The Permeation of Cork Revisited - Journal of Agricultural and Food

DOI: 10.1021/acs.jafc.6b00637. Publication Date (Web): May 1, 2016. Copyright © 2016 American Chemical Society. *(O.T.) E-mail: [email protected]. Note:...
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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]

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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.

158 159

4.

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.

163 164

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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

169 170

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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