# Air leakage in seals with application to syringes

N. Rodriguez  BD Medical-Pharmaceutical Systems, 1 Becton Drive, Franklin
Lakes, NJ 07417, USA  A. Tiwari  PGI-1, FZ Jülich, Germany
MultiscaleConsulting, Wolfshovener str. 2, 52428 Jülich, Germany  B.N.J.
Persson  PGI-1, FZ Jülich, Germany  MultiscaleConsulting, Wolfshovener str. 2,
52428 Jülich, Germany

###### Abstract

We study the leakage of air in syringes with Teflon coated rubber stopper and
glass barrel. The leakrate depends on the interfacial surface roughness, the
viscoelastic properties of the rubber and on the elastoplastic properties of
the Teflon coating. The measured leakage rates are compared to the predictions
of a simple theory for gas flow, which takes into account both the diffusive
and ballistic air flow, and the elastoplastic multiscale contact mechanics
which determines the probability distribution of interfacial separations. The
theory shows that the interfacial air flow (leakage) channels are so narrow
that the gas flow is mainly ballistic (the so called Knudsen limit). The
implications for container closure integrity is discussed.

1 Introduction

All solids have surface roughness, which has a huge influence on a large
number of physical phenomena such as adhesion, friction, contact mechanics and
the leakage of seals[1, 2, 3, 4, 5, 6, 7, 8]. Thus when two solids with
nominally flat surfaces are squeezed into contact, unless the applied
squeezing pressure is high enough, or the elastic modulus of at least one of
the solids low enough, a non-contact region will occur at the interface. If
the non-contact region percolate, open flow channels exist, extending from one
side of the nominal contact region to the other side. This will allow fluid to
flow at the interface from a high fluid pressure region to a low pressure
region.

We consider the leakage of air at the interface between a Teflon coated rubber
stopper and a glass barrel. We have studied this problem in Ref. [9] but here
we extend that study by using a new design of the stopper involving much
higher contact pressures, where the contact is close to the percolation
threshold.

Teflon (polytetrafluorethylen) and other films (e.g., ultra-high-molecular-
weight polyethylene or ethylene-tetrafluorethylene-copolymer) are used in
laminating rubber stoppers and have elastic modulus $100-1000$ times higher
than the typical rubbers stoppers ($2-6\ {\rm MPa}$). Thus, the average
interfacial separation, resulting from the surface roughness, is much larger
than for the uncoated rubber stopper, and the contact area percolation
threshold may not be achieved. Therefore, in any laminated rubber stoppers an
accurate calculation of the interfacial separations as well as design
parameters (contact pressure, geometry, etc…) is particularly important to
assure container closure integrity, low weight losses, functional performance,
and no microbial ingress during the shelf life and use of the product. One way
to study the size of the most narrow constrictions in the open (percolating)
channels at the stopper-barrel interface is by measuring the air leakage rate
in syringes (with closed needle).

2 Experimental results

The Teflon laminated rubber stopper used in this study has three ribs 1-3
which contact the glass barrel. The ribs have (half) circular cross-section
with the front and middle ribs with the diameter $0.4\ {\rm mm}$, and back rib
3 with the diameter $0.8\ {\rm mm}$. We first determine the width and the
average contact pressure acting in the contact region. We also study the
surface topography of the Teflon surface on the ribs. Finally we present the
results of the air leakage experiments.

Figure 1:  Optical picture of the first rib contact region for Teflon
laminated rubber stopper in a glass barrel. The front and the middle ribs, 1
and 2, are nominally identical and with contact width $w\approx 0.42\ {\rm
mm}$ in the axial direction. The third outer rib 3 is wider but the contact
pressure much lower, and this rib has only a negligible influence on the air
leakage rate.

Nominal contact width $w$

Using an optical microscope we have measured the width in the axial direction
of the contact regions between the Teflon laminated rubber stopper and the
glass barrel (see Fig. 1). The two inner ribs, 1 and 2, are nominally
identical and with the contact width $w\approx 0.42\ {\rm mm}$. The third
(outer) rib 3 is wider but the contact pressure is much lower, and this rib
has only a small influence on the air leakage rate. Note that the pressure is
so high as to fully flatten the rib 1 and 2. We will assume below that the
contact pressure is Hertzian-like but this is only a rough estimation because
of the large deformations involved.

The old design of the stopper studied in Ref. [9] had two ribs, the inner rib
with rectangular cross section with the contact width $\approx 1.2\ {\rm mm}$
and back rib circular with the diameter $1\ {\rm mm}$. In this case the
average contact pressure ($\sim 1\ {\rm MPa}$) is much lower than for the new
design.

Figure 2:  Experimental method used to determine the radial force squeezing
the stopper against the barrel. Two steel blocks with a cylinder cavity with
the same radius as the inner radius $R$ of the barrel is squeezed together,
with the rubber stopper inserted, with such a force $F$ that the gap just
closes without any repulsive force from the steel-steel contact. The steel
surfaces are lubricated so negligible shear stress occur in the contact
between the stopper and the steel surface. The average pressure $p_{0}$ acting
in the contact region between the stopper and the steel surface is determined
by $p_{0}Dw=F$, where $w$ is the width in the axial direction of the contact
region and $D=2R$ the inner diameter of the barrel.

Average contact pressure $p_{0}$

We have measured the average contact pressure between the stopper and the
barrel using the set-up shown in Fig. 2. Two steel blocks with a cylinder
cavity with the same radius as the inner radius $R$ of the barrel is squeezed
together with the stopper inbetween, with such a force $F$ that the gap just
closes without any repulsive force at the steel-steel contact. The steel
surfaces are lubricated so negligible shear stress occurs in the contact
between the stopper and the steel surface. If $w$ is the width in the axial
direction of the contact region and if $p_{0}$ denotes the average pressure
acting in the contact region between the stopper and the steel surface then

$F=\int_{-\pi/2}^{\pi/2}d\phi\ wRp_{0}{\rm cos}\phi=2wRp_{0}.$

Thus $p_{0}=F/wD$ where $D=2R$ is the inner diameter of the barrel. Using this
equation we can determine the average pressure acting on each rib by removing
the other ribs. In the present case this shows that for the inner two ribs 1
and 2 (which are nominal identical with contact width $w\approx 0.42\ {\rm
mm}$), the (average) contact pressure $p_{0}\approx 3.2\ {\rm MPa}$, and the
maximal (Hertz) contact pressure $p_{\rm max}=3p_{0}/2\approx 4.8\ {\rm MPa}$,
where we have used $D=6\ {\rm mm}$. The third (outermost) rib 3 is wider but
the contact pressure much smaller, so this rib does not affect the air leakage
rate.

Figure 3:  The surface roughness power spectra obtained from stylus topography
measurements on the inner rib 1 (red) and the next inner rib 2 (blue). The
solid lines are the effective power spectra of the plastically deformed Teflon
surfaces using the Teflon penetration hardness $\sigma_{\rm P}=13\ {\rm MPa}$.

Surface roughness power spectrum

The most important information about the surface topography of a rough surface
is the surface roughness power spectrum. For a one-dimensional (1D) line scan
$z=h(x)$ the power spectrum is given by

$C_{\rm 1D}(q)={1\over 2\pi}\int dx\ \langle h(x)h(0)\rangle e^{iqx}$ $None$

where $\langle..\rangle$ stands for ensemble averaging. For surfaces with
isotropic roughness the 2D power spectrum $C(q)$ can be obtained directly from
$C_{\rm 1D}(q)$ as described elsewhere [10, 11]. For randomly rough surfaces,
all the (ensemble averaged) information about the surface is contained in the
power spectrum $C(q)$ (see Ref. [5, 6]). For this reason the only information
about the surface roughness which enter in contact mechanics theories (with or
without adhesion) is the function $C(q)$. Thus, the (ensemble averaged) area
of real contact, the interfacial stress distribution, and the distribution of
interfacial separations, are all determined by $C(q)$.

We have measured the surface roughness profile using a stylus instrument
[Mitutoyo Portable Surface Roughness Measurement Surftest SJ-410 with a
diamond tip with the radius of curvature $r_{0}=1\ {\rm\mu m}$, and with the
tip-substrate repulsive force $F_{\rm N}=0.75\ {\rm mN}$ and the tip speed
$v=50\ {\rm\mu m/s}$], and calculated the surface roughness power spectrum as
described in detail elsewhere[6]. The dashed lines in Fig. 3 shows the surface
roughness power spectra obtained from stylus topography measurements on the
rib 1 (red) and rib 2 (blue). The solid lines are the effective power spectra
of the plastically deformed Teflon surfaces obtained as described in Ref. [12]
and summarized below. In the calculation we have used the rubber modulus
$E=4.6\ {\rm MPa}$ and Poisson ratio $\nu=0.5$, the Teflon film thickness
$d=15\ {\rm\mu m}$, the Teflon modulus $E=500\ {\rm MPa}$ and Poisson ratio
$\nu=0.4$, and the Teflon penetration hardness $\sigma_{\rm P}=13\ {\rm MPa}$.
The penetration hardness of Teflon (without filler) is typically in the range
$20-30\ {\rm MPa}$, but we interpret the lower penetration hardness we use as
an effective hardness as the Teflon surface is exposed also to shear stresses
and wear processes as it moves in the glass barrel, which smooth the surface.
In addition, some plastic flow occur already when the contact pressure is well
below the penetration hardness[13]. In fact, it has been observed that Teflon
start to flow plastically around $13\ {\rm MPa}$ in wear experiments[14].

Here we note that the Persson contact mechanics theory assumes randomly rough
surface roughness. However, plastic deformation in general result in non-
random roughness. The procedure we use to obtain the power spectrum for
plastically deformed surfaces has been described in detail elsewhere[12] but
is briefly summarized here.

In elastic contact mechanics the contact area decreases continuously as we
increase the magnification and observe shorter wavelength roughness. Hence,
when a solid with surface roughness is squeezed against a flat rigid surface
with the force $F_{\rm N}$ the solid may deform elastically in the contact
regions observed at low magnification, where the contact area $A$ is large and
the contact stress $F_{\rm N}/A$ low, but plastically above some critical
magnification which depends on the penetration hardness. We take this plastic
deformation into account by smoothing (or removing) the short wavelength
roughness. We do this in such a way that the elastic contact area of the
plastically deformed surface will be the same (as a function of magnification)
as that obtained using the Persson elastoplastic contact mechanics theory[15,
16]. But in order for the surface to be randomly rough one must smooth the
surface everywhere, i.e., also in the non-contact area. We have argued
before[5, 17] that this has only a small influence of the interfacial
separation in the open flow channels relevant for the fluid leakage problem.
Nevertheless, the power spectrum obtained this way is an effective power
spectrum designed for special purpose, and it cannot be compared to the real
power spectrum obtained from the surface topography of the plastically
deformed surface, as discussed in our earlier studies[18].

Figure 4:  Experiments performed in order to measure the air leak-rate of
syringes (see text for details).

Leakage rate

We have measured the air leakage in 15 syringes using the same procedure as in
Ref. [9] and summarized in Fig. 4. We first assemble the barrel-stopper in
empty configuration with the stopper pushed to the end of the barrel,
resulting in a small volume $V_{0}$ of gas in the syringe at atmospheric
pressure. Next the needle is closed so no air can penetrate into the syringe
from the needle side, and the stopper is pulled back (retracted) to full fill
position resulting in a volume $L_{0}A_{0}$ of gas at low pressure. In the
first experiment the pull-force is immediately removed, which results in the
stopper moving to a new position $L(0)$ where the gas pressure $p_{\rm b}$ is
such that the pressure force $(p_{\rm a}-p_{\rm b})A_{0}$ (due to the
difference in the gas pressure outside and inside the barrel) is equal to the
stopper-barrel friction force. Next we repeat the experiment except now the
stopper is kept in the pulled back (retracted) position for some time $t_{1}$.
This results (due to air leakage at the barrel-stopper interface) in an
increase in the air pressure, and when the pull force is removed after some
time $t_{1}$ the stopper will move to a new position with $L(t_{1})>L(0)$. The
volume increase $\Delta V=[L(t_{1})-L(0)]A_{0}$ is due to the air leakage into
the syringe. However, the air pressure in the volume $\Delta V$ is not the
atmospheric air pressure $p_{\rm a}$ but is equal to $p_{\rm b}$. Thus, the
$\Delta V$ correspond to a volume $\Delta V_{\rm a}=\Delta Vp_{\rm b}/p_{\rm
a}$ of air of atmospheric pressure. Since no leakage is assumed to occur
during the first experiment we have $p_{\rm a}V_{0}=p_{\rm b}V_{\rm b}$ so
that $p_{\rm b}/p_{\rm a}=V_{0}/V_{\rm b}$. Hence the volume of air of
atmospheric pressure leaking into the syringe per unit time equal

$\dot{V}={\Delta V_{\rm a}\over t_{1}}={\Delta V\over t_{1}}{p_{\rm b}\over
p_{\rm a}}={\Delta V\over t_{1}}{V_{0}\over V_{\rm b}}$

It is easy to study the leakage for each rib separately by introducing a thin
cut in the other ribs through which the air (or fluid) can move with
negligible resistance.

In our experiments, the pressure difference between inside and outside the
syringe is about $1\ {\rm bar}$, and the average air leakage rate about
$2.2\times 10^{-4}\ {\rm mm^{3}/s}$. We repeated the test in five of the
fifteen syringes obtaining very similar leakage result as in the initial
measurements. In an earlier study (see Ref. [9]) with a different design of
the Teflon coated rubber stopper, where the average contact pressure was much
smaller, we observed larger air leakage rates, of order $4.8\times 10^{-3}\
{\rm mm^{3}/s}$, i.e. about a factor of 20 higher than for the new design.

Figure 5:  Diffusive (a) and ballistic (b) motion of the gas atoms (e.g. He)
in the critical junction. In case (a) the gas mean free path $\lambda$ is much
smaller than the gap width $h=u_{\rm c}$ and the gas molecules makes many
collisions with other gas molecules before a collision with the solid walls.
In the opposite limit, when $\lambda>>u_{\rm c}$ the gas molecules makes many
collisions with the solid wall before colliding with another gas molecule. In
the first case (a) the gas can be treated as a (compressible) fluid, while a
kinetic approach is needed in case (b). Figure 6:  Solid and dashed lines are
the calculated leakage rate as a function of the average contact pressure
using the plastically deformed surface (solid lines) and the original surface
(dashed lines). The + symbols are the measured leakage rates using different
(but nominally identical) clean syringes. The square symbol is for the same
type of syringe but with the glass barrel lubricated by silicone oil. The
$\times$ symbols are for a different Teflon coated rubber stopper where the
rib contact regions are much wider and the contact pressure much smaller (see
Ref. [9]). Figure 7:  The surface roughness power spectra obtained from stylus
topography measurements on the inner rib 1 (red) and the next inner rib 2
(blue). The green line is the power spectrum for another design of the
laminated rubber stopper with much wider rib contact regions. For this stopper
both engineering stylus, optical and Atomic Force Microscopy topography was
perform and the green line is a fit to all the measured data (see Ref. [9]).

3 Analysis of experiments

The leakrate of fluids at interfaces between solids with random roughness can
be calculated using the critical junction theory or the Bruggeman effective
medium theory as described in detail elsewhere[21, 22, 23, 24, 25, 26, 27, 28,
29]. In the critical junction theory it is assumed that all the pressure drop
in the fluid occurs at the most narrow constrictions along the biggest open
(percolating) flow channels. The more accurate effective medium theory
includes all flow channels in an approximate way but both theories gives
usually very similar results. The probability distribution of interfacial
separations, which enter in the theory for the leakage rate, is determined
using the Persson contact mechanics theory[15, 30, 31, 22, 32]. In the present
case, with the rubber stopper covered by a thin Teflon film, one must include
the plastic deformations of the Teflon surface roughness profile, see Sec. 2
and Ref. [12].

For the leakage of fluids one can usually assume laminar flow of Newtonian
fluid characterized by a viscosity $\eta$. This description is also valid for
gases if the surface separation at the critical constriction is much larger
than the gas molecule mean free path $\lambda$. However, this is not the case
in the present application where the flow through the critical constriction is
ballistic rather than diffusive, see Fig. 5. In Ref. [33, 34] we have
presented an interpolation formula for gas flow through a narrow pore which
correctly describes both the diffusive (large pore diameter) and ballistic
(narrow pore diameter) limits. In the equation enters the viscosity $\eta$,
the mean free path $\lambda$ and the average velocity $\bar{v}$ of a gas
molecules. We have used $\eta=1.76\times 10^{-5}\ {\rm Pas}$, $\lambda=59\
{\rm nm}$ and $\bar{v}=470\ {\rm m/s}$.

The solid and dashed lines in Fig. 6 shows the calculated leakage rate as a
function of the average contact pressure using the plastically deformed
surface (solid lines) and the original surface (dashed lines). The + symbols
are the measured leakage rates using different (but nominally identical) clean
syringes. The square symbol is for the same type of syringe, but with the
glass barrel lubricated by silicone oil. The silicone oil block air flow
channels and reduces the air leakage rate, and one may have expected an even
larger reduction than observed. Note the crucial influence of the plastic
deformation which for the relevant average contact pressure $p_{0}\approx 3.2\
{\rm MPa}$ reduces the leakage rate by a factor of $\sim 1000$.

The maximum contact pressure for the rib contacts 1 and 2 is very close to the
pressure where the contact area percolate. This result in the large
fluctuations (by a factor of nearly 100) in the measured leakage rate between
nominally identical syringes, and also in a large sensitivity in the
calculated leakage rate to small variations in the system parameters, e.g.,
the penetration hardness.

The $\times$ symbols in Fig. 6 are for a different Teflon coated rubber
stopper studied in Ref. [9], where the rib contact regions are much wider and
the contact pressure much smaller. For this stopper the Teflon coating was
slightly smoother (see Fig. 7) than in this study, and when this is taken into
account the theory prediction agrees very well with the measured data. This is
shown by the green line in Fig. 6, which was calculated using the power
spectrum of the plastically deformed surface obtained in the same way as for
the rib 1 and 2. Note that the fluctuations in the measured data is much
smaller than for the new stopper design, which is consistent with the fact
that the contact pressure is well below the pressure where the contact area
percolate.

Figure 8:  The contact area as a function of the contact pressure for rib 1.
The solid line is for the plastically deformed surface and the dashed line for
the originally (not deformed) Teflon surface. The contact area for a randomly
rough surface percolate when $A/A_{0}\approx 0.42$. The dashed lines indicate
the average and the maximum contact pressure assuming Hertz-like pressure
distribution.

For a randomly rough surface the contact area percolate when $A/A_{0}\approx
0.42$ (see Ref. [23]). When the contact area percolate no open flow channel
occurs at the interface and the leakage rate vanish. Fig. 8 shows the contact
area as a function of the contact pressure for rib 1. The solid line is for
the plastically deformed surface and the dashed line for the originally (not
deformed) Teflon surface. The dashed lines indicate the average and the
maximum contact pressure assuming Hertz-like pressure distribution. Note that
for the maximum contact pressure the contact area nearly percolate.

Container closure integrity is very important for syringes, so that no
microorganism (bacteria and viruses) can penetrate from the outside to inside
the syringe. The diameter of virus is in the range $\approx 20-400\ {\rm nm}$
and it is clear that complete container closure integrity would imply that the
most narrow junction (denoted critical junction) in the largest open (non-
contact) interfacial channel should be at most $20\ {\rm nm}$. When this
condition is satisfied, the fluid leakage is also negligible.

Figure 9:  The calculated surface separation at the critical constriction as a
function of the average contact pressure using the plastically deformed
surface (solid lines) and the original surface (dashed lines). The critical
constriction is the most narrow constriction along the biggest open flow
channels at the Teflon-glass interface.

Fig. 9 shows the calculated surface separation at the critical constriction as
a function of the average contact pressure using the plastically deformed
surface (solid lines) and the original surface (dashed lines). The critical
constriction is the most narrow constriction along the biggest open flow
channels at the Teflon-glass interface. The separation of the surfaces at the
critical constriction is $\sim 2\ {\rm nm}$, which implies container closure
integrity.

The theory predict that the separation between the surfaces at the most narrow
constrictions along the biggest fluid flow channels is so small ($\sim 24\
{\rm nm}$) that also for the old design no bacterial ingress is possible,
which has been confirmed experimentally.

4 Discussion

The study presented in this paper and in Ref. [9, 35] shows the importance of
plastic flow in some applications to seals. The good agreement found here, and
in Ref. [9], support the procedure we use to include the plastic deformation.
The role of plastic deformation was studied for metallic seals in Ref. [35]
where the theory showed that the plastic flow reduce the leakage rate with a
factor of $\sim 8$, resulting in water leakage rates in good agreement with
experiments. We have shown in Ref. [36, 18] that plastic flow may modify the
surface topography in different ways for metals and polymers, but at least for
Teflon and steel the way we include plastic flow gives good results for both
systems.

We note that the interfacial separations predicted by the theory for the new
and old stopper design ($2-25\ {\rm nm}$) are so small that gas leakage
through molecular diffusion in the elastomer material[19, 20] may contribute
in an important way to the measured leakage rate. In fact, for the syringe
lubricated by silicone oil, where $\dot{V}\approx 2.5\times 10^{-6}\ {\rm
mm^{3}/s}$, this might be the dominant leakage mechanism. As a consequence the
traditional way of container closure integrity studies (dye ingress or gas
transport through the interface) is a subject of debate nowadays.

5 Summary and conclusion

We have performed air leakage experiments for the contact between a rubber
stopper, laminated with a thin Teflon film, and a smooth cylindrical glass
barrel. The measured leakrates where compared with theory predictions based on
a theory involving gas flow through narrow constrictions taking into account
the interfacial separation and mean-free path of the gas molecules. We used
the Perssons contact mechanics theory to calculate the probability
distribution of surface separation at the stopper/glass interface, and the
Bruggeman effective medium theory to calculate the fluid flow resistance in
the complex set of interconnected flow channels at the interface. We found
that the plastic deformation of the Teflon surface reduces the interfacial
separation by a factor of $\sim 100$, and result in a reduction of the
leakrate by a factor of $\sim 1000$. The measured leakage rates are in good
agreement with the theory predictions, but exhibit large fluctuations because
the contact is close to the percolation threshold where small variations in
the system parameters can generate large changes in the leakage rate.

Acknowledgments:

We thank Martin Müser for useful comments on the manuscript. We thank Lucile
Gontard for checking some roughness parameters.

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