Subscriber access provided by UNIV OF PITTSBURGH
Correspondence/Rebuttal
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
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
Journal of Agricultural and Food Chemistry is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 11
Journal of Agricultural and Food Chemistry
1
The permeation of cork revisited
2
Orlando M.N.D Teodoro
3
Center for Physics and Technological Research - CEFITEC, Physics Department, Faculty of Sciences
4
and Technology, Universidade Nova de Lisboa, Campus de Caparica, P2829-516 CAPARICA
5
PORTUGAL
6
Corresponding author: Orlando Teodoro,
[email protected] 7
8
Abstract
9
The permeation mechanism of gases through cork is revisited. We show that some of the recent
10
work on this topic describing the permeation as pure “Fickian” diffusion is not well supported. As
11
matter of fact, the permeation through cork is better described by the flow throughout small
12
channels in the walls of cork cells, the plasmodesmata. Calculations show that molecular flow
13
through these channels give flow rates well in the range of the experimental values. The small
14
dependence of the flow rate in the feeding pressure maybe easily explained by a small contribution
15
of viscous flow due to the relatively large diameter of these channels. Moreover, a “Fickian” model
16
fail to explain new experiments on permeation as function of cork compression. However, flow
17
through channels can easily explain the decrease of permeation for compressed cork.
18
19
Introduction
20
In recent papers [1-4], the author and coworkers have shown that the mechanism of permeation
21
for gases in uncompressed cork is different of that for vapors (water and ethanol). Permeation for
22
gases follows the Maxwell-Boltzmann distribution of velocities well in accordance with a transport
23
mechanism based on molecular flow through small channels (called Knudsen flow by some
24
authors). For water and ethanol vapor, the higher permeation rates suggested a second route for
25
permeation. Since sorption for these vapors were much higher, we proposed that sorption followed 1
ACS Paragon Plus Environment
Journal of Agricultural and Food Chemistry
Page 2 of 11
26
by diffusion in the dense cork walls should play the main role in the transport. We believe that
27
these conclusions were well supported by our experiments as extensively discussed in the above-
28
mentioned work.
29
However, other authors working in the same topic reported different findings [5,6] contradicting
30
our previous conclusions. These papers also address the type of flow in cork. Lequin, Lagorce-
31
Tachon and co-authors did sorption and permeability experiments similar to ours. However a
32
serious inaccuracy is found in their theoretical discussion, misleading them in the main conclusions.
33
Therefore, they conclude that transport of oxygen is due to pure diffusion (“Fickian mechanism”),
34
not to molecular flow through small channels, as we proposed before. Moreover, there are no
35
results for other gases than oxygen. Indeed, our results for different gases (He, N2, O2, CO2 and
36
C2H2F4) have shown a very good dependence of the flow on the thermal velocity that is a well-
37
known function of the gas mass and temperature. Only for water and ethanol, vapor and liquids,
38
we found a clear different behavior [2].
39
In this work we argue on the conclusions of Lequin, Lagorce-Tachon and co-authors and we discuss
40
the proper approach to calculate the permeation in porous media and how to properly model the
41
permeation of cork through the known channels in cork cells having diameters in the range of 100
42
nm [7].
43
44
Permeability through porous membranes
45
According to Lequin and Lagorce-Tachon [5,6] the permeation ܲ through a porous membrane in
46
the so-called Knudsen regime is given, in kg/m/Pa/s units, by (Equation 4 in reference 5 and
47
equation 6 in reference 6):
ܲ =
݀ 8ܯ ඨ 3 ߨܴܶ
Eq. 1
48
where d is the tube diameter, R the gas constant, T the absolute temperature and M the molar
49
mass of the gas. Such equation seems to be based on the Knudsen description for the transport of 2
ACS Paragon Plus Environment
Page 3 of 11
Journal of Agricultural and Food Chemistry
50
gases in long tubes under molecular flow, which is well described in many vacuum technology
51
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
52
More accurate descriptions can be provided by more calculations on rarefied flow dynamics as cited
53
in reference [4]. The volumetric flow rate ܳ across this tube under a pressure difference ∆ is
54
given by:
ܳ = = ∆ܥ
55
݀ ଷ ߨܴܶ ඨ ∙ ∆ 3݈ 2ܯ
Eq. 3
And the mass flow rate comes
ܳ = ܳ ∙ ܳ=
ܯ ܴܶ
݀ ଷ ߨܯ ඨ ∙ ∆ 3݈ 2ܴܶ Eq. 4
56
Since permeability may be defined as
ܲ =
57
ܳ∙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
3
ACS Paragon Plus Environment
Journal of Agricultural and Food Chemistry
Page 4 of 11
58
In the above equation we took thickness ݏas the tube length l. If we take the area from the tube
59
cross-section, we will obtain Eq. 1, the same as the equation in the cited papers [5,6].
60
However, this approach is not correct. It assumes that permeation is done through a continuous of
61
tubes in parallel like a honeycomb, what is definitely not the case for cork cell walls. This
62
assumption is not only geometrically impossible (hexagons instead of circular tubes would be
63
required), but also does not takes in account the blanked area of a surface with pores or channels,
64
as is the cork cell wall. Figure 1 illustrates this point for a wall having equally spaced number of
65
tubes of exaggerated diameter.
66
The consequence of this mistaken equation led the authors to formulate wrong results and
67
conclusions in both papers. For example, in paper [1] is stated:
68
“In the other way round, if the mean pore diameter is estimated from our experimental
69
permeability value, the value found for the pore diameter is of 0.4 nm, which is the size of
70
the oxygen molecule. It is therefore too small to obey the Knudsen regime (pore diameter
71
between 1 and 500 nm).” …
72
“Therefore, we can reasonably assume that the transfer of oxygen in cork is essentially
73
controlled by the diffusion through the cork cell walls. This conclusion is not in agreement
74
with a recent work on cork permeability to gases20.”
75
Cited reference 20 is our work [1]. For example, in reference [6] calculations based in the
76
mistaken equation at the end of page 9182 obviously led to an “unrealistic pore diameter”
77
excluding any further consideration of flow through pores.
78
The proper procedure to calculate the permeability would lead to a diameter well in the range of
79
the known plasmodesma diameter, as was argued in [1-4] but immediately excluded by Lequin and
80
Lagorce-Tachon in their papers. Such procedure should take in account the number of pores per
81
unit area, not only the area of each pore.
82
Using the rough approach explained in reference [1] describing the cork cell as a cube of 30 µm of
83
edge having 1 to 3 cylindrical pores per side with diameter in the a range of 30 to 100 nm, we can
84
easily realize that the blanked area of the cell wall is about 105 larger than the porous area. Thus, 4
ACS Paragon Plus Environment
Page 5 of 11
Journal of Agricultural and Food Chemistry
85
it is incorrect to assume that cork cell walls are a continuous of pores as it is assumed in equation
86
1 [5,6].
87
Therefore, the rational to propose “Fickian diffusion” for their experimental results lacks of a
88
theoretical support. Moreover, according to Lagorce-Tachon et.al, [6] the pressure dependence of
89
the diffusion requires the introduction of the quantity “activation volume”. This quantity was
90
calculated for cork by the authors, but the obtained result was several orders of magnitude
91
different from all those cited in the literature. Despite activation volumes in the literature were
92
from very different materials, there is no possible comparison with available data, what makes
93
such description very controversial.
94
95
Permeability dependence of pressure
96
How can we explain the slight increase of the permeability with pressure, as plotted in Fig.2 of
97
reference [6]? If we plot the pressure dependence with a vertical axis starting from zero, slope is
98
not that pronounced. Indeed, in our experiments we also confirmed a similar dependence. To
99
explain this behavior we need to return to the equations of gas flow.
100
Molecular flow in a cylindrical tube occurs when the particle mean free path is larger than its
101
diameter. If it is smaller, then collisions between particles have to be taken into account. If such
102
collisions are more often than collisions with tube walls, flow is in viscous regime. Between the
103
viscous and the molecular regime we have a combined flow regime often called transitional regime
104
(see e.g. reference [8,9]).
105
In the transitional regime, flow is described by kinetic equations as it is well described in references
106
[10,11]. However, in many textbooks (e.g. ref. [8,9]) flow in this regime is approximated by a
107
simple combination of molecular and viscous flow given by (for the volumetric flow rate, ܳ ):
ܳ =
ߨ݀ ସ ݀ ଷ 2ܴܶ ∆ ∙ + ܼ ∙ ඨ ∙ ∆ 128ߟ݈ 3݈ ߨܯ
Eq. 7
5
ACS Paragon Plus Environment
Journal of Agricultural and Food Chemistry
Page 6 of 11
108
Where ߟ is the gas viscosity, is the average pressure along the tube and ܼ is a weighting
109
parameter ranging from 0.865 to 1 (often taken as 1). From equation 7 we can now rewrite
110
equation 6 leading to:
ܲ =
1 ߨ݀ ସ ܯ ݀ ଷ ߨܯ ቌ ∙ +ܼ∙ ඨ ቍ ܣ128ߟ ܴܶ 3 2ܴܶ
Eq. 8
111
The pressure dependence in now obvious in the above equation. For a capillary of 1.5 µm long and
112
100 nm in diameter at a mean pressure of 0.5 atm, the first term inside the brackets is 5% of the
113
total giving noticeable pressure dependence. The larger the capillary diameter or the mean
114
pressure, the larger will be this contribution.
115
Consequently, we can explain the small pressure dependence of the permeability considering that
116
flow through channels is not exclusively molecular but has some viscous contribution. Since there
117
is no sharp transition between flow regimes [7], we should take into account both contributions.
118
119
Compression of cork
120
Another sound argument against the “Fickian” diffusion explanation of Lequin, Lagorce-Tachon and
121
co-authors is the effect of compression of cork. In Fig.2 we plot the relative permeation as function
122
of the sample compression. Cork samples were hold by a technique similar to the one described in
123
[1]. However, a highly porous metallic cylinder was used to compress the cork through the hole
124
against a second similar cylinder in the opposite side. Compression was done in small steps and
125
the permeation was measured.
126
In all samples, permeability was decreased by a factor ranging roughly 10 to 100. These results
127
are in accordance with the well known fact that compressed cork provides a better sealing than
128
uncompressed cork, making it suitable, for instance, to seal pressurized wine in champagne bottles.
129
If the gas flow is performed by “Fickian” diffusion through the dense cell walls, then there is no
130
apparent reason for the permeability decrease with compression. The number of dense walls is 6
ACS Paragon Plus Environment
Page 7 of 11
Journal of Agricultural and Food Chemistry
131
kept constant after compression and the dense route for diffusion is unchanged. The only
132
difference is that the cells volume is decreased and the walls are set in a layer-by-layer
133
arrangement, surely with some corrugation.
134
However, if permeation is mainly due to flow through capillaries then, when cell walls are piled up
135
in a layer-by-layer fashion, capillary entrances are plugged by the opposite cell wall, decreasing its
136
conductance until the capillary entrance is completely closed. The flow rate after total compression
137
is the one that could be related with a pure diffusional mechanism. But this flow is typically 1 or 2
138
orders of magnitude lower than the flow observed for uncompressed cork.
139
Another argument against the flow through pores is that plasmodesmata should be filled by
140
cytoplasm residues. However, we should keep in mind that cork cells are completely hollowed of
141
cytoplasm, being composed of only the cell wall and gas inside. Therefore, the same mechanism
142
used by the tree to absorb or remove the cytoplasm may be used to keep the plasmodesma
143
unclogged.
144
We can conclude that arguments of Lequin, Lagorce-Tachon and co-authors are based in a
145
mistaken equation and not supported by experiments. The gas flow through cork can be properly
146
explained by molecular flow through small capillaries (plasmodesma). This model can explain the
147
small pressure dependence of the permeability as well as the effect of cork compression.
148
The work on cork here presented was supported by FCT-MEC through CEFITEC research grant
149
UID/FIS/00068/2013.
150
References
151
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
156
gas permeation based on the morphology of a natural polymer material. J. Memb. Sci. 2013, 428, 52–
157
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
ACS Paragon Plus Environment
Journal of Agricultural and Food Chemistry
160
5.
161 162
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
Page 8 of 11
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.
165
Teixeira, R. T.; Pereira, H. Ultrastructural observations reveal the presence of channels between cork cells. Microsc. Microanal. 2009, 15, 539–44.
166
8.
167
9. Handbook of Vacuum Technology ed. Karl Jousten, Wiley-VCH Weinheim (2009), p.150.
168
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).
171
8
ACS Paragon Plus Environment
Page 9 of 11
172
Journal of Agricultural and Food Chemistry
Captions:
173
174
Figure 1- Section of a cork wall with some channels to show how permeability can be obtained.
175
176
177 178
Figure 1- Relative permeability of cork as function of compression for 17 samples having thicknesses ranging 2 to 3 mm.
179
180
9
ACS Paragon Plus Environment
Journal of Agricultural and Food Chemistry
Page 10 of 11
181
182
183
184
185
186 This is the area to take in account to calculate the permeability
187
188
The cross section of a single tube cannot be used to calculate the permeability of this wall
189
190
191
192
193
194 Figure 1- Section of a cork wall with some channels to show how permeability can be obtained.
195
196
197
10
ACS Paragon Plus Environment
Page 11 of 11
Journal of Agricultural and Food Chemistry
198
199
200
201
202
203
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
204
205
206
207
Figure 2- Relative permeability of cork as function of compression for 17 samples having
208
thicknesses ranging 2 to 3 mm.
209
11
ACS Paragon Plus Environment
