Measurement and Modeling of Water-Vapor Diffusion ... - Springer Link

Int J Thermophys (2013) 34:412–423 DOI 10.1007/s10765-013-1422-2

Measurement and Modeling of Water-Vapor Diffusion in Elastomers with Impact in Humidity and Vacuum Measurements Janez Šetina · Makfir Sefa · Bojan Erjavec · Domen Hudoklin

Received: 12 April 2010 / Accepted: 22 March 2013 / Published online: 20 April 2013 © Springer Science+Business Media New York 2013

Abstract The dynamics of water-vapor dissolution in Viton O-rings is measured with a gravimetric method using a precise mass comparator. A sample gasket was degassed in high vacuum for a sufficiently long period to remove more than 99 % of the dissolved water vapor. After that, it was exposed to the ambient atmosphere with a controlled temperature, and relative humidity and water-vapor uptake curves were measured gravimetrically with a precise balance. The dynamics of a water-vapor release into vacuum from another sample that was previously saturated with water vapor at room temperature was determined. The sample was placed in a vacuum outgassing rate measurement apparatus. The time dependence of the evolved water vapor was calculated by integrating the measured outgassing rate. The physical process of water absorption can be described by the diffusion equation. The geometry of the samples required solving the diffusion equation in cylindrical coordinates. This was done numerically using a finite-difference method. As a result of the modeling, room temperature values of the diffusion constant D, the solubility s, and the permeability K = D × s of water vapor in the sample material (Viton A-401C) were obtained. For sample 1, we obtained D = 8.0 × 10−8 cm2 · s−1 and s = 6.5 × 10−7 g · cm−3 · Pa−1 , while for sample 2, D = 3.0 × 10−7 cm2 · s−1 and s = 3.5 × 10−7 g · cm−3 · Pa−1 .

J. Šetina · B. Erjavec Institute of Metals and Technology, Ljubljana, Slovenia e-mail: [email protected] M. Sefa Lotriˇc d.o.o., Selca, Slovenia D. Hudoklin (B) Laboratory of Metrology and Quality (MIRS/UL-FE/LMK), University of Ljubljana, Faculty of Electrical Engineering, Ljubljana, Slovenia e-mail: [email protected]

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Keywords Finite-difference method · Fluoroelastomer · Humidity measurements · Uncertainty · Viton · Water content · Water permeation

1 Introduction Vacuum systems are built by joining together several components (e.g., vacuum pumps, connection tubing, isolation valves, process vacuum chamber, vacuum gages, etc.). The joints can be permanent (i.e., welded or brazed, etc.) or demountable in order to allow easy exchange of components or modifications of the system. Demountable joints are based on a pair of flanges and a suitable gasket which is compressed between them. Gasket materials are usually soft metals (mostly copper or aluminum), or elastomers. The main advantages of elastomer-based demountable seals are a lower sealing force and less demanding surface finish of a flange sealing surface, resulting in significantly reduced cost and simplified handling. Elastomer materials are also often used as a sealing material in shut-off valves for pure and humid gas delivery and handling, and as demountable seals (gaskets) in demountable joints in vacuum systems. On the other hand, in contrast to soft metals, elastomer materials exhibit outgassing and some permeability to atmospheric gases, so the minimum pressure for vacuum systems is ∼10−6 Pa. Consequently, the residual gas in vacuum systems using elastomer seals inevitably contains traces of atmospheric gases (i.e., N2 , O2 , and particularly water vapor). Demountable seals with elastomer gaskets have been standardized as ISO-KF (small flange) components (ISO 2861) and ISO-K (clamping flange) components (ISO 1609). The temperature range of both types depends on a gasket material and can extend up to 200 ◦ C. This type of gasket is known as an O-ring and in use is centered between the two flanges with a suitable centering ring. The most commonly used gasket material is Viton (trade mark of DuPont). It is relatively cheap and has adequate vacuum properties (a sufficiently low outgassing rate) for use in high vacuum down to 10−6 Pa. Viton is a fluoroelastomer and was chosen because its heat and chemical resistant properties have made it of great importance in a wide range of industry and laboratory applications. Several types of Viton have been developed to meet specific demands, by changing the chemical composition (fillers and additives) and fluorine content. For vacuum use the main properties are the outgassing rate, which is defined as the amount of gas released per unit of time and the permeation rate, which is the amount of gas diffusing through the gasket from the high-pressure side (atmosphere) into the vacuum per unit of time. There are data available in the literature about the outgassing rate and permeation of atmospheric gases through elastomer gaskets [1–3], but data for water vapor is very limited [4–6]. Yoshimura [7] has shown that the permeation of water vapor from a humid atmosphere has a detrimental influence on the vacuum level that can be achieved in a vacuum system sealed with Viton O-rings. There are different methods for measurement of the outgassing rate [8]. The throughput method gives the instantaneous outgassing rate of the material under test and is based on measurement of the pressure difference across a known conductance. This difference is proportional to the outgassing rate under molecular flow conditions.

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The mass-loss (or weight-loss measurement) method involves exposing the test specimen to a vacuum environment and monitoring the mass loss with a precise balance. This method has been adopted by aerospace industries for testing materials for spacecraft components. Molecular contamination of surfaces due to outgassing poses a serious threat to sensitive devices, such as optics and thermal control surfaces. The method has been standardized in ASTM E595. The aim of our work was to determine the diffusion constant, solubility, and permeability of water vapor in Viton around room temperature. We have made measurements of the mass gain of a fully degassed sample in humid air and of the outgassing rate of a gasket that was saturated with water vapor. Measurements were modeled with a diffusion equation which was solved numerically for appropriate initial and boundary conditions using the finite-difference method (FDM) (see Sect. 2.1). By fitting the solutions to the measured values, we obtained the diffusion constant and solubility of the sample material.

2 Diffusion and Permeation of Gases in Solids Diffusion of gases in solids is covered in many textbooks on vacuum technology [9], or books on diffusion [10]. The diffusion constant and thickness of the material are two parameters which determine the time delay to achieve a steady condition after the change of the water-vapor pressure at the high-pressure or low-pressure side of the sample plate. Diffusion is a “random-walk” process of the dissolved component (gas in liquid or solid). The transport of the diffusive component is driven by the gradient of concentration. In our experiments, the O-ring can be approximated as a segment of an infinite cylinder having a diameter equal to the thickness and a length equal to the mean circumference of the O-ring. So we have to consider diffusion in the radial direction in a cylindrical geometry. The mass flow of the dissolved gas j through a unit area is given by

j = −Dgrad(c) = −D

∂c , ∂r

(1)

where r is the radius, c is the concentration (density) of the dissolved gas, and D is the diffusion coefficient. The concentration is a function of the radius and time and is described by the diffusion equation: ∂c 1 ∂ = ∂t r ∂r

∂c rD . ∂r

(2)

The amount of gas dissolved in the solid when exposed to a gas pressure P usually follows Henry’s law for small concentrations: c = s P (Henry’s law),

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(3)


415

where s is the solubility. Equation 3 is valid for gases which dissolve in solids without dissociation (usual case for nonmetals, including polymers). If j is given in units of g · s−1 · cm−2 , then c has units of g · cm−3 . The water-vapor content W can be obtained by dividing c by the density of the solid material ρ: W =

c ρ

(4)

The product of the diffusion coefficient D and the solubility s is another characteristic quantity of diffusion processes and is named the permeation constant K : K = Ds

(5)

The diffusion coefficient D mainly determines the time delay for establishing equilibrium, whereas the permeation constant K determines the steady-state gas flow density. 2.1 Solving the Diffusion Equation Using the Finite-Difference Method Non-steady-state solutions of Eq. 2 can, in general, be expressed only in a complicated form of an infinite series [10]. Alternatively, Eq. 2 can be solved also numerically. For modeling of our experimental data, we have used the FDM. The FDM is based on dividing the sample (calculation domain) into a finite number of layers and replacing partial derivatives in Eq. 2 at layer boundaries with differential quotients, as described in the book by Crank [10] in Chap. 8. Formulae 8.31 to 8.34 from that book were used to derive explicit finite-difference formulas for calculation of the time evolution of the concentration profile. In our modeling, the calculation domain was divided into 40 layers. Calculations of concentrations at the layer boundaries were performed in an Excel spreadsheet. We considered only radial diffusion from the surface into the cylinder (absorption) and diffusion from the center of the cylinder toward the surface (outgassing). To check if division into 40 layers was enough, we have made additional modeling with (20 and 10) layers. Reducing the number of layers from 40 to 20 had a very small effect (maximum difference of 0.027 mg was obtained around t = 0.3 h and then it was reduced to less than 0.01 mg at t > 5 h). It was also found that the maximum errors were in the ratio 16:4:1 for FDM models with (10, 20, and 40) layers. So, we can conclude that numerical errors of modeling with 40 layers were negligible compared to other measurement errors (Table 2). 3 Experimental 3.1 Sample Description Our samples were Viton fluoroelastomer O-rings that are used with KF16 and KF40 type vacuum flanges. They were purchased from a major vacuum equipment manufacturer, and were claimed to be made from A-401C compound. The O-ring cord diameter

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(thickness) is 5 mm in both cases. Inner/outer diameters (d1 /d2 ) are 18 mm/28 mm for KF16 and 42 mm/52 mm for KF40 gaskets, respectively. The nominal mass of the KF16 gasket was 2.6 g and of the KF40 was 5.3 g. For FDM purposes, the toroidal shape of the gasket was approximated by a straight cylinder with a diameter of 5 mm and a length equal to π × (d1 + d2 )/2. 3.2 Mass Gain Measurements in Air In the first part of our experiment, we measured the absorption of water vapor by the KF16 gasket (sample 1) according to ASTM E595 with the modification of measuring the mass increase continuously for 60 h. Before the measurements were initiated, the sample was kept in high vacuum below 1 × 10−4 Pa for a period of 13 days (1.1 × 106 s) to reach a fully outgassed state. At time t = 0 it was taken out of the vacuum environment, and placed on the balance. Inside the airflow-protecting lid of the balance, we installed a container with liquid water, and calibrated humidity and temperature sensors, which are traceable to national standards [11,12]. The water container was kept inside the lid overnight to reach thermal equilibrium with the surrounding environment and also to saturate the surfaces with water vapor. The humidity sensor showed a stable reading of 98 % ∼10 min after filling the container with water. Measurements were done in a mass calibration lab where the ambient air was controlled to (50 ± 5) %RH. A mean ambient temperature of 21.4 ◦ C with a standard deviation of 0.12 ◦ C over a period of 60 h was measured. The standard deviation of the mean temperature inside the lid was even smaller, 0.05 ◦ C, due to the shielding effect of the lid. Every opening of the protective lid, even for a short time, caused an immediate drop in the relative humidity; however, in a few minutes after closing the lid, the RH reading returned to 98 %RH. The balance for mass increase measurements was an electronic mass comparator (Mettler Toledo MT-5) with a 5 g measuring capability and 0.0001 mg resolution. A check of long-term stability showed that the typical drift over 5000 s was less than 0.003 mg, and over 20 h, less than 0.01 mg. A precise weighing in air requires a correction for the air buoyancy effect [13] where densities of ambient air and the weighed object have to be considered. The density of our Viton A-401C is 1.81 g · cm−3 . We have analyzed the possible influence of variations of the ambient air density on the buoyancy corrections during a 60 h measuring period, when the variations of atmospheric pressure varied less than ±20 hPa. Under these conditions, the buoyancy corrections of our gasket sample varied less than ±0.04 mg, and this value was taken as one of the components of the mass measurement uncertainty. 3.3 Outgassing Measurements in Vacuum In the second part of the experiment, we measured the vacuum outgassing of a gasket KF40 (sample 2) that had previously been exposed to water vapor above the liquid water in a closed Petri dish (≈100 %RH at 22 ◦ C ± 0.5 ◦ C) for 65 h.

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The outgassing measurement apparatus was constructed according to the throughput method described in [8]. It consisted of two vacuum chambers separated by the orifice of known conductance C = 2.3 L · s−1 for H2 O. The sample is placed in the upper chamber, and the turbomolecular vacuum pump with an effective pumping speed S = 40 L · s−1 is connected to the lower chamber. Gas flow from the sample causes a pressure difference across the orifice P = Pu − Pl , where Pu and Pl are the pressures in the upper chamber and lower chamber, respectively. In our measurement setup, a cold cathode ionization vacuum gage was used to measure Pu . The outgassing rate (in units of Pa · dm3 · s−1 ) can be calculated [8] as Q = (Pu − Pl )C = Pl S

(6)

Equation 6 can be rewritten to calculate the gas flow directly from the measured Pu : Q=

Pu C = Pu Ceff (1 + Rp )

(7)

Rp = C/S is the ratio of the orifice conductance and pumping speed. Ceff = is the effective conductance.

C (1+Rp )

4 Results Figure 1 shows the measured mass increase of a KF16 gasket over a period of 60 h. The mass increase was modeled with the FDM using appropriate initial conditions (zero concentration of water vapor inside the gasket) and the boundary condition on the surface of the gasket at t > 0 equal to P(H2 O) = 25.4 hPa ± 0.6 hPa [calculated for (100 ± 2) %RH at a mean ambient temperature of 21.4 ◦ C]. Adjustable parameters in the FDM are the diffusion constant and solubility. Their values were selected to get 2.5

Mass increase, mg

Measured values

FDM model

2.0

1.5

1.0

0.5

0.0 0

10

20

30

40

50

60

70

Time, h Fig. 1 Measured (thick gray line) and modeled (thin black line) increase in mass of a fully outgassed Viton A-401C (size KF16—sample 1) gasket following exposure to 98 %RH at 21.4 ◦ C (see text for details)

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Table 1 Diffusion constant D, solubility s, and permeability K determined by FDM modeling of mass change due to water-vapor adsorption/desorption in a Viton gasket at room temperature D (cm2 · s−1 )

s (g · cm−3 · Pa−1 )

K (g · cm−1 · s−1 · Pa−1 )

T (◦ C) Source

m + (t)

8.6 × 10−8

6.7 × 10−7

5.8 × 10−14

21.4

m − (t)

7.4 × 10−8

6.3 × 10−7

4.6 × 10−14

This work gravimetric measurements

m(t)

8.0 × 10−8

6.5 × 10−7

5.2 × 10−14

±3 %

±11 % 22.0

This work vacuum measurements

3.0 × 10−15

20–30

[4]

1.61 × 10−14

25

[5]

1.64 × 10−14

20

[6]

Modeled curve

Relative uncertainty ±8 % Modeled curve m + (t)

3.6 × 10−7

4.2 × 10−7

1.5 × 10−13

m − (t)

2.4 × 10−7

2.8 × 10−7

6.7 × 10−14

m(t)

3.0 × 10−7

3.5 × 10−7

1.1 × 10−13

Relative uncertainty ±20 % Literature data

±20 %

+44 % / −36 %

To estimate the uncertainty of D and s, three curves were modeled in each experiment: m(t), m(t) + U (m(t)), and m(t) − U (m(t))

the best fit of calculated values and experimental data for mass increase m(t) and are given in Table 1 (with a note “gravimetric measurements”). The vacuum system for the outgassing measurement apparatus is made of stainless steel with CF type [9] flanges and soft copper gaskets. During venting, a certain amount of water vapor also adsorbs on the chamber walls. Because the area of the chamber walls is more than ten times larger than the surface of the sample, it was necessary to check the background outgassing due to desorption from chamber walls. Essentially the same background was obtained after exposing the empty chamber to ambient air at (45 ± 5) %RH for (1, 20, and 60) h. For vacuum measurements, a gasket saturated with water vapor was installed in the test chamber, and at time t = 0, pumping was started. When the pressure dropped below 0.1 Pa, the ionization gage was switched on to record the time dependence of the pressure Pu . Figure 2 shows the calculated outgassing rate (Eq. 7) of the chamber without the gasket (background) and the chamber with the gasket. Outgassing from the gasket is more than one order of magnitude higher than the background. By integrating the measured flow Q, we can calculate the amount of desorbed water vapor:

MH2 O × m = RT

t Qdt 0

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(8)


419

0

Outgassing rate, Pa · L· s-1

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

2

10

4

3

10

10

5

10

Time, s Fig. 2 Vacuum outgassing rate measurement of empty chamber (background) and chamber with Viton A-401C (size KF40—sample 2) gasket previously saturated with water vapor

The calculated amount of desorbed water vapor as a function of time of pumping is shown in Fig. 3, together with a FDM calculation of diffused water out of the gasket. The values of the diffusion constant and solubility which gave the best fit of calculated values and experimental data are given in Table 1 (with a note “vacuum measurements”).

5 Discussion In FDM calculations, the diffusion constant D and solubility s are parameters which are adjusted to obtain minimum deviations of the modeled curve from the measured values. Quite close agreement of measured values m(t) and the FDM curve after selecting appropriate values of D and s in Figs. 1 and 3 indicates that the water-vapor uptake and release in Viton can be well described with the diffusion equation (Eq. 2). The diffusion constant, solubility, and permeability were determined by modeling of the measured mass change caused by water adsorption or desorption. In our FDM modeling, we are not able to evaluate the uncertainty of D, s, and K in a standard way by writing an explicit mathematical model of measurements including different influential quantities in order to calculate sensitivity coefficients by differentiating the model equation. To estimate uncertainties we had to apply an alternative approach, as follows. First, we estimated the uncertainty of the measured mass change U (m). Then we constructed two additional mass change curves m + U (m) and m − U (m). These two curves, denoted as m + (t) and m − (t) represent the upper and lower error boundaries of our mass measurements. We made FDM modeling of these two curves (Fig. 4). Values of parameters D + , D − , s + , s − , K + , and K − can be used to determine the uncertainty interval of the respective quantities. The calculated

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Outgassed amount, mg

3 Measured values

FDM model

2.5 2 1.5 1 0.5 0 0

5

10

15

20

Time, h Fig. 3 Modeled (thick black line) amount of outgassed water and calculated values (thick gray line) from background corrected outgassing rate of sample 2

diffusion constant, solubility, and permeability for the modeled curves are given in Table 1. In gravimetric measurements of sample 1, the uncertainty of the mass change has three main contributions (Table 2). The uncertainty due to buoyancy corrections was discussed in Sect. 3.2 and amounts to 0.04 mg (k = 2). The uncertainty due to the drift of the balance during 60 h of measurements was estimated to be 0.03 mg, based on balance stability measurements which were also discussed in Sect. 3.2. The largest component of uncertainty due to the initial mass of the sample at time t = 0 (time when it was taken out of the vacuum environment). Some water vapor from ambient air was adsorbed already during transfer of the sample to the balance and it can be seen from Fig. 1 that there was quite a fast mass increase at the beginning of measurements. The initial mass was estimated by extrapolation of the first few mass data points to time t = 0. The uncertainty of this extrapolation was estimated to be 0.05 mg. The combined uncertainty from these three components is U (m) = 0.07 mg (k = 2). In vacuum measurements of sample 2, the water-vapor content was calculated by integrating the outgassing rate (Eq. 8). Based on the experience of the time stability, linearity, and hysteresis of the vacuum gage which was used in our experiments (PKR 261 from Pfeiffer Vacuum), we estimated a relative uncertainty of 20 % in the range of the measured pressure. The effective conductance of the orifice in Eq. 7 was calculated from dimensional measurements and the specified pumping speed of the turbo pump. The relative uncertainty of 10 % was estimated for the effective conductance. It can be seen from Fig. 2 that the background is at least one order of magnitude lower than the signal from the sample, so the relative uncertainty of the background correction was estimated to be less than 5 %. By combining these contributions, it follows from Eq. 7 that the relative uncertainty of the measured outgassing rate is 23 %. The relative uncertainty of the integral in Eq. 8 is also equal to this value, since the uncertainty of the time interval in the numerical integration is negligible. Therefore, the resulting relative uncertainty of the calculated mass change with Eq. 8 is 23 %. Also, in this case

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2.5

Mass increase, mg

m+U(m) 2.0

1.5

m-U(m)

1.0

0.5 FDM model

m +/- U(m)

0.0 0

10

20

30

40

50

60

70

Time, h Fig. 4 FDM modeling of mass change curves m + U (m) and m − U (m) to estimate uncertainties of the diffusion constant, solubility, and permeability for sample 1 Table 2 Main contributions to the uncertainty of mass change m in gravimetric and vacuum measurements Uncertainty contribution

Value of uncertainty (k = 2)

Mass gain measurements in air Buoyancy corrections

0.04 mg

Drift of balance

0.03 mg

Initial mass (m 0 )

0.05 mg

Combined: U (m)

0.07 mg

Outgassing measurements in vacuum Vacuum gage (Pu )

20 % (relative)

Effective conductance of the orifice (Ceff )

10 % (relative)

Background correction