# Multilayer Adsorption of Heptane Vapor at Water Drop Surfaces

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

^{−1}to 40 mN m

^{−1}, and less after a certain adsorption time. The observed adsorption kinetics is analyzed by a theoretical model based on multilayer adsorption of alkanes from the vapor phase at the water surface. The model assumes a dependence of the kinetic coefficients of adsorption and desorption on the surface coverage and in equilibrium it reduces to the classical Brunauer–Emmett–Teller adsorption isotherm. The calculated time dependencies of adsorption and surface tension agree well with experimental data and predict a five-layer adsorption of heptane.

## 1. Introduction

_{14}EO

_{8}solutions at the interface to air saturated by pentane, hexane, heptane and toluene, respectively, and for drops of water at these interfaces were discussed [17]. The experimental results were well described by a model which implies a diffusion processes for the C

_{14}EO

_{8}molecules and the existence of a kinetic barrier for the alkane molecules. The desorption process was shown to be slow for heptane and hexane, however, for pentane vapor the desorption was quite rapid, and after the desorption commences the surface tension became equal to that at the interface with pure air.

^{−1}to 30–40 mN m

^{−1}) occurred. The time at which this sharp decrease started became lower with increasing temperature: 10,000–15,000 s at 20 °C, and 500–700 s at 50 °C. This phenomenon was attributed to the formation of heptane adsorption layers with significant thicknesses. The sharp surface tension decrease was observed with errors in fitting the drop profile coordinates being below 5 µm. Therefore, this phenomenon of a sharp surface tension decrease caused by the formation of polylayers does really exist.

## 2. Theory

_{j}is the number of alkane molecules within the layer, S is the total surface area, ω

_{j}is the molar area of alkane molecule in the j-th layer, and β

_{j}and α

_{j}are the kinetic coefficients for the adsorption and desorption process, respectively, in the first and subsequent layers. The formalism involved in Equations (1) and (2) is illustrated schematically by Figure 1. In Equation (2), the adsorption terms are proportional to the factors $({\mathrm{\Gamma}}_{\mathrm{j}-1}{\mathsf{\omega}}_{\mathrm{j}-1}-{\mathrm{\Gamma}}_{\mathrm{j}}{\mathsf{\omega}}_{\mathrm{j}})$, because the adsorption in the j-th layer is possible on the part of the surface which is already covered by the previous layer but is not covered yet by the considered layer, while, in Equation (1), the term $(1-{\mathrm{\Gamma}}_{1}{\mathsf{\omega}}_{1})$ corresponds to the adsorption within the first layer. Similarly, the desorption terms are proportional to the factors $({\mathrm{\Gamma}}_{\mathrm{j}}{\mathsf{\omega}}_{\mathrm{j}}-{\mathrm{\Gamma}}_{\mathrm{j}+1}{\mathsf{\omega}}_{\mathrm{j}+1})$, which means the desorption from the j-th layer can occur on the part of the surface already covered by this layer but not blocked yet from the top by the subsequent layer, because the molecules from the subsequent layer prevent desorption from the previous layer. The kinetic equation for the j-th layer, Equation (2), reduces to the standard Langmuir kinetics equation in the particular case when the molecules of the previous layer cover the whole surface $({\mathrm{\Gamma}}_{\mathrm{j}-1}{\mathsf{\omega}}_{\mathrm{j}-1}=1)$ and the subsequent layer has not begun to form yet $({\mathrm{\Gamma}}_{\mathrm{j}+1}{\mathsf{\omega}}_{\mathrm{j}+1}=0)$. Note that, in the formulation above, it is assumed that the molar area of the adsorbed alkane molecule can be different in different layers; this will allow involving the intrinsic compressibility of the adsorbed molecules, as shown below.

_{j}= β and α

_{j}= α for j ≥ 2; also the molar areas in all layers are assumed to be the same and equal to ω. In this particular case the equilibrium adsorptions in the layers can be obtained as:

_{j}and α

_{j}in Equations (1) and (2) depend on the surface coverage:

_{a,j}and φ

_{d,j}, and the parameters ${\mathsf{\beta}}_{\mathrm{j},0}$ and ${\mathsf{\alpha}}_{\mathrm{j},0}$ for the second to the subsequent layers do not change, thus reducing the number of model parameters to eight: ${\mathsf{\beta}}_{1,0}$, ${\mathsf{\beta}}_{2,0}$, ${\mathsf{\alpha}}_{1,0}$, ${\mathsf{\alpha}}_{2,0}$, ${\mathsf{\varphi}}_{\mathrm{a},1}$, ${\mathsf{\varphi}}_{\mathrm{d},1}$, ${\mathsf{\varphi}}_{\mathrm{a},2}$ and ${\mathsf{\varphi}}_{\mathrm{d},2}$, where the subscript 2 refers to the second and further layers. In this case, when the interaction between alkane molecules within the layer is assumed to exist, the equilibrium state cannot be described by the simple Equations (3)–(5), but obeys a more complicated set of transcendent equations which would have to be solved numerically.

_{j}and α

_{j}as functions of θ

_{j}are given by Equation (6).

_{j}in Equations (7)–(9) to depend on the surface pressure per j-th layer Π

_{j}:

_{0}is the molar area at the very initial stage of the surface coverage, and, similar to [17,18]:

_{0}− Π

_{0}is the surface tension of the pure solvent (water). Equations (11) and (12) assume a quasi-equilibrium state of the adsorption film, which is determined by the set of surface coverages varying slowly with time: θ

_{j}= θ

_{j}(t) according to Equations (6)–(9). Thus, to describe the multilayer adsorption process, we have to solve numerically the set of ordinary differential equations (Equations (7)–(9)), taking into account the additional equations (Equations (6), and (10)–(13)). The calculation algorithm used for the solution is described in some detail in the Appendix A.

## 3. Results and Discussion

^{−1}(first characteristic point shown for 20 °C data at 1400 s by arrow). Subsequently, the surface tension remains almost constant until the second characteristic point at 15,000 s, followed by a sharp decrease.

^{−1}. It is seen in Figure 2a,b that the curves calculated using the proposed model agree reasonably well with the experiments.

_{1,0}(first layer) is larger than β

_{2,0}(second and subsequent layers). With increasing temperature, this difference becomes less pronounced: from two orders of magnitude at 20 °C to one order of magnitude at 50 °C. At the same time, the parameter β

_{1,0}becomes lower, while the product β

_{1,0}× c remains approximately constant, i.e., the heptane adsorption rate in the first layer is independent of temperature. On the contrary, for the second and further layers, both the β

_{2,0}value and the β

_{2,0}× c product in general increase with temperature: the product value becomes higher at 50 °C by one order of magnitude as compared with its value at 20 °C. In addition, d

_{2}is essentially higher than d

_{1}(as shown in Table 1 after and before the slash (“/”), respectively). It should be noted that the number of layers assumed in the calculations influences not only the adsorbed amount, but also the surface tension. In particular, at 30 °C, for a five-layer adsorption, it takes 28,000 s to decrease the surface tension value down to 30 mN m

^{−1}, while, for a two-layer adsorption, the required time is as high as 40,000 s.

^{−3}for the density of liquid heptane. Again, the agreement between the experimental data [6] and the proposed theory is acceptable.

_{2}values. We attribute this to the mutual influence of adjacent layers and, in particular, to a decreased desorption rate due to the screening of lower layers by subsequent layers.

^{−1}yield different total adsorption values for the same time of 80,000 s and systems in which different numbers of layers L are assumed: 0.33 × 10

^{−5}, 0.7 × 10

^{−5}, 1.1 × 10

^{−5}, 1.65 × 10

^{−5}and 2.16 × 10

^{−5}mol m

^{−2}for L = 1, 2, 3, 4, and 5, respectively. That is, the increasing number of adsorbed layers in the model system results in an increase of the adsorbed amount in each layer; in particular, for the five-layer model, the adsorbed amount per each layer is 0.44 × 10

^{−5}mol m

^{−2}. This phenomenon is due to the compressibility effect, i.e., the decrease of the heptane molar area with the increase in surface pressure, cf. Equation (10). Note, however, that a decrease in molar area is accompanied by a respective increase in the thickness of the adsorbed layer, so that the volume per molecule in the layer does not change significantly [22]. For incompressible adsorbed layers (ε = 0) and L = 5 the calculations yield the total adsorption of 1.65 × 10

^{−5}mol m

^{−2}, that is, 0.33 × 10

^{−5}mol m

^{−2}per each adsorbed layer.

_{2}value twice as high as ${\mathrm{b}}_{2}={\mathsf{\beta}}_{2,0}/{\mathsf{\alpha}}_{2,0}$ with ${\mathsf{\beta}}_{2,0}$ and ${\mathsf{\alpha}}_{2,0}$ listed in Table 1. Curve 2 was calculated for d

_{2}= 0 (instead of d

_{2}= 1.4 as given in Table 1). Both the increase of b

_{2}and the decrease of the adsorption and desorption activation energies (related to d

_{2}) quite expectedly result in an increased adsorption rate. Curve 4 corresponds to a

_{2}= 0 (instead of a

_{2}= 1.0 as given in Table 1): this corresponds to the increase of the adsorption activation energy and results in a decreased adsorption rate (note that the decrease of ${\mathrm{a}}_{\mathrm{i}}=({\mathsf{\varphi}}_{\mathrm{d},\mathrm{i}}-{\mathsf{\varphi}}_{\mathrm{a},\mathrm{i}})/2$ corresponds to an increase of ${\mathsf{\varphi}}_{\mathrm{a},\mathrm{i}}$, i.e., to the increase of the adsorption activation energy at the same values of other parameters). Curve 5, which was calculated for systems with only two layers instead of five, shows higher surface tension, and not only a slower kinetics, but also very low adsorption value, as was already discussed above.

_{2}was twice increased as compared to that in Table 1. It is essential that the increase of b

_{2}value not only leads to an acceleration and increase of adsorption in the second and further layers, but also enhances the adsorption in the first layer. This phenomenon, as was already mentioned above, is due to the hampering of the desorption from the first layer because of the enhanced formation of the second (and further) layers.

## 4. Conclusions

^{−1}down to 30–40 mN m

^{−1}) at sufficiently long adsorption times (10,000–15,000 s at 20 °C, and 500–700 s at 50 °C). This phenomenon indicates the formation of heptane adsorption layers with significant thicknesses, much thicker than a monolayer. Such conclusion is confirmed by direct X-ray and neutron reflection measurements of the adsorption film thicknesses of alkanes, according to which the maximum thickness can amount up to 4–5 nm [6].

## Acknowledgments

## Author contributions

## Conflicts of Interest

## Appendix A

_{1}, T

_{2}) is:

_{j}values at both ends of the small interval δT = T

_{2}− T

_{1}, and the integral in the rhs through the integrand values at these ends, one obtains:

_{j}and ${\mathsf{\theta}}_{\mathrm{j}}$ at the temporal step T

_{1}one should solve simultaneously L Equations (A2)–(A4) expressed as determined by Equation (A9), and L Equations (A5) and (A6), to obtain the solution at T

_{2}.

_{a,1}= φ

_{d,1}= 0) and the dependence of molar area on surface pressure (ε = 0) one reduces the set to a single equation:

^{−2}] is calculated via the coverages of all L layers:

_{1}equal to the value obtained from Equation (A18); θ

_{j}= d

_{init}× θ

_{j−1}(j = 2, …, L); p

_{j}= θ

_{j}, (j = 1, …, L). At subsequent nodes the linear extrapolation through the previous two nodes was used to obtain the initial guess. Other controlling parameters of the calculations, Equation (A22), were the initial temporal step dT and τ. The convergence of the procedure was verified by: (i) the comparison of the numerical solution values for L = 1, φ

_{a,1}= φ

_{d,1}= 0, ε = 0, with those calculated by the analytical solution of Equation (A18); and (ii) the comparison of the data obtained by the numerical procedure for the values of controlling parameters differing by a factor of 10. It was found that throughout the relevant domain of calculation parameters the solutions obtained with dT = 0.0001 and τ = 0.002 were consistent to within the 6-th decimal place, and the value d

_{init}= 0.1 ensured the convergence of the initial approximation at the first temporal node.

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**Figure 1.**Schematic of adsorption in a multilayer system. The horizontal bars correspond to the fraction of total surface area covered by molecules in the j-th layer, while the arrows correspond to the adsorption onto and the desorption from a layer. Note that the total surface coverage corresponds to θ = 1.

**Figure 2.**Dynamic surface tension at water drops in air saturated by heptane vapor at different temperatures: 20 and 30 °C (

**a**); and 40 and 50 °C (

**b**). Points, experimental data; curves, theoretical calculations assuming five layers, and values of parameters listed in Table 1. Arrows indicate the positions of characteristic points. For detailed explanation, see text.

**Figure 3.**The dynamics of heptane adsorption on water drop surface at various temperatures: curves, theoretical calculations based on the proposed model; points, experimental data obtained in [6]. For detailed explanation, see text.

**Figure 6.**Influence of certain model parameters on the dynamic surface tension. For detailed discussion, see text.

**Figure 8.**Dynamics of the coverage of Layers 1, 2 and 4 at 30 °C for two different b

_{2}values. For detailed discussion, see text.

**Table 1.**Model parameters of heptane adsorption at different temperatures. Note, instead of four model parameters φ

_{a,i}and φ

_{d,i}, (i = 1, 2), the parameters ${\mathrm{a}}_{\mathrm{i}}=({\mathsf{\varphi}}_{\mathrm{d},\mathrm{i}}-{\mathsf{\varphi}}_{\mathrm{a},\mathrm{i}})/2$ and ${\mathrm{d}}_{\mathrm{i}}=({\mathsf{\varphi}}_{\mathrm{d},\mathrm{i}}+{\mathsf{\varphi}}_{\mathrm{a},\mathrm{i}})/2$ are used in the calculations.

Temperature, °C | с, mol m ^{−3} | ω, 10 ^{5} m^{2} mol^{−1} | β_{1,0}, 10 ^{−9} m s^{−1} | β_{2,0}, 10 ^{−10} m s^{−1} | α_{1,0}, 10 ^{−9} mol m^{−2} s^{−1} | α_{2,0}, 10 ^{−10} mol m^{−2} s^{−1} | a_{1}/a_{2} | d_{1}/d_{2} | ε, m mN ^{−1} |
---|---|---|---|---|---|---|---|---|---|

20 | 1.9 | 3.0 | 7.3 | 0.63 | 4.5 | 1.5 | 0.5/1.5 | 0.5/0.7 | 0.0025 |

25 | 2.5 | 3.0 | 5.6 | 0.88 | 7.0 | 1.6 | 0.7/1/2 | 0.3/1.0 | 0.0025 |

30 | 3.1 | 3.0 | 3.95 | 1.1 | 9.7 | 1.6 | 1.0/1.0 | 0.2/1.4 | 0.0023 |

40 | 4.7 | 3.0 | 3.0 | 2.7 | 17 | 10 | 1.0/0.5 | 0.0/1.6 | 0.0025 |

50 | 7.0 | 3.0 | 1.6 | 2.0 | 3.0 | 8.0 | 0.5/0.3 | 0.4/1.0 | 0.00304 |

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**MDPI and ACS Style**

Kovalchuk, V.I.; Aksenenko, E.V.; Fainerman, V.B.; Javadi, A.; Miller, R.
Multilayer Adsorption of Heptane Vapor at Water Drop Surfaces. *Colloids Interfaces* **2017**, *1*, 8.
https://doi.org/10.3390/colloids1010008

**AMA Style**

Kovalchuk VI, Aksenenko EV, Fainerman VB, Javadi A, Miller R.
Multilayer Adsorption of Heptane Vapor at Water Drop Surfaces. *Colloids and Interfaces*. 2017; 1(1):8.
https://doi.org/10.3390/colloids1010008

**Chicago/Turabian Style**

Kovalchuk, Volodymyr I., Eugene V. Aksenenko, Valentin B. Fainerman, Aliyar Javadi, and Reinhard Miller.
2017. "Multilayer Adsorption of Heptane Vapor at Water Drop Surfaces" *Colloids and Interfaces* 1, no. 1: 8.
https://doi.org/10.3390/colloids1010008