Journal of Colloid and Interface Science 450 (2015) 224–238
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Cooperative adsorption on solid surfaces Shijie Liu Department of Paper and Bioprocess Engineering, State University of New York – College of Environmental Science and Forestry, Syracuse, NY 13210, United States
g r a p h i c a l a b s t r a c t
a r t i c l e
i n f o
Article history: Received 9 February 2015 Accepted 6 March 2015 Available online 17 March 2015 Keywords: Adsorption kinetics Adsorption isotherm Cooperativity Competitive adsorption Multilayer adsorption Nonideal adsorption
a b s t r a c t Adsorptions are commonly of monolayer coverage of adsorbate molecules on adsorbent sites, in particular for chemisorptions, where Langmuir adsorption isotherm equation and kinetics are adequate. The Langmuir adsorption is termed ideal adsorption as the surface active centers are uniformly distributed, the molecules are of point-sizes and the interactions between adsorbate molecules and the adsorbent are uniform. However, there are more cases where Langmuir adsorption isotherm and/or kinetics are inadequate in describing the adsorption behavior. Apparent multilayer adsorption has been shown to be descriptive of both physisorptions and chemisorptions as a means of idealization to nonideal adsorptions. The deviation of adsorption isotherm and/or kinetics from (ideal) Langmuir adsorption is due to cooperative adsorption, or interactions between adsorbates or between adsorbate and adsorbent caused deviation from ‘‘uniform’’ interactions. The multi-layer or apparent multilayer behavior of adsorption is an excellent model to describe cooperative adsorption. Adsorption kinetics and isotherms have been derived for adsorptions without differentiating the different types of adsorptions. The simplistic approach can explain majority of the adsorption isotherms and kinetic behaviors. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Adsorption is the phenomenon of solute(s) in a bulk fluid phase attaching onto an interface or surface. Adsorption can be classified as gas adsorption and liquid adsorption based on the initial state of
E-mail address: [email protected] http://dx.doi.org/10.1016/j.jcis.2015.03.013 0021-9797/Ó 2015 Elsevier Inc. All rights reserved.
solute(s). The solute participating in the adsorption process is also called adsorbate and the material that provides the surface is called the adsorbent. The adsorbed molecules are restricted on the interface and thus lost their mobility akin to fluid. While adsorption can occur on liquid–liquid interfaces, adsorption is commonly known to occur on solid surfaces. Adsorption on solid surfaces is widely applied in separation, clarification and purification of matter. In [1] a mathematical model was derived based on
S. Liu / Journal of Colloid and Interface Science 450 (2015) 224–238
multilayer approximation for adsorption of single species, which was extended to multiple species competitive adsorption in [2]. To illustrate the adsorption process for quantitative description, Fig. 1 exemplifies the steps in adsorption [3]. Adsorbate molecule A in the bulk fluid phase diffuses (through the solvent or mixture) to the external surfaces of the solid matrices (1), next diffuses through the internal pores of the solid matrix, reaching the ‘‘active center’’ r on the solid material (2), becoming attached or ‘‘fixed’’ on the solid material (3). The action of a free adsorbate molecule A in the fluid phase becomes ‘‘fixed’’ on the solid surface is known as adsorption. The kinetics of adsorption is thus controlled by both mass transport (of adsorbate to the active center) and the ‘‘attachment’’ (adsorption). Clearly, steps (1) and (2) are mass transfer steps. This implies that mass transfer can be important in adsorption processes. The rates (or volumetric fluxes) of mass transfer are given by
NAE ¼ kc ðC A C AS Þ
ð1Þ
NAI ¼ kp ðC AS C Ap Þ
ð2Þ
where NA’s are volumetric based transfer flux (or rate) of species A from the bulk fluid phase (with a concentration of CA) to the solid external surface (where concentration A in the fluid phase is CAS) and to the ‘‘end’’ of the internal pore (where the concentration of A in the fluid phase is CAp), at which the molecule is fixed or adsorbed on the surface. The pore permeation or transfer is through a net-work of internal pores of the solid matrix. The pores are of distributed length/ depth, and diameter. Thus, pore permeability or internal mass transfer rate constant kp can appear to be a function of time due to the adsorption occurring first on sites where adsorbate is accessed first. In [1], the pore mass transfer or permeation coefficient is given by
C A0 C Ae kp ¼ kp0 C A0 C A
ð3Þ
where kp0 is constant and CAe is the concentration of species A in the fluid phase at the adsorption equilibrium or infinite residence time. One can show that Eq. (3) is consistent with the Weber–Morris model (of intraparticle diffusion limited adsorption) and the exact solution of adsorption to an isolated sphere with negligible external mass transfer by [4] at short-times. Step (3) exemplifies the approaching of adsorbate molecule (A) to the active center in the solid material. Each active center r can attract and attach one adsorbate molecule. The reverse or desorption occurs in the reverse order.
225
It becomes clear that the last step, step (3), is the key step intrinsic to adsorption. Often, when we speak of adsorption kinetics, we mean the kinetics of step (3). It is most convenient to consider the adsorption as chemical reaction or surface reaction. That is,
A þ r¢A r
ð4Þ
As Eq. (4) indicates, the adsorption is reversible. The reverse of adsorption is desorption. It is the nature of reversibility that made adsorption an effective separation technique for liquids and gases. The Gibbs free energy for the adsorption process is given by
DG ¼ DH T DS
ð5Þ
where DH is the enthalpy change due to adsorption of the molecule on the solid surface or heat of adsorption, DS is the entropy change from the bulk phase (more random) to the adsorbed phase (more orderly), T is the temperature, and DG is the Gibbs free energy change due to adsorption. For the adsorption to occur, DG < 0, therefore, DH = DG + TDS < 0 or the adsorption is exothermic. This condition holds because DS < 0 for the adsorption process if the adsorption does not involve the breaking apart of the adsorbate molecule, as opposes to that in dissociative adsorption. The decrease in entropy is because of the change from a disorderly state (in fluid phase) to an orderly state (‘‘fixed’’ on the interface). For dissociative adsorption, the heat of adsorption can be endothermic as the entropy change is positive when a molecule is breaking into two parts. The kinetics of adsorption follows naturally with the collision theory. Consider the rate at which adsorbate A molecules in a homogeneous gas (or liquid) phase will strike a solid surface, the collision frequency is given by
sffiffiffiffiffiffiffiffiffiffiffiffiffi RT Z cT ðA; surfaceÞ ¼ N AV C A 2pM A
ð6Þ
which for gas adsorption is equivalent to
NAV pA ffi Z cT ðA; surfaceÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pMA RT
ð7Þ
Only after the adsorbate molecule strikes the surface, it could produce adsorption. The rate of adsorption is thus a fraction of the collision frequency. Therefore, adsorption rate can be considered either in concentrations for liquid and gas adsorptions or in partial pressures for gas adsorption. Based on Eq. (6), the adsorption as represented by Eq. (4) can be regarded as elementary reaction. The net rate of this adsorption is given by
r ¼ kad C A C rv kdes C rA
ð8Þ
or
A 1
CA
A 1
External Mass transfer CAS
A 2
2
Internal Diffusion
3 σσσσσσσσσσσσσσσσσ σσσσσσσσσσσσσσσσσ σ σ σ σ σ σ σ σ σA| σAσon σ site σσσσσ σσσσσσσσσσσσσσσσσ σσσσσσσσσσσσσσσσσ σ σ σ σ σ Solid σ σ σ/ σadsorbent σ σ σ σ σsurface σσσ σσσσσσσσσσσσσσσσσ
CAp
Fig. 1. Typical steps of adsorption involving one species (A) in the bulk fluid phase and one active site (r) group in the solid material.
r ¼ kA C A hC r kA hA C r
ð9Þ
where kad = kA is the adsorption rate constant, CA is the concentration of the adsorbate in the bulk fluid phase (near the adsorbent surface, which should be CAp in Fig. 1), Crv is the surface concentration of vacant active centers, CrA is the surface concentration of the active centers that are occupied by the adsorbate molecule A, h is the fraction of the vacant active centers, Cr is the surface concentration of the total possible available positions or sites or active centers where A can be adsorbed, hA is the fraction of the total possible available positions that are occupied by A. Clearly,
h¼
C rv Cr
hA ¼
C rA Cr
ð10Þ
ð11Þ
226
S. Liu / Journal of Colloid and Interface Science 450 (2015) 224–238
The overall kinetic behavior is governed by [1]
NAI ¼ NAI ¼ r A
ð12Þ
That is to say, the external mass transfer flux, internal mass transfer flux, and the net adsorption rate are identical. It has shown excellent agreement in modeling the adsorption kinetics with this model [1]. At this point, we expect that the rate of adsorption is important for its practical applications. However, the study of adsorption kinetics has not always been consistent with the fundamental understanding as such summarized above. Often, the study of adsorption kinetics is purely empirical. Commonly applied kinetic models [5] are (1). Lagergren pseudo-first order [6]; (2). the chemisorptions pseudo-second order [7]; (3). Elovich kinetic model [8]; (4). the intraparticle diffusion model [9]; and (5). The liquid film diffusion model [10]. These models have been applied to analyze adsorption data with varied successes; however, none of these models give rise to an acceptable equilibrium behavior or isotherm. There has been an outcry to fix the incompatible adsorption kinetic analyses [11,12], the use of these empirical models that are not compatible with fundamental understanding of adsorption persists. These models have one common feature: they reduced to an algebraic model of adsorbate concentration vs adsorption time and can be linearized such that experimental data can be put on a straight line plot to determine the parameters. It is especially appealing for researchers do not have access to modern computational tools to use linear plots to ascertain the adsorption behavior. In [13] it was shown that the pseudo-first order (Lagergren model) and the pseudo-second order (Elovich kinetic model) can be simplified from the Langmuir kinetics under extreme conditions, which was reinforced by [14]. In fact, all these empirical models can be reduced from more rigorous kinetic analyses under extreme conditions: low surface coverage vs high surface coverage, short-time vs long time, mass transfer vs adsorption. Therefore, limited successes are ensured if these empirical models are employed to correlate experimental data owing to the range of data covered. The equilibrium behavior is described by the adsorption isotherm. Adsorption isotherm can be obtained by setting the adsorption rate, Eqs. (8) or (9), to be zero. For example, Langmuir isotherm represents a class of adsorptions where the active centers are identical, i.e. the rate constants in Eqs. (8) or (9) are not functions of concentration or surface coverage. The adsorbed molecules have no effect on the active sites that have not been occupied. In this case, reaction (4) can represent the adsorption process in whole and Langmuir isotherm is thus given by
C SA ¼ C rA ¼ C r hA ¼ C r
K ACA 1 þ K ACA
ð13Þ
where KA is the adsorption isotherm constant,
KA ¼
kA kA
ð14Þ
The Langmuir adsorption isotherm, Eq. (13), is well received in the literature. Most applications approximate adsorptions with this adsorption isotherm. Therefore, Langmuir adsorption is standard or ideal adsorption, deviation from Langmuir adsorption is deemed cooperative adsorption. Cooperative adsorption is adsorption where adsorbed adsorbate has an effect on the adsorption of ‘‘new’’ adsorbate molecules. Therefore, cooperativity is common for adsorptions. There has been great number of studies on the equilibrium behavior of adsorption from thermodynamic point of view, although most empirical and/or thermodynamics based isotherms lacking the dynamic or kinetic connections. Besides Langmuir
adsorption isotherm, Freundlich isotherm has been the most common in addition to the BET [15] multi-layer model. In [1] the identical interactions of active centers on surfaces (of Langmuir) were relaxed. The interactions of adsorbate molecule – adsorbate molecules, or the cooperativity, are introduced through multiple layer adsorptions. A multilayer adsorption model derived in [1] for adsorption of single species on surfaces is applicable to both physisorption and chemisorption,
C SA ¼ C r cA K A C A
1 ½1 þ Nð1 K A C A ÞðK A C A ÞN ð1 K A C A Þ½1 ð1 cA ÞK A C A cA ðK A C A ÞNþ1
ð15Þ
where cA is the ratio of adsorption isotherm constant for the base layer over top layers, KA is the adsorption isotherm constant for the top layers, Cr is the base active center concentration (or the saturation monolayer concentration), and N is the number of layers the surface can accommodate to the adsorbate A. Eq. (15) was shown to be reduced to BET equation for physisorptions with infinite number of layers possible, or N ?1. Compared with single-adsorbate isotherms, multi-component isotherms require a tedious procedure to be obtained experimentally; therefore many attempts were carried out to obtain the latter from single-adsorbate equilibrium data. However, multi-component systems themselves are complicated due to the interaction and competition effects involved. The earliest attempt to model the adsorption isotherm for multicomponent adsorption is by Butler & Ockrent in 1930 [16] who extended the Langmuir isotherm to account for competitive adsorption. Butler & Ockrent assumed a homogeneous surface with respect to the energy of adsorption (generic to Langmuir adsorption), no interaction between adsorbed species and that all adsorption sites are equally available to all adsorbed species. As homogeneity is rarely the case, the simple extension of Langmuir adsorption had limited validity and was improved and modified by Jain and Snoeyink [17] who assumed that a portion of adsorption takes place without competition because not all adsorption sites are available to all species. The non-uniformity (i.e. part of the surface adsorption is competitive while others are not) was attributed to the chemical nature of sites and adsorbates or the relatively large molecular size of the latter. The number of sites with noncompetitive adsorption is given by the difference between the maximum loadings of the species. The model of Jain and Snoeyink [17] was valid for bi-solute systems with wide molecular size difference or chemical properties with respect to the adsorbent. However, for more than two components this model is inapplicable. In general, Langmuir adsorption model and its extensions by Butler & Ockrent [16] and Jain & Snoeyink [17] are too simplified for the complexity of multi-component systems. This led Myers and Prausnitz [18] to adopt the spreading pressure principle and apply it for gaseous mixtures, followed by Radke and Prausnitz [19] who applied it for dilute liquid mixtures. The adsorbed molecules are considered to be in a ‘‘liquid’’ state. They proposed that when multi-adsorbates adsorb simultaneously at the same temperature and spreading pressure as each species would, the adsorbed phase forms an ideal solution. This is called the ideal adsorbed solution theory (IAST). Today, the IAST is regarded as a model with the most thermodynamically accepted foundation. The IAST provides a thermodynamically consistent and practical method for predicting binary and ternary adsorption isotherms by incorporating single-adsorbate isotherm data (or model) only [20–22]. The single-adsorbate adsorption isotherms can be Langmuir, Freundlich, or BET. The IAST model is relatively easy to apply to multispecies adsorption systems, and flexible to adapt to different systems by choosing proper isotherms for the single-adsorbate systems. Like
227
S. Liu / Journal of Colloid and Interface Science 450 (2015) 224–238
the ideal liquid solution theories, the validity is still limited. More complicated liquid solution theories have been applied to extend or modify the IAST model. The non-ideal adsorbed solution theory is thus in order [23–25]. Along the line of adsorbed solution theories, the potential theory of adsorption regard the mixture as a heterogeneous substance segregated in the external field emitted by the adsorbent [26–29]. The same standard equation of state is used for both the segregated phase and the bulk fluid phase. The IAST is based on the assumption that the adsorbed mixture forms an ideal solution at a constant spreading pressure pm. The concentration of adsorbate in the bulk fluid phase is related to the concentration of adsorbate adsorbed on the solid surface much the same as the Raoul’s law (of relating solute concentration in vapor phase to that in liquid solution):
Ci ¼
C 0i ðpm ; TÞC Si PNS j¼1 C Sj
1 6 i 6 Ns
ð16Þ
where C 0i ðpm ; TÞ is equilibrium bulk-fluid phase concentration of pure adsorbate i at the same temperature T and spreading pressure pm of the mixture with NS components. Considering the two-dimensional adsorbed phase as an ideal solution leads to NS X
Ci
0 i¼1 C i
¼
NS X
C Si
j¼1
C 0Si
¼1
ð17Þ
that relates the mixture equilibrium adsorbed phase concentration C 0Si
Z 0
C 0i
C 0Si C 0i
0
dC i
1 6 i 6 NS
rij ¼ kij ðhi1 hi ÞC rj C j kij ðhi hiþ1 ÞC rj /ij
ð18Þ
where as is external surface area per unit mass of adsorbent; and R is the ideal gas constant. Integration of Eq. (18) may be done analytically through the use of various isotherms fitted to single-solute data. Although many isotherm equations are commonly used for modeling adsorption isotherms, however following Henry’s law equation at low coverage is very important when predicting multi-component adsorption systems by the IAST (see Eq. (16)) because of the integration limits. The draw-back of the solution and/or potential theory of adsorption presented in the previous paragraphs is that they completely lacks the one-to-one adsorbate–adsorbent site adsorption illustrated in Fig. 1 and may be applied only to physisorption conceptually. The vacancy solution theory [30–35], however, incorporated the active site concept in the solution theory and thus extended the utility. The essence of these thermodynamic approaches is treating the bulk fluid as a continuum and the adsorbent as another continuum (of point sources). While the solution theories (IAST, nonideal adsorbed solution, potential theory of adsorption and vacancy solution) are thermodynamically sound approaches to the complicated multispecies adsorption systems, they are not directly linked to kinetic analysis. Adsorption rates are important for applications due to the requirement of equipment design/sizing and performance evaluations. Therefore, thermodynamic based approaches or isotherms derived without relevance in kinetics are not convenient for applications. Multispecies adsorption on adsorbent surfaces can be described kinetically by extending the multiple layer adsorption theory of single species adsorption [2]
ð19Þ 1
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Cooperative adsorption on solid surfaces Shijie Liu Department of Paper and Bioprocess Engineering, State University of New York – College of Environmental Science and Forestry, Syracuse, NY 13210, United States
g r a p h i c a l a b s t r a c t
a r t i c l e
i n f o
Article history: Received 9 February 2015 Accepted 6 March 2015 Available online 17 March 2015 Keywords: Adsorption kinetics Adsorption isotherm Cooperativity Competitive adsorption Multilayer adsorption Nonideal adsorption
a b s t r a c t Adsorptions are commonly of monolayer coverage of adsorbate molecules on adsorbent sites, in particular for chemisorptions, where Langmuir adsorption isotherm equation and kinetics are adequate. The Langmuir adsorption is termed ideal adsorption as the surface active centers are uniformly distributed, the molecules are of point-sizes and the interactions between adsorbate molecules and the adsorbent are uniform. However, there are more cases where Langmuir adsorption isotherm and/or kinetics are inadequate in describing the adsorption behavior. Apparent multilayer adsorption has been shown to be descriptive of both physisorptions and chemisorptions as a means of idealization to nonideal adsorptions. The deviation of adsorption isotherm and/or kinetics from (ideal) Langmuir adsorption is due to cooperative adsorption, or interactions between adsorbates or between adsorbate and adsorbent caused deviation from ‘‘uniform’’ interactions. The multi-layer or apparent multilayer behavior of adsorption is an excellent model to describe cooperative adsorption. Adsorption kinetics and isotherms have been derived for adsorptions without differentiating the different types of adsorptions. The simplistic approach can explain majority of the adsorption isotherms and kinetic behaviors. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Adsorption is the phenomenon of solute(s) in a bulk fluid phase attaching onto an interface or surface. Adsorption can be classified as gas adsorption and liquid adsorption based on the initial state of
E-mail address: [email protected] http://dx.doi.org/10.1016/j.jcis.2015.03.013 0021-9797/Ó 2015 Elsevier Inc. All rights reserved.
solute(s). The solute participating in the adsorption process is also called adsorbate and the material that provides the surface is called the adsorbent. The adsorbed molecules are restricted on the interface and thus lost their mobility akin to fluid. While adsorption can occur on liquid–liquid interfaces, adsorption is commonly known to occur on solid surfaces. Adsorption on solid surfaces is widely applied in separation, clarification and purification of matter. In [1] a mathematical model was derived based on
S. Liu / Journal of Colloid and Interface Science 450 (2015) 224–238
multilayer approximation for adsorption of single species, which was extended to multiple species competitive adsorption in [2]. To illustrate the adsorption process for quantitative description, Fig. 1 exemplifies the steps in adsorption [3]. Adsorbate molecule A in the bulk fluid phase diffuses (through the solvent or mixture) to the external surfaces of the solid matrices (1), next diffuses through the internal pores of the solid matrix, reaching the ‘‘active center’’ r on the solid material (2), becoming attached or ‘‘fixed’’ on the solid material (3). The action of a free adsorbate molecule A in the fluid phase becomes ‘‘fixed’’ on the solid surface is known as adsorption. The kinetics of adsorption is thus controlled by both mass transport (of adsorbate to the active center) and the ‘‘attachment’’ (adsorption). Clearly, steps (1) and (2) are mass transfer steps. This implies that mass transfer can be important in adsorption processes. The rates (or volumetric fluxes) of mass transfer are given by
NAE ¼ kc ðC A C AS Þ
ð1Þ
NAI ¼ kp ðC AS C Ap Þ
ð2Þ
where NA’s are volumetric based transfer flux (or rate) of species A from the bulk fluid phase (with a concentration of CA) to the solid external surface (where concentration A in the fluid phase is CAS) and to the ‘‘end’’ of the internal pore (where the concentration of A in the fluid phase is CAp), at which the molecule is fixed or adsorbed on the surface. The pore permeation or transfer is through a net-work of internal pores of the solid matrix. The pores are of distributed length/ depth, and diameter. Thus, pore permeability or internal mass transfer rate constant kp can appear to be a function of time due to the adsorption occurring first on sites where adsorbate is accessed first. In [1], the pore mass transfer or permeation coefficient is given by
C A0 C Ae kp ¼ kp0 C A0 C A
ð3Þ
where kp0 is constant and CAe is the concentration of species A in the fluid phase at the adsorption equilibrium or infinite residence time. One can show that Eq. (3) is consistent with the Weber–Morris model (of intraparticle diffusion limited adsorption) and the exact solution of adsorption to an isolated sphere with negligible external mass transfer by [4] at short-times. Step (3) exemplifies the approaching of adsorbate molecule (A) to the active center in the solid material. Each active center r can attract and attach one adsorbate molecule. The reverse or desorption occurs in the reverse order.
225
It becomes clear that the last step, step (3), is the key step intrinsic to adsorption. Often, when we speak of adsorption kinetics, we mean the kinetics of step (3). It is most convenient to consider the adsorption as chemical reaction or surface reaction. That is,
A þ r¢A r
ð4Þ
As Eq. (4) indicates, the adsorption is reversible. The reverse of adsorption is desorption. It is the nature of reversibility that made adsorption an effective separation technique for liquids and gases. The Gibbs free energy for the adsorption process is given by
DG ¼ DH T DS
ð5Þ
where DH is the enthalpy change due to adsorption of the molecule on the solid surface or heat of adsorption, DS is the entropy change from the bulk phase (more random) to the adsorbed phase (more orderly), T is the temperature, and DG is the Gibbs free energy change due to adsorption. For the adsorption to occur, DG < 0, therefore, DH = DG + TDS < 0 or the adsorption is exothermic. This condition holds because DS < 0 for the adsorption process if the adsorption does not involve the breaking apart of the adsorbate molecule, as opposes to that in dissociative adsorption. The decrease in entropy is because of the change from a disorderly state (in fluid phase) to an orderly state (‘‘fixed’’ on the interface). For dissociative adsorption, the heat of adsorption can be endothermic as the entropy change is positive when a molecule is breaking into two parts. The kinetics of adsorption follows naturally with the collision theory. Consider the rate at which adsorbate A molecules in a homogeneous gas (or liquid) phase will strike a solid surface, the collision frequency is given by
sffiffiffiffiffiffiffiffiffiffiffiffiffi RT Z cT ðA; surfaceÞ ¼ N AV C A 2pM A
ð6Þ
which for gas adsorption is equivalent to
NAV pA ffi Z cT ðA; surfaceÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pMA RT
ð7Þ
Only after the adsorbate molecule strikes the surface, it could produce adsorption. The rate of adsorption is thus a fraction of the collision frequency. Therefore, adsorption rate can be considered either in concentrations for liquid and gas adsorptions or in partial pressures for gas adsorption. Based on Eq. (6), the adsorption as represented by Eq. (4) can be regarded as elementary reaction. The net rate of this adsorption is given by
r ¼ kad C A C rv kdes C rA
ð8Þ
or
A 1
CA
A 1
External Mass transfer CAS
A 2
2
Internal Diffusion
3 σσσσσσσσσσσσσσσσσ σσσσσσσσσσσσσσσσσ σ σ σ σ σ σ σ σ σA| σAσon σ site σσσσσ σσσσσσσσσσσσσσσσσ σσσσσσσσσσσσσσσσσ σ σ σ σ σ Solid σ σ σ/ σadsorbent σ σ σ σ σsurface σσσ σσσσσσσσσσσσσσσσσ
CAp
Fig. 1. Typical steps of adsorption involving one species (A) in the bulk fluid phase and one active site (r) group in the solid material.
r ¼ kA C A hC r kA hA C r
ð9Þ
where kad = kA is the adsorption rate constant, CA is the concentration of the adsorbate in the bulk fluid phase (near the adsorbent surface, which should be CAp in Fig. 1), Crv is the surface concentration of vacant active centers, CrA is the surface concentration of the active centers that are occupied by the adsorbate molecule A, h is the fraction of the vacant active centers, Cr is the surface concentration of the total possible available positions or sites or active centers where A can be adsorbed, hA is the fraction of the total possible available positions that are occupied by A. Clearly,
h¼
C rv Cr
hA ¼
C rA Cr
ð10Þ
ð11Þ
226
S. Liu / Journal of Colloid and Interface Science 450 (2015) 224–238
The overall kinetic behavior is governed by [1]
NAI ¼ NAI ¼ r A
ð12Þ
That is to say, the external mass transfer flux, internal mass transfer flux, and the net adsorption rate are identical. It has shown excellent agreement in modeling the adsorption kinetics with this model [1]. At this point, we expect that the rate of adsorption is important for its practical applications. However, the study of adsorption kinetics has not always been consistent with the fundamental understanding as such summarized above. Often, the study of adsorption kinetics is purely empirical. Commonly applied kinetic models [5] are (1). Lagergren pseudo-first order [6]; (2). the chemisorptions pseudo-second order [7]; (3). Elovich kinetic model [8]; (4). the intraparticle diffusion model [9]; and (5). The liquid film diffusion model [10]. These models have been applied to analyze adsorption data with varied successes; however, none of these models give rise to an acceptable equilibrium behavior or isotherm. There has been an outcry to fix the incompatible adsorption kinetic analyses [11,12], the use of these empirical models that are not compatible with fundamental understanding of adsorption persists. These models have one common feature: they reduced to an algebraic model of adsorbate concentration vs adsorption time and can be linearized such that experimental data can be put on a straight line plot to determine the parameters. It is especially appealing for researchers do not have access to modern computational tools to use linear plots to ascertain the adsorption behavior. In [13] it was shown that the pseudo-first order (Lagergren model) and the pseudo-second order (Elovich kinetic model) can be simplified from the Langmuir kinetics under extreme conditions, which was reinforced by [14]. In fact, all these empirical models can be reduced from more rigorous kinetic analyses under extreme conditions: low surface coverage vs high surface coverage, short-time vs long time, mass transfer vs adsorption. Therefore, limited successes are ensured if these empirical models are employed to correlate experimental data owing to the range of data covered. The equilibrium behavior is described by the adsorption isotherm. Adsorption isotherm can be obtained by setting the adsorption rate, Eqs. (8) or (9), to be zero. For example, Langmuir isotherm represents a class of adsorptions where the active centers are identical, i.e. the rate constants in Eqs. (8) or (9) are not functions of concentration or surface coverage. The adsorbed molecules have no effect on the active sites that have not been occupied. In this case, reaction (4) can represent the adsorption process in whole and Langmuir isotherm is thus given by
C SA ¼ C rA ¼ C r hA ¼ C r
K ACA 1 þ K ACA
ð13Þ
where KA is the adsorption isotherm constant,
KA ¼
kA kA
ð14Þ
The Langmuir adsorption isotherm, Eq. (13), is well received in the literature. Most applications approximate adsorptions with this adsorption isotherm. Therefore, Langmuir adsorption is standard or ideal adsorption, deviation from Langmuir adsorption is deemed cooperative adsorption. Cooperative adsorption is adsorption where adsorbed adsorbate has an effect on the adsorption of ‘‘new’’ adsorbate molecules. Therefore, cooperativity is common for adsorptions. There has been great number of studies on the equilibrium behavior of adsorption from thermodynamic point of view, although most empirical and/or thermodynamics based isotherms lacking the dynamic or kinetic connections. Besides Langmuir
adsorption isotherm, Freundlich isotherm has been the most common in addition to the BET [15] multi-layer model. In [1] the identical interactions of active centers on surfaces (of Langmuir) were relaxed. The interactions of adsorbate molecule – adsorbate molecules, or the cooperativity, are introduced through multiple layer adsorptions. A multilayer adsorption model derived in [1] for adsorption of single species on surfaces is applicable to both physisorption and chemisorption,
C SA ¼ C r cA K A C A
1 ½1 þ Nð1 K A C A ÞðK A C A ÞN ð1 K A C A Þ½1 ð1 cA ÞK A C A cA ðK A C A ÞNþ1
ð15Þ
where cA is the ratio of adsorption isotherm constant for the base layer over top layers, KA is the adsorption isotherm constant for the top layers, Cr is the base active center concentration (or the saturation monolayer concentration), and N is the number of layers the surface can accommodate to the adsorbate A. Eq. (15) was shown to be reduced to BET equation for physisorptions with infinite number of layers possible, or N ?1. Compared with single-adsorbate isotherms, multi-component isotherms require a tedious procedure to be obtained experimentally; therefore many attempts were carried out to obtain the latter from single-adsorbate equilibrium data. However, multi-component systems themselves are complicated due to the interaction and competition effects involved. The earliest attempt to model the adsorption isotherm for multicomponent adsorption is by Butler & Ockrent in 1930 [16] who extended the Langmuir isotherm to account for competitive adsorption. Butler & Ockrent assumed a homogeneous surface with respect to the energy of adsorption (generic to Langmuir adsorption), no interaction between adsorbed species and that all adsorption sites are equally available to all adsorbed species. As homogeneity is rarely the case, the simple extension of Langmuir adsorption had limited validity and was improved and modified by Jain and Snoeyink [17] who assumed that a portion of adsorption takes place without competition because not all adsorption sites are available to all species. The non-uniformity (i.e. part of the surface adsorption is competitive while others are not) was attributed to the chemical nature of sites and adsorbates or the relatively large molecular size of the latter. The number of sites with noncompetitive adsorption is given by the difference between the maximum loadings of the species. The model of Jain and Snoeyink [17] was valid for bi-solute systems with wide molecular size difference or chemical properties with respect to the adsorbent. However, for more than two components this model is inapplicable. In general, Langmuir adsorption model and its extensions by Butler & Ockrent [16] and Jain & Snoeyink [17] are too simplified for the complexity of multi-component systems. This led Myers and Prausnitz [18] to adopt the spreading pressure principle and apply it for gaseous mixtures, followed by Radke and Prausnitz [19] who applied it for dilute liquid mixtures. The adsorbed molecules are considered to be in a ‘‘liquid’’ state. They proposed that when multi-adsorbates adsorb simultaneously at the same temperature and spreading pressure as each species would, the adsorbed phase forms an ideal solution. This is called the ideal adsorbed solution theory (IAST). Today, the IAST is regarded as a model with the most thermodynamically accepted foundation. The IAST provides a thermodynamically consistent and practical method for predicting binary and ternary adsorption isotherms by incorporating single-adsorbate isotherm data (or model) only [20–22]. The single-adsorbate adsorption isotherms can be Langmuir, Freundlich, or BET. The IAST model is relatively easy to apply to multispecies adsorption systems, and flexible to adapt to different systems by choosing proper isotherms for the single-adsorbate systems. Like
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the ideal liquid solution theories, the validity is still limited. More complicated liquid solution theories have been applied to extend or modify the IAST model. The non-ideal adsorbed solution theory is thus in order [23–25]. Along the line of adsorbed solution theories, the potential theory of adsorption regard the mixture as a heterogeneous substance segregated in the external field emitted by the adsorbent [26–29]. The same standard equation of state is used for both the segregated phase and the bulk fluid phase. The IAST is based on the assumption that the adsorbed mixture forms an ideal solution at a constant spreading pressure pm. The concentration of adsorbate in the bulk fluid phase is related to the concentration of adsorbate adsorbed on the solid surface much the same as the Raoul’s law (of relating solute concentration in vapor phase to that in liquid solution):
Ci ¼
C 0i ðpm ; TÞC Si PNS j¼1 C Sj
1 6 i 6 Ns
ð16Þ
where C 0i ðpm ; TÞ is equilibrium bulk-fluid phase concentration of pure adsorbate i at the same temperature T and spreading pressure pm of the mixture with NS components. Considering the two-dimensional adsorbed phase as an ideal solution leads to NS X
Ci
0 i¼1 C i
¼
NS X
C Si
j¼1
C 0Si
¼1
ð17Þ
that relates the mixture equilibrium adsorbed phase concentration C 0Si
Z 0
C 0i
C 0Si C 0i
0
dC i
1 6 i 6 NS
rij ¼ kij ðhi1 hi ÞC rj C j kij ðhi hiþ1 ÞC rj /ij
ð18Þ
where as is external surface area per unit mass of adsorbent; and R is the ideal gas constant. Integration of Eq. (18) may be done analytically through the use of various isotherms fitted to single-solute data. Although many isotherm equations are commonly used for modeling adsorption isotherms, however following Henry’s law equation at low coverage is very important when predicting multi-component adsorption systems by the IAST (see Eq. (16)) because of the integration limits. The draw-back of the solution and/or potential theory of adsorption presented in the previous paragraphs is that they completely lacks the one-to-one adsorbate–adsorbent site adsorption illustrated in Fig. 1 and may be applied only to physisorption conceptually. The vacancy solution theory [30–35], however, incorporated the active site concept in the solution theory and thus extended the utility. The essence of these thermodynamic approaches is treating the bulk fluid as a continuum and the adsorbent as another continuum (of point sources). While the solution theories (IAST, nonideal adsorbed solution, potential theory of adsorption and vacancy solution) are thermodynamically sound approaches to the complicated multispecies adsorption systems, they are not directly linked to kinetic analysis. Adsorption rates are important for applications due to the requirement of equipment design/sizing and performance evaluations. Therefore, thermodynamic based approaches or isotherms derived without relevance in kinetics are not convenient for applications. Multispecies adsorption on adsorbent surfaces can be described kinetically by extending the multiple layer adsorption theory of single species adsorption [2]
ð19Þ 1
