Extended adsorbing surface reach and memory effects on the diffusive behavior of particles in confined systems

International Journal of Heat and Mass Transfer 151 (2020) 119433

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Extended adsorbing surface reach and memory effects on the diffusive behavior of particles in confined systems M.V. Recanello a, E.K. Lenzi b, A.F. Martins c, Q. Li d, R.S. Zola a,∗ a

Department of Physics, Federal University of Technology (UTFPR), 86812-460 Apucarana, Paraná, Brazil Department of Physics, Universidade Estadual de Ponta Grossa - Ponta Grossa, PR 87030-900, Brazil c Laboratory of Materials, Macromolecules and Composites, Federal University of Technology (UTFPR), 86812-460 Apucarana, Paraná, Brazil d Chemical Physics Interdisciplinary Program and Liquid Crystal Institute, Kent State University, OH 44242, United States of America b

a r t i c l e

i n f o

Article history: Received 29 October 2019 Revised 27 January 2020 Accepted 27 January 2020

Keywords: Diffusion in confined systems Adsorption-desorption Memory effect

a b s t r a c t The adsorption-desorption phenomena play a pivotal role in several industrial and scientific processes, deserving attention from research groups across several fields of knowledge. In this article, we study adsorption-desorption of neutral particles scattered in a confined liquid when memory effects are present in both processes, the adsorption and desorption phenomenon. On contrary to previous studies, in which kernels representing memory effects were added to kinetic equations in the desorption term, resulting in considerable change in the dynamics of adsorbed particles, in this work we study the insertion of memory effects on the adsorption process. Such memory means that the preceding state of the particle in the bulk is relevant to the adsorption process, thus heavily affecting how particles are distributed in the bulk and making the reach of the surfaces far greater than when these effects are not considered. We firstly study a case where memory effect occurs only on the adsorption case, and then we analyze situations in which memory effects are included on both adsorption and desorption, and the distinct diffusion regimes in these systems. Our results could be potentially applied and easily adapted to model systems with complex geometries where diffusion and adsorption are present, such as occurs in slit pores. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction Separation of mixed quantities, which is unfavorable according to the second law of thermodynamics, is an ubiquitous process not only in industrial applications, but in several fields of research [1]. One important manner of achieving separation is through selective adsorption, where at least one kind of material (the adsorbat) adhere to a solid surface (the adsorbent) through chemical or physical forces [2]. In general, adsorption processes are very important in soft matter systems, including liquid crystals [3], related to anchoring energy [4–6] and degradation of display performance [7], polymers in solution, for pharmaceutical applications [8], biophysics [9] and nanocomposite materials [10]. Furthermore, it is used to capture colloidal particles [11], measure properties in fluidfluid interfaces [12,13], and for CO2 separation [14]. In several cases, the limiting surfaces of confined systems are responsible for the adsorption process. In such confinement, the diffusion of particles is heavily affected by the dynamics occurring at the adsorbing substrates [15]. Indeed, measuring diffusion time

∗

Corresponding author. E-mail address: [email protected] (R.S. Zola).

https://doi.org/10.1016/j.ijheatmasstransfer.2020.119433 0017-9310/© 2020 Elsevier Ltd. All rights reserved.

is important [16] because it often probes porous media [17,18] and memory effects [15]. For example, the study of adsorption and diffusion of particles through slit pores is of particular importance in basic and applied research [19]. In these systems, besides potential applications in separation processes and materials development, new physical phenomena arise from finite-size effects, varying dimensionality, and surface forces [19]. The consequences of the augmented reach of the surfaces is the drastic modification observed in the bulk densities when compared to large confined systems (consequently, with reduced surface effect systems), as reported in many different published studies [14,19–22]. Of course, the mathematical description of such systems is often cumbersome, involving sophisticated modeling and computational simulations [20,21]. In this work, we model the adsorption-desorption process of confined neutral particles dissolved in an isotropic liquid. In this model, the adsorbing surfaces are treated by means of kinetic equations [23] in the presence of kernels that modify both the adsorption and the desorption processes. In a previous publication, we dealt with kernels in the desorption process, which introduced memory effects in the desorption of particles [24,25]. Depending on the kernel’s choice, different kinds of surface dynamics were observed, associated with either chemisorption, physisorption or

2

M.V. Recanello, E.K. Lenzi and A.F. Martins et al. / International Journal of Heat and Mass Transfer 151 (2020) 119433

a combination of both. More recently, we studied the effect of non-identical adsorbing surfaces on neutral [26] and charged particles [27]. In both cases, we found that the adsorption dynamics heavily influences the bulk dynamics, in such a way that small changes in the characteristic times of adsorption-desorption produces large changes in the diffusive regimes of the particles in the bulk. Here, we generalize the problem, extending the importance of the surfaces even further by considering non-singular kernels on the adsorption term. Therefore, the adsorption of a giving particle depends on its preceding state in the bulk, hence extending the reach of the substrates. Thus, the adsorbing substrates under the presence of such term can be related to systems where the surface plays major roles, that is, the surface has an augmented reach when compared to ordinary situations, as it is the case in several systems. Examples of augmented surfaces, in which the present model might be applied, are in soft adhesion systems [28], such as in polymer-coated colloids being adsorbed and suffering preadsorption processes [29], diffusion of colloidal particles functionalized with DNA on coated surfaces [30], gas transport in porous coal matrix [31], diffusion dynamics through porous media [32,33] and diffusion in living cells, such as the diffusion in membranes [34– 36]. Furthermore, it can also be used on systems that present adsorption and diffusion of particles with complex adsorbents [37– 40]. In all these references [29–40], particles diffusing upon the action of complex surfaces occur, which ultimately implicates in nonordinary behaviors observed in the diffusive process and densities of surface and bulk particles. We further generalize the analysis by considering, simultaneously, kernels on both adsorption and desorption processes. The dynamics of bulk and surfaces as well as the diffusive regimes, in connection with applied cases, are presented and discussed. This work not only brings new insights about diffusive regimes in confined systems but also generalize the problem to have, on demand, control/model of the nature of the adsorption process as well as the reach of the surfaces. In this generalized mode, the nature of the adsorbing substrates, determining the nature of the adsorption process (chemisorption, physisorption or a combination of both) and the overall structure of the bulk (presence of traps, molecular crowding, changing diffusion coefficient, etc.) determine the dynamical surface and bulk distribution of particles. 2. Model Let us consider a confined sample in the shape of slab filled with an isotropic liquid with dispersed neutral particles. We assume that the only relevant direction is z, and that the system is bounded by two identical surfaces separated by a distance d and located at z = ±d/2. The density of particles in the bulk, ρ (z, t), obeys the classical diffusion equation, that is

∂ρ ∂ ρ =D 2, ∂t ∂z 2

(1)

where D is the diffusion coefficient of the neutral particles through the isotropic medium. As the diffusing particles reach the substrates, they may be adsorbed and desorbed. The density of adsorbed particles, at any time, is given by σ (t), and obeys the following kinetic equation

dσ = dt



0

t

K (t − t  )ρ (±d/2, t  )dt  −



t 0

S(t − t  )σ (t  )dt  .

(2)

Eq. (2) is a balance equation written with the aid of kernels (K(t) and S(t)) controlling the dynamics of adsorption and desorption. It was first introduced without the kernels [41] (i.e, dσ (t )/dt = κρ (±d/2, t ) − 1/τ σ (t )). In a few words, it states that the time variation of particles adsorbed by the surfaces is equal the number of particles right in front of the surface, available to be adsorbed, minus the number of desorbed particles [4,23]. The use

of kernels, such as in Eq. (2), represents more complex scenarios in the adsorption and desorption process, related to memory effects, such as non-Debye relaxation [42], often used in dielectric relaxation, diffusion-controlled relaxation in liquids, liquid crystals, and amorphous polymers. The use of kernels was first introduced to differentiate mechanisms of desorption (last term of Eq. (2)), in which the choice of the kernel would be able to model chemissorption, physisorption or a combination of both [24,25]. Similar equations such as (2) have been used before in other contexts, such as to describe chemical reactions with variable rates [43,44], but it has never been applied in the context described here. The use of two different kernels, such as in Eq. (2), allows the introduction of memory processes in both adsorption and desorption phenomena. In addition to Eqs. (1) and (2), we must invoke the conservation of the number of particles, given by

2σ +



d/2

ρ (z, t )dz = ρ0 d,

−d/2

(3)

where ρ (z, 0 ) = ρ0 , that is, we assume a homogeneous distribution as initial condition. It is worth mentioning that Eq. (3) is a direct consequence of considering that the surfaces and the bulk are connected by the boundary condition:

D

  d ∂ ρ (z, t ) = ± σ (t ) . ∂z dt z=±d/2

(4)

Eqs. (1)–(3) (or Eq. (4)) give the behavior for ρ (z, t) and σ (t) according the dynamics considered. In order to obtain them, we use the Laplace transform, starting from Eq. (1), which, in the Laplace space, yields

ρ (z, s ) =

ρ0 s

+ A(s ) sinh(



s/Dz ) + B(s ) cosh(



s/Dz ),

(5)

∞

where ρ (z, s ) = 0 ρ (z, t )e−st dt = L {ρ (z, t)}. Since the substrates are identical, we identify that A(s ) = 0. The other term, B(s), is found by using Eq. (2), in the Laplace space, or

sG(s ) − σ (0 ) = K (s )ρ (0, s ) − S(s )G(s ),

(6)

where G(s) is the Laplace transformation of σ (t) (L{σ (t )} = G(s )) and it is assumed that initially there are no particles adsorbed on the surfaces (σ (0 ) = 0). By combining Eqs. (3)–(6), we may determine B(s) and, consequently, arrive, in the Laplace’s space, at the density of particles on the surface:

G (s ) =

K ( s ) ρ0



s s + S (s ) + K (s )

s

D

coth( d2

 s . ) D

(7)

This result for the surface particle density is very interesting and allows to obtain, after some calculations, the following equation:



d d σ (t ) + dt dt

0

where K(t ) =



F(t ) = L

−1

t

t 0

dt  F(t − t  )σ (t  ) = K(t )ρ0 −



0

t

dt  S(t − t  )σ (t (8) )

dt  K (t  ) and

K (s ) √ coth sD

 d 2

s D

(t ) .

(9)

Eq. (8) explicitly shows the influence of the diffusion process of the particles in the bulk on the adsorption-desorption processes through an additional term in the kinetic equation, which can be related to a fractional differential operator with the kernel given by Eq. (9). For the bulk density, it is possible to show that

  ρ0 K (s ) Ds cosh(z Ds ) ρ (z, s ) = −    . s K (s )s Ds cosh( d2 Ds ) + s(s + S(s )) sinh( d2 Ds ) ρ0

(10)

M.V. Recanello, E.K. Lenzi and A.F. Martins et al. / International Journal of Heat and Mass Transfer 151 (2020) 119433

3

From Eqs. (7) and (10), one can clearly see that the dynamics of both, surface and bulk, heavily depends on the choice of the kernels K(s) and S(s). In order to find ρ (z, t) and σ (t), we must perform the inverse Laplace transform, which will be done according to the choice of the kernels present related to the adsorption and desorption processes. To make it easier to the reader, in Appendix A we present a table describing all the symbols used in the calculations presented in this article.

3. Results and discussion In a previous study [24], the cases in which the kernel repre− t

senting S(t) is either S(t ) = δ (t/τ )/τ 2 or S(t ) = 1/(τd τa )e τa were studied. In the first case, the kernel is a localized function of time, that is, it has a brief effect. This kernel results in surface dynamics that are often observed in the process of chemisorption, where there is a loss of memory from the preceding state, the desorption process occurs with characteristic time τ [24] and the preceding state of the particle at the surface is not relevant. On the other hand, the second kernel represents a typical relaxation process, which means the molecule on the surface has a memory of its preceding state. This memory, in the desorption context, is related to loss of energy during collision with the surface in a previous adsorption process. Such energy loss occurs once the molecule “falls” into an adsorbing well, but is then desorbed back to the bulk with a memory of the preceding state, so it can be absorbed again and successively until a saturation of the adsorbing wall is reached [24,25]. The use of such kernel yields behaviors that are very similar to physisorption processes. Therefore, as previously studied [24,25], kernels like the ones presented above in the desorption process are related to chemisorption and physisorption processes, respectively. Here, we study the effect of adding a kernel to the adsorption term and generalize it to the situation where kernels on both terms are present. We first identify that if a kernel such as K (t ) = δ (t ) is used, the same results as reported before [23,24] are recovered, that is, we could have either surface densities that represent chemisorption or physisorption (depending on the choice of S(t)), without considerable change in the bulk density. This case indicates an adsorption process where information of the preceding state is not important, so there is no information of the preceding state of the particles in the bulk for the surfaces to adsorb them. In order to understand the effect of K(t), we start by considering − t

the following situation: S(t ) = δ (t/τ )/τ 2 and K (t ) = κ /τκ a e τκ a , where κ is related to adsorption time while τ κ a is the memory time, that is, it represents the time width of the memory effect. This choice of kernels represent long-range memory on the adsorption term, thus meaning the surface has more importance in the bulk particles, and short memory on the desorption process, which recovers the usual kinetic equation presented in reference [23]. Therefore, a term like this on the adsorption term states that the preceding state of the particle in the bulk is important for the surface to make the adsorption process. This addition should be relevant when modeling complicated bulk structures such as diffusion in porous media [18,19], with molecular crowding [45,46], molecular obstruction [47] and diffusion through distinct geometries [48]. It could also be potentially important in living cells, where the adsorption phenomena involved in the molecular trapping and transport at biomembranes play crucial role in separating particles important to the cell in such crowded environment. By choosing these two kernels, Eqs. (7) and (10) become:

G (s ) =



κρ0 τ tanh

d s  D

d  s

2

s (sτ + 1 )(sτκ a + 1 ) tanh

2

D

+ κτ

 s , D

(11)

Fig. 1. Normalized surface and bulk densities vs. t∗ (surface) and vs. Z = 2z/d (bulk) t when K (t ) = κ /τκ a e− τκ a and S (t ) = δ (t/τ )/τ 2 . a) is the surface density while the inset is the bulk density calculated for several values of t∗ when τD /τ = 13.3, τk /τ = 0.3 and τκ a /τ = 6.7. Figure b) shows two situations, for τD /τ = 5, and τκ a /τ = 1.25, when τk /τ = 1 (solid line) and τk /τ = 16 (circles). The bulk distribution (inset) is calculated for t ∗ = 5.0.

and

ρ (z, s ) =

ρ0 s

−

ρ0 κτ



s D

d s 

sech

2

D

s (sτ + 1 )(sτκ a + 1 ) tanh

s

cosh z

d s  2

D

D

+ κτ

 s  . (12) D

Now, we must find the inverse of the Laplace transform. It must be performed with the Bromwich’s integral in the complex plane with the residue technique [49]. This procedure is done by finding the poles in each equation. From Eqs. (10) and (11), one can observe that the denominator of G(s) and the second term denominator of ρ (z, s) are the same, in which the zeros of the denominator result in the poles of both functions. The poles are located in s = 0 and are periodically arranged according to

tan[Xn ] = −

τk 4 τ

Xn2

τ τD2 Xn   − τD 4 τκ a X 2 − τD

(13)

where we introduced the new times, τD = d2 /D, which is the diffusion time and τκ = d/2κ , the adsorption time; Xn are the roots of Eq. (12). Notice that, if τκ a = 0, we recover know results presented elsewhere [24]. Therefore, the inversion procedure for Eqs. (10) and (11) result in a series expansion summed over Xn (solutions of the periodic Eq. (12)). The expression is given in Appendix B. Fig. 1a shows the normalized adsorbed concentration (2σ (t)/ρ 0 d) vs. t ∗ = 4t/τD for τD /τ = 13.3, τk /τ = 0.3 and τκ a /τ = 6.7. This initial example displays a classical adsorption dynamics, where the surface density increases as more and more particles are adsorbed until a

4

M.V. Recanello, E.K. Lenzi and A.F. Martins et al. / International Journal of Heat and Mass Transfer 151 (2020) 119433

saturation point is achieved. However, the curve presents a concavity change in the initial moments that has not been observed in models without kernel in the adsorption term. It resembles a breakthrough curve, a region where the adsorption zone in a fixed bed multilayer adsorbant starts to diminish until the saturation point is reached, behavior often observed in porous absorbants. This behavior highlights the importance of the surface in the bulk particles, here represented by the kernel in the adsorption term, since it can recreate a behavior often observed in porous media. With respect to the model, this result can be understood once we observe the values of the characteristic times, all scaled with respect to the desorption time. The diffusion time is the largest of all, meaning the particles take some time to diffuse their way to the substrates. On the other hand, the adsorption time is shorter than the desorption time, so initially, the surfaces start adsorbing a large amount of particles without desorbing them. The long (τ κ a ), however, introduces a memory effect to the adsorption process, acting as if layers of the adsorbing bed would be gradually filled. Eventually the desorption time kicks in and the adsorbant starts to desorb some of its particle and the curve changes concavity, until it reaches a stationary state. The inset of Fig. 1 shows the bulk distribution, ρ (z, t) vs. Z = 2z/d for different values of t∗ , indicating that the amount of particles in the bulk decreases over time as the surfaces adsorb the free particles. Fig. 1b shows the cases in which τD /τ = 5, and τκ a /τ = 1.25 for two different values of τ k /τ . The solid line depicts τk /τ = 1 while the circles represent τk /τ = 16. For the first case, the desorption time is equal the adsorption time. As a consequence, the adsorption is limited in accumulating particles by the fast desorption and, therefore, the surface density quickly reaches the saturation point. It is interesting however to observe how the memory time causes an accumulation of particles near the substrates, resulting in more particles near the substrates than in the center of the sample. On the other hand, for the second case, the desorption time is much shorter than the adsorption time. Consequently, the particles are quickly dessorved and the surface concentration is even lower. The bulk density in this case ends up having accumulation in the central part of the sample, as shown in the inset figures, both calculated at t ∗ = 5.0. Such bulk densities are not observed unless the memory effect in the adsorption process is present. Fig. 2 shows the case in which τD /τ = 30.0, τk /τ = 0.8 and τκ a /τ = 1.0. Since the adsorption time is faster than desorption time, the substrates quickly adsorb the particles nearby, reaching an stationary value after t∗ ≈ 1.0. However, the memory effect associated to the slow diffusion time causes the accumulation

near the substrates, as it is easy to observe near the substrates. This accumulation, combined with the slow diffusion time causes a slightly higher concentration in the center of the cell compared to regions closer to the substrates (the two minimums observed in the inset of the figure). The unusual shape of ρ (z, t) is due to the influence of the surfaces represented by the memory time τ κ a . The surfaces act within a certain range inside the bulk, given by zs = κτ [4], in such a way that particles within zs participate in the adsorption-desorption process. The presence of τ κ a indicates the time the surfaces “remembers” (or weights) the states of particles within the range zs to be adsorbed. Therefore, combinations of τ κ a and large zs (every time τ κ < τ , zs is larger than d, so all the particles participate in the adsorption process, being limited only by the diffusion time τ D ) may result in accumulation near the substrates. Now we explore a more general case, in which both, adsorption and desorption terms are written with kernels. As previously discussed, the situation in which K (s ) = κ and S(s ) = 1/τ have been explored before [23] and the case where K (s ) = κ and S(s ) = 1/(s + 1/τb )τb (exponential decay), where τ b is the memory time in the desorption time, has been explored in reference [24]. Now, we consider the case in which both processes have memory effect, that is

K (t ) =

κ −t/τa e τa

S(t ) =

and

1

τ τb

e−t/τb ,

(14)

therefore, the preceding state of the particle is important for both the adsorption and the desorption process. A kernel such as given by (14), in the adsorption term, has the physical meaning as previously discussed. In the desorption term is related to the loss of energy caused by successive collisions of the particles with the surfaces. In other words, if the particle is in a weak bound state with the surface, the thermal motion may cause it to be desorbed, returning to the bulk with lower energy; eventually, the particle collides again with the surface, until it reaches a state where it sticks to the substrate [50], so there is a memory effect related to the state of particle with respect to the surface. Therefore, the combination of both kernels as Eq. (14), represents a very general case, in which both memory effects, in the adsorption and desorption may be at play; furthermore, these effects vanish once τ a → 0 and τ b → 0. Therefore, by using Eq. (14), Eqs. (7) and (10) become G (s ) =



ρ0 κτ (sτb + 1 ) sinh 

d s 

s (sτa + 1 ) s2 τ τb + sτ + 1 sinh

2

D

d s  2

+ κτ

Ds D

(sτb + 1 ) cosh

d  s  , 2

D

(15) and

ρ (z, s ) =

ρ0 s

−

s

  (sτb + 1 ) cosh z Ds 

d s 

  . s (sτa + 1 ) s2 τ τb + sτ + 1 sinh 2 D + κτ Ds (sτb + 1 ) cosh d2 Ds 

ρ0 κτ



D

(16) Again, the denominators of Eqs. (15) and (16) (second term on the right-hand side), are the same, so the infinite poles on which the series from both σ (t) and ρ (z, t) are summed, are the same and given by:

 τ τD2 Xn τD − 4τb Xn2  , tan[Xn ] = − τκ τD − 4τa Xn2 τD2 + 16τ τb Xn4 − 4τ τD Xn2

∗

Fig. 2. Normalized surface and bulk densities (inset) vs. t (surface) and vs. Z = t 2z/d (bulk) when K (t ) = κ /τκ a e− τκ a and S (t ) = δ (t/τ )/τ 2 . These curves are calculated for τD /τ = 30.0, τk /τ = 0.8 and τκ a /τ = 1.0.

(17)

where we have again used τD = d2 /D and τκ = d/2κ . With the series expansion, we can use numerical values for the characteristic times to study σ (t) and ρ (z, t). Fig. 3 shows two examples of the densities of adsorbed particles and the distribution in the

M.V. Recanello, E.K. Lenzi and A.F. Martins et al. / International Journal of Heat and Mass Transfer 151 (2020) 119433

Fig. 3. Normalized surface and bulk densities (inset) vs. t∗ (surface) and vs. Z = 2z/d for (bulk) for several values of t∗ when K (t ) = τκa e−t/τa and S (t ) = τ1τ e−t/τb . b a) is calculated for τD /τ = 20.0, τk /τ = 0.05, τa /τ = 0.5 and τb /τ = 6.0. Figure b) shows σ (t) and ρ (Z, t) for τD /τ = 10.0, τk /τ = 1.0, τa /τ = 1.0 and τb /τ = 5.0.

bulk. Fig. 3a shows the case in which τD /τ = 20.0, τk /τ = 0.05, τa /τ = 0.5 and τb /τ = 6.0. We first notice that the dynamics at the surfaces are quite different from the previous results. Indeed, such curves are typical of physisorption process or a mixed process where the physisorption precedes the chemisorption process, as demonstrated for siloxane polymers adsorbing to alumina [2,51]. Such behavior has also been observed in the adsorption of propyltrimethoxysilane on iron and aluminium oxide surfaces [52], in which the surface dynamics is explained only if adsorption and desorption processes are considered altogether. Therefore, we explain the behavior of σ (t) due to the very short adsorption time, which means a fast accumulation of particles, and the long memory on the desorption process, which indicates that once the particles return to the volume, they can be adsorbed again, just as reported for two adsorption process occurring together [51]. It is quite unusual, however, the behavior of the bulk particles. The presence of a memory term in the adsorption causes an accumulation of particles near the substrates. Since the substrates adsorb very fast in the initial moments and the diffusion time is long, two regions of low density form near the substrates (the two minima in the inset of Fig. 3a). After a few moments, the substrates start to desorb, sending particles back to the bulk, which interferes on the drift current [26] of particles to the substrates, thus increasing the concentration in the central regions. It is important to notice that such behavior is observed only if memory effects are present on both, adsorption and desorption process. A curve such as the inset of Fig. 3a resembles particle distribution in slit pores [53–55] and

5

Fig. 4. (z)2 vs. t∗ for the same set of parameters used in Fig. 1. Figure a) shows the variance vs. t∗ for the parameters used in Fig. 1a. The dashed lines represent the exponent of t∗ in three different regions, showing the subdiffusive nature of the process. Figure b) uses the same parameters as in Fig. 1b, demonstrating a case where the bulk distribution increases due to accumulation of particles near the substrates (solid curve) and a situation where the distribution initially spreads out and then shirinks once the particles start to accumulate near the substrates and more particles are adsorbed.

therefore, could potentially be applied to describing the dynamics in such systems. Fig. 3b shows the case in which τD /τ = 10.0, τk /τ = 1.0, τa /τ = 1.0 and τb /τ = 5.0. On contrary to the figure a, the desorption time occurs more rapidly, which can be understood due to the faster desorption time of this system when compared to the adsorption time (of figure a). Also, here the diffusion time is faster and the desorption time is equal the adsorption time and the memory time on adsorption, so the bulk density decreases without forming the two minima. Interestingly, the bulk density at t ∗ = 5.0 is slightly higher in the middle of the sample than at t ∗ = 1.0, which is consequence of the surfaces still be deserving when t ∗ = 1.0. Next, we look at the variance of z in order to determine the diffusive regimes present in this cell. This is an important parameter because it is related to spreading of the distribution, which, on the other hand, characterizes the time dependent diffusion time (D(t)) [56]. In other words, the variance is a direct measure of the diffusive regime of the bulk particles, which are affected by the immobilization caused by the surfaces. The variance is given by (z )2 = (z − z )2 , and we here look how it evolves with time. In general, (z)2 ~ tγ , where γ is related to the kind of diffusion. If γ = 1, the diffusion is called usual. If 0 < γ < 1 or γ > 1, the diffusion is called anomalous, being either sub-diffusive for the first case or supperdiffusive for the second case [57,58]. Fig. 4a and b

6

M.V. Recanello, E.K. Lenzi and A.F. Martins et al. / International Journal of Heat and Mass Transfer 151 (2020) 119433

show (z)2 vs. t∗ for the same set of parameters shown in Fig. 1a and b. Fig. 4a shows that there are several diffusive regimes in the process depicted in Fig. 1a, all of which representing sub-diffusive regimes. Since the distribution is decreasing, (z)2 decays with time. Initially, there is a fast diffusion process, which corresponds to the first change in concavity represented in Fig. 1a, then followed by a slower diffusion process (until t∗ ~ 0.5) that is related to fast adsorbing process depicted in Fig. 1a. Once the substrates reach a near saturation point, the diffusive regime becomes even slower (t∗ ~ 1.5) and eventually the diffusion ceases to exist near t∗ ~ 3.0. Notice that a curve such as shown in Fig. 4a closely resembles some diffusive characteristics found in living cells [59]. In particular, as observed in the diffusion of gold-labeled dioleoylPE in the plasma membrane of fetal rat skin keratinocyte cells [34], the variance shows anomalous subdiffusion with similar behavior as Fig. 4a. Such behavior, which has been attributed to the nested compartmentation of the membrane by fences [34,47] and later modeled in terms of a finite hierarchy of traps [47], is a clear evidence of augmented surface influence on the diffusive behavior of confined systems. Fig. 4b on the other hand depicts the two systems shown in Fig. 1b, where the solid curve represents the case τκ /τ = 16.0 while the doted curve represents τκ /τ = 1.0. In both cases, the system reaches saturation within the first moments (t∗ ~ 0.5). Nonetheless, the accumulation of particles due to the memory effect causes the distribution spread to grow in the initial moments for both cases. For τκ /τ = 16.0, as previously discussed, the desorption time is much shorter than adsorption, so the particles accumulate in the center, meaning (z)2 increases (distribution is opening) and quickly reaches a saturation point. For τκ /τ = 1.0, on the other hand, there is an initial spreading of the distribution followed by a regime where the distribution of particles start to decrease, in accordance with Fig. 1b. In both cases, the diffusive regime is essentially sub-diffusive, demonstrating the importance of the adsorption process on the diffusion of bulk particles. Finally, Fig. 5 shows (z)2 vs. t∗ for the two situations depicted in Fig. 3, where kernels are used in both, adsorption and desorption processes. The solid curve represents what is depicted in Fig. 3a, where a fast adsorption process occurs followed by slow desorption, which results in accumulation both near the substrates and in the central regions of the cell. The variance, in this case, shows a fast decrease in the initial moments, corresponding to the initial adsorption process, where the distribution in the bulk decreases. It, however, changes concavity after t∗ ~ 0.2, once the slow desorption process takes place, meaning particles are return-

Fig. 5. (z)2 vs. t∗ for the same set of parameters used in Fig. 3. The main figure represents the case depicted in Fig. 3a while the inset shows the case retracted in Fig. 3b.

ing to the bulk and therefore the distribution in the bulk is increasing and saturating only for long times (t∗ ~ 10.0). The inset figure (dashed), corresponds to the same set of parameters shown in 3-b. In this case, the initial adsorption process is slower when compared to the case represented in 3-a, so (z)2 decreases in the initial moments. Then, the variance starts increasing, indicating that the desorption process takes place and the distribution in the bulk increases. Interestingly, (z)2 decreases again near t∗ ~ 0.8, indicating that the adsorption process is still occurring. Such information is not easily observed in the σ (t) curve shown in Fig. 3b, which only represents the net amount of adsorbed particles. It is also important to notice that, throughout this whole process, several diffusive regimes occur, all of which with γ < 1, that is, in a sub-diffusive manner. It is important to stress that the unusual bulk distributions reported here may not be easily measured in an experimental setup. Although tracking algorithms are often used to probe time dependent distributions, in the situations involving limited-size samples, where the surface has augmented reach, tracking the particles is very difficult. Indeed, most of works that present bulk distributions in porous media for example, involve either analytical models or computer simulations [19]. However, the effects of adsorbing substrates on bulk distributions can be observed indirectly, for example, when measuring time dependent diffusion coefficient, which is closely related to the variance. For example, polymer-coated colloids being adsorbed on a horizontal substrate may suffer a complex sequence of preadsorption processes [29]. This soft adhesion dynamical process results in a time dependent diffusion coefficient that closely resembles the variance curve presented in Fig. 5. Similar effects are observed in colloidal particles functionalized with DNA sticky ends diffusing on a complementary coated surface [30]. Therefore, a possible way of using our results to some experimental results is to adapt the calculations to the geometry of systems with complex adsorption process as presented in reference [29]. Another interesting fact is the diffusion of particles through cell membranes. As mentioned earlier, the diffusive behavior in the plasma membrane of fetal rat skin keratinocyte cells [34] is subdiffusive. Recently, molecular dynamics simulations of small molecules diffusing through lipid bilayers, have shown a subdiffusive behavior, which is associated with voids in the center of the bilayer, possible related with the lateral diffusion of lipids [36]. The lipid bilayer forms a complex network within the membrane, starting from the hydrophilic, fully hydrated, heads until the dehydrated interior. A diffusing particle has to go through these two media and diffuse through voids occurring inside the membrane. As a result, the diffusing process is subdiffusive and the particle distribution at any given time is not uniform, but rather shows accumulation at certain points, which is quite similar to the behavior observed in our system. Thus, besides subdiffusive behavior, this system presents particle distributions that resembles the distributions found here. Therefore, as a possible application, from a phenomenological point of view, our model could be adapted for modeling diffusion through membranes, if the polar heads of a lipid bilayer are viewed as the surfaces that have a long reach within the bulk, thus influencing the diffusive behavior across the membrane. Furthermore, one may design confined slab samples with highly porous adsorbing wall (or a membrane) and track diffusing particles in the bulk. Other possibilities would involve the study of diffusion in channels (backbone structure), or, in the case of charged particles in electrolytic cells with complex adsorbing walls, measure the impedance spectroscopy to probe the diffusive regimes [60]. In any case, as discussed here, simple modifications to kinetic equations may lead to drastic changes to bulk effects. In a phenomenological manner, such modifications can be used in several different scenarios to access measurable quantities such as the characteristic times governing the problem or time dependent diffusion coefficients.

M.V. Recanello, E.K. Lenzi and A.F. Martins et al. / International Journal of Heat and Mass Transfer 151 (2020) 119433

7

Table 1 List of symbols and definitions used in this article.

4. Conclusions In conclusion, we have investigated a confined system of neutral particles dispersed in an isotropic fluid in the presence of the adsorption-desorption phenomena. The model, which couples diffusion with kinetic equations, is written in the presence of kernels representing memory effects in the adsorption and desorption terms, thus generalizing the possibilities of phenomena present in the adsorption-desorption process when compared to previously published models. In a first analysis, we considered the presence of long range memory effects only in the adsorption process, thus the preceding state of the particle in the bulk becomes relevant for the adsorption by the substrates. By simply changing the characteristic times of the system, a rich variety of bulk and surface distributions is found, including surface dynamics unseen before with such models. In a second case, long range memory effects are considered in both, the adsorption and desorption processes, which makes it possible to recreate the behavior observed in chemisorption and physisorption and, potentially, more complicated systems such as diffusion in slit pores and other complex geometries. Analysis made calculating the variance show that the diffusive regimes are essentially sub-diffusive in nature, thus making evident the important role the adsorbing surfaces have on the diffusion of bulk particles. We stress that the generalized model presented here could be easily adapted (by adding more adsorption sites or changing the geometry of the problem, for example) to describe more complicated situations such as diffusion and adsorption processes occurring in living cells and porous media. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Symbol

Definition

ρ (z, t)

Bulk density of diffusing particles. It is a function of space (z) and time (t) Density of adsorbed particles. It is a function of time (t) Surface density in the s space Diffusion coefficient Cell thickness Bulk distribution at t = 0 Non-singular kernel used in the adsorption process Non-singular kernel used in the desorption process Complex frequency variable Adsorption rate constant Desorption time Diffusion time Adsorption time Memory time Dimensionless length Dimensionless time Variance Anomalous diffusion coefficient Roots of periodic poles Simplified roots of periodic poles

σ (t) G(s) D d

ρ0

K(t) S(t) s

κ τ τD = d2 /D τκ = d/2κ τ κa, τ a, τ b

Z = 2z/d t ∗ = 4t/τD ( z ) 2

γ βn Xn

The residues are calculated by taking the following limits (shown below for G(s)):

Res(s = 0 ) = lim sG(s )ets s→0

Res(s = −βn2 ) = lim (s + βn2 )G(s )ets ,

(19)

s→−βn2

Acknowledgement The authors thank CNPq and CAPES for financial support. This work was partially supported by the National Institutes of Science and Technology of Complex Fluids – INCT-FCx (R. S. Z.) and Complex Systems – INCT-SC (E. K. L.). Appendix A. Table of symbols In this appendix, we present to the reader a table with symbols used to represent the main quantities explored in this manuscript (Table 1). Appendix B. Inversion procedure We here describe the inversion procedure used to find ρ (Z, t) and σ (t). The inversion procedure of Eqs. (11) and (12) (and (15) and (16)) is performed by means of the Bromwich’s integral in the complex plane [49]. This is done with the aid of the residue theorem, which is calculated once the poles of the equations are found. Eq. (12) is composed of two terms. The first term on the right-hand side is easily inverted, and results in ρ 0 . The second term has exactly the same denominator as Eq. (11). The poles are found once one sets the denominators equal to zero. Clearly, there is one pole in s = 0 and periodic poles in s = −βn2 , found once we make



(sτ + 1 )(sτκ a + 1 ) tanh

d 2

s D



+ κτ

s = 0. D

(18)

where n = 1, 2, ... + ∞. Therefore, for S(t ) = δ (t/τ )/τ 2 and K (t ) = t

κ /τκ a e− τκ a , we arrive at



 2τ τD2 1 − e−t Xn sin(2Xn ) 2σ (t )

2  , (20) =− 2 ρ0 d n Xn τ τD + α1 + α2 sin (2Xn ) + τ τD cos (2Xn ) ∗

2

and

 ∗ 2  4τ τD2 1 − e−t Xn cos(Xn ) cos(Xn Z ) ρ (Z, t ) =1+ , ρ0 τ τD2 + α1 + α2 sin(2Xn ) + τ τD2 cos(2Xn ) n

(21)

where

  α1 = 2τκ τD − 4τ Xn2 τD − 4τκ a Xn2

 α2 = 8τκ Xn 8τ τκ a Xn2 − τD (τ + τκ a ) . In the second case studied here, for S(t ) = 1/(τ τb )e K (t ) = κ /τa e

− τt

a

, we arrive at

(22)

− τt

b

and

8

M.V. Recanello, E.K. Lenzi and A.F. Martins et al. / International Journal of Heat and Mass Transfer 151 (2020) 119433



2 3 1 − e−t Xn sin(2Xn ) 2σ (t ) 



 , = 2 2 2 2 ρ0 d n Xn 1 + 2 sin (2Xn ) − 4 + 8τD τκ τa Xn τD − 4τ Xn − τD 2τD τκ + τ τD − 8τκ Xn ∗

2

(23)

and

ρ (Z, t ) = 1− ρ0

 n



4 3 1 − e−t

∗

X2



cos(Xn ) cos(Xn Z )

 2

 , 2

(24)

4 − 1 − 2 sin(2Xn ) + τD 2τD τκ τD − 4τa Xn + τ τD2 + 32τκ τa Xn4 − 8τD τ kXn

where

 1 = 4τ τb X 2 3τD2 + 32τκ τa Xn4 − 8τD τκ Xn2

 2 = 8τa Xn τD2 τa + τ τD2 + 48τa τb Xn4 − 8τD Xn2 (τa + τb )

 3 = τ τD2 τD − 4τb Xn2

 4 = τ τD2 cos(2Xn ) τD − 12τb Xn2 .

(25)

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