Article pubs.acs.org/JPCB
Diffusion Influenced Adsorption Kinetics Toshiaki Miura and Kazuhiko Seki* National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 5, 1-1-1 Higashi, Tsukuba, Ibaraki 305-8565, Japan S Supporting Information *
ABSTRACT: When the kinetics of adsorption is influenced by the diffusive flow of solutes, the solute concentration at the surface is influenced by the surface coverage of solutes, which is given by the Langmuir−Hinshelwood adsorption equation. The diffusion equation with the boundary condition given by the Langmuir−Hinshelwood adsorption equation leads to the nonlinear integro-differential equation for the surface coverage. In this paper, we solved the nonlinear integrodifferential equation using the Grünwald−Letnikov formula developed to solve fractional kinetics. Guided by the numerical results, analytical expressions for the upper and lower bounds of the exact numerical results were obtained. The upper and lower bounds were close to the exact numerical results in the diffusion- and reaction-controlled limits, respectively. We examined the validity of the two simple analytical expressions obtained in the diffusion-controlled limit. The results were generalized to include the effect of dispersive diffusion. We also investigated the effect of molecular rearrangement of anisotropic molecules on surface coverage.
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exp(−b2√t),13,15−17 where b1 and b2 are constants. The latter expression takes into account the saturation of the surface coverage. However, to confirm that the latter expression is superior to the former, the accuracy of both of these expressions needs to be investigated in the same diffusioncontrolled limit. Under such circumstances, distinction between reaction- and diffusion-controlled limits depends on the choice of analytical expressions in the diffusion-controlled limit. Similarly, the values of the diffusion constants obtained by applying either analytical expression in the diffusion-controlled limit cannot be verified. In this paper, we first numerically solve the coupled equation of the Ward−Tordai equation and the Langmuir−Hinshelwood adsorption equation using the Grünwald−Letnikov formula. The Grünwald−Letnikov formula has been developed for the same kernel as that in the Ward−Tordai equation.18,19 Guided by the numerical solutions, the upper and lower bounds of surface coverage are obtained. These bounds reduce to the known results by taking the reaction- and diffusion-controlled limits. We examine the validity of the two simple analytical expressions obtained in the diffusion-controlled limit. The results are further generalized for the case when the diffusion is dispersive. We also investigate the effect of the molecular rearrangement of anisotropic molecules on the surface.
INTRODUCTION Diffusion-influenced adsorption kinetics has been studied in many fields, including adsorption of polymers, adsorption of surfactants at the air/water interface, and catalytic reactions.1−7 When a fresh surface is introduced into a solution, surface coverage of solutes on the surface develops with time until the equilibrium is attained. To theoretically determine the kinetics, the coupled equations of the surface coverage and the solute concentration need to be self-consistently solved. The available surface area for the solutes depends on the surface coverage, and the increasing rate of surface coverage depends on the solute concentrations. Previously, the self-consistent equation was derived by Ward and Tordai.8 The adsorption kinetics can be described by the Langmuir−Hinshelwood adsorption equation. The Ward− Tordai equation coupled with the Langmuir−Hinshelwood adsorption equation leads to a nonlinear equation for the surface coverage and the solute concentration at the surface. The nonlinear equation can be numerically solved using the trapezoidal rule or other procedures.9−11 Analytical solutions can be obtained only in the reaction-controlled and diffusioncontrolled limits, and these analytical solutions can then be used to determine whether the adsorption processes are diffusion-controlled (diffusion-limited) or reaction-controlled (barrier-limited).12,13 In the reaction-controlled limit, the surface coverage normalized by the maximum coverage at time t is approximated as θ(t) ≈ 1− exp(−At), where A is a constant.1,12 In the diffusion-controlled limit, two different expressions are frequently used: θ(t) ≈ b 1 √t 12,14 and θ(t) ≈ 1 − © XXXX American Chemical Society
Special Issue: Biman Bagchi Festschrift Received: January 20, 2015 Revised: April 14, 2015
A
DOI: 10.1021/acs.jpcb.5b00580 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
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The inverse transformation can be expressed as21−23
ADSORPTION KINETICS We consider the adsorption kinetics of solutes in the aqueous phase by diffusion. The concentration of solutes at distance x from the adsorption plane is assumed to be homogeneous in any plane parallel to the surface of the adsorption sites. The concentration of solutes c(x, t) in the aqueous phase obeys the diffusion equation d d2 c(x , t ) = D 2 c(x , t ) dt dx
=
∫−∞ dk exp(ikx)Ĉ(k , t )
∫0
∞
∫−∞ dk ∫0
∞
(1)
where D denotes the solute diffusion coefficient. The surface coverage of solutes Γ(t) increases with adsorption of solutes, and the adsorption rate is proportional to the concentration of solutes at the surface c(0, t) in the aqueous phase. By taking into account the effect of c(0, t), the Langmuir−Hinshelwood adsorption−desorption equation for the surface coverage of solutes (Γ(t)) can be written as5 d Γ(t ) = kac(0, t )[Γm − Γ(t )] − kd Γ(t ) dt
∞
1 2π 1 = 2π
C(x , t ) =
∞
dx1 exp[ik(x − x1)]C(x1, t )
⎧C(x , t ) for x > 0 ⎪ dx1δ(x − x1)C(x1, t ) = ⎨C(0, t )/2 for x = 0 ⎪ ⎩0 for x < 0 (10)
Using the Fourier−Laplace transformation in space, the solution is9,20,23 C(0, t ) = C0 − = C0 −
∞
1 π
∫−∞ dk ∫0
∫0
t
dt1
t
dt1 exp[− Dk 2(t − t1)]
d θ(t1) dt1
1
d θ(t1) πD(t − t1) dt1
(2)
(11)
where ka, kd, and Γm denote the adsorption rate, desorption rate, and maximum surface coverage, respectively. When the current to the surface equals the rate of change of the surface coverage, the boundary condition consistent with adsorption− desorption is given by the conservation law5,6,20
Equation 11 is equivalent to the Ward−Tordai equation where θ(t) is expressed in terms of C(0, t).6,8 By substituting eq 11 into eq 4 and re-expressing the integrodifferential equation using the Laplace transformation, we obtain a closed equation for θ(t):
d d Γ(t ) = D c(x , t ) dt dx x=0
⎡ d d θ = K a ⎢C 0 − ⎢⎣ dt dt
(3)
The initial condition is c(x, 0) = c0, and the other boundary condition is limx→∞ c(x, t) = c0. It is convenient to introduce variables θ = Γ(t)/Γm, C(x, t) = c(x, t)/Γm, and Ka = kaΓm. Using these variables, eq 2 can be rewritten as d θ(t ) = K aC(0, t )[1 − θ(t )] − kdθ(t ) dt
(12)
DC0 Dc0 t t , τ = K a 2 = ka 2 Γm 2, = 2 D Ka D ka Γm D D κd = 2 k d = 2 2 k d Ka ka Γm
r= (4)
⎡ d d θ = ⎢r − ⎢⎣ dτ dτ
∫0
τ
dτ1
⎤ θ(τ1)⎥(1 − θ ) − κdθ ⎥⎦ π (τ − τ1) (14)
(6)
It should be noted that the integro-differential equation can be numerically evaluated using the Grünwald−Letnikov formula18,19 d dτ
(7)
∫0
τ
dτ1
1 π (τ − τ1)
θ(τ1) =
∫0
[τ / h]
∑ ωjθ(τ − jh) j=0
where [τ/h] denotes the integer part of τ/h and the coefficient ωj can be obtained from
(8)
ω0 = 1,
⎛ 3 ⎞⎟ ωk = ⎜1 − ωk − 1 ⎝ 2k ⎠
(16)
In general, eq 14 with eq 15 can be numerically integrated using standard numerical methods. When κd = 0, the numerical results can be successively obtained from the difference equation
∞
exp( −ikx)C(x , t ) dx
1 h
(15)
ϵ is a small constant, and the limit of ϵ→ 0 will be taken when we calculate C(0, t). The closed form solution can be obtained by introducing the Fourier−Laplace transformation in space, which is defined as C ̂ (k , t ) =
(13)
1
with the perfectly reflecting boundary condition
d C(x , t )|x = 0 = 0 dx
and
Equation 12 can be rewritten in dimensionless form as
(5)
For convenience, we express eqs 5 and 6 equivalently by using a sink term as d d2 d C(x , t ) = D 2 C(x , t ) − θ(t )δ(x − ϵ) dt d t dx
⎤ θ(t1)⎥(1 − θ ) ⎥⎦ πD(t − t1) 1
To systematically solve eq 12, we introduce the dimensionless variables
and the initial condition is given by C0 = c0/Γm. The boundary condition can be expressed as d d θ(t ) = D C(x , t ) dt dx x=0
dt1
− kdθ
The normalized concentration obeys the diffusion equation d d2 C(x , t ) = D 2 C(x , t ) dt dx
∫0
t
(9) B
DOI: 10.1021/acs.jpcb.5b00580 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B θ(τ + h) = 1 − (1 − θ(τ )) ⎡ ⎛ 1 exp⎢ −h⎜⎜r − ⎢ h ⎣ ⎝
[τ / h]
∑ j=0
⎞⎤ ωjθ(τ − jh)⎟⎟⎥ ⎥ ⎠⎦
In the opposite limit of ka2Γm2t/D > 1, the result can be interpreted as diffusion-limited adsorption13,15−17 Γ(t ) = Γm − Γm exp[−2(c0/Γm) Dt /π ] (17)
Compared with eq 21, eq 26 exhibits saturation and solute adsorption by diffusion is hindered by the solutes already covering the surface. Saturation to a maximum surface coverage is approximately taken into account. In Figures 1−4, we show the numerical results calculated using eq 17. In the same figures, we show the approximate
where eq 14 is integrated first and the result is discretized. h is taken to be sufficiently small: h = 0.0005 for r ≥ 0.1 and h = 0.05 for r = 0.01. We checked the numerical accuracy by changing the value of h. Below, we show the results when the initial surface coverage is given by θ(0) = 0. We compare the exact numerical results obtained using eq 17 with approximate results by ignoring dissociation (κd = 0). Except for the last period of surface coverage leading to saturation, the surface coverage can be expressed as d d θ≈r− dτ dτ
∫0
τ
dτ1
1 π (τ − τ1)
(26)
θ(τ1) (18)
The solution in the Laplace domain is θ(̂ s) ≈
r 1 ss+ s
(19)
After the inverse Laplace transformation
Figure 1. Dimensionless surface coverage θ as a function of dimensionless time τ for r = 10. The thick solid line denotes the exact numerical solution of eq 14. The long and short dashed lines represent the approximate results of eqs 24 and 20, respectively. The circles represent the result of the reaction-controlled limit given by eq 25.
⎞ ⎛ c D ⎛ ⎛ k 2 Γ 2t ⎞ t ⎞ t Γ(t ) ≈ 0 ⎜⎜exp⎜ a m ⎟ erfc⎜ka Γm − 1⎟⎟ ⎟ + 2ka Γm ⎝ ka Γm ⎝ ⎝ D ⎠ D⎠ D ⎠ (20)
where erfc(x) denotes the complementary error function 24 2 defined by (2/√π)∫ ∞ In the diffusion-conx exp(−t ) dt. trolled time domain expressed by ka2Γm2t/D > 1, the surface coverage reduces to the conventional form12,14 Γ(t ) = 2c0 Dt /π
(21)
Another approximate solution can be obtained by partly taking into account the effect of saturation given by [1 − θ(t)] in eq 4. Equation 14 with κd = 0 can be formally integrated, and the result can be expressed as ⎡ 1 θ(t ) = 1 − exp⎢ −rt + π ⎣
∫0
t
dτ1
θ(τ1) 1/2
(τ − τ1)
⎤ ⎥ ⎦
(22)
By denoting the Laplace transformation of f(t) as 3(f (t )), we obtain24 ⎛ 1 3⎜⎜ −rt + π ⎝
∫0
t
dτ1
θ(τ1)
⎞ θ(̂ s) r ⎟=− 2 + s s ⎠
1/2 ⎟
(τ − τ1)
Figure 2. Dimensionless surface coverage θ as a function of dimensionless time τ for r = 1. The thick solid line denotes the exact numerical solution of eq 14. The long and short dashed lines represent the approximate results of eqs 24 and 20, respectively. The circles represent the result of the reaction-controlled limit given by eq 25.
(23)
By substituting eq 19 into eq 23 and applying the inverse Laplace transformation, from eq 22, we obtain
results of eqs 20 and 24. Note that eq 20 is the upper bound and eq 24 is the lower bound of the exact numerical results. Equation 20 is obtained from eq 12 by ignoring the suppression factor given by 1 − θ on the right-hand side. Because the suppression factor is ignored, eq 20 is the upper bound. Equation 24 is obtained by taking into account 1 − θ in eq 12 but ignoring 1 − θ in calculating C(0, t). Because the surface coverage is not suppressed by ignoring 1 − θ, the current of solutes toward the surface is increased and the approximate C(0, t) decays faster than the exact numerical results. If the approximate C(0, t) value is less than the true value, increase of the approximate θ value obtained from eq 4 is suppressed.
⎧ ⎪ c0D ⎡ ⎛ ka 2 Γm 2t ⎞ ⎢exp⎜ ⎟ Γ(t ) = Γm − Γm exp⎨ − 2 ⎪ ⎩ ka Γm ⎢⎣ ⎝ D ⎠ ⎤⎫ ⎛ ⎪ t ⎞ t erfc⎜ka Γm − 1⎥⎬ ⎟ + 2ka Γm ⎝ ⎥⎦⎪ D⎠ D ⎭
(24)
This equation covers both reaction- and diffusion-limited adsorption. In the limit of ka2Γm2t/D < 1, reaction-limited adsorption is recovered:1,12 Γ(t ) = Γm − Γm exp( −kac0t )
(25) C
DOI: 10.1021/acs.jpcb.5b00580 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
the intrinsic reaction rate is much higher than the diffusive flow rate to the distance Γm/c0 from the adsorption plane, Dc02/Γm2 = 0.01kac0. In this limit, the solute concentration in the vicinity of the adsorption plane is close to zero until the adsorption surface is almost entirely covered by solutes. The saturation effect can be ignored for a wide range of time. Although the maximum surface coverage is taken into account in eq 26, the exact numerical results follow the simplified approximate solution given by eq 21 rather than that given by eq 26 until the result of eq 21 exceeds 1.
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EFFECT OF DESORPTION So far, we have not taken into account desorption. In the presence of desorption, the surface coverage can be obtained from the recursive equation combining Euler’s method and the Grünwald−Letnikov formula (eq 15):
Figure 3. Dimensionless surface coverage θ as a function of dimensionless time τ for r = 0.1. The thick solid line denotes the exact numerical solution of eq 14. The long and short dashed lines represent the approximate results of eqs 24 and 20, respectively. The dash-dotted line denotes one of the results of the diffusion-controlled limit given by eq 26. The crosses represent another result of the diffusion-controlled limit given by eq 21.
⎛ 1 θ(τ + h) = θ(τ ) + h⎜⎜r − h ⎝
[τ / h]
⎞
j=0
⎠
∑ ωjθ(τ − jh)⎟⎟
× [1 − θ(τ )] − κdhθ(τ )
(27)
In Figure 5, we show diffusion-controlled adsorption under the influence of various values of the desorption rate (κd). h is taken
Figure 4. Dimensionless surface coverage θ as a function of dimensionless time τ for r = 0.01. The thick solid line denotes the exact numerical solution of eq 14. The long and short dashed red lines represent the approximate results of eqs 24 and 20, respectively. The dash-dotted line denotes one of the results of the diffusion-controlled limit given by eq 26. The crosses represent another result of the diffusion-controlled limit given by eq 21.
Therefore, θ obtained from eq 12 using the approximate C(0, t) value is the lower bound. When the adsorption is reaction-controlled (r ≥ 1), the lower bound solution given by eq 24 is a better approximation than the upper bound solution given by eq 20. When the adsorption is diffusion-controlled (r ≪ 1), the upper bound solution given by eq 20 approximates the exact results better than the lower bound solution given by eq 24 before the surface coverage saturates. When the adsorption is strongly reaction-controlled (r ≫ 1), the simplified expression given by eq 25 is better than that given by eq 26. Equation 25 is valid for reaction-controlled adsorption and considerably overestimates the exact numerical results when r = 1. When 1 > r ≥ 0.1, both eq 21 and eq 26 are valid only at short times. In Figure 4, we show the results in the diffusion-controlled limit. The result of eq 21 overlaps with that of eq 20, and the result of eq 26 overlaps with that of eq 24. When such overlap takes place, the result of eq 21 is close to the exact result. Although the saturation effect is partly taken into account in eqs 24 and 26, eqs 20 and 21 are better approximations to the exact numerical results than eqs 24 and 26 in the diffusioncontrolled limit expressed by r ≤ 0.01. r = 0.01 indicates that
Figure 5. Dimensionless surface coverage θ as a function of dimensionless time τ for r = 0.01. The thick solid lines denote the exact numerical solution of eq 14. The thin solid line denotes the approximate result without taking into account κd given by eq 20. The dashed lines represent the approximate results of eq 28.
to be sufficiently small so that the results become insensitive to the value of h. The results of Figure 5 were obtained by setting h = 0.0005 for κd = 10, h = 0.005 for κd = 1, and h = 0.01 for κd = 0.1. An analytical approximate result can be obtained as20,25 r θ(τ ) ≈ [1 − exp(κd 2τ ) erfc(κd τ )] κd (28) by the inverse Laplace transformation of r 1 θ(̂ s) ≈ s s + κd
(29)
The asymptotic time dependence is obtained from eq 28 as θ (τ ) ≈ D
r ⎛ 1 ⎞ ⎜1 − ⎟ κd ⎝ κd τ ⎠
(30) DOI: 10.1021/acs.jpcb.5b00580 J. Phys. Chem. B XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry B where the asymptotic expression given by erfc(z) ≈ exp(−z2)/z is introduced.24 The asymptotic time dependence is different from that given by eq 26 obtained for irreversible adsorption. As shown in Figure 5, eq 28 reproduces the asymptotic time regime of numerical results for κd = 0.1−10. The initial growth is approximately given by eq 20. By applying the short time expansion to eq 28, eq 21 can be obtained.20 Equation 21 is derived from eq 20 under the condition of τ = ka2Γm2t/D > 1. As shown in Figure 5, the initial growth given by eq 20 smoothly connects to the asymptotic growth given by eq 28 at τ > 1 when κd < 1. The adsorption and desorption rates have been studied experimentally by changing the bulk concentrations of adsorbing species.26−29 In principle, the adsorption processes change from diffusion-controlled to reaction-controlled by increasing the bulk concentration. The change of the controlling processes can be characterized by the dimensionless quantity given by r = Dc0/(kaΓm2). When r ≪ 1, the adsorption mechanism is diffusion-controlled for most of the time of the coverage growth except the initial period given by t < D/ (ka2Γm2). When r ≫ 1, the adsorption mechanism is reactioncontrolled except for the final period of saturation. When r ∼ 1, the controlling processes are mixed. The estimated values of the adsorption rates of polymers (n-alkanethiols, thiophene) on the gold substrates were around 1 cm3/(mol s), and those of desorption rates were equal to or smaller than 10−7 s−1.26,27 The maximum packing concentrations of monolayers (Γm) and the diffusion constants (D) were 10−9−10−10 mol/cm2 and 10−5−10−6 cm2/s, respectively.26,27 The concentration of nalkanethiols was in the range between 0.5 and 20 μmol/L, and that of thiophene was in the range between 0.1 and 10 mmol/ L.26,27 By using these values, we obtain r ≥ 500 and the results are consistent with the conclusion that the adsorption mechanism is reaction-controlled for the concentration range studied by the experiments. Contrary to the above results for polymer adsorption on the gold substrates, the higher adsorption rates of 107 cm3/(mol s) were reported for the adsorption of alkyl-polyethoxylate (C12Ej) surfactants to the air/aqueous surface.28,29 The concentration was smaller than 10−8 mol/cm3. The desorption rates vary (1−102) × 10−4 s−1.28,29 The values of Γm and those of D were in the same order of those of polymers. By using these values, we obtain r ≤ 10−3 and the results indicate that the adsorption mechanism is diffusion-controlled for the concentration range studied by the experiments. More precisely, when τ = (t/D)ka2Γm2 > 1 (t > 0.01 s), the coverage increases by a diffusion-controlled mechanism until it saturates. For the adsorption of surfactants, the diffusion-controlled growth asymptotically approaches to the maximum surface coverage given by the adsorption− desorption equilibrium. The asymptotic time dependence can be given by eq 30.
the surface.30 As the surface coverage of lying molecules increases, rearrangement of adsorbed molecules occurs and the rod-like molecules take a perpendicular orientation with respect to the surface.30 The rearrangement is induced by interaction between rod-like molecules because the long axes are aligned parallel to each other under crowding.30 As the rearrangement from the lying to the standing configuration proceeds, the available uncovered surface increases and further adsorption is possible. The adsorption processes are schematically shown in Figure 6.
Figure 6. Schematic diagram of the adsorption of rod-like molecules. The molecules initially adsorb with their long axis parallel to the surface. As the surface coverage of lying molecules increases, rearrangement of adsorbed molecules occurs and the rod-like molecules take a perpendicular orientation with respect to the surface.
We define the surface coverage of lying molecules as θl(τ) and that of standing molecules as θs(τ). The total surface coverage can be expressed as θ(τ) = θl(τ) + θs(τ). The area covered by a single rod-like molecule decreases with rearrangement from the lying to the standing configuration, and the ratio is denoted as rls. We will not consider the desorption of molecules from the surface and focus on the effect of molecular rearrangement on the adsorption kinetics. The boundary condition is still given by eq 6. The adsorption of rod-like molecules is influenced by the free unoccupied area given by 1 − θl(τ) − rlsθs(τ), and the integro-differential equation (eq 14) changes to ⎡ d d θ l(τ ) = ⎢r − dτ dτ ⎣⎢
∫0
τ
dτ1
⎤ θ(τ1)⎥ ⎥⎦ π (τ − τ1) 1
[1 − θ l(τ ) − rlsθs(τ )] − κ2θ l(τ )2
(31)
where the last term represents the rearrangement of molecules from the lying to the standing configuration because of the interaction between molecules aligned parallel to each other. The second order rate constant associated with the rearrangement due to interaction between aligned molecules is denoted by κ2 in the dimensionless representation. The increase of standing molecules can be expressed as d θs(τ ) = κ2θ l(τ )2 dτ
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(32)
when ignoring direct adsorption with a perpendicular configuration. The ratio of surface area covered by a standing molecule to that of a lying molecule (rls) is small enough to ignore direct adsorption of molecules with a perpendicular configuration. The total surface coverage can be obtained by summing eqs 31 and 32:
EFFECT OF REARRANGEMENT OF ANISOTROPIC MOLECULES The adsorption of rod-like molecules can exhibit complex kinetics related to rearrangement of the molecules at the surface.30 In this manuscript, when the long axis of a molecule is parallel to the surface, the molecular configuration will be referred to as “lying”. When the long axis of a molecule is perpendicular to the surface, the molecular configuration will be referred to as “standing”. The rod-like molecules adsorb on the surface with their long axis parallel to the surface; that is, the molecules initially lie on
⎡ d d θ (τ ) = ⎢r − ⎢⎣ dτ dτ
∫0
τ
dτ1
⎤ θ(τ1)⎥ ⎥⎦ π (τ − τ1) 1
× [1 − θ l(τ ) − rlsθs(τ )] E
(33)
DOI: 10.1021/acs.jpcb.5b00580 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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d d2 d C(x , t ) = 0 Dt1 − α Dα 2 C(x , t ) − θ(t )δ(x − ϵ) dt dt dx
The time evolution of the total surface coverage reduces to eq 14 by setting rls = 1.0. The increase of the total surface coverage is approximately given by eq 20 until the surface is saturated. In this section, we investigate the effect of surface rearrangement of rod-like molecules on the adsorption kinetics by changing the value of the rate of molecular rearrangement (κ2). The numerical results were calculated by discretizing eqs 31 and 32 using Euler’s method and the Grünwald−Letnikov formula, as shown in eq 27. The results are shown in Figure 7
(35)
where the perfectly reflecting boundary condition expressed by eq 8 is imposed as before. Using eq 35, eq 11 is generalized to C(0, t ) = C0 −
1 Dtα /2θ(t ) Dα 0
(36)
Equation 36 is eq 11 for the case of dispersive diffusion. θ(t) can be easily expressed in terms of C(0, t), and the result can be interpreted as the generalization of the Ward−Tordai equation for dispersive diffusion.8 As shown previously,32−34 the diffusive flow at the boundary interferes with adsorption when the diffusion is dispersive characterized by the memory kernel in eq 34. Because the adsorption interferes with the long-tailed diffusive flow expressed by the memory kernel, eq 14 is hardly generalized if (1 − θ) is included. The details on the difficulty are explained in the Supporting Information. As long as the surface coverage is small satisfying θ < 1/2, eq 12 is generalized to (see the Supporting Information for the derivation)32−34 d θ = 0 Dt1 − α KαC(0, t ) − kdθ dt
(37)
α 32
where Kα has units of 1/time . By introducing dimensionless quantities given by
Figure 7. Dimensionless surface coverage θ as a function of dimensionless time τ for r = 0.1. The thick solid lines denote θl(τ), and the thick dashed lines denote θs(τ). The thin black solid line denotes θ(τ) for κ2 = 0.1. The lines of θ(τ) for κ2 = 1 and 10 are almost the same as that for κ2 = 0.1. The black, blue, and red lines represent κ2 = 0.1, 1, and 10, respectively.
r=
D1 − α c(x , t ) = 0 t
∂ ∂t
∫0
t
dt1
Dα1/ α
t
and
d θ = 0 Dτ1 − α (r ) − 0 Dτ1 − α /2(θ ) − κdθ dτ 1−α α/2 0Dτ 0Dτ
(39) 1−α/2 . 0Dτ
where we have used the relation = This relation can be proved in the Laplace domain using 3[0 Dτα f (t )] = s αf ̂ (s), where 3[···] denotes the Laplace transformation. The solution of eq 39 is expressed as r θ(̂ s) = α s [s + s1 − (α /2) + κd] (40)
EFFECT OF DISPERSIVE DIFFUSION In this section, we generalize the above approach to the case when the diffusion is dispersive. When multiple time scales are involved in the diffusive motion of solutes, the mean square displacement of solutes increases sublinearly with time, ⟨x2(t)⟩ ∼ tα with α < 1, and the motion of solutes is called subdiffusive. The concentration of solutes undergoing subdiffusion can be described by the fractional-diffusion equation
where Dα is the fractional diffusion coefficient and Riemann−Liouville operator18,19
C0 , τ =
⎛ D ⎞1/ α κd = ⎜ α2 ⎟ kd ⎝ Kα ⎠
the time evolution equation for the surface coverage is approximately given by
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in the Laplace domain. At short times when the desorption can be ignored, eq 40 can be approximated as r θ(̂ s) ≈ α s [s + s1 − (α /2)] (41) The time evolution of θ in the reaction-controlled limit and that in the diffusion-controlled limit are expressed as
(34) 1−α 0Dt
Kα(2/ α) − 1
Kα2/ α
(38)
using a sufficiently small value of h (h = 0.01). The surface coverage of lying molecules increases with increasing time and reaches a maximum. After the maximum surface coverage of lying molecules is reached, the surface coverage of lying molecules decreases with time. The maximum value and the time to reach the maximum decrease with increasing κ2. The surface coverage of standing molecules gradually increases with increasing time, and the rate increases by increasing κ2. A similar time evolution of θl(τ) and θs(τ) was recently obtained by molecular dynamics simulations of polymer molecules.31
∂ c(x , t ) = 0 Dt1 − α Dα ∇2 c(x , t ) ∂t
Dα1/ α
θ (τ ) ≈ r
τα Γ(1 + α)
θ (τ ) ≈ r
τ α /2 Γ(1 + (α /2))
is the
1 1 c(x , t1) Γ(α) (t − t1)1 − α
for
τ1
(43)
respectively. The surface coverage increases with time by a power law (∼τα/2) in the diffusion-controlled limit. At long times when the surface coverage is limited by desorption, eq 40 can be approximated as
where Γ(z) is the gamma function.24 Equation 7 is generalized to F
DOI: 10.1021/acs.jpcb.5b00580 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B θ(̂ s) ≈
r α 1 − (α /2)
s [s
+ κd ]
In this manuscript, we show that the surface coverage is largely affected by the dispersive character of bulk diffusion and the time dependence of the desorption rates.
(44)
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After the inverse Laplace transformation, we obtain θ(τ ) ≈ rτ α /2E1 − (α /2),1 + (α /2)( −κdτ1 − (α /2))
CONCLUSION We solved the nonlinear integro-differential equation describing diffusion-influenced adsorption using the Grünwald−Letnikov formula. The numerical procedure is fast and simple. The stability of the numerical method can be systematically analyzed.19 Guided by the numerical results, the upper and lower analytical bounds of the exact numerical results were obtained for the fundamental adsorption−desorption processes described by the time-dependent Langmuir−Hinshelwood equation coupled to the diffusion equation. The numerical method can be applied when the simple Langmuir−Hinshelwood equation is replaced by that for the more complicated model such as the Kisliuk model or its modified model,26,37−39 as long as the rate of the coverage change is a linear function of the adsorbate concentration. We considered the case that the simple Langmuir−Hinshelwood equation was replaced by that describing the rearrangements of anisotropic molecules. When the time-dependent Langmuir−Hinshelwood equation is coupled to the diffusion equation, the upper and lower bounds of the exact numerical results were obtained. The upper bound solution (eq 20) is close to the numerical results in the diffusion-controlled limit (r ≪ 1) until the saturation occurs, while the lower bound solution (eq 24) is close to the numerical results in the reaction-controlled limit (r ≥ 1) including the saturation period. We show that the conventional expressions given by eqs 21 and 26 are valid only at short times when 1 > r ≥ 0.1. When r ≪ 0.1, eq 21 is a better approximation than eq 26. In short, θ(t) ≈ b1√t is better than θ(t) ≈ 1− exp(−b2√t) in the limit of extremely diffusioncontrolled until the coverage saturates. We share the conclusion of ref 16 that the latter equation is appropriate only when the surface coverage is low in the diffusion-controlled limit. When desorption occurs, the more precise equation given by eq 20 reproduces the initial growth obtained by numerical calculations. In the presence of desorption, the asymptotic time dependence is given by 1/√t which is different from that given by θ(t) ≈ 1 − exp(−b2√t) . It should be mentioned that eq 24 could be appropriate when the surface coverage process is neither reaction-controlled nor diffusion-controlled under irreversible adsorption. The results were generalized to include the effect of dispersive diffusion. We also investigated the effect of molecular rearrangement of rod-like molecules on the surface. The kinetics of the surface coverage of lying rod-like molecules exhibits a maximum as a function of time. The maximum value and the time to reach the maximum decrease with increasing rate of molecular rearrangement.
(45)
where Ea,b(z) denotes the generalized Mittag−Leffler function given by18 ∞
Ea , b(z) =
∑ j=0
zj Γ(aj + b)
and we used the identity
(46)
18
⎡ sa−b ⎤ ⎥ = τ b − 1Ea , b( −dτ a) 3−1⎢ a ⎣s + d⎦
(47)
̂ where 3−1[ f ̂ (s)] denotes the inverse Laplace transform of f(s). We can rewrite eq 45 by noticing sa−b 1 s 2a − b 1 ⎛ a−b s 2a − b ⎞ s ⎟ ⎜ = = − sa + d sa sa + d d⎝ sa + d ⎠
(48)
and using the inverse-Laplace transform of eq 48 given by ⎤ 1 ⎡ t b−a−1 ⎢ − t b − a − 1Ea , b − a( −dt a)⎥ d ⎣ Γ(b − a) ⎦
(49)
Equation 45 can be expressed as θ (τ ) =
=
⎤ r ⎡ 1 − E1 − (α /2), α( −κdτ1 − (α /2))⎥ 1−α ⎢ ⎦ κdτ ⎣ Γ(α) ⎤ r ⎡⎢ 1 1 1 ⎥ − 1−α Γ((3α /2) − 1) κdτ1 − (α /2) ⎥⎦ κdτ ⎢⎣ Γ(α) (50)
where we used the asymptotic expression of Ea,b(z) given by18 Ea , b(z) ∼ −
1 1 Γ(b − a) z
for
z→∞
(51)
The asymptotic time dependence has a long-time tail given by θ ∼ 1/τ1−α. The slow time dependence originates from the interference between the adsorption and the dispersive diffusion. The asymptotic time dependence of the coverage changes from θ ∼ 1/τ1−α to θ ∼ 1/τ1/2 when the diffusion is normal expressed by α = 1. When α < 1, the surface coverage slowly decays to zero, which is consistent with the steady state solution of eq 37. The situation changes when the desorption is also dispersive. The rates of desorption can be distributed if the activation energy for the desorption is distributed. By assuming the exponential distribution for the activation energy E given by exp[−E/(kBT0)], the desorption time density with a long-time tail given by ∼1/tα′+1 is obtained, where α′ = T/T0 and T is the temperature.35 In the Laplace domain, the desorption rate is proportional to s1−α′. The surface coverage θ (τ) approaches a finite value and the asymptotic time dependence is given by ∼1/τα/2 when α = α′ (see the Supporting Information for details). Recently, it has been pointed out that the dispersive diffusion changes the kinetics of rebinding to a surface.36 Unbinding time measurements and first passage measurements were suggested.
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ASSOCIATED CONTENT
S Supporting Information *
Figure showing a schematic picture of one-dimensional continuous time random walks with the adsorption surface, derivation of eq 37 and modification of eqs 37 and 50 when desorption rates are distributed. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b00580. G
DOI: 10.1021/acs.jpcb.5b00580 J. Phys. Chem. B XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry B
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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DOI: 10.1021/acs.jpcb.5b00580 J. Phys. Chem. B XXXX, XXX, XXX−XXX
