equilibrium of the molecule movement may be described kinematically by the following formulae:where k _B stands for a Boltzmann constant.

Let us assume that the mass center of this particle moves within a space of volume V _fi (free space) and may be found at any phase-space variables v with the same probability (i.e. v ≤ V _fi ). Hence, time averages t over the particle movement after a suitable lapse of time, frequently longer than t _int , correspond to macro scale ensemble averages and the volume in space is preserved, see (Hansen and McDonald 2006). Thereby, kinetic energy \(k_{B} \tau \left( t \right)\) averaged in time may be equivalently represented by an energy k _B τ _i averaged over the movement space in volume v. Thus, the kinetic energy is expressed in units [K] and τ _i denotes an average particle “temperature”:

Subsequently, the average solid–fluid potential U _avge (t) and boundary potential U _bdry (t) may be equivalently described by analogous formulas leading to the U _avge (V _fi ) and U _bdry (V _fi ).

At the boundary, when considering individual particle, the potential is generated by the solid–fluid adhesive forces producing component U _bdry (V _fi ). Nevertheless, when considering the macro scale, at the boundary, potential is also generated by possible collisions of adsorptive particles in the volatile phase and depends on an external pressure p (i.e. expressed as a pV potential).

Total effects of interactions of the adsorbate particle with volatile phase, represented by the pressure p, may be partitioned into subsystems using a collision probability s _pi /s _p of an individual i-th particle. Kinematically, interactions with other particles represented by the pV potential may be related to the free volume V _fi . Hence, the interaction force p·s _pi should be recalculated to an effective pressure p _fi , and then multiplied by the volume V _fi :where s _pi is an open surface, limiting the particle movement in volume V _pi and corresponding to the open surface s _fi , limiting the particle mass center movement in volume V _fi . For a convenience, the relationship s _fi /V _fi may be denoted as a specific surface area of the space of particles mass center a _fi .

Hence, the kinematic equilibrium of the i-th particle takes the form:

The values for the particle volume V _pi and the surface ratio s _pi /s _fi may be roughly evaluated by replacing the real free space of the adsorbate molecule with the sphere of the same volume V _fi and corresponding diameter d _fi . Then the volume V _pi is calculated as a sphere with the diameter d _pi , increased by the adsorbate hard sphere core diameter σ _ff :

In our study solid is represented by a set of carbon atoms spaced like in a graphite sheet and forming a nanostructure of a predefined shape (nanotube, nanocone or nanocubic niche etc.). Hence, the carbon-fluid potential at a given k-th position of adsorbate molecule mass center is calculated by summing up the LJ_12-6 potential u _C-fk (Lennard-Jones 1931). The latter is performed over all set of carbon atoms forming the solid wall, with a corresponding carbon-fluid intermolecular distance r _C-fk :where ε _C-fk and σ _C-fk are the potential well-depth and intermolecular diameter, respectively, both calculated applying Lorentz-Berthelot (Berthelot 1898; Lorentz 1881) mixing rules from pure-species quantities.

The boundary potential U _bdry (V _fi ) is taken arbitrary as a fixed value of \(\sum\limits_{C} {u_{{C - f_{k} }} }\). Hence, the average potential U _avge (V _fi ) is:

The values for V _fi , corresponding to a given U _bdry , may be evaluated with analytical or Monte-Carlo calculations. The analytical calculations run initially by searching consecutive points k at an equipotential surface s _fi of a given U _bdry . If \(\sum\limits_{C} {u_{{C - f_{k} }} }\) ≈ U _bdry the k-th point is accepted as belonging to the surface s _fi (more precisely, to a volume δs _fi ·δr _C-fk ). Then, the volume V _fi is determined, as a volume of space limited by all points found at the s _fi surface. Starting with a minimum potential U _bdry for which V _fi  ≈ 0, the calculations run step by step with an increasing U _bdry by a small value δU _bdry . In the Monte-Carlo technique u _C-fk is calculated many times for randomly chosen points k (shots). The point k is accepted as an admissible n _adm if respects geometrical constraints of the solid and as a successful n _v if additionally fulfills the relation \(\sum\limits_{C} {u_{{C - f_{k} }} }\) ≤ U _bdry . Having an appropriately large number of admissible shots n _adm  > 10⁶ with a number n _v of successful shots, one can calculate the volume V _fi corresponding to a given U _bdry : V _fi  ≈ n _v /n _adm . In both techniques the values for U _avge (V _fi ) are calculated by numerical integration of the U _bdry (V _fi )– see Eq. (7). Subsequently the corresponding particle volume V _pi is calculated according to Eq. (5).

So far as a movement of a single particle is considered only, one should assume that there are no particles in volatile phase, i.e. p = 0. In this case, having the values for U _avge (V _pi ) and U _bdry (V _pi ) one can calculate with Eq. (4) the values for τ _i corresponding to a given V _pi . For a convenience, the quantities in Eq. (4) may be also related to one mole of particles, enabling to calculate the kinematic equilibrium temperature τ _i (corresponding to a given V _fi , V _pi , and so to U _avge (V _pi ), U _bdry (V _pi )):andwhere N _A is an Avogadro constant and R is a gas constant.

The functions U _m (V _mpi ) and U _madh (V _mpi ) are schematically shown in Fig. 1. Subsequently, the inverse relationship V _mpi (τ _i ) calculated with Eq. (8) is schematically shown in Fig. 2.

Let us denote V _ind as a maximum volume of a free movement space (e.g. close to an averaged volume of particle movement at a critical point), thus it is not likely that this space may be shared with other molecules. The corresponding temperature τ _ind is found with Eq. (8) and τ _ind  = arg {min {V _mpi (τ _ind ) = V _ind }}. When the particle temperature is appropriately low, i.e. τ _i  ≤ τ _ind and so V _mpi (τ _i ) ≤ V _ind , one may assume that it occupies a localized site, so the localized monolayer adsorption takes place. If the temperature is τ _i  > τ _ind , but is still appropriately low, that the particle may be captured into an interaction potential well, one may assume that the particle may easily occupy an individual spaces V _mpi (τ _i ) or joint a space V _mpi (τ _i ) containing j molecules. Thus, maximum volume occupied by individual particles j (depending on pressure p) may be expressed in general form V _max  = V _mpi (τ _i )·j, and so the corresponding maximum temperature τ _max  = arg{V _mpi (τ _i ) = V _max  ≫ V _ind }. If the space of volume V _max covers a large fraction of the solid surface, adsorption mechanism may be seen as a purely mobile, otherwise it may be closer to the clustering based or to locally mobile adsorption (based on forming a local 2D fluid). Nevertheless, the derived formulas are legitimate only for characterization of localized adsorption mechanism properties. Thus they don’t decide on the mechanism in the temperature range τ _ind  < τ _i  ≤ τ _max (i.e. it may be mobile or clustering based mechanism). Finally, particles of the temperature τ _i  > τ _max are treated as belonging to the volatile phase (not adsorbed).

While accepting the abovementioned criteria, one should consider that the total volume V _mpi and potentials U _madh (V _mpi ), U _m (V _mpi ) are additive at appropriately low temperature range τ _i  ≤ τ _ind . Moreover, excess energy evolved due to the interaction of the captured particle with solid wall is totally withdrawn to keep the constant temperature T in isothermal adsorption. Hence, the quantities in Eq. (4) may be averaged over all set of molecules (e.g. containing N particles) with Maxwell–Boltzmann distribution function f(τ _i ,T) at a given temperature T. If the individual particles temperature is τ _ind  < τ _i  ≤ τ _max , the additivity of the total volume of adsorbed individual particles is no longer valid. Nevertheless, for a consistency of the proposed description it is useful to evaluate the number of particles j sharing occupied volume also in the range V _ind  < V _mpi (τ _i ) ≤ V _max . Assuming, that the volume V _ind corresponds to the maximum separated volume of one mole of individual particles, the upper bound for a volume V _mpi (τ _i ) of the number of particles, may be evaluated with simple expression j = V _mpi (τ _i )/V _ind . As well, the potentials U _madh (V _mpi ) and U _m (V _mpi ) may be seen as proportionally shared in terms of the number of j particles. Thus, the following expressions are valid:

Hence, it enables for the averaging of the quantities in Eq. (4) also up to the limit τ _i  = τ _max .

The integration of the distribution function f(τ _i ,T) up to the limits τ _ind and τ _max leads to the fraction w _ind and w _max of volatile phase particles, at the particular temperature T. Following adsorption mechanism identification criteria w _ind is a localized mechanism fraction and so w _max correspond to the other mechanisms, e.g. possible mobile mechanism (see Fig. 3):

where the remaining part (1 − w _max ) is the fraction of one mole of individual particles of volatile phase π. The values obtained in Eq. (11) refer to the idealized adsorptive volatile phase fractions that may be adsorbed with particular mechanism, but it does not specify liquidlike phase particles properties.

At the intervals between collisions, each i-th particle in the adsorption system is at the kinematic equilibrium and the average temperature of adsorbed molecules is expressed as:where N _ind is a number of particles corresponding to the particular temperature τ _i , fulfilling the constraint τ _i  ≤ τ _ind .

Hence, by virtue of Eqs. (8) and (12) the following equation is valid:where \(\bar{ \cdot }\) refers to the quantities at the given temperature T _k , averaged over the interval τ _i   \(\in\)  (0; τ _ind ). The averaging is performed for all quantities in analogy to the following formulae:

Moreover, Eq. (13) implies that relationship V _mpi (T _k , p = 0) may be seen as the upper bound for a volume V _mpi (T _k , p). It may be easily examined by solving nonlinear system, of Eq. (13) and equilibrium vapor pressure estimation with quasipolynomial Riedel equation (Vetere 1991). Thus, enabling to find corresponding temperature T _k at the pressure p:where p is expressed in units [kPa] and A, B, C, D are empirical constants that may be taken from KDB (CHERIC 2015; Kang et al. 2001).

Let us assume that τ _ind  = ∞ and adsorption system, held at a constant temperature T, consists of N _ind separated subsystems, each containing one molecule i. The volume of any molecular system at its equilibrium state minimizes free enthalpy G of the system:where U _ads , S _ads , V _p denote the total energy, entropy and volume of adsorbate molecules, respectively.

For such a system internal entropy and energy are additive, so the following equations are valid:where S _conf denotes the configurational entropy of the system, S _i stands for the i-th particle internal entropy (it was assumed that finding of an individual particle in any infinitesimal volume δV is equally probable), S ₀ is a constant of no importance and U _avge (V _fi ) is defined in Eq. (7).

The configurational entropy does not depend on adsorbate molecules volume, so ∂S _conf  / ∂V _p  = 0. In turn, effects of any change in free volume V _fi of an i-th particle are limited to this particle internal entropy and temperature. Hence, the entropy term and its total differential should be written in the following form:

Thus, by virtue of Eqs. (7), (17) and (18), free enthalpy total differential may be expressed:

Let us take that any change in the total volume V _p splits into the volumes V _pi and V _fi in proportion to the surfaces s _p , s _pi and s _fi of corresponding spaces (i.e. any change in volume may be expressed by small change dr). Hence, the differentials fulfill the relations:

Moreover, the total effect of interactions of adsorbate particles N _ind with volatile phase, represented in Eq. (19) by the pressure p, may be partitioned onto the subsystems using the collision probability s _pi /s _p of an individual i-th particle:

For a convenience, let us again relate the quantities obtained in Eq. (19) to one mole of particles according to Eq. (9). By virtue of Eqs. (20) and (21) into Eq. (19) the thermodynamic equilibrium may be expressed in the following form:

Subsequently, taking into account quantities defined in Eq. (9), the thermodynamic equilibrium temperature T _t may be obtained:

In order to evaluate the adequacy of kinematic considerations to the analysis of adsorbed particles properties, let us calculate the thermodynamic temperature T _t . Thus, let us assume that the distributions of the quantities U _madh (V _mpi ) and U _m (V _mpi ) and corresponding s _pi and a _mfi is the same as in the kinematic equilibrium, at the temperature T _k (see Eq. (13)):where N stands for the number of the available τ _n , found with Eq. (8).

The relationship V _mp (T _t ) may be viewed as a realistic description of adsorbate volume in real localized adsorption systems. Moreover, the comparison of Eqs. (24) and (13) provides information that the particle temperature distribution in real systems is the Maxwell–Boltzmann distribution, modified with a _mfn weights.

In real adsorption system, it is not likely that adsorption will take place only with purely localized or purely mobile mechanism. In turn, intuitively both mechanisms are expected, even if single heterogeneity is considered as a model (e.g. cavity, corner). Hence, the particular mechanism domination may be evaluated based on the molecules fractions w _ind and w _max , see Eq. (11). One may consider the ratio of the obtained fractions, as follows:

The relationships in Eq. (25) are exceptional cases and most likely in real adsorption systems they might not be observed. Nevertheless, at the given temperature T of experimental adsorption measurements, it is possible to select the most appropriate mathematical adsorption model based on \({{w_{ind} } \mathord{\left/ {\vphantom {{w_{ind} } {w_{max} }}} \right. \kern-0pt} {w_{max} }}\) ratio (see Fig. 3).

The localized mechanism in uniBET model, elaborated in our team, may be considered as a mixing of molecules being in the reference liquidlike state, with cells placed at the adsorbent surface, see e.g.: (Duda et al. 2013; Milewska-Duda et al. 2000; Milewska-Duda and Duda 2001). In such a view a key parameter is an effective adhesion energy E _adh , related to a measurable monolayer adsorption energy Q _A with a Huggins formula (Flory 1953):where E _vap is the cohesion (evaporation) energy of the adsorbate in its reference state and ζ is a surface texture parameter describing possible contact of adsorbate with neighboring adsorbent molecules (intermolecular effective contact ratio). Adhesion energy E _adh is evaluated for a pair of molecules being in a full contact, by combining, via Berthelot rule, adsorbate and adsorbent cohesion energies:where δ _c is a solubility parameter and E _vap is evaluated with PVT relationship for V _m , a molar volume of adsorbate in its reference state (for vaporous substances it is the liquid state), see (Milewska-Duda and Duda 2009).

In order to get an insight into adequacy of the kinematic evaluations in Eq. (13) one may calculate the ζ parameter. Thus, taking the values for E _adh (V _mp ) found with Eq. (27), V _mp and corresponding effective adhesion energy U _madh (V _mp ) calculated with Eq. (10) and averaged with Eq. (14):

Despite the simplifications involved in Eq. (10) for the volume V _ind  < V _mpi  < V _ind , the parameter ζ found with Eq. (28) may be viewed as a basis for validation of the proposed approach, when compared to empirical values published e.g. in (Duda et al. 2013).