# Diffusion Transport of Water and Methanol Vapors in Polyvinyltrimethylsilane

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### Abstract

The specifics of diffusion of water and methanol vapors in nonporous polymer films based on polyvinyltrimethylsilane (PVTMS) have been studied. The vapor diffusion coefficients have been determined by measuring the kinetics of unsteady flow through the membrane (differential method) and subsequent processing the results by functional scaling . The kinetic curves have been found to deviate from those described by classical Fick’s law. It has been theoretically shown that such deviations can be due to the formation of associates of penetrant molecules inside the membrane, and a modified method for calculating diffusion coefficients has been proposed for this case. The behavior of the diffusion coefficients of water and methanol vapors in PVTMS in the temperature range of 50–90°C and the vapor activity range of 0.3–0.9 has been studied. The activation energies of diffusion of water and methanol vapors in PVTMS have been determined to be 23 and 44 kJ/mol, respectively, and the effective kinetic diameters of the molecules have been calculated to be 0.29 and 0.37 nm, respectively. The proposed approach opens the possibility for systematic studies of the diffusion kinetics of vapors of different organic compounds with an assessment of their kinetic contribution to the membrane permselectivity.

## Keywords:

vapor diffusion in polymers PVTMS diffusion kinetics unsteady mass transfer## INTRODUCTION

Currently, one of the developing areas in the field of membrane separation is membrane vapor separation. In contrast to gas separation, the field of selective vapor transport in nonporous polymer membranes and membrane materials has been scarcely studied. As in the case of gas separation, vapor transport in membranes occurs via the solution–diffusion mechanism [1]. This mechanism suggests that mass transfer through the membrane is described by the diffusion of penetrant molecules and is characterized by a permeability coefficient, which is the product of diffusion and solubility coefficients. The permeability coefficient is the main transport characteristic of a component in the membrane material, and it is always determined during a study. In turn, the diffusion and solubility coefficients, which are the terms of the permeability coefficient, are determined less frequently, since they are usually not considered in the case of general comparison of the separation properties of membrane materials and or modeling membrane separation processes, with the exception of unsteady processes [2, 3]. Nonetheless, knowledge of diffusion and solubility coefficients is undoubtedly required for basic research, prediction of membrane gas transport properties for a wide range of penetrants [4], understanding complex and unusual mass transfer effects, and studying and modeling kinetic (unsteady-state) separation processes [5, 6]. There are several methods to measure experimentally the diffusion coefficient of gases and vapors in membrane materials. One of the first and most widely used methods is the Daynes–Barrer method [7, 8, 9] due to its simplicity and the possibility of determining the diffusion coefficients and permeability in one experiment. The calculation of the diffusion coefficient in this method is based on determining the time lag by extrapolating the linear portion of the experimental curve, which represents the steady state of mass transfer, to the time axis. This method is suitable for the case when there is diffusion transport described by classical second Fick’s law (the differential equation of system (1)). However, in the case of any deviation of mass transfer from the classical transfer, for example, the dependence of the diffusion coefficient on the concentration, significant nonlinearity of the sorption isotherm, facilitated transport, etc., this method will give an error [10] and, moreover, will not always allow the unambiguous detection of the presence of deviations. Unlike gases, vapor transport is more likely to have special features and deviations from the classical case, since vapors are condensable and may exhibit stronger interaction both with the membrane material and with one another. Therefore, in this work the differential permeation method was used to determine the diffusion coefficients of water and methanol vapors. As shown below, this method combined with the corresponding mathematical processing makes it possible to unambiguously detect the presence of deviations of the diffusion transport from the classical second Fick’s law, and the proposed modification of the calculation method allows one to obtain reliable values for the diffusion coefficients.

## THEORETICAL

The diffusion coefficients of gases in membrane polymer materials are determined by various kinetic methods that allow calculations using, as a rule, analytical solutions (or their approximations) of the equation of unsteady-state diffusion transport (second Fick’s law) with the corresponding initial and boundary conditions:

The most common technique is the Daynes–Barrer (integral permeation) method, which makes it possible to determine the gas diffusion coefficient and membrane permeability in one experiment. The method consists in recording the time dependence of the volume of gas passed through the membrane with a stepwise change in the concentration of the test gas on the upstream surface of the membrane (as a rule, gas is fed after evacuation of the membrane). The diffusion coefficient in this method is calculated from the time lag θ determined by extrapolating the linear part of the experimental curve representing the steady-state mass transfer to the time axis (Fig. 1). The diffusion coefficient is related to the time lag by the following expression:

The method of measuring the time dependence of unsteady flow of gas passing through a membrane with a stepwise change in concentration on the upstream surface of the membrane (differential permeation method) is less frequently used. The method consists in recording the permeation rate curve (Fig. 2) followed by its processing by various mathematical methods [11]. One of the most accurate processing methods is functional scaling. It consists in a nonlinear transformation of the normalized permeation rate curve leading to a linear function *u*(*t*) in the classical case (Fig. 3), after approximation of this function by a linear relationship *u*(*t*) = *b*_{1} + *b*_{2}*t*, the diffusion coefficient can be calculated from the slope *b*_{2} as follows:

Any deviations of diffusion transport from the classical second Fick’s law (for example, because of specific interaction of penetrant molecules with the membrane material due to the presence of sorption centers or its effect on the diffusion coefficient, etc.) will generally lead to distorted rate curves. Deviations are especially likely to occur during vapor transport, since vapors are condensable components and, in addition to the interaction with the diffusion medium, can interact with each other. In contrast to the time-lag method, the use of the differential measurement method combined with functional scaling is convenient in this case because it allows the unambiguous detection of distortions of the kinetic curve, since the transformed curve will deviate from the linear dependence. Such a deviation was revealed when studying the diffusion of water and methanol vapors in PVTMS (Fig. 5).

This deviation may be primarily due to the formation of associates of the penetrant molecules in the polymer [12], since water and methanol vapors are condensable in the temperature range studied. A part of the unsteady flow penetrating into the membrane is consumed for the formation of associates, which distorts the rate curve, extending it into the region of longer times and increasing the time to reach the steady state. At the same time, the kinetic curve transformed by functional scaling significantly deviates from the linear dependence and its approximation by a linear function obviously becomes incorrect.

It was suggested that the initial part of the transformed kinetic curve can be used to calculate the diffusion coefficient, since it is least subjected to distortion caused by the formation of associates. To verify this assumption, the unsteady-state transport of the component through the membrane to form associates of the penetrant molecules was simulated. When building the mathematical model, the following assumptions were made: (i) the diffusion coefficient of component molecules is constant; (ii) the associates consist of two component molecules; (iii) the formation of associates is reversible; (iv) the associates have low mobility compared to individual molecules and their diffusion can be ignored (slight contribution of diffusion of associates to the total flow, i.e. *D*_{2} = 0); (v) the formation and decay of associates is determined by kinetic constants.

The corresponding system of differential equations with initial and boundary conditions will be as follows:

The system was solved numerically using the finite difference method. The kinetic constants for the formation and decay of associates (*k*_{1} and *k*_{–1}) were chosen to be such that the calculated rate curve was as close as possible to the experimental curve (Fig. 4).

The processing of the obtained theoretical rate curve by functional scaling showed a deviation from the linear dependence similar to the experimental data (Fig. 5). To calculate the diffusion coefficient, the initial part of the transformed curve was approximated by a linear function; the value of the diffusion coefficient calculated from the slope coincided with the value specified when solving the system of Eqs. (4). Thus, it has been theoretically shown that the diffusion coefficient of components can be determined from the initial part of the kinetic curve in the case of transport specifics such as the formation of associates of penetrant molecules. Note that deviations will occur in the same way in case of vapor transport in the presence of sorption centers in the membrane material (the case of a significant contribution of sorption according to the Langmuir law); the proposed method for calculating the diffusion coefficient from the initial part of the kinetic curve will also give reliable results. This method for calculating diffusion coefficients was used to process the experimental kinetic curves obtained in this work.

For comparison, Fig. 6 shows the theoretical kinetic curve obtainable using the Daynes–Barrer method, as calculated from the solution of the system of Eqs. (4). It can be seen that the time lag is 14.68 s, which is significantly longer than 7.72 s found by solving the system of Eqs. (1) with the same diffusion coefficient (Fig. 1). Thus, the diffusion coefficient calculated using this value of time lag will be almost two times less than that set in the simulation: 2.26 × 10^{–10} instead of 4.27 × 10^{–10} m^{2}/s.

## EXPERIMENTAL

To study the vapor diffusion coefficients, an experimental procedure was developed that makes it possible to provide the required initial and boundary conditions on the membrane and record the unsteady vapor flow through the membrane. The experimental setup is shown in Fig. 7. Distilled water and methanol of the special purity grade were used in the experiments. The vapor diffusion was studied using homogeneous PVTMS membranes samples of 40 and 141 μm in thickness prepared in laboratory by casting from a PVTMS solution in toluene (5–7 wt %) onto cellophane. The choice of PVTMS as a polymer for research was due to several reasons: (i) the test penetrants (water and methanol) have no effect on the polymer; (ii) PVTMS-based membrane samples have sufficiently high permeability, good reproducibility, and stability of gas transport properties over time; (iii) there is a large amount of experimental data on the diffusion coefficients, solubility, and permeability in PVTMS for a wide variety of gases, which allows for the subsequent correlation analysis of the results obtained. In addition, laboratory composite membranes based on PVTMS have been tested and showed good results in the vapor-phase separation of water–alcohol mixtures [13].

A membrane sample was placed in a flow-through membrane cell (there are feed and draw channels upstream and downstream of the membrane); the working area of the sample in the cell was 25 cm^{2}. To prevent vapor condensation, the cell, bubbler, circulation pump, rotameter, and corresponding connecting pipelines of the circulation loop were placed in an air thermostat. The required vapor activity was ensured by maintaining the temperature of the liquid phase (bubbler) below the cell temperature using an additional circulation loop with an external heat exchanger (HE_{1}-HE_{2}) by adjusting the flow of coolant (water) with a liquid pump (LP). The temperature of the cell and bubbler was monitored by temperature sensors (TS). Before starting the experiment, the bubbler was purged with a carrier gas (nitrogen); for this purpose, valve V was opened and the carrier gas flow was directed through the bubbler with flow switch FS_{2}. Then, the flow switch FS_{1} directed the carrier gas stream into the membrane cell above the membrane, valve V was closed, the flow of carrier gas downstream of the membrane was turned on by opening pressure regulator PR_{2}, the vacuum pump was turned on, the thermal conductivity detector (TCD) was powered, and the stationary signal of the TCD baseline was waited to appear. During the start of the experiment, a circulation pump (CP) was turned on, which ensures the flow of the vapor–gas mixture (carrier gas with vapors) circulated through the bubbler and the cell; the flow of vapor passing through the membrane was measured with TCD (detector temperature 100°C, current 60 mA), the signal from which was recorded using the EKOCHROM software and hardware complex for further processing.

## RESULTS AND DISCUSSION

Our study of the diffusion of water and methanol vapors through PVTMS-based film samples was carried out in the temperature range of 50–90°C and the vapor activity range of 0.3–0.9.

The experimental differential permeation rate curves for methanol vapor (Fig. 8a), as in the case of water vapor transport, demonstrate a deviation of the diffusion kinetics from the classical second Fick’s law as evidenced by the nonlinearity of the dependence after transformation by functional scaling (Fig. 8b). To calculate the diffusion coefficient of methanol vapor, the initial portion of the converted kinetic curve was also used.

It was found that the diffusion coefficient of vapors is independent of their activity in the interval studied (Fig. 9), i.e. the assumption that it is constant is true. This indirectly confirms the fact that the deviation of the vapor diffusion kinetics from the classical second Fick’s law is due to associates formed by penetrant molecules.

Based on the obtained temperature dependences of the diffusion coefficients of the vapors (Figs. 9 and 10), we calculated the activation energies of diffusion (*E*_{D}) of water and methanol vapors to be 23 and 44 kJ/mol, respectively.

*d*

_{EF}) of water and methanol molecules. The effective diameters were calculated using the correlation approach [14] that is practiced in the field of membrane gas separation to assess the transport characteristics of penetrants according to the following relationship:

*K*

_{1}and

*K*

_{2}are isothermally constant coefficients, individual for each polymer. It is assumed that the effective cross section of the diffusing molecule expressed as the square of its effective diameter is constant for each gas and is independent of the diffusion medium (polymer) under consideration. Figure 11 shows the dependence described by Eq. (5) for PVTMS at 25°С, which is built based on data on the diffusion coefficients of inert gases [15].

The calculated values of the effective diameters of the water and methanol molecules are listed in Table 1, where the values calculated earlier [16] from the data on the diffusion coefficients of these components in PVC are also given.

Comparison of the effective diameters of the water and methanol molecules

Component |
(present results) |
(results reported in [16]) |
---|---|---|

Н | 0.29 | 0.26 |

CH | 0.37 | 0.34 |

The good agreement between the values obtained for polymers that substantially differ in membrane properties indicates that the correlation approach is applicable to prediction of the transport characteristics of vapors. Thus, the obtained data on the effective diameters of water and methanol molecules can be used in the future to assess the transport characteristics of these components in other polymeric materials. This is especially true for water, since its vapor is present in many gas mixtures, which are separated using the membrane method.

## CONCLUSIONS

In this work, significant deviations of the kinetics of diffusion of water and methanol vapors in PVTMS from the classical second Fick’s law have been revealed. It has been assumed that the deviations are due to associates formed by penetrant molecules in the polymer. It has been shown theoretically the differential method for measuring the diffusion coefficient should be used in combination with the transformation of permeation rate curves by functional scaling for reliable detection of deviations. The unsteady-state transport of water vapor through the PVTMS film with allowance for the formation of associates has been simulated. A method for calculating the diffusion coefficient based on distorted transformed rate curves has been proposed and the possibility of applying the proposed method has been theoretically proved.

Experimental results have shown that the vapor diffusion coefficient is constant for each component and does not depend on the vapor activity in the studied range.

## FUNDING

This work was supported by the Russian Foundation for Basic Research, grant no. 16-08-01187.

| |

| concentration of diffusant, mol/m |

| diffusion coefficient, m |

| effective molecule diameter, m |

| activation energy of diffusion, kJ/mol |

| kinetic constant of formation of associates, m |

| kinetic constant of decay of associates, s |

| correlation coefficients |

| membrane thickness, m |

| time, s |

| temperature, K |

| longitudinal coordinate, m |

θ | time lag, s |

## Notes

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