Multicomponent mass transfer kinetics in nanocomposite (NC) bifunctional matrixes: NC selectivity and diffusion concentration waves

1 Introduction

The aim of this review is to consider the theoretical approaches, and fundamental aspects of the multicomponent mass transport of the ion exchange (IEX) kinetics in various bifunctional matrixes of the nanocomposites (NC) on the basis of the author’s NC model [1–5].

The new contemporary NC model is developed for investigations of the multicomponent mass transfer kinetics in the bifunctional NC materials [1–6]. The proposed postulates of the presented author’s NC model give the possibility to include into the theoretical modeling the bifunctional nature of the NC matrix with the multicomponent diffusion mass transport of the i components, which are characterized by the D_i diffusivities.

The detailed description of the properties of such type novel NC materials with the example: “NC metal-ion exchangers” including the methods of their synthesis has been published recently in the Russian monograph [7]. Theoretical aspects for the synthesis of the NC matrix have been discussed in detail including a number of practical applications [7]. The NC material includes numerous nanoparticles (NP) embedded inside the matrix medium of the NC. The details of the NC synthesis with zero valent metal (Me⁰) in the NC ion exchanger matrix are presented [7]. Synthesized NC with the bifunctional matrixes have practical advantages in comparison with the usual IEX materials, as for example, NC “Me⁰-ion exchangers” [7, 8].

Figure 1A, B shows the obvious experimental micrography (A) including the drawing (B) of the NC structure with the embedded NP agglomerates. The corresponding picture (Figure 1B) is presented as the illustration of the embedded NP into the NC matrix [7]. Figure 1A shows one of the experimental microphotography of the structure for the NC “Me⁰-ion exchangers” including Me⁰ NPs [Figure 1B (5), Figure 2; NP-shaded] inside the bifunctional NC matrix. The microphotography of the experimentally synthesized NC (Figure 1A) was obtained by Prof. T. Kravchenko group from Voronezh State University [7]. Many experimental samples are presented as the illustrations of the NC media in the monograph [7] and in the author’s various joint publications [2–5, 8, 9]. The Me⁰-NPs in the NC matrix (Figure 1A, NP light points; Figure 1B, R⁰, shaded regions) play the role of the active R⁰ nanocenter sites located at the bifunctional NC matrix in the scheme of the contemporary model of the NC kinetics (Figure 2, shaded) [1–6].

Figure 1

(A, B) Mass transfer inside bifunctional NC matrix: (A) experimental NC microphotography, magnification: 10,000, (B) illustration of NC r-bead (ro-fiber); 2 micropores, three fixed groups R, four counterions, 5-Me⁰-NP agglomerates (shaded, B) [7].

Figure 2

Scheme of mass “Transformations” inside bifunctional NC matrix at the active nanosites (R⁰, Me⁰): (Ia) – association stage of MAL_S reactions (left); (Ib) – dissociation stage of MAL_S reactions (right): (I) Selectivity; (II) multicomponent diffusion mass transfer along NC pores: (Figure 1B, 2). Fluxes J_k describe “sink” or “source” for mass transformations.

2 Contemporary NC model for multicomponent mass transfer inside bifunctional matrix of nanocomposites

The modern theoretical NC model for multicomponent NC kinetics has been designed for the bifunctional NC matrix with the key conception including the two coexisting routes: (I)+(II) (scheme, Figure 2) for the multicomponent diffusion mass transfer inside the bifunctional NC matrix medium. In the bifunctional NC matrix, these two coexisting routes, (I)+(II), reflect the two simultaneous processes:

1. The sorption (Ia) – desorption (Ib) reactions onto the active nanosites R⁰, (shaded, Figure 2, I – route,) are described by mass action laws (MAL_S) [10] for the i components of the NC bifunctional matrix together with the second (II, co-route) multicomponent diffusion (D_i) process.

2. The multicomponent diffusion (D_i) of the i components occurs in the pores of the NC matrix (Figure 2, II – route).

The conception (I) of the NC model describes the equilibria of the dissociation-association MAL_S reactions (Ia, Ib) onto the active R⁰ nanosites (Figure 2, R⁰, shaded) inside the NC matrix. In compliance with the scheme in Figure 2, there have been proposed the two coexisting routes: (I, Selectivity) and (II, Diffusion) for the multicomponent NC mass transfer inside the bifunctional NC matrix [1–5]. Along with the multicomponent diffusion (D_i) mass transfer for route (II), the “dissociation-association ” reactions (I) equilibria are realized simultaneously in the NC model via the mechanism of the “sink and sources” (Ia,b) with the corresponding J_k mass fluxes (Figure 2, I – route). These reactions’ equilibria are described quantitatively by MAL_S on the basis of the non-irreversible thermodynamics approach [10].

Figure 2 represents the scheme illustration of the NC matrix [1–5] with all accompanying mass transfer elements. The I – route onto the active R⁰ nanosites of the NC with the “sinks” (Ia) and “sources” (Ib) represents the mass transformation mechanism for the i, j components with the [ij] “transformation”: [ij]↔[i]+[j] (Ia,b). Figure 2 shows the simultaneous second route II – the multicomponent diffusion mass transfer of the i components (with D_i diffusivities) in the NC matrix pores [1–5]. The property of the route (I) for the multicomponent i, j component mass transformations: (i+j→ij), (scheme, Figure 2) may be interpreted and determined as the “selectivity” factor. The examples of the [ij] – “mass transformation” (Eq. 1) are presented below in Sections 4–6 including the corresponding MAL_S relations (Eq. 2) on the I – route.

The various i, j components participate in the MAL_S reactions (Figure 2, stages Ia, Ib) simultaneously together with the multicomponent diffusion (D_i, D_j) mass transfer (scheme, Figure 2, II) of the i components inside the pores [Figure 1B (2)] of the NC bifunctional matrix during the whole NC kinetic process. The mechanism of the “association-dissociation” reactions equilibria (Ia,Ib; Figure 2) retard the whole kinetic multicomponent mass transfer process. This retardation of the i components may be interpreted as the selectivity (I) for the multicomponent NC kinetics depending on the MAL_S parameters, K_S values (Eq. 2) of the reactions (I). The examples of the simulation of the multicomponent NC mass transfer kinetics with the route (I) are considered in detail (Sections 4–6) [3–5].

Typical composition of the multicomponent D_i diffusion NC kinetics may include various i components: ions, complexes, and neutral substances with zero charges (see the examples in Sections 4–6). In addition, it should be taken into consideration that the immovable m components arose as the consequence of the MAL_S mass transformation (route I). The p component and fixed R⁰ nanosites (with D_R=0) may be associated with the formation of the immovable _m(R⁰p) component. Consequently, in this case, the m component _m(R⁰p) acquires zero diffusivity (D_m =0) in the bifunctional NC matrix. (Here and elsewhere, the k number of any k component is represented by the left index of the component sign.)

The created modern multicomponent NC model [1–6] discussed here has common points with the preceding approaches published previously [11–19] especially at the end of the last century [16–20]. However, for the IEX kinetics, the previous kinetic models have been applied only to the IEX kinetic processes, including the cases accompanied by the chemical reactions in the usual IEX resins (besides for r-beads only) [11–20], but not for the new bifunctional NC matrixes [1–6, 21]. In contrast, the obtained theoretical results of the extensive computerized investigation for the multicomponent NC kinetics on the basis of the author’s NC model [1–5] are new and obtained for the first time including, in addition to the consideration of the three various shapes of the NC matrix: spherical r-bead, cylindrical ro-fiber, and planar L-membrane.

The newly created author’s NC model for the multicomponent mass transfer kinetics [1–6] is applied for the computer simulation into the bifunctional NC matrix of the three various shapes mentioned. Here the modern bifunctional NC: “Me⁰-ion exchanger” [7], is used as the real example for the contemporary NC model applications. For the generalization of the simulated NC kinetics examples (Sections 4–6), the active R⁰ nanosites (scheme, Figure 2), hereinafter, are labeled as the k component: _kR⁰. So for any NC kinetics, the R⁰ symbol means the NC nanosites with absolutely the same meaning as Me⁰ for the example “Me⁰-ion exchanger” in Figures 1A, B; 2 (the “shaded regions” are the R⁰ nanoparticles agglomerates).

In the author’s NC model [1–5], the fixed groups R of the bifunctional NC-IEX matrix [Figure 1B (3)] are not taken into consideration in the kinetic multicomponent NC mass transfer process – only R⁰ nanosites are active (Figure 2, R⁰ shaded region). The reasons of this neglect of the fixed R groups’ [Figure 1B (3)] participation in the NC mass transfer process are:

1. The NC sites (R⁰, Me⁰) are formed near the fixed R groups of the bifunctional NC matrix during the NC synthesis process. The free fixed R groups consist of the minority of the NC matrix content;

2. In the author’s opinion the possible inclusion of the fixed R groups’ influence gives the change of the multicomponent diffusivity factor (D_i, II) during the multicomponent mass transfer NC process inside the bifunctional NC matrix, but not on the change of the influence of the “selectivity factors” (K_s, I). From the qualitative estimations, the principal influence of the selectivity factor (I – route) as the consequence of the bifunctional structure of the NC matrix should not be changed. The author’s main conclusion (Section 9) for the NC model: is the combined influence of the two multicomponent factors (I, Selectivity+II, Diffusivity) in the NC mass transfer process is in analogy to the chromatographic factors: HETP_i+isotherms’ influence (see Table 1);

3. In principle, it is possible to include the fixed IEX R groups into the consideration in the expanded NC model with the expansion of the number of the i components. It gives, however, the essential complexity of the NC model (especially because of the increase in the N number of the i components), and the corresponding complexity in the numerical modeling (including the problem of the stability of the numerical computerized counting).

The modern multicomponent NC model with the key conception (I, II – co-routes for the NC mass transfer kinetics, Figure 2) is included into the mass transfer partial differential Eq. (3) including “sinks and sources” mechanism with the completion by the additional relations (Section 3) [1–6]. The adequate computerized description of the mass transfer in the bifunctional NC matrix is implemented into the computerized modeling by using finite differences for the numerical solution of the partial differential mass balance Eq. (3), (details in Section 3).

Herewith, the effect of the MAL_S reactions [Eq. (2), Section 3] onto the R⁰ nanosites (I, scheme, Figure 2) on the kinetic behavior of the bifunctional multicomponent NC system may be crucial: the whole multicomponent diffusion NC kinetics rate may be decreased by 1 or 2 orders of magnitude with the decisive dependence of the NC process on the Selectivity, Eq. (2) – MAL_S, K_S values (route, I).

Therefore, distinctive changes in the mechanism of the kinetic mass transport process may occur due to the (Ia,Ib) MAL_S reaction factors – K_S (or in other words – “Selectivity”) influence. Consideration of the active _kR⁰ nanosites influence in the bifunctional NC matrix [subroute (Ia,Ib), Figure 2] is especially relevant in the case of the NC mass transport processes via the diffusion of the i, j components in the new bifunctional NC matrixes with the R⁰ nanosites availability (Figure 2) [1–6].

It means in this approach that the multicomponent diffusion (D_i) NC kinetics of the mass transport are described by the set of the nonlinear multicomponent diffusion mass transfer differential equations [Eq. (3), Section 3]. The generalization of the NC mass balance is realized by taking into the consideration of the additional internal J_k fluxes, which are expressed by the “sinks” and “sources” terms for the masses of the corresponding k components. These k sinks or sources are expressed via the MAL_S relations, Eq. (2), that are inherent to the association-dissociation reactions (Ia, Ib) onto the R⁰ nanosites in the bifunctional NC matrix (scheme, Figure 2). The multicomponent set of the generalized mass transport equations are represented mathematically by the nonlinear diffusional partial differential equations, Eq. (3) with the corresponding boundary (9) and initial (10) conditions [Eqs. (9), (10), Section 3].

Such nonlinear rather complex multicomponent NC system [partial differential Eq. (3) with the accompanying relations: (4)–(10), Section 3] might be solved only by the computerized simulation. For the generalization of the theoretical results, the most appropriate consideration is fulfilled via the dimensionless variables: X_i concentration distributions along the length (r, ro,L distances) in the course of the dimensionless T time (see Nomenclature).

All the computerized calculations with the results of the author’s simulation for the i component X_i concentration “waves” distributions are presented below as the distance-dependent distributions in the course of T time: X_i {distance; T}. Here, in the multicomponent diffusion NC kinetics (with D_i diffusitivities), there is used additionally another key concept (W⁺) of the X_i concentration waves, that propagate along the short distances (radius for r-bead, ro-fiber, or thickness for L-membrane) inside the NC matrixes. The computer numerical calculations during the simulation give the dependence: X_i{distance; T} – distributions expressed in the discrete T_s – time moments.

The calculated X_i concentration distributions “waves” may be transformed into the picture frames in the course of time (T_s), which illustrate the propagation of the X_i waves. This illustration is visual and fairly understandably good. Then, the frames may be assembled in the author’s calculated scientific animations (see comments, Section 10) showing clearly the propagation of the diffusion X_i concentration i waves for each i component of the multicomponent mass transfer NC kinetics in the bifunctional NC matrix media and additionally in the theory of multicomponent chromatography (examples in [2–5, 22, 23]).

3 Simulation of multicomponent mass transfer kinetics in bifunctional NC matrix: generalized mass balance equations

The generalized phenomenological approach is described here on the bases of the non-equilibrium thermodynamics [10] with the partial differential equations, Eq. (3), which has been used together with the new contemporary NC model approach for the multicomponent mass transfer in the NC-IEX bifunctional matrixes (scheme, Figure 2) of various r, ro, L shapes [1–6].

The approaches for the various IEX multicomponent kinetic models (but not for the NC ones) had been used previously during the long-time period (for around 50–60 years) starting with an early F. Helfferich book [11] through his Reviews [12, 13] and emphasizing the state-of-the-art report [14] until the 1980–1990 [15–20]. A number of the additional investigations for the multicomponent IEX, NC kinetics with the i concentration waves are cited in the author’s previous articles [1–6, 20, 21].

The mathematical approach is presented below in this section 3 with the list of the positions: the equations and relations considered in the computerized modeling of the multicomponent IEX kinetic mass transport inside the bifunctional NC matrix containing the nanosites R⁰ (Figures 1, 2) [1–5]. The list includes the association-dissociation MAL_S reactions equilibria, Eq. (1) onto the R⁰ nanosites (Figure 2) with the phenomenological MAL_S relations, Eq. (2). The diffusion mass transfer partial differential equations include the additional terms – fluxes S_i, Eq. (3). These S_i terms describe all fluxes (in the arbitrary volume V, Figure 3) of the differential Eq. (3) of the considered multicomponent NC system, which are conditioned by the “sinks-sources” mechanism mentioned (route I, Figure 2). It is natural that the total balance for all participating substances with the internal J_k fluxes (k=1, 2, …) is maintained during the computerized simulations in the considered NC systems [1–5].

Figure 3

Illustration of mass balance partial differential Eq. (3), with its physical sense including J_i mass fluxes in V volume: input (left) J_{i in}^-l, output (right) J_{i out}^+r. Eq. (3) is completed by inner “sink and source” J_k Fluxes presented in Figure 2.

3.1 Dissociation-association reactions, MAL relationships, mass transformations onto the nanosites

The association-dissociation reactions onto the R⁰ nanosites (route I, Figures 1 and 2) may be recorded in the general form [10]

(1)∑i{(ms)Mi}∑j{(ns)Mj}, (1)

where M_i are the symbols of the i components participating in the association-dissociation reactions (1) with the “sink-sources” mechanism on the R⁰ nanosites (sub-routes Ia, Ib; Figure 2) included in the created modern NC model postulates [1–5]. In correspondence with the reactions (1), the mass transformations, Eq. (2) (I route, Figure 2), bring the redistributions of the masses of the participating i components. For the further advance of the NC model considered here, the equilibria of the chemical reactions Eq. (2), (route I, Figure 2) have been taken into consideration by using the classical MAL_S relationships [10, 17]

(2)∏i{[Xj]ms[Xi]ns}=KS, s=1, 2, … (MALS), (2)

where the expression Π_i(i≠j) is the product of the dimensionless concentrations {[X_j]^ms (i=1, 2, …); ns, ms (negative or positive) are the stoichiometric coefficients of the chemical s reactions; s is the corresponding index; the K_S-MAL_S constants of the chemical reactions equilibria (1) are dependent on the rates of the association-dissociation (Ia, Ib, Figure 2) stages [10].

The MAL_S relations (2) together with the “sinks” and “sources” mechanism at the R⁰ nanosites in the mass balance partial differential Eq. (3) present the phenomenological description of the created NC model [1–5]. The MAL_S reactions equilibria Eqs. (1, 2) onto the NC R⁰ nanosites (route I, Figure 2) take the corresponding internal J_k fluxes (Figure 2, scheme) into consideration.

In principle, the stoichiometric coefficients (ms, ns) for the MAL_S chemical reactions equilibria, Eqs. (1, 2) might be fractional [10, 17]. During the computerized simulations [2–5], all the ms, ns values are assumed to be (±1). However, indeed, all these values may be easily used in the contemporary NC model [2–5], as fractional or more than one (unity). The modern approach with all mentioned postulates and equations of the created contemporary NC model [2–5] has been realized by modern computer modeling. The numerical solution of the multicomponent diffusion partial differential Eq. (3), together with the corresponding boundary and initial conditions [Eq. (9), Eq. (10)], has been applied for the computerized calculations of the propagating X_i concentration i wave profiles. This numerical technique is based on the implicit finite difference formulation (with sweeping procedure) for parabolic partial difference diffusion equations [Eq. (3)], with the iteration technique including the multicomponent mathematical matrix calculations approach [2–5, 15, 17, 20, 21].

The set of the author’s corresponding computer FORTRAN programs has been composed for the simulation of various multicomponent mass transfer in the NC kinetic systems, describing its kinetic behavior inside the bifunctional NC matrix [2–5]. The behavior of the multicomponent NC systems is described effectively by the multicomponent propagating concentration i waves: X_i distributions [the framework of the various considerations is W⁺ (wave) concept]. Thus, the computerized investigations of the influence of the MAL_S reactions equilibria parameters (K_S) have been realized in cooperation with the multicomponent diffusion (with D_i diffusivities, II route) of the i components for the mass transfer NC kinetics inside the various bifunctional NC matrixes for the three NC shapes (L, ro, r).

The basic properties (I, Selectivity, K_S, and II, Multicomponent Diffusion, D_i) in the development of the created generalized NC model accounting for the multicomponent and nonlinear character of the NC system are introduced [1–5]. Included are the new properties of the bifunctional NC matrix based on the proposed key concept with the two co-routes (I and II) including “association-dissociation” MAL_S reactions onto the introduced nanosites: R⁰ (scheme, Figure 2). The corresponding “sink”-“sources” mechanism onto the R⁰ nanosites with the J_k fluxes (Figure 2) has been introduced into the NC model approach [1–5].

For further consideration, let us simplify the generalized approach for the MAL_S equilibria, Eq. (2), to be closer to the multicomponent examples (Sections 4–6), which have been realized by modern computers [1–6]. The mathematical realization of the multicomponent diffusion (D_i) NC kinetics and the relations for the chemical MAL_S reactions equilibria Eqs. (1, 2) inside the bifunctional NC matrix are based on the approach with the application of the simple MAL_S form (Figure 2) in the modern NC model for the arbitrary i, j, ij component concentrations: [i], [j], [ij]. The corresponding simple scheme, as for example for the simple reactions i+j↔ij may be represented by the simple MAL_S relations for the monovalent components: [i]*[j]=K_S *[ij] (scheme, Figure 2)_.

The specific case is realized for the p components, which are transformed into the mth complex (mth component): _m(R⁰p) with zero diffusivity (D_m=0). Meanwhile, such p components may participate in the MAL_S association-dissociation transformation _i p→_m(R⁰p): _i p+_kR⁰↔_m(R⁰p) with the following “transformation change” of the masses. It will be shown (Section 5) that due to this p component transformation at the nanosites (R⁰, Figure 2), the formed _kR⁰ component concentration wave propagates (though D_kR =0, explanation is below) in the bifunctional NC matrix (Sections 4 and 5, Figures 4 and 5; _5,6R⁰ waves, brown solids). Besides, for the second NC six (6)-component system (Sections 4 and 5, ₆Variant 2, Figure 5), even the three propagating ₆R⁰, ₂RH⁺, ₃RHCl⁰ concentration waves are formed (though for all these waves, D_2,3,6=0). Such _m(R⁰p) wave propagation is possible due to the two MAL_S reactions, Eq. (2), onto the active R⁰ nanosites with the corresponding [ij] mass transformations including the following interference of the other movable concentration i waves (D_i≠0, here, D_1,4,5≠0, Section 5).

Figure 4

(A, B) Variant 1. Five (5)-component system, (Anion SO₄^2-). Comparison of propagating ₅R⁰ waves (5-solid, brown, D₅=0!) in NC matrixes: L-membrane (a, A_12L, b, G_12L); ro-fiber (a, B_12ro, b, H_12ro); r-bead (a, C_12r, b, S_12r). Kb=36 (a, up, A_1,2L-C_1,2r); Kb=398 (b, down, G_1,2L-S_1,2r). D_3HSO4=0.0085<D_1SO4=0.01<D_4H=0.03. «Triangles» on abscissa axes show CM_5R positions in L, ro, r matrixes (i components numbers near curves: ₁SO₄^2- -pink, dashed; ₂R⁰H⁺-brown, dashed; ₃HSO₄^--green, dotted; ₄H⁺-blue, ₅R⁰-brown, solid). T₁ =4 (left) to T₂ =10 (right).

Figure 5

(A, B)₆Variant 2. Six (6)-component system (anion Cl^- ). Propagation of ₆R⁰ wave (6, brown, solid) in NC matrix of various shapes: L-membrane (a, up, A, B, C); ro-fiber (b, down, G, H, S); (i components: ₁Cl^-pink, dashed; ₂RH⁺-brown, dashed; ₃RHCl-green, dashed; ₄H⁺-blue; ₅HCl-black, dotted). K₁=0.6; K₂=25; K₃=1.T_L^1-3=3(A), 5 (B), 14 (C); T_ro^1-3=2(G), 4(H), 10(S). D_5HCl=0.01=D_1Cl, D_4H=0.03. «Triangles» on abscissa show the CM_6R positions.

The obtained results of the computerized simulations on the basis of the created contemporary NC kinetic model [1–6] are presented with the well-known “multicomponent concentration waves” W⁺ concept. The multicomponent “wave” theoretical approach has been used especially effectively for the dynamic (chromatographic) systems [22, 31].

In the excellent monograph [24], the concentration waves’ behavior is described in the multicomponent chromatographic (dynamic) ideal systems (i.e., without the dispersion factors for the concentration waves) for the multicomponent competitive Langmuir sorption (or IEX) isotherms. The mathematical theory of the hyperbolic partial differential equations with application of the mathematical h transformation (in other mathematical terminology – Rhieman invariants) approach is effectively used for the propagating multicomponent waves presentation [24].

Here, the X_i concentration wave behavior is presented for the multicomponent diffusion NC mass transport kinetics. The difference and similarity between these two kinds of dynamic (in chromatography) and NC kinetic systems is described in Section 9 (Conclusion, with Table 1). The visual n components behavior (n=5, ₅Variant 1, and n=6, ₆Variant 2) of the X_i concentration NC kinetic waves with their interference are presented in Sections 4–6 on the basis of the theoretical computerized simulation for the multicomponent diffusion (D_i) NC kinetics. These results for the multicomponent diffusion NC kinetics are based on the created author’s contemporary NC model with the numerical computerized solutions [1–6].

Kinetic X_i concentration waves arose with the subsequent propagation along the distance [r-, ro-radius, (R₀), or L thickness] inside the bifunctional NC matrixes during the multicomponent diffusion NC mass transfer. The multicomponent diffusion i waves with their propagation and interference of the X_i waves in the bifunctional NC matrix play the decisive role in the description of the multicomponent mass transfer diffusion NC kinetics. Therefore, additionally to the short review above, this well-known and widely used “wave” approach (W⁺) in the multicomponent NC and IEX systems [1–5, 19–32] is shortly considered in Section 4, below.

4 Concentration waves concept in multicomponent NC kinetics: selectivity and multicomponent diffusion

The obtained results of the computerized simulation on the basis of the created contemporary NC model [1–5] are presented via the well known “multicomponent concentration waves” W⁺ concept [14–17, 19–35].

The X_i concentration waves arise and propagate along the dimensionless distance (r-, ro-radius, or L-thickness) inside the NC matrix during the multicomponent mass transfer. The multicomponent i waves with their propagation in the bifunctional NC matrix play the decisive role in the description of the multicomponent NC kinetics of the mass transfer (Sections 4–6). Therefore, this well-known and widely used “wave” approach (especially in the theory of multicomponent chromatography) [21–33] is shortly reviewed in Section 4.1.

The key W⁺ concept of “multicomponent waves” is widely used in the theoretical description for many scientific fields of the multicomponent transport for various kinetic and dynamic systems. The “multicomponent waves” concept has wide area for applications in such research fields as percolation processes [29–31], mechanics of liquids, gas dynamics [33], theory of burning, and even street traffic [30, 34, 35]. The term “wave” (W⁺ concept) has been used in all these publications [1–6, 14–17, 19–35] including the mentioned excellent monograph [24] and the books [33–35] concerned with shock waves, car traffic, and kinematic waves. There are phenomenological concepts potentially common to all filtration processes, which can also be extended to a whole series of migration phenomena such as chromatography, sedimentation, electrophoresis, and some others [22–35]. The review [31] and presentations [22, 23] published by the author in cooperation with W. Hoell (Karlsruhe Research Center, Germany) include the application of the multicomponent dynamic wave concept with the description of the surface complexation theory (SCT) model. The SCT model for the multicomponent IEX equilibrium had been elaborated by Prof. W. Hoell group at the end of the last century (Refs. in the review [31]).

Coming back to the theme of this review, the described postulates of the mathematical modern NC model created [1–5], where all the relationships have been realized including multicomponent mass transfer NC kinetic partial differential equations, electro-neutrality relations, classical Nernst-Plank equations for the J_i fluxes of the i components, and MAL_S relations for the chemical reactions equilibria (Section 3, Figures 2 and 3) [1–5]. All the systems describing the multicomponent kinetic behavior in the bifunctional NC matrix of various shapes (r-bead, ro-fiber, L-membrane) were involved in the simulation with a number of the author’s computer programs for the multicomponent mass transfer NC kinetics.

The computer calculations have been fulfilled for a number of variants with the different values of the diffusion coefficients (D_i diffusivities) including various K_S constants of the MAL_S chemical reactions [Eqs. (1, 2), Figure 2]. The results of the simulations for the multicomponent NC kinetics are presented below (Sections 4–9). All the computer calculations have been obtained by using the dimensionless values, including X_i concentrations; diffusion coefficients D_i; constants of chemical association-dissociation MAL_S reactions (K_S) (see Nomenclature). The mass transfer kinetics in the NC model include the multicomponent X_i concentration wave propagation along the dimensionless distance: r, (ro, or L) in the course of the dimensionless time (T=D₀ t/r₀²).

4.3 Concentration i waves in simulation of multicomponent diffusion mass transfer NC kinetics: mass transformation for the nanosite wave inside the NC matrixes

It is very productive for the quantitative estimation and description of the traveling _kR⁰ concentration wave behavior to use the well-known integral characteristics: Center of Mass-CM_k(T) and Dispersion-Disp_k(T) [2–5]. Such integral characteristics are very useful and have widespread occurrence in mathematical statistics [36]. For computer modeling, these integral CM_k(T), Disp_k(T) parameters may be easily calculated during the computerized simulation of the NC kinetic process. The CM_R(T) dependence describes the velocity of the _kR wave along the distance in the NC matrix: radius for r-bead (ro-fiber), or through the thickness (L) of the planar L-membrane. The Disp_R(T) dependence describes the change of the width of the X_kR concentration (frontal) wave in correspondence with its physical meaning. The calculated X_i concentration ₅R⁰ waves (n=5, ₅Variant 1) are represented by smooth curves of the X_5R distribution (Figure 4A, B, brown, 5); therefore, these integral CM_k(T), Disp_k(T) characteristics are applicable with rather strong validation to the _kR⁰ wave (k=5, 6) considered in Figure 4A, B (₅Variant 1) and Figure 5A, B (n=6, ₆Variant 2).

Figure 6

(A, B) ₅Variant 1. NC kinetics total time dependences for the integral parameters: CM_5R^r,ro,L(T), (solids); Disp_5R^r,ro,L(T), (dashed). Propagation of the ₅R⁰ wave in the NC matrix: L-membrane (brown), ro-fiber (gray), r-bead (green). Ordinate axes: distance from matrix boundary (L₀, ro₀, r₀=1) until the “zero” point (L, ro, r=0): 1>L, r, ro>0. D_1SO4=0.01; D_3HSO4=0.0085; D_4H=0.03. Kb=36 (A); 398 (B). T^fin(A)<T^fin(B).

Figures 4A, B and 5A, B show the calculated kinetic X_i concentration waves with the corresponding integral CM_R (T), Disp_R (T) parameters (Figures 6A, B and 7) in the applications for the two multicomponent (n=5, ₅Variant 1; n=6, ₆Variant 2) mass transfer NC kinetic systems in the bifunctional NC matrixes of various shapes (r-bead, ro-fiber, L-membrane).

Figure 7

₆Variant 2. Six (6)-component system (anion Cl^-). Generalized dependences for integral parameters: CM_6R^L,ro (T), (solids); Disp_6R^L,ro (T), (dashed). Propagation of ₆R⁰ wave: L-membrane (brown), ro-fiber (gray). Ordinate axes: distance from matrix boundary (L₀, ro₀=1) until “zero” point (L, ro=0): 1>L, ro>0. D_1Cl=0.01; D_5HCl=0.01;D_4H=0.03. K₁=0.6; K₂=25; K₃=1.

Figure 4A, B presents the X_i concentration i wave profiles (1–5, colored curves) time behavior, and smooth ₅R⁰ concentration waves with the integral CM₅ positions. These CM₅ positions are indicated by the brown triangles in r (ro or L) abscissa. The results of the computer simulations for the propagating X_i concentration waves are presented for the two time moments: T₁=4(left); T₂=10 (right) in Figure 4 (A, A_1,2L–C_1,2r ); (B, G_1,2L–S_1,2r ) selected for the first ₅Variant 1.

Figures 5A, B present the X_i concentration wave profiles (1–6, colored) time behavior and the integral ₆R⁰ waves (though not very smooth) with the CM₆ positions. These CM₆ positions are indicated by the brown triangles in L (or ro) abscissa. The results of the computer simulations for the propagating X_i concentration waves are presented for the three time moments 2<T_L,ro^1–3<14 in Figure 5A (L, A, B, C), and Figure 5B (ro, G, H, S) selected for the second ₆Variant 2.

Figures 6A, B presents the time dependences for both the integral parameters: CM₅(T), (solids), and Disp₅(T), (dashed) for the first ₅Variant 1 with various L, ro, r matrixes. Figure 7 presents the time dependences for both the integral parameters: CM_6R(T) (solids) and Disp_6R(T) (dashed) for the second ₆Variant 2 with various L,ro -matrixes.

These results of the computerized investigations [1–5] give the possibility to analyze and estimate the X_i concentration wave behavior for the multicomponent selective IEX, NC mass transfer kinetics. The estimations especially concern the mentioned analogy in X_i concentration wave behavior between the multicomponent NC kinetics and the theory of multicomponent chromatography [see Conclusion (Section 9) and Table 1].

The computerized investigations show that the time dependences of the two integral parameters: CM_R(T), Disp_R(T), (Figures 6A, B and 7) are suitable for the description of the behavior of the k component _kR⁰ wave (k=5, 6), where the _kR⁰ wave means the nanosites R⁰ distribution in the r, ro, L matrix during the NC kinetic process. Examples of such description will be presented in this section (Section 4.2) for the two various cases of the NC kinetics for the _kR⁰ nanosites: ₅Variant 1 with one dissociation-association MAL_1b reaction (1b) (Figures 4 and 6) and ₆Variant 2 with the two MAL_2,3 reactions (2.2), (2.3) (Figures 5 and 7) correspondingly. The lists of the i components in the bifunctional NC matrix are shown below for each case: ₅Variant1 and ₆Variant2. The formation and propagation of the _kR⁰ concentration k wave inside the bifunctional NC r, ro, L matrixes are presented in Figure 4A, B (brown solids, k=5, ₅Variant1) or in Figure 5A, B (brown solids, k=6, ₆Variant2).

The _5,6R⁰ wave movement propagation occurs as the effect of two factors:

1. the MAL_S reaction (I route, Figure 2) (1b) or (2.2) with mass transformation for the ₄H⁺ component: ₄H⁺↔_kRH⁺ at R⁰ nanosites with the corresponding fixed (D_k=0) _kR⁰ component;

2. the propagation of the diffusion (D₄≠0) ₄H⁺ concentration wave in the NC matrix.

This ₄H⁺ transformation, together with the simultaneous diffusion propagation of the ₄H⁺ wave, leads to the change in the [_kR⁰] concentration due to the MAL_S mass transformations: ₄H⁺↔_kRH⁺(1b) or (2.2) (Section 4.2.2, below) and, consequently, to the propagation of the forming [_kR⁰] wave (k=5, ₅Variant 1, Figure 4A, B), or (k=6, ₆Variant 2, Figure 5A, B).

4.3.1 Five (5)-component system: sorption NC kinetics of H₂SO₄ acid inside the bifunctional NC matrix with one MAL_S reaction (1b) at the active nanosites

₅Variant 1. Kinetics of H₂SO₄ acid for the five-component NC System. The i components (numbers – indexes to the left of the symbols) with the corresponding diffusivities D_i: ₁SO₄^2--acid anions (sulfates) with the diffusivity (D_1SO4=0.01) in the NC pores (2, Figure 1B);

• ₂RH⁺ – immovable 2nd component (D_2RH+=0), formed by R⁰ nanosites, “protonated” by acid cations (₄H⁺), due to the MAL_1b reaction (1b), (I route, Figure 2);

• ₃HSO₄^- – anions of the acid [MAL-reaction (1a)] with the diffusivity (D₃=0.008) inside NC pores;

• ₄H⁺ – cations of H₂SO₄ acid with the diffusivity (D_4H+=0.03) in the NC pores (2, Figure 1B);

• ₅R⁰ – zero valent fixed nanosites (D_5R=0), generated by the NP agglomerates (I, Figure 2).

The IEX, NC system (Ia; Ib) is characterized by n=5 components, three diffusion coefficients (D_1SO4; D_3HSO4; D_4H+) with two MAL_S reactions: in the NC pores (Eq. 1a) and MAL_1b reaction at active nanosites: ₅R⁰ (Eq. 1b) correspondingly:

(1a)for NC pores, 3HSO4-↔4H++1SO42-;[3HSO4-]=Ka∗[4H+]∗[1SO42-]. (Ka, MALa), (1a)

(1b)at active nanosites 5R0, 2RH+↔5R0+4H+;[2RH+]=Kb∗[5R0]∗[4H+], (Kb, MALb) (1b)

The relationships in Eqs. (1a) and (1b) are presented by MAL_S with the constants Ka, Kb.

It follows for the ₅Variant 1 that the relation (1c) holds in equilibrium at the nanosites ₅R⁰ for the [₅R⁰] concentration:

(1c)[5R0]+[2RH+]=1→[5R0]+Kb[5R0]∗[4H+]=1→[5R0]=1/(1+Kb[4H+]) (1c)

The sorption of H₂SO₄ acid is described in the ₅Variant 1 with the sulfate anions (An^-=₁SO₄^2-) including [X_i] concentrations of all five components (i=1, 2, …, 5). Such five (5)-component NC diffusion process (routes I+II, Figure 2), in the bifunctional NC matrix is accompanied by the two MAL_S relations: (1a), (1b) with (D_1SO4; D_3HSO4; D_4H+) diffusivities of i=1, 3, 4 components.

The relationship (1c) shows that for ₅Variant 1, the propagation of the ₅R⁰ wave depends on the [₄H⁺] concentration with the Kb value (1c). This dependence is displayed in Figure 4A, B for the ₅R⁰ concentration wave. The results are calculated on the basis of the NC modern model (Sections 2 and 3) [1–5] with the computerized simulation by using the system of mass transfer partial differential equations [Section 3, Eq. (3)].

4.3.2 Six (6)-component system: sorption NC kinetics of HCl acid inside the bifunctional NC matrix with the two MAL_S reactions (2.2; 2.3) at the active nanosites

₆Variant 2. Kinetics of HCl acid for the six-component NC system. The i components with the diffusion D_i coefficients:

• ₁Cl^- – anions of acid (chlorides) with diffusivity (D_1Cl=0.01) in the NC pores (2, Figure 1B);

• ₂RH⁺ – immovable (D_2RH+=0) 2nd component, formed by ₆R⁰ nanosites protonated by acid cations -₄H⁺ due to the MAL₂ reaction K₂, (2.2);

• ₃RHCl⁰ – immovable (D_3RHCl=0) 3^d component, generated by the MAL reaction K₃, (2.3);

• ₄H⁺ – cations of HCl acid with diffusivity (D_4H+=0.03) in the NC pores (Figure 1B);

• ₅HCl⁰ – acid in MAL₁ reaction (K₁, 2.1) with D_5HCl diffusivity in the NC pores;

• ₆R⁰ – zero valent fixed (D_6R=0) nanosites generated by the NP agglomerates (I, Figure 2).

The NC system (2.1–2.3) is characterized by n=6 components, three diffusion coefficients (D_1Cl; D_4H+; D_5HCl) with three MAL_S reactions: one in the NC pores (2.1) with additional two association-dissociation MAL_S reactions (2.2), (2.3) at the active R⁰ nanosites correspondingly:

(2.1)For NC pores, 5HCl↔4H++1Cl-;[5HCl]=K1∗[4H+]∗[1Cl-] (K1, MAL1), (2.1)

(2.2)at active nanosites 6R0,1st step: 2RH+↔6R0+4H+;[2RH+]=K2∗[6R0]∗4H+ (K2, MAL2), (2.2)

(2.3)at active nanosites 6R0,2nd step: 3RHCl0↔2RH++1Cl-;[3RHCl]=K3∗[2RH+]∗[1Cl-] (K3, MAL3), (2.3)

The second relations in (2.1), (2.2), and (2.3) are described by MAL_S relations with the corresponding MAL_S constants: K₁, K₂, K₃.

Described in ₆Variant 2 is the sorption of HCl acid with chloride anions (An^-=₁Cl^- ) including [X_i] concentrations of all six (6) i components (i=1, 2,…, 6). Such diffusion six-component process in the bifunctional NC matrix is accompanied by the three MAL_S relations (2.1)–(2.3); (D_1Cl, D_4H+, D_5HCl) diffusivities of the i=1, 4, 5 components.

The effect of two successive steps of the two “association-dissociation” MAL_S reactions – (2.2), (2.3) – on the [₆R⁰] concentration wave may be easily expressed by using them with the result: it follows (₆Variant 2) that expression (2.4) holds in the equilibrium at the R⁰ nanosites for the [₆R⁰] concentration

(2.4)[6R0]=1/(1+K2[4H+]+K2K3[4H+][1Cl-]) (2.4)

The relationship (2.4) shows that the profile of the ₆R⁰ wave for ₆Variant 2 depends on the combination of the two [₄H⁺], [₁Cl^-] concentrations together with the two MAL_S constants K_2,K₃. The K₂ influence is more essential than K₃ (subject to that in (2.4), where the 2nd term is larger than the 3^d one).

Thus, the comparison of the total (1c) and (2.4) relations shows that the propagation of the integral ₆R⁰ wave (₆Variant 2 with two MAL_S constants: K_2,K₃) is more complex, and more variable than for the integral ₅R⁰ wave (₅Variant 1) with one Kb, MAL_1b constant (1b). Taking the relationships (1a–1c), and (2.1–2.4) into account and comparing ₅Variant 1 and ₆Variant2, it is clear that from the two considered NC systems, ₆Variant 2 is more complex [4] and has more degrees of freedom (i.e., the ability for more variations) for ₆Variant 2 than for ₅Variant 1 [2–5].

5 Multicomponent wave behavior in the modeling of NC mass transfer kinetics in bifunctional NC matrixes

The multicomponent X_i concentration wave behavior in the bifunctional NC matrix is described here on the basis of the created modern NC model (Sections 2 and 3) [1–6] with the computerized modeling including the multicomponent wave key W⁺ concept. The two various different multicomponent NC kinetic mass transfer systems are considered: five (5)-component NC system with one reaction (1a) in the pores of the NC matrix and with one more MAL_1b reaction at the R⁰ nanosites (1b) (₅Variant 1, Section 5.1); the other NC kinetics in the six (6)-component NC system is characterized by one reaction (2.1) in the pores of the NC matrix with two more MAL_S reactions at the R⁰ nanosites (2.2), (2.3), (₆Variant 2, Section 5.2). The lists of the i components are presented in Section 4.2.1, and Section 4.2.2.

6 Discussion of concentration wave behavior for two multicomponent NC, IEX systems

The computerized simulation of the mass transfer kinetics inside the NC matrix (Sections 4 and 5) discovers the clear analogies of the multicomponent NC kinetics [2–5] with the main theoretical basis of the theory of nonlinear multicomponent chromatography [24–30, 32]. These analogies concern the multicomponent X_i concentration wave behavior in the bifunctional NC matrix [1–5]. In the bifunctional NC matrix, the physical sense meaning of the reasons for the explanation of the propagating X_i concentration wave behavior (Figures 4 and 6) is described by the joint influence of the two coexisting factors: MAL_S reactions Selectivity (I) and multicomponent Diffusion (II) (scheme, Figure 2). Such type of the same joint effect: reaction equilibrium (I, Selectivity)+(II), Diffusion effects is represented analogically in the theory of chromatography by the two terms also: multicomponent isotherms for equilibria (I) and set of the broadening factors for the concentration waves in columns (II). Factor (II) is described in the theory of chromatography as the well-known effective HETP value for the chromatographic columns [24–30, 32]. The analogy of the two mass transfer (NC kinetic and dynamic) processes is summarized in the Table 1.

The first part of Sections 5 and 6 concerns mainly the ₅Variant1: mass transfer of the five components in the bifunctional NC matrix with one association-dissociation MAL_1b (Kb) reaction (1b) at the R⁰ nanosites (scheme, Figure 2), ₅Variant1. The influence of the first factor (I Reaction, Kb Selectivity) is determined by the Kb Selectivity value: the more the Kb is (1b), the less is the Dispersion (Disp_R) of the 5th concentration ₅R⁰ wave. In other words, the width of the wave profile becomes narrower with the large Selectivity value (Kb=398, Figure 4B).

Besides, the comparison of the CM_R^A curves (Figure 6A, brown solids) for the small selectivity Kb=36 (A) with the CM_R^B curves (Figure 6B, brown solids) for the large selectivity Kb=398 (B) shows that the more the Kb value is (from A to B), the slower is the ₅R⁰ wave propagation.

The comparison of the dispersion curves Disp_R(T), in Figure 6A (dashed) with Figure 6B (dashed) shows visually and distinctly the Selectivity effect: the Disp_R^L,ro,r (T) narrowing with the asymptotic trend to the permanent Dispersion value. The influence of the second factor (II multicomponent Diffusion) gives, as usual, the widening of the concentration ₅R⁰ waves with the increase in the multicomponent D_i diffusivity. The nontrivial, specific effect for the NC system should be emphasized here: there is no diffusivity for the _5,6R⁰ component (D_5R;6R=0); nevertheless, the propagation of the _5,6R⁰ waves takes place (Figures 4A, B, 6A, B or Figures 5A, B and 7). The physical reason of such propagations is not the D_5R;6R diffusivity (D_5R;6R=0) but the MAL_S reaction (₅Variant1 (1b) or ₆Variant2 (2.2), (2.3)) influence: the transformation for the I components in MAL_S reactions including the D_i diffusivities of other moveable components (D_1SO4,(1Cl); D_3HSO4;(5HCl); D_4H) bring the resulting mass transfer and the propagation of the [₅R⁰], [₆R⁰] concentration _kR⁰ waves (Figures 4 and 5) of the fixed R⁰ nanosites (Figures 1 and 2).

One more interesting result with the evidence of the above marked analogy can be seen in Figures 4, 6 and 7. In these cases, the typical behavior of the Disp_R (Dispersion) in the course of time (T, abscissa) takes place: the dispersion (Disp_R) of the _5,6R⁰ waves tends asymptotically to the permanent value at the end of the T abscissa (see dashed curves behavior along the abscissa T with the asymptotic tendency to the permanent value, Figures 5A, B and 7, dashed curves).

In the theory of chromatography, the same effect takes place for the X_i concentration waves in column, when the favorable isotherm factor (I) is used to compensate the unfavorable influence of the broadening HETP factors (II) for the concentration waves in columns: a favorable equilibrium (I) compensates the widening of the concentration wave with the same type of the asymptotic tendency [24–30].

Thus, more variable ₆Variant 2 (6th – component NC system) has a more complex behavior for the i concentration waves. For ₆Variant 2, the number of the determining parameters (including concentrations) is larger than for ₅Variant 1. In the mentioned analogies between multicomponent chromatography and the NC kinetics, ₅Variant 1 on the basis of the NC contemporary model [1–5] is not so visual and becomes more complex for ₆Variant 2.

The main reason is the influence of the two MAL_S constants: K₂, K₃ (2.2, 2.3; ₆Variant 2) for the n=6 system in contrast to the simple influence of the one K_b constant (1b; ₅Variant 1, n=5).

7 Ternary nonselective IEX (1A, 2B, 3C) diffusion kinetics: new displacement effect for three interfering concentration waves

One important partial case of the NC model [1–5] for the nonselective ternary (i=₁A, ₂B, ₃C ionic components) diffusion IEX kinetics with the results of the computer simulation [21] is reported in Sections 7 and 8 (below) of this review. In the nonselective IEX kinetics, the influence of only one multicomponent factor, the three-component diffusion (Figure 2, D_iDiffusivity, route II, i=1, 2, 3) is significant and should be taken into the consideration [21]. The influence of the first factor I (route I, Selectivity) of the created author’s NC model [1–6] (Section 1) may be reasonably neglected for the rather simple nonselective ternary IEX system. Such simplified variants of the ternary diffusion (D_i, i=₁A, ₂B, ₃C) nonselective IEX have been used on the basis of the computerized investigation with the monofunctional IEX matrix(without the Selectivity: I – route). The modeling of this partial case of the previously elaborated bifunctional IEX NC matrix with the ternary diffusion kinetics of the i components (D_i diffusivities: II – route only, Figure 2) gives the new displacement effect for the interfering X_i concentration waves [21].

There are some common points in the several previous publications [17–19, 37] with the author’s theoretical considerations for the ternary nonselective IEX mass transfer kinetics [2–5, 21] presented here.

The main essential difference between the approaches [17–19, 21, 37] is that, in the author’s theoretical considerations [2–5, 21] (Sections 7 and 8), the interference of the three X_i concentration waves (i=₁A, ₂B, ₃C) is realized with the inclusion of the corresponding kinetic F_i(T) curves calculated simultaneously in parallel and “abreast” (see Figure 8A, B) [21], while in the papers [17–19, 37], only integral characteristics, kinetic F_ir (T) curves in the r-bead resin are experimentally (and partly theoretically) investigated without the study of the interference of the traveling X_ir concentration waves.

Figure 8

(A, B) Coupled pictures. New displacement effect (₃C, ₂B waves): propagation of ₁A, ₂B, ₃C waves (left, 1-pink, 2-blue, 3-dotted) in r, ro matrixes: X_ir (T_r^k) (a, B, C, left); X_iro (T_ro^k); (b, H, S, left). Nonmonotonic kinetic curves: F_2r (T),(a, right); F_2ro (T), (b, right). Maximum accumulations: F_2r,ro^max(T_mr,ro ) – color (green) points: T_mr =0.2 (a, right); T_mro =0.51 (b, right). “White” k-points (…) in curves F_2r,ro (T), (right): T^k=T_r,ro^1-3 current moments for X_ir,ro (T^k) profiles (left). T_r,ro^1-3=0; 0.4; 0.8. Diffusivities: D₁=0.20, D₂=0.25>>D₃=0.01. Location of displacement effect for ₃C,₂B waves (left): up arrows (a, left: B, C); dashed verticals (b, left: H, S).

The most general approach was demonstrated in the author’s publication [21] with the multicomponent diffusion X_ir,ro concentration waves (Section 4) together with the study of the interference of the i waves. This author’s approach [21] brings the original results including the new displacement effect mentioned. Therefore, the wave W⁺ concept (Section 4), together with the consideration of the i wave interference, is very fruitful for the investigation of the nonselective ternary IEX mass transfer kinetics considered [21].

During the mass transfer process for the ternary nonselective IEX system, (₂B⁺+₃C⁺)_solution/(R₁A)_resin, the two ₂B, ₃C concentration diffusion waves come from the outer solution into the resin matrix, which is initially in the ₁A form (R₁A_resin). In the result of the computerized simulation of this important kinetic mass transfer process [21], it has been found out and demonstrated visually that the new displacement effect for the two incoming ₂B, ₃C concentration waves is conditioned by the interference of these ₂B, ₃C waves [21].

Discussing the anomalous, nonmonotonous kinetic F_2Br,ro curve behavior for the most mobile incoming ₂B ionic component, the four variants of the available published results [17–19, 21, 37] should be reviewed in the computerized simulation of the such type ternary (i=₁A, ₂B, ₃C ionic components) nonselective intraparticle mass transfer IEX diffusion kinetics. Three variants of the nonselective IEX kinetics have been considered previously in the publications [17–19, 37] with several cases for the unusual, nonmonotonic F_2Br (T) curve behavior in the r-bead of the IEX resins.

In the modern author’s paper [21], the theoretical simulation for the many cases of the ternary nonselective IEX kinetics has been extended additionally for the three various IEX matrix shapes: r-bead, ro-fiber, and L-membrane. The generalized approach with the application of the partial differential mass balance equations [Section 3; Eq. (3) with S_i=0, and Nernst-Plank Eq. (4)] have been used in Refs. [17–19, 21] for the three ionic i components (i=₁A, ₂B, ₃C) including the electro-neutrality and the absence of the electric current relations (5), (6).

In the first publication for the simulation of the ternary nonselective IEX kinetics [17], only one variant of the nonmonotonous F_2Br (T) curve behavior has been calculated by the modeling approach with the certain diffusivities relation D_A:D_B:D_C=1:5:0.2. However, in this paper [17], no variants of the modeling calculations for the propagating ₁A, ₂B, ₃C concentration i waves have been presented at all. Unfortunately, the discussions of any reasons were given neither for the X_ir concentration waves nor for the nonmonotonous kinetic F_2Br (T) curve behavior [17]. Therefore, the new displacement effect obtained in [21] for the incoming and interfering ₃C, ₂B concentration waves has not been displayed in [17] and naturally has not been discussed. Besides, no conclusions and comments concerning the possibility of the displacement effect were presented [17] for the propagation of the three X_ir concentration waves.

The obtained experimental results [18] is important, together with the subsequent theoretical analyses by the authors [19], for the ternary (₁A, ₂B, ₃C) nonselective IEX diffusion mass transfer kinetics. The experimental results [18] include the nonmonotonous integral kinetic F_2Br (T) curves, where, in the experiments with the applied cell, the ₂B (H⁺ or Na⁺) accumulation of the ionic “mean” ₂B concentration has been obtained [18]. The experiments with the nonmonotonic kinetic F_2Br curves have been done for the intraparticle ternary diffusion IEX kinetics (during sorption stage): (₂Na⁺+Zn²⁺)/R^-₁H⁺ and (₂H⁺+Zn²⁺)/R^-₁Na⁺ for various types of the IEX Dowex and Diaion resins [18]. The experimental dependences have been found [18] for the ₂B accumulation of the (₂H⁺ or ₂Na⁺) ions presented as the moderate (₂Na⁺, or ₂H⁺) kinetic peaks in the two integral F_2Br (T) curves for the three D_i diffusivities with the relation: D_H>D_Na>>D_Zn. In the authors’ publication [19], the theoretical analyses have been applied to the previous experimental kinetic dependencies [18]. In the article [19], the X_2B concentration profiles (₂Na⁺, or ₂H⁺) have been calculated for two various “sorption cases”: (a), (₂Na⁺+₃Zn²⁺)/R₁H, or (b), (₂H⁺+₃Zn²⁺)/R₁Na. The ₂Na⁺ or ₂H⁺ accumulation has been explained on the basis of the ₂H⁺ (or ₂Na⁺) concentration gradient (4) in comparison with the electric potential (Φ) gradient influence, Eq. (4) [18, 19].

The new displacement effect has been discussed in detail in the author’s publication [21] and presented here additionally in (Section 8) for the interfering X_ir,ro concentration waves in the ternary nonselective IEX kinetics. In the previous publications mentioned [17–19], this phenomenon neither have been displayed nor considered.

The coupling effect of the electric field on the IEX mass transfer is taken into consideration [17–19] via the Nernst-Plank relationship (4) as well as in the other considerations mentioned later [37]. In all these papers, the displacement effect with its cause has not been mentioned.

This new displacement effect as well as the cause for it has been demonstrated only in the author’s publication [21] (see also this review, Figure 8A, B, Section 8).

It ought to be mentioned that there is another result concerning the nonmonotonous kinetic F_2Br (T) curve behavior in the ternary nonselective intraparticle IEX diffusion kinetics [37]. The publication [37] shows the experimental nonmonotonous behavior of the kinetic F_2Br (T) curve for the real ternary IEX kinetics (₁A=Na⁺, ₂B=K⁺, ₃C=Sr²⁺) in the r-bead of the Russian KU-2x8 resin [37].

For the theoretical calculations [37], the authors’ phenomenological multispecies IEX kinetics model named by authors as “MIE” (the macroscopic model of the IEX kinetics) has been used [37]. The experimental nonmonotonous behavior of the kinetic F_2Br curves is explained on the basis of the authors’ theory [37] for the ternary IEX (₁A=Na⁺, ₂B=K⁺, ₃C=Sr²⁺) kinetics including the electric field (Φ) effects, Eq. (4). The MIE model [37] is based on the usual differential equations but not on the partial differential Eq. (3) (here in Section 3) for the nonselective multicomponent (₁A, ₂B, ₃C) IEX kinetics. Therefore, in the phenomenological MIE model [37], there is no possibility, in principle, to consider the propagating ₁A, ₂B, ₃C concentration wave behavior inside the IEX resin matrix. Thus, the study of the nonselective IEX kinetics by the such type (MIE) authors’ phenomenological model [37] is accessible only in the study of the restrictive region of the kinetic F_ir(T) curve behavior, but not for the X_ir,ro concentration wave consideration for the multicomponent nonselective intraparticle IEX diffusion kinetics.

The new displacement effect as the result of the X_i wave interference may be investigated only on the basis of the interference of the propagating three (or more) i wave behaviors as shown in Section 8.1.1 below (X_ir,ro – left, Figure 8A, B) [21]. The author’s approach has been applied for the consideration of the ₂B accumulation together with the unusual nonmonotonous kinetic F_2Br,ro curve behavior [21]. These computerized author’s simulations [21] give the possibility to display the reason for the nonmonotonous kinetic F_2Br,ro curve behavior [21]. In the previous approaches [17–19, 37], it has been used as a comparison for the J_i fluxes of the X_i concentration gradients’ influence together with the effects of the electric field (Φ) gradient, Eq. (4), (Section 3). In contrast to the previous publications [17–19, 37] in the modern author’s approach [21], the concept of the interferences of the X_i concentration waves has been used. In this approach, the grad Φ in Eq. (4) is excluded (mathematically) from the consideration by the superposition of the j concentration gradients (grad X_j) via the relation in Eq. (7) (“diffusion potential” [10], Section 3). Thus, in the such phenomenological approach [21], the electric field (Φ) effects are substituted by the equivalent phenomenological effects of the X_i concentration wave interference [see the remark after the Eq. (8) relations, Section 3]. Owing to the consideration of the wave interference effects of the propagating ₁A, ₂B, ₃C concentration waves [in Section 3, Eq. (8)], the new displacement effect with the 2B accumulation has been displayed visually including the calculated nonmonotonous behavior of the kinetic F_2Br,ro curves [Figure 8A (r), B (ro), right] for the matrixes of the r-bead, ro-fiber shapes [21].

It is important to note that such new displacement effect for the nonselective ternary IEX kinetics is not calculated in the simulation for the mass transfer in the L-membrane matrix. The physical sense of the displacement effect availability in the r, ro-matrixes, with its absence in the L-membranes is given later (Section 8.1).

Before the consideration of the ternary nonselective IEX diffusion kinetics, let us demonstrate the one i component (i=1) concentration wave behavior (Figure 9A, B) for the two simplest binary IEX cases. This case of the one component IEX kinetics corresponded to the binary (i/j) exchange [20] in the r-bead matrix. In Figure 9A, B, the two experimental micropictures are shown as the illustration of the propagating diffusion X_ir concentration (i=1) wave profile for the binary (i/j) IEX inside the r-bead matrix. Figure 9A, B shows the binary Ni⁺ waves (by the light lines) for the two types of the binary IEX: (A) nonselective IEX: Ni⁺/H⁺, left; and (B) selective IEX: Ni⁺/Na⁺, right [20]. The micropictures (A – left, B -right) show the propagation of the two types of the incoming Ni concentration waves. The experimental IEX micropicture (Figure 9B, right) presents additionally the illustration of the visual structure of the selective IEX matrix.

Figure 9

(A, B) Experimental SCM micropictures, binary i/j IEX: propagation of incoming Ni concentration waves to the center of the r-bead: to the left (A); to the right (B), VPC ampholyte resin (Russia). Light lines: Ni⁺ concentration waves-profiles. Micropictures, two kinds of IEX: spreading Ni⁺ wave (nonselective IEX, A); sharp Ni⁺ wave (selective IEX, B) [20].

The intraparticle multicomponent nonselective IEX kinetics include the influence of only one multicomponent Diffusion factor (route-II, Figure 2): the ternary diffusion (D_i) mass transfer of the ionic i components with D_i diffusivities, i=₁A, ₂B, ₃C. The nonselective IEX matrix is considered [21] as the partial (though very important) case without the Selectivity factor variant of the IEX mass transfer kinetics on the basis of the author’s generalized multicomponent NC model considered above (Sections 1 and 2) [1–6, 21]. The investigation of the nonselective ternary IEX mass transfer kinetics gives the possibility to estimate separately the influence of only one multicomponent D_i diffusion, II factor (without the selectivity factor, I).

Naturally, in this nonselective ternary IEX, the key wave, W⁺ concept (Section 4) plays even the more important role for the multicomponent D_i diffusion IEX model. The wave concept in the multicomponent nonselective IEX mass transfer kinetics gives the possibility to estimate the new displacement effect arising due to the mutual interference of the ₁A, ₂B, ₃C concentration diffusion (D_i) i waves in the IEX r, ro matrixes.

It is shown in Figure 8A (r), B(ro) (X_i – left; F_i- right) that only for the definite conditions, the new displacement effect may appear (Section 8.2.1) for the interfering ₂B, ₃C concentration incoming I waves [21]. The conditions for the appearance of the displacement effect are investigated and presented in the discussions of the simulation results (Section 8.2) [21].

This new displacement effect (Section 8.1) appears for the two interfering ₂B, ₃C concentration waves and affects the anomalous and nonmonotonous behavior of the kinetic F_2B(T) curve. The displacement of the fast ₂B wave by the second (slow), ₃C wave brings the accumulation of the ₂B component in the propagating ₂B wave. In the fast ₂B wave displaced (X_2Br,ro , left,Figures 8A, B), this ₂B accumulation leads to the unusual nonmonotonous F_2Br,ro curve behavior (Figure 8A, B, right, F_2Br,ro^max). The ₂B accumulation is increased initially, then, it is decreased in the F_2Br,ro (T) curves (Figure 8A, B, F_2Br,ro, right) with the availability of the F_2Br,ro^max values.

The consideration of the ternary nonselective IEX kinetics by the mathematical approach in [2–5, 21] has the obvious advantage in comparison with the previous results in the publications [17–19, 37]. The author’s results in the modern publication [21] give the possibility to consider the multicomponent IEX kinetic process not via the kinetic F_i curve behavior only, but on the more high and detailed level: via the study of the propagation of the multicomponent interfering X_i concentration wave behavior in the IEX matrix of the various shapes: r.ro matrixes, or L-membranes (X_2Br,ro , left, Figure 8A, B) [21].

Besides the wide diapason of the D_i diffusivities for the nonmonotonous F_2Br,ro curve behavior and its association with the new displacement effect (Section 8.2) has been determined by the detailed mathematical modeling used in [21]. This conclusion is shown with the evidence due to the original presentation of the propagating X_i concentration diffusion waves together and abreast with the kinetic F_ir,ro curves in the “coupled pictures”: X_i, left∣F_i, right [see Figure 8A (r), B (ro); Section 8.1].

Figure 8A (up, r), B(down, ro) shows the displacement effect in the r, ro matrixes for the interfering ₂B, ₃C waves (left) abreast with the F_2Br,ro (T) curve behavior (right) in “the coupled pictures” (Section 8.1).

The author’s results obtained [21] are new and displayed for the first time. The discussion of the computerized simulation for the X_i concentration waves (Section 8) produces the proofs for the interference of the X_i concentration waves with the subsequent origination for the new displacement effect [Figure 8A (up, r), B (down, ro)] [21].

8 Modeling of ternary mass transfer nonselective IEX diffusion kinetics with new displacement effect

In the result of the computerized simulation [2–4, 21], the propagations of the X_i concentration waves have been calculated for a number of variants of the ternary nonselective diffusion IEX kinetics. The corresponding kinetic F_i(T) curves have been calculated simultaneously with the X_i concentration waves during the simulations for all variants with wide D_i diapason (Section 8.2) [21]. In the investigation, the most interesting kinetic F_2Br,ro (T) curves have been obtained [21] with the unusual nonmonotonous behavior (Figure 8A (r), B (ro), right). In this nonstandard cases, the kinetic maximum F_2Br,ro^max is available in the kinetic ₂B component F_2Br,ro (T) curves for the r-beads and ro-fibers (Figure 8A (r), B (ro), right). Such variants of the nonselective ternary IEX kinetics were calculated [2–4, 21] on the basis of the author’s computerized NC model (without the I Selectivity property).

In the author’s investigations [21], the previous studies [2–5] have been continued for the wider diapason of the D_i diffusivities (i=₁A, ₂B, ₃C) including the comparative consideration of the interference of the X_i concentration diffusion waves for the nonselective IEX kinetics inside r-bead or ro-fiber [21]. The results obtained [21] allow to make more precise conclusions and to come definitely to the real reasons for the unusual, nonmonotonic behavior of the kinetic F_2Br,ro (T) curves with F_2Br,ro^max value availability.

8.1 New displacement effect conditioned by interference of ₁A, ₂B, ₃C concentration waves in ternary nonselective diffusion IEX kinetics

It is shown (Section 8.1.1) that the cause (and the physical sense) for such unusual, nonmonotonic F_2Br,ro (T) curve behavior is the new displacement effect during the interference of the two incoming ₃C, ₂B waves: fast ₂B wave is displaced along r,(ro) distance in the IEX matrix by the other slow ₃C wave during the IEX kinetic process with the propagation of these ₃C, ₂B waves to the r,(ro) center [21].

The computerized simulation of the multicomponent mass transfer in the nonselective IEX kinetics is realized [2–5, 21] by the numerical solution of the partial differential mass balance Eq. (3) in the author’s model [1–6]. For the simulation of the nonselective IEX kinetics, the S_i terms in Eq. (3) are deleted [21]. Thus, the computerized solution of Eq. (3) brings the calculation of the multicomponent ₁A,₂B,₃C concentration X_ir,ro wave distributions inside the r-bead or ro-fiber. The diffusion ₂B,₃C waves propagate to the r,(ro) center along the dimensionless r,(ro) radius of the IEX matrix in the course of T time (T=D₀ t/r₀²). The interference of the three propagating X_ir,ro (T) concentration i waves (i=₁A,₂B,₃C) takes place during the ternary nonselective IEX diffusion kinetics with the appearance of the new displacement effect for the X_ir,ro (T) waves propagating in T time (Figures 8A (r, B, C); B(ro, H, S), X_ir,ro -left). This effect is investigated and discussed [21] on the basis of the visual “coupled pictures” (Figure 8A, B) with the results of the computer modeling (Section 8.1.1).

8.1.1 Simulation of IEX diffusion kinetics: original method of presentation of X_i concentration waves abreast and together with the kinetic F_i curves by the coupled pictures {X_i – left∣right – F_i}

In the computerized simulations of the nonselective IEX mass transfer kinetics behavior, there were modeled a number of variants calculated for the wide diapason of the D_i diffusivities of the ionic i components [21]. The results of the computerized calculations are presented here in Figure 8A (r), B (ro) via “the coupled pictures”: {X_ir,ro (T)-left ∣right-F_ir,ro (T)}. The T_mr,ro positions of the F_2r,ro^max(T_mr,ro ) accumulations are shown by color points in the F_2Br,ro (T) curves (right, Figure 8A, B). For the two considered examples in Figure 8A, B, the color points in the F_2Br,ro (T) curves corresponded to the T_mr,ro time values: T_mr =0.2(a, right); T_mro =0.51(b, right).

The features and advantages of the “coupled pictures” (Figure 8A, B, left∣right) are obvious: the X_ir,ro (T) waves (left) are placed abreast and together with the F_ir,ro (T) curves (right) [21]. The “white” points (‥) {T_r,ro^k; F_2r,ro^k} in the integral kinetic F_2r,ro (T) curves (Figure 8A, B, right) show the current ₂B filling (F_2Br,ro^k) in the r(or ro) matrix. The X_ir,ro (T^k) waves (distributions) corresponded to the current (T_r,ro^k) kth moments for X_ir,ro (T^1–3) profiles in Figure 8A, B (left). Such “coupled pictures” {X_i-left∣right-F_i} (Figure 8A, B) are very suitable for the detailed demonstration of the new displacement effect in progress with T time for the ternary nonselective IEX mass transfer kinetics in the r,(ro) matrixes [21].

For this purpose, in Figure 8A (r-bead); B (ro-fiber), the original method of presentation of the results of the nonselective multicomponent IEX kinetics simulation is used in the T time series (T¹(0)<T²<T³). The abreast {left -X_i∣F₂-right} demonstration in Figure 8A, B {X_i-left∣F_i-right} gives the possibility to detect with confidence the cause and location of the new displacement effect, which occurs during the movement of both the incoming ₂B, ₃C waves to the center of the r-bead or ro-fiber. (The simulations show that for the L-membrane, the displacement effect do not appear during the nonselective ternary IEX kinetics [2–5, 21].)

Meanwhile, for the intermediate T² time of the r,(ro) kinetic IEX processes, the relations T_mr<T²<T_mro hold. In the F_2B(T) curve, the T position (T_mr,ro ) is marked by the color point {T_mr,ro , F_2r,ro^max}. The F_2r,ro^max values in Figure 8A, B-right and in Figures 10 and 11 depend on the relationships for the D_B∼D_A∼D_C diffusivities. The comparison of the total T_r,ro^fin time for the kinetic F_ir,ro (T) curves (including the T_mr , T_mro ) shows (Figure 8A, B, F-right) that the IEX kinetic process is two times faster for the r-bead (a) than for the ro-fiber (b).

Figure 10

(A, B) D_B^1-3 various, nonmonotonous F_2r,ro (T) curve behavior: r-bead (a, 1r-3r), T_2mr =0.19 (A), 0.25 (B), 0.48 (C); ro-fiber (b, 1ro-3ro). T_2mro =0.53 (G), 0.61 (H), 1.15(S). D_A=0.20; D_C=0.01; {T_mr ; F_2r^max}-point marked by green color in F_2r,ro (T) curves.

Figure 11

(A, B) D_A^1-3 various, nonmonotonic F_2r,ro (T) curve behavior: r-bead (a, 1r-3r), T_2mr =0.24 (A), 0.31(B), 0.45 (C); ro-fiber (b, 1ro-3ro), T_2mro =0.66 (G), 0.92 (H), 1.47(S). D_B=0.25; D_C=0.01. {T_mro ; F_2ro^max}-point marked by green color in F_2r,ro (T) curves.

In the course of T time from T=0 until T^fin, the X_i wave propagation along the r,(ro) distances (Figure 8A, B, left) and the current {T^k, F^k} white points “sliding” along the F_2B(T) curve (Figure 8A, B, right) are interconnected during the whole kinetic process (0<T<T^fin). In other words, the propagation of the X_ir,ro (T) concentration waves in Figure 8 (X_i, left; (r)A-C, or (ro)G-S) corresponds to the white point “sliding-transition” along the F_2B(T) curves in Figure 8A, B (F_2r,ro curves, right). The dynamic process of the multicomponent IEX kinetics for the X_ir,ro (T) wave profiles (left) with the “white point sliding” along the F_2Br,ro (T) curves,(right) is especially visual and understandable in the author’s animations with the series of the “coupled pictures” (Figure 8A, B) showing all the process in T time visually [21].

These author’s animations are assembled by the frames – “coupled pictures” calculated (Figure 8A, B: {left, X_i∣F_i, right}). In the animations, the “coupled pictures” are formed abreast by the two pictures: X_i profiles with the kinetic F_i(T) curves including “white points” {T^k; F^k} (Figure 8A, B). All calculated curves are transformed into the {X_i(r, ro; T), F_i(T)} – “coupled pictures,” which are obtained from the results of the numerical modeling.

8.1.2 The details of the new displacement effect in the author’s “coupled pictures”

The “coupled pictures” {X_ir,ro (T),left ∣F_ir,ro (T),right} in Figure 8A, B show the series of X_ir,ro wave propagation for the three selected time moments: T¹(0)<T²<T³. The presented time diapason T¹(0.)-T²(0.4)-T³ (0.8) covers all the time region (0<T_mr,ro <T³) around T_mr (a)_; T_mro (b) values. The details of the considered displacement effect (for ₃C, ₂B waves) are shown by the X_ir,ro concentration ₂B, ₃C wave behavior shown on the left side of the “coupled pictures,” (Figure 8A, r, up; B, ro, down). The reason of this displacement effect is specified mainly by the interference of the two ₃C, ₂B waves (₂B-fast and ₃C-slow: D_2B>>D_3C, Figure 8A, B), which propagate to the r, ro centers.

In both r, ro matrixes, the displacement effect is located in the region of the small ₃C concentration values (X_3r,ro , left, Figure 8A, B). The region of the X_2r,ro accumulation (left) with the corresponding color points (F_2r,ro^max, right) is shown in the X_2r waves by the small vertical up arrows (↑) in Figure 8A (B, C; r-bead) or in the X_2ro waves by the vertical dashed () lines in Figure 8A (H, S; ro-fiber).

In both cases (Figure 8A, B), the location of the ₂B-accumulation shows that the cause for the accumulation of the ₂B concentration in the fast ₂B wave is the displacement of the ₂B wave by the incursion of the second slow incoming ₃C concentration wave (with X_3r,ro profiles, 3 left).

Simultaneously on the right side of the “coupled pictures” (Figure 8A, B, right) are shown the F_ir,ro curves, where the “white” points mark the ₂B filling, which corresponded to the current time T^1–3 moments of the X_ir,ro (T^1–3) wave profiles (left, Figure 8A, B).

Such original presentation with the “coupled pictures” {X_ir,ro (T), left∣F_ir,ro (T), right} like in Figure 8A, B gives the effective and visual interpretation of the results of the computerized modeling in the course of T time: T¹(0.) <T²=0.4<T³=0.8.

The comparison of the details of the displacement effect for r-bead (a, left) and ro-fiber (b, left) shows the difference between the results (compare X_ir in Figure 8A with X_iro in Figure 8B). Figure 8A, B show that the X_2r wave (in r-bead) comes to the T_mr moment with the X_2r jump (2-blue, X_2r , left; a, B, C; up arrows ↑), while the X_2ro wave (in ro-fiber) comes to the T_mro moment without the X_2ro -jump (2-blue, X_2ro , left; b, H, S; dashed verticals: ), i.e., gradually.

The corresponding marks for X_2r,ro (T^2,3) profiles, (left) are presented in Figure 8A (r-bead); b (ro-fiber). They show that the ₂B accumulation for the ro-fiber is not so intensive as for the r-bead (T_mr <T_mro , Figure 8A, B, right). Nevertheless, the integral accumulations (a) F_2r^max_, and (b) F_2ro^max values for the r, ro matrixes are approximately the same (Figure 8A, B, right). Thus, these (almost equal) integral ₂B accumulations in the X_2Br,ro waves are detected as the kinetic F_2Br,ro^max peaks in the F_2Br,ro curves (F_2Br,ro^max, Figure 8A, B – right).

The time series (T^1–3) of the “coupled pictures” {X_ir,ro (T), left∣right, F_ir,ro (T)} in Figure 8A (A-C); B (G-S) represent the small fragment of the author’s animations prepared for the visual interpretation of the computerized simulations in the oral computer presentations.

The “coupled pictures” with {X_ir,ro (T^k)∣ F_ir,ro (T^k)} curves (Figure 8, left, right) are included into the animations as the frames for the successive T^k time moments: T_r,ro^k. The author’s animations of the whole kinetic process including the frames-series (T¹_,T²…T^fin) of the “coupled pictures” (as in Figure 8A, B left, right) show visually the interference of the three i concentration ₁A, ₂B, ₃C waves, including the location of the displacement effect for ₂B wave in the course of the ternary nonselective IEX diffusion kinetics. In the animations, the sliding-transition of the “white” points is seen visually along the kinetic F_2B(T) curves (right) together with the X_2r,ro (T) displacement in the course of T-time for the ₂B waves (left) displaced by the slow ₃C waves (Figure 8A, B).

The new displacement effect for the fast ₂B wave (Figure 8A, B, left; dashed-dotted blue, X₂ profiles: r-B, C; ro-H, S) takes place for the definite conditions for the D_i diffusivities (see Section 8.2).

For the given diffusivity relations D_2B=0.25>D_1A=0.20>>D₃c=0.01, there is obtained a considerable displacement effect for the most mobile X_2Br,ro concentration ₂B wave (Figure 8A, B, left) together with the simultaneous accumulation F_2r ,_ro^max (Figure 8A, B, right).

The physical sense of the ₂B concentration jump (up arrows ↑, X_2r , left, Figure 8A, series T_r^2,3) with the peak in the F_2r curve (9A, right) for the r-bead should be explained by the steep decrease dV_r of the current r volume for the diffusion into the r-sphere. It is obvious that the dV_ro volume for the diffusion into the ro-fiber does not show the steep decrease in dV_ro (gradual only). Therefore, the F_2ro^max accumulation for the ro-fiber is gradual, and the ₂B concentration jump is absent visually (X_2ro , left, Figure 8B, 2-blue dashed lines, series T², T³).

For the L-membrane, such displacement effect is not calculated in the simulations – the interference for ₃C, ₂B waves is not strong as the dV_L volume is permanent and not changed during the IEX diffusion to the other side of the L-membrane. Besides, the main additional cause for the absence of ₂B accumulation in the L-membrane concludes the rather obvious fact that there is the sink for the diffusion of the masses in L-membrane at its left side (L=0). It is evident that there is no such sink for the I masses at the r,(ro) centers (r,(ro)=0), Eq. (9), (Section 3).

So the displacement effect for the ₂B wave in the r-bead is more intensive, but T_mr =0.2 value is shorter in comparison with T_mro =0.51 value for the ro-fiber. Thus, in the result, the integral effect in the F_2r,ro (T) curves for r, ro matrixes F_ir,ro^max×T_m*Intensity is displayed as almost the same (Figure 8A, B, right). For the r, ro matrixes in Figure 8A, B (right), the kinetic F_2r,ro (T) curves show almost the same maximum: F_2r^max(a)∼F_2ro^max(b).

Among two ₂B, ₃C waves incoming into the IEX resin, the ₃C concentration wave moves much slower (in this case D_B>>D_C) than the fast ₂B wave, but the ₃C wave invasion with the displacement effect followed stipulates the nonmonotonous kinetic F_2Br,ro(T) curve behavior. The larger the difference between D_B-D_C is, the more effective is the displacement shown for the ₂B wave [21].

The results of the simulations in Figure 8A, B demonstrate that this condition (D_B>>D_C) for the D_i diffusivities of the ₂B, ₃C waves is not enough for the occurrence of the distinct kinetic peak (F_2B^max value). The F_2B^max peak value depends essentially on the D_1A diffusivity of the ₁A component wave outgoing from the IEX matrix: the larger the D_1A diffusivity is, the higher is the peak in the F_2Br.ro (T) curves. The faster ₁A wave moves out of the IEX matrix, the smaller will be the influence of the ₁A wave on the total wave interference in the ternary IEX system considered.

It means that the interference of all three diffusion X_i concentration waves (i=₁A, ₂B, ₃C) in the IEX matrix plays the main role in the behavior of the kinetic F_2Br,ro (T) curves (right). In addition, Figure 8A, B shows that the shapes of the matrix (r-bead or ro-fiber) introduce the significant effect into the integral F_2Br,ro (T) curve behavior. The shapes of the IEX matrix influence mainly the time position (T_mr,ro ) of the kinetic maximum (F_2Br,ro^max) together with the time for the accomplishment of the final stage (T^fin) of the IEX nonselective kinetic process.

Therefore, the ₂B concentration wave reaches the center of the r-bead much faster than for the ro-fiber. The comparison of the kinetic curves F_2r,ro (T) (Figure 8A, B, right) shows that the duration (T^fin) of the whole diffusion kinetic process is much shorter for the r-bead than for the ro-fiber (Figure 8A, B, right, F_2r,ro ): (a),T_r^fin∼1<T_ro^fin∼2, (b).

The necessary conditions for the intensive displacement effect are determined by the inequality relation for the all three D_i diffusivities: (D_2B, D_1A)>>D_3C. Thus, under the certain conditions (i.e., for the big difference in the diffusivities D_2B>>D_3C), the ₂B component is accumulated exactly in the region of the front part of the invading slow ₃C wave (see X_2r ; X_2ro profiles (curves 2, dashed blue) for the ₂B component waves, Figure 8A, B, left). For this ternary IEX kinetics, this accumulation effect for ₂B waves brings the appearance of the kinetic maximum for both cases (F_2r^max∼F_2ro^max, color points, Figure 8A, B, right) with the nonmonotonic F_2r,ro (T) curves (Figure 8A, B, right, F_2r,ro^max). The detailed estimations of the D_B, D_ADiffusivities influence are presented in the next section (Section 8.2).

9 Conclusions

The modern kinetic NC model for the multicomponent mass transfer in the novel NC materials has been created [1–5]. There are realized computerized simulations of the nonlinear multicomponent NC systems on the basis of the created NC model with the two routes I, II (scheme, Figure 2) reflecting the bifunctionality of the NC matrix. The numerical solution of the mass balance partial differential Eq. (3) (Section 3) brings the new results describing the behavior of the multicomponent concentration waves (W⁺) in the bifunctional NC matrixes for the three various NC shapes: r-bead, ro-fiber, or L-membrane.

The results demonstrate rather comprehensive analogy between the theory of nonlinear chromatography and the multicomponent NC model for mass transfer kinetics in the bifunctional NC matrix. This analogy is used for the description of the multicomponent X_i concentration wave behavior. In the interpretation of the marked analogy (Table 1), there are the decisive influences of the two characteristic parameters: the equilibrium parameters of the MAL_S reactions (I, Selectivity) and the multicomponent diffusion effects (II, D_iDiffusivities) on the propagation of the X_i concentration waves inside the bifunctional NC matrix during the NC multicomponent mass transfer kinetics. (The details are formed in Table 1 and presented in discussion (Section 4) with Figures 4A, B, 5A, B 6A, B and 7 above.

The quantitative estimations of the X_i concentration i wave behavior are obtained by using the two integral parameters of the _5,6R⁰ wave distributions: “Center of Mass-CM_R” and “Dispersion-Disp_R.” There are obtained the corresponding conclusions concerning co-influence of the MAL_S reactions (I, K_i) and multicomponent diffusion (II, D_i) coefficients inside the bifunctional NC matrix on the generalized _kR⁰ concentration wave behavior for the R⁰ nanosites.

The larger the main selectivity MAL_S parameter is (Kb (1b) for ₅Variant 1, or K₂ (2.2) for ₆Variant 2), the steeper is the X_kR concentration wavefront for the NC _kR component. The behavior of the generalized wave width, Dispersion, the _5,6Disp_R parameter is in the correspondence with the marked analogy, in the course of time (T), the Disp_R tends asymptotically (for Kb>1) to the constant Disp_R⁰ value (Figures 6A, B and 7). This final Disp_R⁰ value depends on the MAL_SSelectivity (I, K_i) factors and Diffusivity (II, D_i) values.

The “multicomponent concentration wave” W⁺ concept is very effective in the study of the multicomponent mass transfer NC kinetics in the bifunctional NC matrix with the two factors (I,R⁰) MAL_S reactions I (K_i, Selectivity) for the active nanosites R⁰ and II (D_i, multicomponent Diffusion of the i components) in the NC medium (scheme in Figure 2; I, R⁰+II, D_i).

All obtained results are presented in terms of the additional effective key W⁺ concept: propagating multicomponent X_i concentration waves in the bifunctional NC matrix. The effective “multicomponent concentration wave” concept brings the clear and understandable treatment of the multicomponent mass transfer NC kinetics. This new products are obtained in the result of the study of the multicomponent NC mass transfer kinetics in the IEX NC systems for the bifunctional NC matrixes of various shapes: r-bead, ro-fiber, L-membrane.

The understandable and rather visual treatment of the propagating and interfering X_i concentration wave behavior becomes possible during the multicomponent mass transfer NC and IEX kinetics inside the r, ro, L matrixes. The influence of the two crucial factors Selectivity (K_S, I) and the multicomponent Diffusion (D_i, II) is demonstrated visually via the author’s computerized scientific animations.

Two new important results of the multicomponent mass transfer for the obtained NC (Sections 4–6) and IEX (Sections 7 and 8) kinetics including the X_i wave interference should be marked especially:

1. For the NC mass transfer kinetics (Sections 4–6): the propagation of the _kR⁰ and _j(R⁰p) concentrations k, j waves in the NC matrix takes place, though the corresponding D_k, D_j diffusivities are equal to zero (D_k,j=0). At the first glance, these i wave movements look rather paradoxical; however, the physical sense of the k, j wave propagation is quite clear. The propagation of these k, j waves is conditioned by the two reasons: (1) by the various mass transformations in the MAL_S reactions onto the R⁰ nanosites; (2) the propagation of the movable X_i concentration waves (with D_i>0 diffusivities) for the corresponding changeable [X_i] concentrations of the rest i components (Sections 4–6, Table 1).

2. For the nonselective IEX kinetics (Sections 7 and 8): the availability of the calculated original, new displacement effect for the interfering propagating ₁A, ₂B, ₃C concentration i waves inside the IEX matrix during the nonselective ternary IEX kinetic mass transfer (“coupled pictures,” Figure 8A, B, Sections 7 and 8) is found. The kinetic F_2B(T) curve with the unusual nonmonotonous behavior takes place in dependence on the Diffusivities relations (D_A∼D_B∼D_C, Figures 10 and 11).

10 Author’s scientific computerized animations: presentation of the moving X_i concentration waves in T time for the multicomponent NC and IEX kinetics

The author’s computerized animations have been prepared by the author of this review for the visual demonstrations of the obtained results of the computerized modeling. The author’s scientific animations may show obviously the visual illustrations of the propagation of the multicomponent X_i concentration waves in the oral computer presentations or in posters via the notebook. The frames in the scientific animations are prepared as the separate pictures from the calculated X_i concentration profile waves obtained in simulations for the successive time (T^k) moments [2, 22, 23]. The propagations of the traveling concentration waves in the NC bifunctional and nonselective IEX matrixes (Sections 4–8) may be demonstrated visually via the scientific animations with the frames pictures (examples of the “frames pictures” are shown in Figures 4A, B, 6A, B and 8A, B). The same approach has been used for the ternary nonselective IEX kinetics with the use of the frames “coupled pictures” (examples in Figure 8A, B) in the author’s animations with the demonstration of the new displacement effect for the propagating X_i concentration waves visually.

The modern approach with the application of the calculated scientific animations has been used in the author’s presentations in more than 10 years at many International Conferences. Such type of the computerized visual presentations of the author’s theoretical results with the propagating multicomponent dynamic (in theory of chromatography) and the NC kinetici waves have been used by the author repeatedly (including the sessions of the well known “IEX 2004, 2008, 2012” Conferences) [2, 22, 23]. The prepared scientific animations with the propagating multicomponent i waves are perceived by the audience easily.

In the nonselective ternary IEX system, the availability of the new displacement effect (Sections 7–9) for the ₃C, ₂B concentration waves (i=₁A, ₂B, ₃C) is demonstrated visually by the author’s scientific computerized animations representing propagation of the X_2B,3C(r, ro; T) concentration waves in the r-bead or ro-fiber of the nonselective IEX kinetics. The new displacement effect for the interfering X_2B, X_3C concentration waves is shown in the scientific animations with the “coupled pictures” together and abreast with the kinetic F_2Br,ro (T) curves showing their unusual nonmonotonic behavior. The origin, location, and visual presentations of the new displacement effect in the “coupled pictures” have been demonstrated effectively by the author’s scientific animations.

The high level of the state-of-the-art computerization, nowadays, brings such visual and well understandable possibility for the contemporary theoretic scientific investigations (in particular, for the multicomponent chromatography (dynamics) [22, 23, 31] and NC kinetic [2, 21] processes).