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The permeation of cork revisited Orlando MND Teodoro J. Agric. Food Chem., Just Accepted Manuscript • DOI: 10.1021/acs.jafc.6b00637 • Publication Date (Web): 01 May 2016 Downloaded from http://pubs.acs.org on May 10, 2016
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Journal of Agricultural and Food Chemistry
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The permeation of cork revisited
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Orlando M.N.D Teodoro
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Center for Physics and Technological Research - CEFITEC, Physics Department, Faculty of Sciences
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and Technology, Universidade Nova de Lisboa, Campus de Caparica, P2829-516 CAPARICA
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PORTUGAL
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Corresponding author: Orlando Teodoro,
[email protected] 7
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Abstract
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The permeation mechanism of gases through cork is revisited. We show that some of the recent
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work on this topic describing the permeation as pure “Fickian” diffusion is not well supported. As
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matter of fact, the permeation through cork is better described by the flow throughout small
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channels in the walls of cork cells, the plasmodesmata. Calculations show that molecular flow
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through these channels give flow rates well in the range of the experimental values. The small
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dependence of the flow rate in the feeding pressure maybe easily explained by a small contribution
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of viscous flow due to the relatively large diameter of these channels. Moreover, a “Fickian” model
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fail to explain new experiments on permeation as function of cork compression. However, flow
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through channels can easily explain the decrease of permeation for compressed cork.
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Introduction
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In recent papers [1-4], the author and coworkers have shown that the mechanism of permeation
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for gases in uncompressed cork is different of that for vapors (water and ethanol). Permeation for
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gases follows the Maxwell-Boltzmann distribution of velocities well in accordance with a transport
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mechanism based on molecular flow through small channels (called Knudsen flow by some
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authors). For water and ethanol vapor, the higher permeation rates suggested a second route for
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permeation. Since sorption for these vapors were much higher, we proposed that sorption followed 1
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by diffusion in the dense cork walls should play the main role in the transport. We believe that
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these conclusions were well supported by our experiments as extensively discussed in the above-
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mentioned work.
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However, other authors working in the same topic reported different findings [5,6] contradicting
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our previous conclusions. These papers also address the type of flow in cork. Lequin, Lagorce-
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Tachon and co-authors did sorption and permeability experiments similar to ours. However a
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serious inaccuracy is found in their theoretical discussion, misleading them in the main conclusions.
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Therefore, they conclude that transport of oxygen is due to pure diffusion (“Fickian mechanism”),
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not to molecular flow through small channels, as we proposed before. Moreover, there are no
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results for other gases than oxygen. Indeed, our results for different gases (He, N2, O2, CO2 and
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C2H2F4) have shown a very good dependence of the flow on the thermal velocity that is a well-
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known function of the gas mass and temperature. Only for water and ethanol, vapor and liquids,
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we found a clear different behavior [2].
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In this work we argue on the conclusions of Lequin, Lagorce-Tachon and co-authors and we discuss
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the proper approach to calculate the permeation in porous media and how to properly model the
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permeation of cork through the known channels in cork cells having diameters in the range of 100
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nm [7].
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Permeability through porous membranes
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According to Lequin and Lagorce-Tachon [5,6] the permeation ܲ through a porous membrane in
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the so-called Knudsen regime is given, in kg/m/Pa/s units, by (Equation 4 in reference 5 and
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equation 6 in reference 6):
ܲ =
݀ 8ܯ ඨ 3 ߨܴܶ
Eq. 1
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where d is the tube diameter, R the gas constant, T the absolute temperature and M the molar
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mass of the gas. Such equation seems to be based on the Knudsen description for the transport of 2
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gases in long tubes under molecular flow, which is well described in many vacuum technology
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books. E.g. in reference [8], conductance C of a long tube of length ݈ under molecular flow is
=ܥ
݀ ଷ ߨܴܶ ඨ 3݈ 2ܯ
Eq. 2
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More accurate descriptions can be provided by more calculations on rarefied flow dynamics as cited
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in reference [4]. The volumetric flow rate ܳ across this tube under a pressure difference ∆ is
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given by:
ܳ = = ∆ܥ
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݀ ଷ ߨܴܶ ඨ ∙ ∆ 3݈ 2ܯ
Eq. 3
And the mass flow rate comes
ܳ = ܳ ∙ ܳ=
ܯ ܴܶ
݀ ଷ ߨܯ ඨ ∙ ∆ 3݈ 2ܴܶ Eq. 4
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Since permeability may be defined as
ܲ =
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ܳ∙ݏ ∆ ∙ ܣ
Eq. 5
where ݏis the thickness and ܣthe permeation area, we can derive the permeability as
ܲ =
݀ ଷ ߨ ܯ1 ඨ ∙ 3 2ܴܶ ܣ
Eq. 6
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In the above equation we took thickness ݏas the tube length l. If we take the area from the tube
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cross-section, we will obtain Eq. 1, the same as the equation in the cited papers [5,6].
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However, this approach is not correct. It assumes that permeation is done through a continuous of
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tubes in parallel like a honeycomb, what is definitely not the case for cork cell walls. This
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assumption is not only geometrically impossible (hexagons instead of circular tubes would be
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required), but also does not takes in account the blanked area of a surface with pores or channels,
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as is the cork cell wall. Figure 1 illustrates this point for a wall having equally spaced number of
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tubes of exaggerated diameter.
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The consequence of this mistaken equation led the authors to formulate wrong results and
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conclusions in both papers. For example, in paper [1] is stated:
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“In the other way round, if the mean pore diameter is estimated from our experimental
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permeability value, the value found for the pore diameter is of 0.4 nm, which is the size of
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the oxygen molecule. It is therefore too small to obey the Knudsen regime (pore diameter
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between 1 and 500 nm).” …
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“Therefore, we can reasonably assume that the transfer of oxygen in cork is essentially
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controlled by the diffusion through the cork cell walls. This conclusion is not in agreement
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with a recent work on cork permeability to gases20.”
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Cited reference 20 is our work [1]. For example, in reference [6] calculations based in the
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mistaken equation at the end of page 9182 obviously led to an “unrealistic pore diameter”
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excluding any further consideration of flow through pores.
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The proper procedure to calculate the permeability would lead to a diameter well in the range of
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the known plasmodesma diameter, as was argued in [1-4] but immediately excluded by Lequin and
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Lagorce-Tachon in their papers. Such procedure should take in account the number of pores per
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unit area, not only the area of each pore.
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Using the rough approach explained in reference [1] describing the cork cell as a cube of 30 µm of
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edge having 1 to 3 cylindrical pores per side with diameter in the a range of 30 to 100 nm, we can
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easily realize that the blanked area of the cell wall is about 105 larger than the porous area. Thus, 4
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it is incorrect to assume that cork cell walls are a continuous of pores as it is assumed in equation
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1 [5,6].
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Therefore, the rational to propose “Fickian diffusion” for their experimental results lacks of a
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theoretical support. Moreover, according to Lagorce-Tachon et.al, [6] the pressure dependence of
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the diffusion requires the introduction of the quantity “activation volume”. This quantity was
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calculated for cork by the authors, but the obtained result was several orders of magnitude
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different from all those cited in the literature. Despite activation volumes in the literature were
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from very different materials, there is no possible comparison with available data, what makes
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such description very controversial.
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Permeability dependence of pressure
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How can we explain the slight increase of the permeability with pressure, as plotted in Fig.2 of
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reference [6]? If we plot the pressure dependence with a vertical axis starting from zero, slope is
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not that pronounced. Indeed, in our experiments we also confirmed a similar dependence. To
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explain this behavior we need to return to the equations of gas flow.
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Molecular flow in a cylindrical tube occurs when the particle mean free path is larger than its
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diameter. If it is smaller, then collisions between particles have to be taken into account. If such
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collisions are more often than collisions with tube walls, flow is in viscous regime. Between the
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viscous and the molecular regime we have a combined flow regime often called transitional regime
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(see e.g. reference [8,9]).
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In the transitional regime, flow is described by kinetic equations as it is well described in references
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[10,11]. However, in many textbooks (e.g. ref. [8,9]) flow in this regime is approximated by a
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simple combination of molecular and viscous flow given by (for the volumetric flow rate, ܳ ):
ܳ =
ߨ݀ ସ ݀ ଷ 2ܴܶ ∆ ∙ + ܼ ∙ ඨ ∙ ∆ 128ߟ݈ 3݈ ߨܯ
Eq. 7
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Where ߟ is the gas viscosity, is the average pressure along the tube and ܼ is a weighting
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parameter ranging from 0.865 to 1 (often taken as 1). From equation 7 we can now rewrite
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equation 6 leading to:
ܲ =
1 ߨ݀ ସ ܯ ݀ ଷ ߨܯ ቌ ∙ +ܼ∙ ඨ ቍ ܣ128ߟ ܴܶ 3 2ܴܶ
Eq. 8
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The pressure dependence in now obvious in the above equation. For a capillary of 1.5 µm long and
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100 nm in diameter at a mean pressure of 0.5 atm, the first term inside the brackets is 5% of the
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total giving noticeable pressure dependence. The larger the capillary diameter or the mean
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pressure, the larger will be this contribution.
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Consequently, we can explain the small pressure dependence of the permeability considering that
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flow through channels is not exclusively molecular but has some viscous contribution. Since there
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is no sharp transition between flow regimes [7], we should take into account both contributions.
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Compression of cork
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Another sound argument against the “Fickian” diffusion explanation of Lequin, Lagorce-Tachon and
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co-authors is the effect of compression of cork. In Fig.2 we plot the relative permeation as function
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of the sample compression. Cork samples were hold by a technique similar to the one described in
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[1]. However, a highly porous metallic cylinder was used to compress the cork through the hole
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against a second similar cylinder in the opposite side. Compression was done in small steps and
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the permeation was measured.
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In all samples, permeability was decreased by a factor ranging roughly 10 to 100. These results
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are in accordance with the well known fact that compressed cork provides a better sealing than
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uncompressed cork, making it suitable, for instance, to seal pressurized wine in champagne bottles.
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If the gas flow is performed by “Fickian” diffusion through the dense cell walls, then there is no
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apparent reason for the permeability decrease with compression. The number of dense walls is 6
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kept constant after compression and the dense route for diffusion is unchanged. The only
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difference is that the cells volume is decreased and the walls are set in a layer-by-layer
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arrangement, surely with some corrugation.
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However, if permeation is mainly due to flow through capillaries then, when cell walls are piled up
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in a layer-by-layer fashion, capillary entrances are plugged by the opposite cell wall, decreasing its
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conductance until the capillary entrance is completely closed. The flow rate after total compression
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is the one that could be related with a pure diffusional mechanism. But this flow is typically 1 or 2
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orders of magnitude lower than the flow observed for uncompressed cork.
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Another argument against the flow through pores is that plasmodesmata should be filled by
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cytoplasm residues. However, we should keep in mind that cork cells are completely hollowed of
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cytoplasm, being composed of only the cell wall and gas inside. Therefore, the same mechanism
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used by the tree to absorb or remove the cytoplasm may be used to keep the plasmodesma
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unclogged.
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We can conclude that arguments of Lequin, Lagorce-Tachon and co-authors are based in a
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mistaken equation and not supported by experiments. The gas flow through cork can be properly
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explained by molecular flow through small capillaries (plasmodesma). This model can explain the
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small pressure dependence of the permeability as well as the effect of cork compression.
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The work on cork here presented was supported by FCT-MEC through CEFITEC research grant
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UID/FIS/00068/2013.
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References
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1.
152 153
Food Chem. 2011, 59, 3590–3597. 2.
154 155
Faria, D. P.; Fonseca, A. L.; Pereira, H.; Teodoro, O. M. N. D. Permeability of cork to gases. J. Agric.
Fonseca, A. L.; Brazinha, C.; Pereira, H.; Crespo, J. G.; Teodoro, O. M. N. D. Permeability of Cork for Water and Ethanol. J. Agric. Food Chem. 2013, 61, 9672–9679.
3.
Brazinha, C.; Fonseca, A. P.; Pereira, H.; Teodoro, O. M. N. D. Gas transport through cork: Modelling
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gas permeation based on the morphology of a natural polymer material. J. Memb. Sci. 2013, 428, 52–
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62.
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Teodoro, O. M. N. D.; Fonseca, A. L.; Pereira, H.; Moutinho, A. M. C. Vacuum physics applied to the transport of gases through cork. Vacuum 2014, 109, 397–400. 7
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5.
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Lequin, S.; Chassagne, D.; Karbowiak, T.; Simon, J.-M.; Paulin, C.; Bellat, J. P. Diffusion of oxygen in cork. J. Agric. Food Chem. 2012, 60, 3348–56.
6.
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Lagorce-Tachon, A.; Karbowiak, T.; Simon, J.; Bellat, J. Diffusion of Oxygen through Cork Stopper: Is It a Knudsen or a Fickian Mechanism? J. Agric. Food Chem. 2014, 62, 9180–9185.
7.
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Teixeira, R. T.; Pereira, H. Ultrastructural observations reveal the presence of channels between cork cells. Microsc. Microanal. 2009, 15, 539–44.
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8.
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9. Handbook of Vacuum Technology ed. Karl Jousten, Wiley-VCH Weinheim (2009), p.150.
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10. Sharipov, F., Seleznev, V. Data on internal rarefied gas flows. J. Phys. Chem. Ref. Data. 27(3), 657-706
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Foundations of Vacuum Science and Technology, Edited by J.M. Laferty, John Wiley & Sons 1998, p.88.
(1998). 11. Sharipov, F. Rarefied Gas Dynamics. Fundamentals for Research and Practice. (Wiley-VCH, 2016).
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Captions:
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Figure 1- Section of a cork wall with some channels to show how permeability can be obtained.
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Figure 1- Relative permeability of cork as function of compression for 17 samples having thicknesses ranging 2 to 3 mm.
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186 This is the area to take in account to calculate the permeability
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The cross section of a single tube cannot be used to calculate the permeability of this wall
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194 Figure 1- Section of a cork wall with some channels to show how permeability can be obtained.
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1.00E+01
∆Pk/Pk
1.00E+00
1.00E-01
1.00E-02 0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
∆L/L
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Figure 2- Relative permeability of cork as function of compression for 17 samples having
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thicknesses ranging 2 to 3 mm.
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