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Langmuir 2002, 18, 112-119
Gas Permeation through Ultrathin Liquid Films Jianjun Li,†,‡ Heidrun Schu¨ring,† and Ralf Stannarius*,† Faculty of Physics and Geosciences, Institute of Experimental Physics, University of Leipzig, Linne´ strasse 5, D-04103 Leipzig, Germany, and Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun, China Received April 30, 2001. In Final Form: July 16, 2001
We report measurements of gas permeation through ultrathin liquid layers. The gas flux penetrating nanometer thick free-standing smectic films is determined experimentally as a function of membrane thickness and temperature. We observe an unusual film thickness dependence of the permeation in very thin films. For its interpretation, we introduce a molecular model that considers sorption and diffusion of gas molecules in the liquid film as well as the dynamic exchange between the film and the external gas phase. From the fit of experimental data, we obtain the ratio of the solubility of gas in the smectic phase and the diffusion coefficient for gas molecules in the liquid and an energy barrier for gas molecules to enter the liquid film. The measured permeation is strongly temperature dependent; at the transition to lowtemperature smectic modifications with in-plane molecular positional order, the films become almost impermeable.
Introduction The permeation of gases through thin fluid membranes is relevant in many physical, biological, and technological aspects, for example, in breathing, chemical sensing, or gas separation (see, for example, refs 1-3 and references therein). Especially in the latter context, experiments that directly measure the gas flow through thin liquid films have been reported (e.g., see refs 4-9). From technical aspects, gas sorption, diffusion, and permeation play an important role for polymers.10-14 Special membrane techniques have been developed15-17 for quantitative investigations. A study of the gas transport through freely suspended soap films has been reported by Sujatha et al.,7 who investigated the exchange of gases between different compartments of a tube separated by soap films. The movement of the films in the tube due to changes in the volumes of the gas compartments was recorded, and a molecular model for gas permeation through the liquid * Corresponding author. † University of Leipzig. ‡ Chinese Academy of Sciences. (1) Fendler, J. H. J. Mater. Sci. 1987, 30, 323. (2) Moriizumi, T. Thin Solid Films 1988, 160, 413. (3) Tieke, B. Adv. Mater. (Weinheim, Ger.) 1991, 3, 532. (4) Rose, G. D.; Quinn, J. A. J. Colloid Interface Sci. 1968, 27, 193; Science 1968, 159, 636. (5) Li, N. N. AIChE J. 1971, 17, 459. (6) Albrecht, O.; Laschewsky, A.; Ringsdorf, H. Macromolecules 1984, 17, 937; J. Membr. Sci. 1985, 22, 187. (7) Sujatha, K.; Das, T. R.; Kumar, R.; Gandhi, K. S. Chem. Eng. Sci. 1988, 43, 1261. (8) Cook, R. L.; Tock, R. W. Sep. Sci. 1974, 9, 185. (9) Ramachandran, N.; Didwania, A. K.; Sirkar, K. K. J. Colloid Interface Sci. 1981, 83, 94. (10) Shieh, J.-J.; Chung, T. S. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 2851. (11) Zhang, Z. J. Polym. Sci., Part B: Polym. Phys. 2000, 38, 1833. (12) Yeom, C. K.; Lee, J. M.; Hong, Y. T.; Choi, K. Y.; Kim, S. C. J. Membr. Sci. 2000, 166, 71. (13) Merkel, T. C.; Bondar, V. I.; Nagai, K.; Freeman, B. D.; Pinnau, I. J. Polym. Sci., Part B: Polym. Phys. 2000, 38, 415; Macromolecules 1999, 32, 8427. (14) Molyneux, P. J. Appl. Polym. Sci. 2001, 79, 981. (15) Mogri, Z.; Paul, D. R. J. Membr. Sci. 2000, 175, 253. (16) Rallabandi, P. S.; Ford, D. M. J. Membr. Sci. 2000, 171, 239. (17) Obuskovic, G.; Poddar, T. K.; Sirkar, K. K. Ind. Eng. Chem. Res. 1998, 37, 212.
films has been provided.7,8 In this system, the surfactant serves basically as a means to stabilize the thin films while the gas permeation is determined by the water phase. The experiment is particularly sensitive to differences in permeation coefficients of different gases, but because the determination of the film thickness in this system is nontrivial, the authors in ref 7 have not studied the important role of film thickness for gas permeability. Films of controlled thickness for permeation measurements can be produced by Langmuir-Blodgett (LB) techniques.4 Tieke3 has reported the usability of LB membranes for gas separation and liquid permeation. The potential of LB films for membrane application is mainly based on the extremely low thickness and macroscopically ordered structure, which offers the chance to combine a high permeation rate with a highly selective transport. Moreover, the film thickness can be precisely controlled. However, because LB films are of rather poor mechanical stability, permeability studies can only be carried out on films that are built up on stable, porous support material. Naturally, the permeability of gases through LB films is influenced not only by the film itself but also by the characteristics of its porous substrate. So, a direct quantitative study of the gas permeation is not easily achieved. We have chosen a system that provides stable thin and uniform liquid layers and the opportunity to control the film thickness. The formation of free-standing films with an exceptionally high ratio of lateral dimensions and film thickness is one of the fascinating properties of fluids with a one-dimensionally layered structure. Soap films and soap bubbles represent well-known examples of such structures, but similarly, free-standing planar films (see, for example, reviews in refs 18-20) and spherical bubbles21-24 can be produced with smectic liquid crystals. They prove very robust, and their film thickness, i.e., the (18) Stoebe, T.; Huang, C. C. Int. J. Mod. Phys. B 1995, 9, 2285. (19) Bahr, Ch. Int. J. Mod. Phys. B 1994, 8, 3051. (20) Pieranski, P.; Beliard, L.; Tournellec, J.-Ph.; Leoncini, X.; Furtlehner, C.; Dumoulin, H.; Riou, E.; Jouvin, B.; Fe´nerol, J.-P.; Palaric, Ph.; Heuving, J.; Cartier, B.; Kraus, I. Physica A 1993, 194, 364. (21) Oswald, P. J. Phys. (Paris) 1987, 48. (22) Stannarius, R.; Cramer, Ch. Europhys. Lett. 1998, 42, 43. (23) Stannarius, R.; Cramer, Ch. Liq. Cryst. 1997, 24, 371.
10.1021/la010627a CCC: $22.00 © 2002 American Chemical Society Published on Web 12/07/2001
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Table 1. Phase Transition Temperatures of the Investigated Materials in the Bulk DOBAMBC HOPDOB
Cr 74.6 °C (SmI* 62.0 °C) SmC* 94.0 °C SmA 117 °C I (35.0 °C) Cr 62.5 °C SmE 38.0 °C (SmB 44.5 °C) SmC 77.5 °C SmA 83.3 °C N 88.9 °C I
number of smectic layers across the film, can be adjusted during preparation. Thin freely suspended smectic films can be used as simple model membranes for studies of the physics of thin liquid layers and biological membranes. In this study, we are particularly interested in the penetration of gases through very thin smectic liquid crystalline films. We measure gas permeation through the surface of smectic bubbles in dependence on film thickness, temperature, and mesophase. Smectic bubbles can be prepared with uniform film thicknesses between two and several thousand smectic layers. The layers are perfectly stacked parallel to the film surface. Bubble radii can be up to 6 orders of magnitude larger than the film thickness. Only surface tension stabilizes the membranes, and the pressure difference in permeation experiments has to be kept relatively small. On the other hand, because of the Laplace pressure of a curved membrane, the bubbles themselves provide an easily controllable pressure source for the experiment, and the gas volume loss is directly measurable from the bubble radii. Within the layer plane, smectic A and C phases behave liquidlike, without molecular positional order. Because the layer structure itself will not influence the transport processes qualitatively but mainly change the quantitative results for the diffusion coefficients, we can consider the smectic films as convenient model systems for thin liquid films in general. We have organized the paper as follows: In the next section, the experimental setup and the investigated materials are introduced, followed by the presentation of the experimental results in the third section. We provide experimental data of gas permeation through thin smectic films as a function of film thickness and temperature. The data are fitted within a phenomenological model by two parameters. In the fourth section, a molecular model is developed that considers the kinetics of the gas phase, an energy barrier for the gas particles to enter the film, and the diffusion of gas particles in the liquid matrix. A discussion of the experimental results and a relation of the phenomenological quantities to molecular parameters follow, and finally, a summary is given. Experimental Section Experiments have been performed with two standard smectic materials DOBAMBC (4-decyloxybenzylidence-4′-amino-2methylbutylcinnamate) and HOPDOB (4-hexyloxyphenyl-4′-decyloxybenzoate). Their mesophase sequences and phase transition temperatures are given in Table 1. DOBAMBC has been chosen as a well-characterized substance25 with a broad smectic range and rich polymorphism. Whereas in the two high-temperature smectic A and C* modifications the films are positionally disordered in the smectic layers, an additional hexagonal lattice within the layer planes is formed in the smectic I* phase. The second material, HOPDOB, is also extensively characterized.25 It has been shown recently that the material exhibits a surface phase transition:26,27 before reaching the low-temperature crystal B modification, the substance forms a monomolecular higher (24) Stannarius, R.; Cramer, Ch.; Schu¨ring, H. Mol. Cryst. Liq. Cryst. 1999, 329, 1035. (25) Liquid Crystal database. http://licryst.chemie.uni-hamburg.de. (26) Schu¨ring, H.; Thieme, C.; Stannarius, R. Liq. Cryst. 2001, 28, 241.
Figure 1. Experimental setup for the generation and observation of smectic bubbles. ordered layer on the smectic C film. Thus, it offers the opportunity to study effects of a monolayer on the liquid surface on the gas permeation properties. The bubbles are produced by means of a technique described earlier.22 Their film thicknesses are in the nanometer range, and bubble radii are of the order of several millimeters. For the preparation, we used tapered glass capillaries with openings of a few millimeters. One end of the glass capillary is connected to a syringe and pressure gauge for difference-pressure measurements with a relative accuracy of 0.1 Pa and an absolute accuracy of about 2 Pa. The apparatus is enclosed in a sample container for temperature control and protection against acoustic disturbances and ambient airflow (Figure 1). After a freely suspended film has been drawn on the tapered open end of the capillary, a controlled volume of air is slowly injected into the capillary, and the film is blown up to a bubble. The spherical shape of the bubble is a consequence of the minimization of the film surface by the action of the surface tension (σ) of the smectic fluid at the air/film interfaces. The film curvature produces an inner excess pressure (∆p) of the gas in the bubbles, which is directly related to the bubble radius R by the well-known Laplace-Young relation:
∆p )
4σ R
(1)
The bubble is illuminated with parallel 546-nm monochromatic light. Transmission images are taken with a video camera and a HAMAMATSU controller and processed digitally to determine the bubble radii and curvatures. By studying the shrinking of the bubble over a period of a couple of minutes to hours (depending on the efficiency of gas permeation), we can record the volume loss of gas in the smectic bubbles. The permeation process itself is composed of different physical mechanisms: solution of the penetrant in the liquid matrix and its diffusion in the matrix. The permeability coefficient relates the flux of the penetrant through a membrane to the upstream and downstream (feed and permeate) pressures. In a simple phenomenological description of the gas permeation process, we consider three essential parameters. The first one is the pressure difference between both sides of the membrane, which is exactly the Laplace pressure in our system. The difference pressure is recorded permanently during the experiments, and in principle, we can use the measured ∆p(t) curve directly. However, it is more convenient to exploit eq 1, which relates ∆p(t) to the easily accessible bubble radius (R) by the surface tension (σ) of the smectic. Then, the radius is a direct measure for the inner excess pressure and the gas volume loss. (27) Veum, M.; Pettersen, C.; Mach, P.; Crowell, P. A.; Huang, C. C. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2000, 61, R2192.
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in Figure 4, where helium was used to inflate the bubble. This speeds up the shrinking process by a factor of about four as compared to air. The dynamics of the shrinking process are easily derived for a spherical bubble. For a complete sphere, the gas volume loss
V˙ ) 4πR2R˙ should be proportional to the surface, S ≈ 4πR2. Because the difference pressure (∆p) between both sides of the membrane is very small, we assume a linear dependence of the volume loss on ∆p. In this approximation, one finds that
Figure 2. Surface tensions of DOBAMBC (diamonds) and HOPDOB (crosses) determined from the difference pressure and bubble radius by means of eq 1. The temperature dependence of σ is very small except in a pretransition range of HOPDOB near the low-temperature smectic B (crystal) phase. We use these surface tensions to calculate the excess pressure in the bubbles directly from radius data. The determination of σ(T) has been described elsewhere.26 It has been performed with essentially the same setup as used here. The temperature dependence (σ(T)) for the complete smectic mesophase range is shown in Figure 2. Another important parameter is the thickness of the smectic films. It can be controlled during bubble inflation.22 If the bubbles are blown up slowly to a sufficient degree, one achieves a uniform film thickness, which is dependent upon the inflation speed. We have taken care that only bubbles with a final uniform thickness have been selected for evaluation. We determine the film thickness (d) from optical transmission profiles of the bubbles.24 The accuracy of this method is about 5%; for very thin films, it is accurate to about one smectic layer. The third essential parameter is the temperature of the system. It has significant influence on the transport coefficients in the smectic material, the kinetic energy of the gas molecules, and the very structure of the mesophase. Therefore, the setup is enclosed in a thermobox, which can be controlled from room temperature to about 450 K with an accuracy of 0.2 K.
Measurements After inflation of the bubble, the syringe position is fixed, and we measure the gas volume loss through the smectic membrane from the radius changes by means of image processing software. The time scale of the shrinking process depends on membrane thickness, bubble size (because of the radius/excess pressure relation), and temperature. For example, a very thin bubble with a radius of less than 1 mm at 100 °C has a lifetime of only a few minutes, whereas a thick bubble (micrometer film thickness) with several millimeters diameter at low temperature is tight enough to persist for a couple of days with only slight volume loss. Figure 3 shows a sequence of four images of a DOBAMBC bubble during a period of about 4.5 h. During the slow shrinking process, the membrane thickness in general remains constant. The excess smectic material flows back into the meniscus. This is confirmed by measurements of the thickness (d) at different stages of the shrinking process. Somtimes, as shown in Figure 3b, additional smectic layers grow from the meniscus and create a region of increased film thickness with a sharp boundary at the bottom. Such bubbles have been excluded from the analysis. A peculiarity of the system is that during gas loss the excess pressure in the bubble increases. The shrinking process becomes more efficient with decreasing bubble volume, even though the membrane area decreases. A typical change of the bubble radius with time is depicted
4σ 4πR2R˙ ≈ -γ4πR2 R For the moment, γ(d, T) is a phenomenological parameter describing the volume of gas passing the film per pressure difference and area. Later, we will relate its value to molecular quantities (see Molecular Model for Gas Permeation through Smectic Membranes). The solution of this differential equation is
R2(t) ≈ R20 - 8σγt ) R20 - gt
(2)
with the initial radius R0 ) R(0) and the fit parameter g(d, T) ) 8σγ, which is introduced for convenience to describe the bubble radius change with time. This square dependence holds as long as the bubble is large as compared to the radius of the capillary opening (rc). A first-order correction takes into account that the area of the capillary opening must be subtracted from the bubble surface. We approximate it by a disk with radius rc; this yields a corrected surface, S ) 4πR2 - πr2c , and the analytical solution
R2 +
(
)
(
)
r2c r2c r2c r2c ln R2 ) R(0)2 + ln R(0)2 - 8σγt 4 4 4 4 (3)
Figure 4 compares the fit corresponding to eq 3 (solid curve) to experimental data, and the dotted curve shows the solution of the approximation of eq 2 for a complete sphere with the same fit parameters. When R approaches rc, both analytical approximations slightly overestimate g, and if one is interested in the complete description of the shrinking process, one has to replace eq 3 with the exact solution for a spherical cap, which we have calculated numerically. In the experiments, we have chosen the initial radii R0 large enough (>3rc) so that eq 3 is practically identical with the exact numerical solution, and eq 3 was used to obtain g from the R(t) fit. We note that one can distinguish gas loss through a potential leak in the apparatus unambiguously from gas permeation through the film, since the radius change is qualitatively different in that case, R4 - R40 ∝ t. We have carefully established that there is no gas loss through leakages. The tightness of the apparatus can be tested preferentially with very thick (micrometer) bubbles, where gas flux through the smectic membrane is very small. Figure 5 shows the film thickness dependence of the gas permeation at constant temperature. The experimental data have been obtained from various bubbles of different film thicknesses. The film thickness can vary only in steps of the smectic layer thickness of about 3 nm, but our fit of the optical transmission profiles, which gives the film thickness, has been performed with a continuous fit parameter (d). Its unambiguous assignment to an exact
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Figure 3. Sequence of images of a DOBAMBC bubble at 88 °C that shrinks due to the penetration of gas molecules through the bubble membrane. The outer capillary diameter is 1.61 mm. Corresponding times and difference pressures: (a) 0 s, 29.1 Pa; (b) 1 h 42 min, 37.3 Pa; (c) 3 h 29 min, 58.5 Pa; (d) 4 h 8 min, 103.3 Pa. The bubble has uniform and constant film thickness of about 80 nm, except in panel b where some excess layers have grown at the bottom, which disappear again later.
in the film, and consequently, g should be inversely proportional to the film thickness d. This is a good approximation for thick films, but it obviously does not agree with the data obtained for small d (dotted line in Figure 5a). The dashed curves shown in Figure 5a,b are fits to the equation
g(d) )
Figure 4. Radius of a helium-filled DOBAMBC bubble at T ) 92 °C recorded over a period of 22 min. The solid line is a fit to eq 3 with rc ) 2.01 mm; the dots represent the approximation R2 ) R20 - gt.
number of smectic layers was not always possible; therefore, we have plotted the permeation data against the fit parameter, which is not always an integer multiple of the smectic layer thickness and which is accurate to about (1 nm. Each datum point g(d) is obtained from a R(t) fit similar to that shown in Figure 4. As expected, the gas permeation rapidly decays with increasing film thickness. If we interpret the data with a sorption/diffusion model as done, for example, in ref 7, the permeation flow should be proportional to the gas concentration gradient
G0 d + d0
(4)
which differs from the conventional formula by the term d0 > 0, and this term becomes relevant for thin films (d ≈ d0). From the fits of the thickness dependences at constant temperature, we find the parameters G0 ) 0.11 × 10-6 mm3/s and d0 ) 8.0 nm for HOPDOB at 82 °C and G0 ) 0.45 × 10-7 mm3/s and d0 ) 8.4 nm for DOBAMBC at 80 °C. The simple sorption/diffusion model is applicable for d . 8 nm with g(d) ≈ G0/d but fails if the film thickness comes down to the order of a few molecular layers. This will be discussed in the next section in detail, and a relation to molecular parameters will be established. The temperature dependence at constant membrane thickness has been collected in Figures 6 and 7. The data presented in Figure 6 have been measured with two DOBAMBC bubbles of equal membrane thickness within the accuracy of thickness determination. After measuring the gas loss at one temperature, each bubble was carefully inflated again and cooled or heated to the next temperature, preserving the film thickness. Crosses (measured during cooling) and diamonds (measured during heating)
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Figure 7. Temperature dependence of the permeation coefficient g for HOPDOB, taken from several individual bubbles with film thickness d ) 11 ( 1 nm.
Figure 5. Film thickness dependence of the permeation coefficient g. (a) HOPDOB at 82 °C and (b) DOBAMBC at a temperature of 80 °C. The dashed fit curves correspond to the parameters given in the text, the dotted curve in panel a symbolizes a simple inverse film thickness dependence, fitted to the permeation of thick films.
Figure 6. Temperature dependence of the permeation coefficient g for two DOBAMBC bubbles with film thickness of 11 ( 1 nm. The bulk transition temperature SA - SC is depicted. Data from the first bubble during cooling and heating are denoted by crosses and diamonds, respectively. Data of the second bubble are denoted by triangles.
represent measurements of the first bubble. They differ in the vicinity of the smectic I*-C* transition, which is monotropic. In the range from 62 to 74 °C, crosses reflect measurements in the C* phase, while the diamonds give the permeation of the I* phase. The I* phase is clearly identified in the bubbles optically, whereas we have not
resolved the smectic A to smectic C* transition in the freestanding films (textures are not observable on the curved membranes). We expect that transition to be shifted toward higher temperatures in thin films; however, tilted boundary layers may exist in the smectic A phase as well. The smectic A to cholesteric transition is clearly identified because the bubbles burst immediately when melting into the cholesteric mesophase. In the temperature ranges of the smectic C* and A (bulk) phases, the gas permeation increases continuously with increasing temperature. This is expected from the temperature dependence of the thermally activated diffusion of gas molecules in the film and the higher solubility of gases at higher temperatures. Other influences may be the increasing kinetic energy of the gas molecules, the structure and order of the smectic mesogens, and the exchange rates discussed in the next section. The solid line in Figure 6 suggests that the gas permeability follows a rather linear temperature dependence in the S/C mesophase, increasing with the distance from the (monotropic) phase transition into the S/I mesophase at ≈62 °C. In the high-temperature range of the S/C phase, the data follow (within the experimental accuracy) a continuous curve at the transition from S/C to SA. The DOBAMBC curve reaches almost impermeability near the transition to the I* phase, and the permeability of the membrane is more than 1 order of magnitude smaller than in the C* phase at 90 °C. Data for HOPDOB are presented in Figure 7. Here, the results of many bubbles of equivalent film thickness have been assembled. Again, the temperature curve is rather steep, and the films reach almost impermeability at the transition to the low-temperature crystalline B phase. As in DOBAMBC, no peculiarities appear near the (bulk) smectic A to smectic C transition, which is indicated in the figure. We cannot perform measurements in the nematic range where the stabilizing smectic layering is absent. In the crystalline B phase, no permeation experiments can be performed either because the bubbles burst when cooled to this phase. Molecular Model for Gas Permeation through Smectic Membranes So far, the interpretation has been based on a phenomenological description, but within a suitable model, one can compare the experimental findings to molecular parameters. The standard model considers sorption of gas in the liquid and the diffusion of gas molecules in the film, and an inverse proportionality of gas permeation and film
Gas Permeation through Ultrathin Liquid Films
Figure 8. Sketch of the film geometry and assignment of the parameters in the gas permeation model. Ni is the number of gas molecules in volume i (1, inside the bubble; 2, membrane left border; 3, membrane right border; 4, outside the bubble). v is the instant thermal velocity of the gas particle; corresponding particle densities per volume are n1 ... n4.
thickness is obtained. It has been shown in the previous section that the results of the simple sorption/diffusion model are qualitatively correct in thick films but that there are characteristic deviations in the experiments performed with thin films of a few nanometers. The fit curves reach a saturation of the permeation parameter g(d) for d f 0, which is obviously a consequence of the limited feed of particles entering the smectic membrane from the gas phase per unit time. This limiting factor can be neglected in micrometer thick films where diffusion in the liquid is the bottleneck for gas transport. The model introduced here considers two effects: first the molecular diffusion of gas particles in the liquid film, which mediates exchange between the concentrations of gas molecules at the two membrane surfaces, and second, the dynamic exchange of gas molecules between the gas phase and the liquid. Other effects, like adsorption of gas molecules at the liquid surface, play no role here because they are in equilibrium and do not influence the transport between the two gas volumes inside and outside the bubble sphere. For the transport problem, film curvature is negligible in bubbles of millimeter size. On a submicrometer scale, the film can be considered planar. The problem is treated as one-dimensional because only transport perpendicular to the film surfaces (chosen as the x-coordinate) matters. In this geometry, we calculate the momentary number of gas particles transported through the film per unit area and time interval to arrive at a relation similar to eq 4. Inside the film, the transport is governed by diffusion; the resulting concentration gradient will be linear; and the concentration profile can be described by two parameters, the gas concentrations at the two film surfaces, inside the liquid. Outside the film, we can assume a uniform concentration of gas particles per volume, i.e., constant pressures inside and outside the bubble. Exchange of gas molecules inside the bubble volume is much faster than gas loss through the membrane. Four variables characterize the situation depicted in Figure 8: the particle numbers in the gas phase at both surfaces of the membrane inside the liquid and those outside the film on both sides.
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In our model, the number of gas particles hitting the film surface per unit area and time is limited, and consequently, there is also a limited amount of gas molecules entering the film. This assumption is necessary to explain the limited flux through ultrathin films (of a few molecular layers). Otherwise, in the pure diffusion model,7 the gas permeation would diverge in the thin film limit. An analysis of the experimental data and extrapolation to d ) 0 shows that only a small fraction of gas particles hitting the film surface will actually penetrate the air/ film interface. We will account for this hindrance by introducing an energy barrier that has the effect that only molecules with sufficiently high kinetic energy can enter the liquid phase. In equilibrium, the relation between gas concentrations in liquid and in gas phase is given by the solubility of the gas in the liquid (Henry’s law). In a thin film exposed to different pressures on both surfaces, the gas concentrations will not be in equilibrium but are determined by the dynamics of exchange with the gas phase and molecular diffusion. As shown in Figure 8, Ni and ni are the total numbers of gas molecules and concentrations (number of gas molecules per volume), respectively, in volume i. The number N /1 of gas molecules hitting one side of the membrane per unit time is given by
∫0∞vxw(vx)n1 dvx
N *1 ) S
(5)
Here, S is the film area; vx is the component of the thermal velocity of the gas particle normal to the surface; and
w(vx) )
( )
mv2x m exp 2πkT 2kt
x
is the Maxwell velocity distribution integrated over the two velocity components, vy and vz; and m is the mass of the gas molecule. Therefore where
N*1 ) An1 A)S
kT x2πm
(6)
is related to the most probable velocity (vav) in the Maxwell velocity distribution by A ) Svav/x4π. The rate of gas molecules entering the film is proportional but not equal to the number (N*) of molecules hitting the film surface from the gas phase. We allow for an energy barrier for the gas particles penetrating the air/film interface by multiplying the rate N* with a factor β ) exp(-Ea/(NAkT)), where k is the Boltzmann constant and NA is Avogadro’s number. In addition, there is particle flux in opposite direction from the film into the gas phase, proportional to the density of gas molecules in the liquid. The rate equation
N˙ 1 ) -An1β + Bn2
(7)
describes the change in the particle number N1 per unit time. The parameter B in our model is not free but has to be chosen such that, in the equilibrium N˙ 1 ) 0 (semiinfinite liquid phase or thin film with equal gas pressures on both surfaces), particle concentrations are consistent with sorption data, and Henry’s law, p ) Hξ, between the external pressure (p) and the molar concentration (ξ) of the gas in the liquid is reproduced. H stands for Henry’s constant. The concentration (ξi), the density (FL), and the
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molar mass (ML) of the liquid are related to the particle density (ni) introduced above by This yields the relation
ξi )
ML n (i ) 2, 3) FLNA i
N˙ 2 ) +An1β - Bn2 + Cn3 - Cn2
( )(
0 0 Aβ -Aβ
βSDA/B d + 2DS/B(n4 - n1)
V˙ 1 )
βDSA/B ∆p d + 2DS/B p0
(12)
Comparison with eq 4 yields two equations
)( ) n1 n2 n3 n4
G0 )
(10)
(10)
The number N1 of gas particles in the bubble is directly related to the bubble volume, and n1 - n4 is proportional to the pressure difference, ∆p; thus, eq 11 has the same structure as the phenomenological eq 4. For very thick films (d . 2DS/B), the inverse film thickness dependence is reproduced, and the model coincides with the standard model, while in the thin film limit (d f 0) a finite value
1 N˙ 1 ) Aβ(n4 - n1) 2 is reached, independent of the diffusion constant. By comparison of eqs 4 and 11, the experimental parameters G0 and d0 can now be related to the molecular quantities.
8σDNAkTFL MLHp0
(13)
and
d0 )
We will solve it for a constant pressure difference (∆p) between both sides of the membrane. This corresponds to the case when the bubble volume is maintained by continuous injection of air to replace the gas loss by permeation. In practice, the pressure difference increases when the bubble shrinks, but the change of ∆p in our experiment can be neglected on the time scale of molecular dynamics (diffusion of a gas molecule across a submicrometer film occurs on the millisecond scale). The dynamic equilibrium for N2 and N3 in the thin film will be adjusted much faster than for N1 and N4. Therefore, we can assume that the fluxes and concentrations of gas particles in the film are always in dynamic equilibrium and solve the system of equations for short-time periods where the molecular concentrations inside the membrane are constant, N˙ 2 ) N˙ 3 ) 0. From the particle number conservation, N ) N1 + N2 + N3 + N4 ) constant, it follows that N˙ 1 ) -N˙ 4 and that the solution of eq 10 is
N˙ 1 )
Because the pressure difference (∆p) is at least 3 orders of magnitude smaller than the absolute (atmospheric) pressure (p0), we set p1 ≈ p4 ) p0 everywhere in the equations except for pressure difference terms. From the ideal gas equation, one derives the relations V˙ 1 ) kT/p0N˙ 1 and ∆p ) (n1 - n4)kT, and we obtain from eq 11
(9)
Here, C ) S(D/d), and D is the diffusion coefficient of gas molecules in the liquid, to be exact, the coefficient D⊥ for diffusion perpendicular to the smectic layers in the anisotropic smectic material. Similarly, expressions are derived for N˙ 3 and N˙ 4, and we have to look for the stationary solution of the linear ODE system:
0 C -B -C B
g S∆p 8σ
(8)
Now we consider the change of the number (N2) of gas molecules in the smectic membrane near the left film boundary (Figure 8) by molecules leaving into the gas volume 1 and molecules diffusing in the film toward the opposite surface, following Fick’s first law of diffusion. The change of N2 can be described by
B -B -C C 0
The phenomenological coefficient g has been introduced in the Experimental Section as a proportionality factor in the equation
V˙ 1 ) -
Aβ NAkTFL ) B MLH
N˙ 1 -Aβ N˙ 2 Aβ N˙ 3 ) 0 N˙ 4 0
Discussion
kT x2πm
2DS 2DNAkTFL ) B MLHβ
(14)
with three unknown quantities (β, D, and H); all other parameters are known. One important quantity that can be determined from the experiment is the ratio G0/d0, which is the extrapolation value of g for infinitely thin films. This ratio can be used to estimate
β)
x
p0 4σ
kT G0 2πm d0
the ratio of molecules entering the membrane divided by the total number of molecules hitting it. The experimental value of HOPDOB at 82 °C for this ratio is G0/d0 ) 1.38 × 10-8 m2 s-1 corresponding to β ) 1.3 × 10-4 and an activation energy of 26.4 kJ/mol. In DOBAMBC at 80 °C, we find that G0/d0 ) 5.36 × 10-9 m2 s-1, β ) 5.0 × 10-5, and EA ) 29.0 kJ/mol. The second material parameter, derived from the experimental d0, is the ratio D/H between diffusion coefficient and Henry’s constant. With the known material parameters of HOPDOB (density and molar mass) inserted in eq 13, one obtains a ratio D/H ) 10-17m2 s-1 Pa-1. For DOBAMBC, one finds a slightly lower value D/H ) 4.4 × 10-18 m2 s-1 Pa-1. There are only few data available for the gas solubility and diffusion in liquid crystalline materials. Because our experiments have been performed with air, and in particular the solubilities of nitrogen and oxygen may differ substantially, we do not expect to obtain an accurate estimate for the diffusion coefficients or solubilities for both gases from our measurement. For nitrogen and the nematic liquid crystal MBBA, both parameters have been determined in an experimental investigation by Chen and Springer.28 From their data obtained at high pressures, one can estimate a value of H ≈ 7.2 × 107 Pa for N2 in the nematic material MBBA at room temperature. The solubility of N2 strongly increases with temperature. The diffusion coefficient of N2 in MBBA at room temperature (28) Chen, G.-H.; Springer, J. Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 2000, 339, 31.
Gas Permeation through Ultrathin Liquid Films
is about 1.7 × 10-11 m2 s-1.28 The ratio D/H is 2.4 × 10-19 m2 s-1 Pa-1, more than 1 order of magnitude lower than our result for the smectics. With increasing temperature, both the solubility and the diffusion coefficient increase strongly; at 50 °C (MBBA already isotropic) D is about 2.5 × 10-11 m2 s-1, H ≈ 5.2 × 106 Pa (at atmospheric pressure), and D/H ≈ 4.8 × 10-18 m2 s-1 Pa-1 in the same range as the ratios determined in our experiments. Of course, the data cannot be compared directly since DOBAMBC and HOPDOB are in higher ordered mesophases and no corresponding data are available to our knowledge for smectic materials, but the satisfactory agreement with the MBBA data strengthens confidence in the model presented. Because of the strong temperature dependence of several of the involved parameters in eqs 13 and 14 (D, β, and H), the permeation curves also exhibit a pronounced temperature dependence. However, the temperature dependence of g cannot be reduced to simple thermal activation. On approaching the low-temperature smectic I* phase in DOBAMBC, molecular packing in the layers changes dramatically, and the free volume decreases. In Sm I*, the hexagonal lattice will tighten the film and make it practically impermeable for gas molecules. Therefore, the dominating effect will be a decreasing solubility of the gases in the smectic mesophases with lower temperatures, in particular when low-temperature smectic mesophases are approached. The same holds qualitatively for HOPDOB; the accuracy of the experimental data at present is not sufficient to resolve unambiguously possible influences of a higher ordered surface layer above the smectic C to smectic B (crystallization) transition. A qualitative comparison with the nematic is possible. The tendency to become much less soluble in MBBA when the temperature approaches the crystallization point of the nematic has also been reported, at least for nitrogen, in ref 28. In the same way as the permeation coefficients measured in our smectic materials, the sorption curves in MBBA have a linear trend with increasing temperature. Summary Gas permeation through ultrathin films of smectic liquid crystals has been studied by measuring the gas loss through the membrane of self-supporting smectic bubbles. A molecular diffusion model has been presented, which
Langmuir, Vol. 18, No. 1, 2002 119
relates the macroscopic fitting parameter g for the radius shrinking in the experiments to microscopic quantities, the diffusion coefficient of gas particles in the liquid crystalline matrix, and sorption of gas particles in the smectic films. For thick films (.10 nm), diffusion and gas sorption determine the permeation of air through the smectic membranes, while at low film thicknesses another effect limits the permeation. The permeation increases much slower with decreasing film thickness than expected from the standard model. Judging from our fit curves, a saturation value is approached. Of course, the film thickness can only be varied in units of the smectic layer thickness of about 3 nm for both investigated compounds, and films of less than two layers are instable. Therefore, this limit cannot be reached practically. The model introduced here to explain the observed deviations assumes a bottleneck effect at the air/film interface. Because the exchange of gas particles by diffusion between the opposite air/film interfaces becomes very efficient in films of a few molecular layers, the limiting factor is the supply of gas particles from the gas phase. In the limit d f 0, half of the total number of gas particles that enter the membrane will pass it. However, only a small fraction (10-4) of the gas molecules hitting the film surface will solve in the liquid. A corresponding energy barrier for gas particles to enter the liquid crystalline membrane (of the order of 26 ... 29 kJ/mol for the two investigated compounds) has been introduced in our model. The presented method is hardly suitable to determine diffusion coefficients and solubilities of gases in liquid crystals. For a more quantitative investigation of the involved material parameters, in particular of the solubilities, the experiment should be repeated with pure gases, which requires a completely new preparation technique and has not been in the scope of this work. However, the qualitative results for thin films, in particular the film thickness dependence of permeation, should be of general validity and of importance for the description of permeation effects in ultrathin liquid layers. Acknowledgment. This study was supported by the DFG with SFB 294. LA010627A
