Permeation, diffusion, and solution of cyclopropane in silicone rubber

Permeation, Diffusion, and Solution of Cyclopropane in Silicone Rubber

M. S. SUWANDI*, T. HIROSE, and S. A. STERN,+ Department of Chemical Engineering and Materials Science, Syracuse University,

Syracuse, New York 13244

Synopsis

Permeability, solubility, and diffusion coefficients have been determined for cyclopropane (c-C,H,) in silicone rubber a t temperatures between - 8 and 70°C at relative pressures from 0.04 to 0.30. The permeability coefficients, p, are of the order of cm3 (STP) . cm/(s . cm2 . cmHg). p increases slightly with increasing penetrant pressure and decreases with increasing temperature, the energy of activation for permeation being -1.27 kcal/gmol a t zero pressure. The solubility of cyclopropane in silicone rubber can be represented over the experimental concentration range by the Flory-Huggins equation. The solubility decreases with increasing temperature and the partial molar heat of solution is -4.95 kcal/gmol. The solubility coefficient in the Henry's law limit, S(O), for cyclopropane and many other gases and vapors can be correlated with (T, /T)*, where T and T, are the experimental and critical temperatures, respectively. The mutual diffusion coefficients, D, increase with increasing concentration and temperature, the energy of activation for diffusion being 3.68 kcal/gmol. The pressure dependence of p is described satisfactorily by a free-volume model proposed by Fujita and extended by Stern, Frisch, and coworkers. The permeability, diffusion, and solubility behavior of cyclopropane in silicone rubber is similar to that of propane (C,H,).

INTRODUCTION

The transport of gaseous clinical anesthetics in and through silicone rubber is of interest in some biomedical applications, such as the continuous in vitro analysis of anesthetic concentration in blood by mass spectrometry. In a previous study, Suwandi and Stern' determined solubility, diffusion, and permeability coefficients for halothane (CF3CHC1Br) and methoxyflurane (CHCl,CF,OCH,) in silicone rubber a t temperatures from 17 to 60°C and over a wide range of relative pressures. More recently, Fielding and Salamonsen reported values of solubility coefficients for seven anesthetic gases a t 37"C,' as well as solubility, diffusion, and permeability coefficients for halothane and ethane (enflurane) (CHFClCF,OCHF,) at temperatures from 22 to 50"C3

In the present study, solubility, diffusion, and permeability coefficients have been determined for another gaseous anesthetic, namely cyclopropane (c-C,H,), between 4 and 70°C over a range of subatmospheric pressures. The study had the additional objective of investigating the mechanism of transport of organic vapors in silicone rubber, a polymer with a large free volume

*Present address: Dept. of Chemical and Process Engineering, National University of Malaysia,

'To whom correspondence should be addressed. Bangi, Selangor, Malaysia.

Journal of Polymer Science: Part B: Polymer Physics, Vol. 28, 407-423 (1990) 0 1990 John Wiley & Sons, Inc. CCC 0887-6266/90/030407-17$04.00

408 SUWANDI, HIROSE, AND STERN

and very flexible backbone chains. Relatively few detailed studies have been published on this subject.

EXPERIMENTAL TECHNIQUES

Apparatus and Procedure

1. Mean permeability coefficients, P, were determined from steady-state permeation rates, 4, which were measured by the “time-lag” m e t h ~ d . ~ , ~ The apparatus and experimental procedures employed have been described else- where.’

The measurements were made with silicone rubber membranes in the form of flat sheets. Values of p were calculated from the following expression:

where A and 6 are the effective membrane area and thickness, respectively; T is the absolute temperature; P h is the (constant) penetrant gas pressure applied on the membrane; p , (<< P h ) is the (variable) pressure of the penetrant permeating through the membrane into a “re~eiving~~ volume, V; t is the time; and A p , / A t is the rate of increase in the pressure p , at steady state. p is reported here in units of cm3 (STP) - cm/(s . cm2 + cmHg), while the other quantities in eq. (1) are given in appropriate units.

It should be noted that the rates of permeation of cyclopropane were measured with different receiving volumes, V. No significant differences were observed either in the values of or in those of the diffusion time lag, L, for a given T and p h . This indicates that the permeability and time-lag measurements were not significantly affected by the adsorption of cyclopropane on the walls of the receiving volumes (which were made of borosilicate glass).6 The experimental error in p was estimated to be f6%.

2. The equilibrium concentration (solubility), c, of cyclopropane dissolved in silicone rubber a t a pressure p and a t a given temperature was measured by means of a McBain sorption balance, which was described in an earlier study.’

Solubility coefficients, S, were calculated from the relation

S(C) = c /p; (2)

S is reported here in units of cm3 (STP) penetrant/(cm3 polymer . cmHg). The polymer sample used was observed to swell with increasing relative pressure p / p * , where p* is the vapor pressure of the penetrant a t the experimental temperature. Therefore, S was based on the unswollen volume of the polymer. The error in S was estimated to be f 8%.

3. Diffusion coefficients have been determined by many investigators from time-lag measurements, such as made in the present s t ~ d y . ~ , ~ , ~ However, the permeation time lag, L, can be significantly altered by the presence of fillers in polymers. This can occur when the fillers are not completely “wetted” by the polymer, thus giving rise to gaps between polymer and filler particles. Such gaps may affect the transport of the penetrant gases in and through the polymers, thereby modifying the values of the time-lags and diffusion coefficients compared with their values in filler-free polymers.8

PERMEATION, DIFFUSION, AND SOLUTION 409

Since the samples of silicone rubber employed in this work contained a substantial amount of silica filler (see following section), diffusion coefficients were determined from the steady-state permeability coefficients, P, and the solubility coefficients, S. Accordingly, mean diffusion coefficients, ost, were calculated from the isothermal relation4, 5:

- Dst = P / S , (3)

where Bst is valid over the concentration range from c h to c,( << Ch); ch and c, are the concentration of the penetrant (c-C,H,) dissolved at the “upstream” and “downstream” membrane interfaces, respectively, when the penetrant pressures a t these interfaces are ph and p,. Solution equilibrium is assumed to exist at all times between the penetrant in the gas phase at an interface and the penetrant dissolved in the membrane at that interface. If ph >> p,, as is the case in time-lag measurements, s in eq. (3) is evaluated at pressure p h (5). When p and S are given in the above mentioned units, D will be expressed in cm2/s, based on the unswollen (penetrant-free) polymer frame of referen~e.~,

In order to study the mechanism of cyclopropane transport in silicone rubber it was necessary to determine the mutual (differential) diffusion coefficient, D( c). This diffusion coefficient was obtained from the relation4:

D(at c = ch) = ost + ch( dBSt/dch), (4)

For some correlations, discussed later, it was found more convenient to express o,, in terms of the volume-fraction concentration, v , of the penetrant rather than of its mass concentration, c. D was then determined from the following relation4.’:

where v is the volume fraction of cyclopropane corresponding to the concentration ch. The error in D was estimated to be within *16%. It should be noted that eq. (5) in Ref. 1 should be identical with eq. (5) above, but contains an extraneous (1 - v) term. As a result, the values of D in Figures 7 and 8 of Ref. 1 are too high at the higher values of v (by about 15% at the highest penetrant concentrations studied, v = 0.3).

The values of permeability, diffusion, and solubility coefficients reported herein are for the polymer containing the amount of filler stated below, unless otherwise specified.

MATERIALS

Membranes

The membrane samples employed were cut from a sheet of medical-grade Silastic silicone rubber manufactured by Dow Corning Corp. of Midland, Michigan. The nominal thickness of the sheet was 1,016 pm (40 mil) and its density was 1.138 g/cm3. According to the manufact~rer,~ the raw Silastic was prepared from 100 parts by weight of SGM-11 (a dimethyl vinyl-endblocked dimethyl methyl-vinyl siloxane copolymer) and 40 parts by weight (28.6 wt %) of Dow Corning Type A silica filler; it was vulcanized with 1.10% of 2,4-dichloro-


robenzoyl peroxide. The raw Silastic was cured for 3 h a t 300"F, and the membrane sheet made from this material was postcured for 4 h a t 400°F.

The average thickness of each membrane sample used in the permeability measurements was determined with an Ames thickness gage which could be read to 0.1 mil. The thickness of such a sample, which had an effective diameter of 1.45 in,, was the average value of 20-30 random measurements. The relatively thick membranes were required because of the high permeation rates of the penetrant studied.

Penetrant Gas

Cyclopropane (c-C,H,) was supplied by Thomas A. Edison Industries, Medical Gas Division, Stuyvesant Falls, NY. The purity of the gas was about 99% (by volume). The impurities were stated to be CO, (0.03%), halogens as propyl chloride (0.025%), and unsaturated hydrocarbons (0.9%).

The vapor pressure, p*, of cyclopropane was calculated as a function of temperature from the data of Booth and Morris'' and of Hafemann and Miller." The following equation is a least-squares fit of their data:

1046.83 t ( "C) + 273.15

log p* (TOIT) = 7.236 -

This relationship is valid for the temperature range from -32 to 115°C. The normal boiling point of cyclopropane is - 34°C.

RESULTS

Permeability Coefficients

The permeability of silicone rubber to cyclopropane vapor was measured a t temperatures from 4 to 70°C over a range of subatmospheric pressures of the penetrant. The experimental results are presented in Figure 1 in the form of semilogarithmic plots of F versus the applied penetrant pressure, ph. The plots exhibit the following features:

1. The permeability of silicone rubber to cyclopropane is very large, the values of cm3 (STP) . cm/(s . cm2 . cmHg);

2. F increases exponentially with increasing Ph; and 3. decreases with increasing temperature, as is the case for gases and

The data of Figure 1 can be represented by the isothermal relation:

being of the order of

vapors that exhibit a high solubility in polymers.

l o g P = n + map, , (7)

where n and m are constants. Values of n and m at different temperatures are listed in Table I; these values were obtained from least-squares fits of eq. (7) to the experimental data. The temperature dependence of F is shown in Figure 2 in the form of semilogarithmic plots of at constant pressure versus the reciprocal absolute temperature, l/T. The energy of activation of the permeation process, E,, was found to be - 1.27 kcal/gmol a t zero pressure.


x

1;. 40°C

i z w

270°C ~

nr\ 370'C s - 5009: - 700°C

h

A N n "

> ,. Y

t- -I 0 8 - a a z 06-

a I I I I I 2 0 100 200 300 400 500 600

w n ,. - - " "

W

Q W APPLIED PRESSURE, p&( TORR) 2

Fig. 1. Mean permeability coefficients for cyclopropane as a function of applied pressure. The plotted isotherms are least-squares fits of eq. (7) to the experimental data.

TABLE I Dependence of Mean Permeability Coefficients for Cyclopropane in Silicone Rubber

Temperature t ("C)

Parameters in Eq. (7)

n m x lo4

4 27 37 50 70

- 5.922 - 6.002 - 6.034 - 6.069 - 6.113

2.013 1.004 0.783 0.595 0.392

Units: P[cm3 (STP) . cm/(s . cm2 . cmHg)]; ph(Torr); n(dimension1ess); rn(Torr-').

n* i5

8

I-'

u LL lL W

I I I I I I

Fig. 2. Mean permeability coefficients for cyclopropane in silicone rubber as a function of temperature. The isobars were obtained by cross plotting the curves of Figure 1.


RELATIVE PENETRANTPRESSURE. PIP*

Solubility of cyclopropane in silicone rubber as a function of relative pressure. Fig. 3.

Solubility Coefficients

The results of the solubility measurements are presented in Figure 3, where the equilibrium concentration of cyclopropane, c, is plotted as a function of the relative vapor pressure, p / p * (or, approximately, the activity). The plot is seen to be independent of temperature in the range of relative pressures and temperatures investigated. A similar behavior has been observed also with halothane and methoxyflurane.' This type of plot is useful because solubility coefficients can be evaluated a t any other temperature and pressure from a knowledge of p* alone, at least in the range of validity of the plot.

The dependence of the solubility coefficient, S, on the penetrant concentration, c, is shown in Figure 4. The plots in this figure can be represented by empirical relations of the form:

logs = log S(0) + uc, (8)

where S(0) is the solubility coefficient in the limit c -, 0, and a is a parameter that characterizes the dependence of S on c; u depends only on the nature of the penetrant/polymer system and the temperature. Values of S(0) and u for cyclopropane in silicone rubber were determined from the intercepts and slopes, respectively, of the plots in Figure 4 and are listed in Table 11. The data are not corrected for filler content.

The magnitude of the S(0) values are of the order expected from the high critical temperature of cyclopropane (T, = 397.8 K). It has been shown previously that, for a large number of gaseous penetrants in silicone rubber, log S(0) is a unique linear function of (T,/T)2, where T, is the critical temperature of


0 10 20 30 PENETRANT CONCENTRATION, C [ ck$Ei:. , ]

Fig. 4. Solubility coefficients for cyclopropane in silicone rubber as a function of concentration. Symbols correspond to those in Figure 3.

the penetrant.' If cyclopropane is included, one obtains the empirical relation:

logS(0) = 1.0446(Tc/T)2 - 0.7108 (9)

with a correlation coefficient of 0.99400 and a standard deviation of 0.07557; S(0) is here in units of cm3 (STP)/(cm3 filler-free polymer . atm), while T, and T are in K. A slightly different expression, derived from thermodynamic considerations, represents the above data almost equally well":

log[S(O) . p c ] = 1.0161(Tc/T)2 + 0.9254 (10)

TABLE I1 Values of Parameters in Eqs. (8), (?.I), and (24)

From Eq. (8) From Eqs. (21) and (24) Temperature t ("C) S(0) U X 5(0) U

- 8.0 0.502 0.00485 0.319 0.477 0.00331 4.3 0.328 0.00475 0.297 0.317 0.00334

27.0 - - 0.230 0.166 0.00336 37.0 - - 0.178 0.131 0.00331 50.0 - - 0.175 0.0926 0.00343

Units: S(0) [cm3 (STP)/(cm3 polym.. cmHg)]; u[cm3 polym./cm3/(STP)]; X(dimension1ess).


3c

m

2.0 ? - 5. 0 a aJ 9 L = 1.0

\c

aJ - .+ r)

E Y

c m

- a - " E

c 00

- 9 m

0 Ln

J

- - 8 - 1 0

- 2 0 00 10 20 3.0 40

(TCIT)' Fig. 5. Correlation of solubility coefficients in Henry's law limit for different gases and vapors

in silicone rubber. Plot of eq. (9). Key: (v) nitrogen"; (e) argon"; (A) oxygen"; (a) methane2'~27; (0) carbon d i ~ x i d e ~ ' . ~ ~ ; (Q) propanez7; (A) cyclopropane (Ref. 2 and this study); (A) i~o-butane'~; (0) n-b~tane'~~''; (v) neo-pentaneZ5; (A) ether'; (0) n-pentane2.3'25; (0) ethane (Enflurane)2s3; (0) halothane',3; (m) chloroform'; (a) methoxyflurane'r2; (v) Trilene'.

where p, is the critical pressure of the penetrant in atm.; eq. (10) has a correlation coefficient of 0.98882 and a standard deviation of 0.1008. Equation (9) is represented graphically in Figure 5. Neither of the last two equations is applicable to the quantum gases H,, He, and (probably) Ne.

The data of Figure 4 show that the solubility of cyclopropane decreases with increasing temperature. Hence, the solution of cyclopropane in silicone rubber is an exothermic process. This is the case for most gases and vapors in polymers, except perhaps for the quantum gases. The partial molar heat of solution of cyclopropane in silicone rubber at zero concentration was calculated to be - 4.95 kcal/gmol from the slope of a plot of log S(0) versus 1/T, which is linear.

Diffusion Coefficients

Mutual diffusion coefficients, D( v ) , for cyclopropane in silicone rubber were calculated by means of eq. (5). The results of these calculations are presented


2o - PENETRANT VOWME-FRACllON,V

Fig. 6. fraction.

Mutual diffusion coefficients for cyclopropane in silicone rubber as a function of volume

in Figure 6 in the form of semilogarithmic plots of D versus the volume fraction, 0, of the penetrant at five different temperatures. The plots are linear and show that D increases exponentially with increasing z). However, it should be noted that the data cover a range of volume fractions of less than 0.06; higher penetrant concentrations could not be attained at the experimental pressures of this study ( p < 55 cmHg).

The diffusion coefficient also increases markedly with increasing temperature. The energy of activation for diffusion, Ed, was determined from the slope of an Arrhenius-type plot of logD(0) versus 1/T. D(0) is the diffusion coefficient at zero penetrant concentration and was calculated from the relation:

where P(0) was obtained from extrapolations of least-squares fits of the log versus ph plots to zero pressure. The value of Ed was found to be 3.68 kcal/gmol.

DISCUSSION

Permeability

The high permeability of silicone rubber to cyclopropane is due both to the high solubility and high diffusivity of this penetrant in the polymer. The high solubility is expected, as mentioned earlier, from the relatively high critical temperature of cyclopropane; as seen from Figure 5, the critical temperature is a “scaling” factor for the solubility of different penetrants in a given polymer, cf. Ref. 12. The high diffusivity of cyclopropane in silicone rubber is a consequence of the large free volume and flexibility of the Si-0 linkages of


the polymer. This is reflected in the low energy of activation for diffusion of cyclopropane in silicone rubber (see previous section).

The pressure dependence of the permeability coefficients, P, is very likely also due to the high solubility of the penetrant. This dependence increases with decreasing temperature, i.e., with increasing solubility, as can be seen from Figure 1. For many gas/polymer systems, the dependence of on penetrant gas pressure can be described by an extension of Fujita’s free-volume model,13 which was proposed Stern, Frisch, and their coworker^.'^-'^ Accord- ing to the extended model, can be expressed as a function of pressure by the isothermal relation:

where

and where P h and p I ( < p h ) are the penetrant pressures at the two membrane interfaces; R is the universal gas constant; T is the absolute temperature; A, and & are characteristic constants; v is the volume-fraction concentration of the penetrant in the polymer; vf is the fractional free volume (i.e., the volume fraction of the free volume) of the penetrant/polymer system at temperature T and pressure p ; and y is defined below.

The fractional free volume vf is obtained from the relation

uf = u,(T,, p, , 2, = 0) + a(T - TJ - pG, - pJ + yv = u;(T, p , 2, = 0) + yu,

(15)

where vfs is the fractional free volume of the pure polymer ( v = 0) at a reference temperature T, and reference pressure p,; v: is the value of ofs at temperature T and pressure p ; a[= (avJdT) , ] is a “thermal expansion” coefficient; /3[ 3 (avJap) , ] is a “compressibility” coefficient; and y [ = (av , /dv ) , ] is a “concentration” coefficient which defines the dependence of the free volume of the penetrant/polymer system on v , i.e., the effectiveness of the penetrant as a plasticizer. When the pure polymer is considered, ps and p are taken to be hydrostatic pressures; the only hydrostatic pressure on the penetrant/polymer system is that exerted by the penetrant.14

The pressure p and the volume-fraction concentration u of the penetrant are related by the isothermal expression


where S ( v ) is a solubility coefficient; v is related to c by the expression

v = Ac/(l + A c ) , (17)

where A = VL/22,400; VL is the liquid molar volume of the penetrant, in cm3/gm01; c is the penetrant concentration, in cm3 (STP)/cm3 unswollen polymer; and 22,400 is the number of cm3 (STP) per gmol. Equation (17) assumes that the excess volume of mixing of the penetrant/polymer system is zero.

to the thermodynamic variables T, p , and v , and to the three parameters A,, B,, and y. The first two parameters, A, and Bd, are taken to be independent of both penetrant concentration and temperature, whereas the third parameter, y , is a function of t empera t~ re . ’~ , ’~ ’~~ Equation (12) assumes that the transport of penetrant gases in polymers obeys Fick’s laws and that the membranes are homoge- neous, isotropic, and planar. In applying eq. (12), the standard state for the fractional free volume vfs was taken at T, = Tg and p , = 1 atm, where Tg is the glass transition temperature of the polymer; according to the WLF theory, the glass transition is regarded as an “iso-free-volume” Some details of the calculations are given in the Appendix.

The following values were obtained for the parameters A,, B d , and y between 20 to 60°C from the permeability measurements: A, = 2.209 x lo-” m2 . gmol/(sec . J), Bd = 0.36 (dimensionless), and y = 0.155 (dimensionless). The value of Bd must lie between 0 and 1. Parameter y is commonly dependent on the temperature, but is essentially constant under the present experimental conditions. This may be due to the assumptions made in calcu- lating the fractional free volume (see Appendix).

Equation (12) relates the permeability coefficient

Solubility

The solubility data shown in Figure 3 can be represented satisfactorily by means of the Flory-Huggins equation. For the case when the excess volume of mixing of the penetrant/polymer system is negligible, this equation takes the form

P Ac 1

P* l n a = ln- =In

where x is the Flory-Huggins interaction parameter, a is the penetrant activity, and the other symbols have been defined previously. For subcritical penetrants at sufficiently low pressures, a = p / p * . The values of x calculated from eq. (18) in conjunction with the solubility data of Figure 3 are listed as a function of temperature in Table 11. It is seen that x varies from about 0.32 a t -8°C to 0.17 at 50°C; these low values reflect the solubility of cyclopropane in silicone rubber under the experimental conditions. The fact that the solubility data at a given temperature could be represented by the


Flory-Huggins equation with a single value of x over the entire concentration range suggests that the silicone rubber used was not highly crosslinked.'

The solubility coefficient S( = c / p ) can be expressed in terms of the Flory- Huggins equation by the relation

1

1 + A c (19) 1nS = ln(1 + A c ) - ___ -

If (Ac)' < 1, the first three terms on the right-hand side of the above equation can be expanded in series. A summation of terms then yields the expression :

m I n s = -(1 + x) - ln(Ap*) +

n = l

At very low concentrations ( c + 0), eq. (19) reduces to

In S(0) = - (1 + x ) - ln(Ap*) = - (1 + x ) - ln(Vp*/22,400) (21)

Substituting eq. (21) into eq. (20) yields

m

1nS = lnS(0) + ( - l )n- l (n + 1) - + x (Ac)" (22) n= 1 i: I

In cases where Ac is small, the higher terms in the summation on the right-hand side of eq. (22) can be neglected. Equation (22) can then be written in exponential form as

S = S(O)exp[2(1 + X)Ac] (23)

A comparison of eq. (23) with the empirical eq. (8) shows that:

u = 2(1 + x)A/2.303 = 3.8769 X 10-5(1 + x)V (24)

Values of S(0) and u calculated from eqs. (21) and (24), respectively, using the x s obtained at corresponding temperatures from eq. (18), are listed in Table 11. This table also lists values of S(0) and u determined from the solubility data of Figure 4 and eq. (8) at -8.0 and 4.3"C. The agreement between the values of S(0) obtained by the two methods a t these temperatures is satisfactory, but the values of u derived from eq. (24) are significantly lower than those obtained by fitting eq. (8) to the solubility data. However, the values of (I are very small and therefore subject to a significant experimental error; as mentioned earlier, the parameter u characterizes the concentration dependence of S a t a given temperature.


It should be noted that the dashed curves in Figure 4, denoting the dependence of S on c at 27.0, 37.0, and 5O.O0C, were obtained from eq. (8) and the values of S(0) and u derived from eqs. (21) and (24) as described above. The solubility of cyclopropane in silicone rubber was very low at these higher temperatures and only a few measurements were made; hence, S(0) and u could not be determined directly from experimental data.

Solubility coefficients for other gases and vapors in silicone rubber, and the dependence of these coefficients on concentration (or pressure) and temperature, can now be predicted from the above considerations by the following method:

1. First, from a knowledge of the critical temperature T, of the penetrant gas or vapor, a value of S(0) is obtained from eqs. (9) or (10) at the desired temperature T. If T, is not known, it can be estimated by various methods, e.g., ref. 21.

2. Second, parameter x is calculated from eq. (21) using the vapor pressure p* and liquid molar volume V of the penetrant at temperature T; the values of p* and V can also be estimated, if necessary.22

3. Third, the concentration c of the penetrant is calculated at the desired pressure p and temperature T from eq. (18). The solubility coefficient, S( c) =

c /p , can then be obtained from the calculated value of c. The above method is based on the assumption that the Flory-Huggins

equation is obeyed and that x is independent of concentration. Moreover, the method is limited to the concentration range where Ac is small and to subcritical penetrant temperatures (T < T,).

Diffisivity

The presence of fillers may cause differences between the values of the mean diffusion coefficients obtained from time-lag measurements, Dt, and those calculated from permeability and solubility coefficients, Dst, cf. eq. (3). As mentioned earlier, these differences have been attributed to the formation of gaps between polymer and filler particles. In order to ascertain whether such gaps were present in the silicone rubber used in this study, a comparison was made between DL(0) and Dst(0) at different temperatures. DL(0) and DSt(O) are the values of DL (or of DL) and of D,t (or of Dst), respectively, extrapo- lated to zero penetrant Concentration. DL was obtained from the relation4r5

where S is the membrane thickness and L is the time lag. The results of the above calculations are reported in Table 111. The

differences in the values of DJ0) and DL(0) range from a minimum of only 2.5% at 4°C to a maximum of 14.3% at 70°C. These differences are within the experimental error of both Dst(0) and DL(0). The increasing difference between the two diffusion coefficients with increasing temperature is probably due to an increasing error in the solubility coefficients, S, since the solubility of cyclopropane decreases as the temperature is raised. The data of Table I11


TABLE I11 Diffusion Coefficients at Zero Penetrant Pressure

Temperature t ("C) DI,(0) x lo6 DSt(O) x lo6

4 27 37 50 70

3.625 5.697 7.004 8.358

1 1.29

3.717 5.954 7.189 9.147

13.18

suggest that the samples of silicone rubber employed did not contain any gaps, a t least of sizes that would significantly affect the permeability measurements.

The concentration dependence of the mutual diffusion coefficient, D, for c-C3H, in silicone rubber is shown in Figure 6 as a function of the penetrant volume-fraction7 0.

Michaels and B i ~ l e r , ~ have shown that ln[D(0)/d2] is a linear function of d, where d is the molecular diameter of the penetrant. This correlation is shown graphically in Figure 7 for 11 penetrant gases and vapors in silicone rubber; d was taken to be Lennard-Jones size parameter determined from viscosity data.24 It is seen that He and H, do not fit the above correlation, a t least for the selected molecular diameters. In principle, it should be possible to estimate values of D(0) for other penetrants in silicone rubber from Figure 7.

Comparison with Propane

Values of P(O), D(O), and S(0) for cyclopropane (c-C,H,) and propane (C3H,) in silicone rubber are compared at 35°C in Table IV. The comparison is made a t zero penetrant concentration (or pressure) in order to exclude possible plasticization effects. Table IV shows that the value of S(0) for cyclopropane is larger than that for propane, as expected from the higher critical temperature of the former penetrant [T,(c-C,H,) = 397.8 K as compared with q(C,H,) = 369.8 K], cf. eqs. (9) or (10).

The value of D(0) for cyclopropane is about 31% lower than that for propane, although the molecular diameter ofothe former penetrant is smaller than that of the latter [a(c-C,H,) = 4.807 A as compared with a(C,H,) = 5.118 A]. However, this difference is within the combined limits of error of the D(0) values for the two penetrants a t 35°C. Moreover, the Lennard-Jones size parameters may not be satisfactory indexes of the relative magnitudes of D(0) because they do not account for differences in molecular shapes.

Finally, the value of P(0) for cyclopropane is 26% higher than that of propane, because of the higher solubility of the former penetrant in silicone rubber.

It should be noted that P(O), S(O), and hence also D(O), for propane were determined with a filler-free poly(dimethy1 siloxane)." Consequently, the corresponding coefficients obtained in the present study with cyclopropane


1000 800 600

20

10

4

Fig. 7. Correlation of diffusion coefficients at zero penetrant concentration with molecular diameter of penetrant at 26-28°C. Values of the molecular diameters d of halothane and methoxyflurane were calculated from liquid molar volumes; Values of d for other penetrants were obtained from Ref. 24

TABLE IV Comparison of Permeability, Diffusion,

and Solubility Coefficients for Cyclopropane and Propane in Silicone Rubber a t 35.0"C

Penetrant P(O) x lo7 ~ ( o ) x lo5 S(0) Reference

c-C H a C$,b6

11.62 0.700 0.166 This work 8.58 1.01 0.0849 (28)

Units: P(0) [cm3(STP) cm/(s . cm2 . cmHg)]; D(0) (cm2/s); S(0) [cm3(STP)/(cm3 polym

"P(0) and S(0) are corrected for the filler content of the polymer. bMeasurements were made with filler-free polymer.

.cmHg)].

were corrected for the filler content of silicone rubber by dividing by the amorphous volume-fraction of polymer (0.809). Such a correction is only an approximation for P(0) and D(0).

APPENDIX: CALCULATION OF FRACTIONAL FREE VOLUME OF PURE POLYMER

The term u:(T, p , 21 = 0) in eq. (15) can be written in the form

.f"(T, p , z, = 0 ) = u,(T,l, 21 = 0) - p ( p - l ) , (-4.1)


where $(T, p , v = 0) is the fractional free volume of the pure polymer ( u = 0) a t temperature T and (hydrostatic) pressure p ; 2, is the volume-fraction concentration of the penetrant in the polymer: vf(T, 1, v = 0) is the fractional free volume of the pure polymer a t T and ambient pressure ( p is taken as 1 atm); and P [ = (8vf/8p),] is a compressibility coefficient. The value of vf(T, 1,O) can be determined from the Williams-Landel-Ferry (WLF) relation”:

where u,(T,, 1, o = 0) is the fractional free volume of the pure polymer a t the glass-transition temperature, Tg, of the polymer and a t ambient pressure; and af = a, - ag, where a, and ag are the coefficients of thermal expansion of the polymer in the rubbery and glassy states, respectively.

When the actual values of v,(Tg, 1, v = 0) and af for a given polymer are not available, it is possible to use instead the “universal” values u,(T, 1, z, =

0) = 0.025 and af = 4.8 X 10-40C-’. However, in the latter case, eq. (A.l) is valid only in the temperature range Tg < T < Tg + 100°C. Since the Tg of silicone rubber is about 146 K, i.e., Tg + 100°C = 246 K, and the present permeability measurements were made in the temperature range from 277 to 343 K, eq. (A.l) was modified by a method proposed by Kulkarni and Stemz2 According to this method, q(T, 1 , v = 0) becomes

(0.025)(0.809) + /T(0.8CB)( a, - 0.975ag) dT

1 + L:.809a,dT (44.3)

Tg vOf(T,1,0) =

where 0.809 is the volume-fraction of polymer corrected for filler content. The following values have been reported for the coefficients of linear

expansion of poly(dimethy1 si10xane)~~: 2.87 X 10-4”C-’ for the rubbery state ( t = - 30 to - 10°C) and 1.28 X 10-40C-1 for the glassy state ( t = - 150 to - 130°C). Hence, assuming that these coefficients are independent of temperature and that the thermal expansion is isotropic, one obtains a, = (3)(2.87 x

= 8.61 X 10-40C-’ and ag = (3)(1.28 X l op4) = 3.84 X 10-40C-1. It is interesting to note that a, - a = 4.77 X 10-40C-’, which is very close to the “universal” value of 4.8 X 10-940C-1.

The “compressibility” P [ = ( 8vf /8p) , ] in eqs. (15) and (A.l) has been obtained from the following relation:

where P’ is the conventional compressibility coefficient, V is the specific volume of the polymer, and 0: is the fractional free volume of the pure polymer a t temperature T and p = 1 atm, cf. eq. (A.1); in a previous study v;“ was taken at p = 0, but the difference is negligible.16.17 The value of P’ was reported to be 8.67 X lop5 atm-’ for filler-free poly(dimethy1 s i l o ~ a n e ) ~ ~ ;


consequently, this value was multiplied by 0.809 to correct (approximately) for the filler content of the silicone rubber used in this study.

Financial support from the Center for Membrane Engineering and Science at Syracuse Univer- sity is gratefully acknowledged.

References 1. M. S. Suwandi and S. A. Stern, J . Polym Sci., Polym. Phys. Ed., 11, 663 (1973). 2. R. Fielding and R. F. Salamonsen, J. Membrane Sci., 5, 319 (1979). 3. R. Fielding and R. F. Salamonsen, J. Membrane Sci., 5, 327 (1979). 4. J. Crank, The Mathematics of Diffusion, 2nd ed., Clarendon Press, Oxford, 1975. 5. H. L. Frisch and S. A. Stem, Crit. Revs. Solid State and Mats. Sci., 11, 123 (1983). 6. S. A. Stern and G. W. Britton, J. Polym. Sci., A-2,10, 295 (1972). 7. H. L. Frisch, J. Chem. Phys., 61, 93 (1957). 8. R. M. Barrer and H. T. Chio, J. Polym. Sci., C , 10, 111 (1965). 9. Dow Corning Corporation, Midland, MI, personal communication, cf. ref. 1.

10. H. S . Booth and W. C. Morris, J. Phys. Chem., 62, 875 (1958). 11. D. R. Hafemann and S. L. Miller, J. Phys. Chem., 73, 1392 (1969). 12. S. A. Stern and S.-P. Shiah, Molec. Pharmacol., 19, 56 (1981). 13. H. Fujita, Fortschr. Hochpolym. Forsch., 3, 1 (1961). 14. S. A. Stern, S.-M. Fang, and H. L. Frisch, J. Polym. Sci., A-2, 10, 201 (1972). 15. S.-M. Fang, S. A. Stern, and H. L. Frisch, Chem. Eng. Sci., 30,773 (1975). 16. S. A. Stem, S. S. Kulkami, and H. L. Frisch, J. Polym. Sci., Polym. Phys. Ed., 21, 467

17. S. A. Stern, S. R. Sampat, and S. S. Kulkami, J. Polym. Scz., Polym. Phys. Ed. 24, 2149

18. M. L. Williams, R. F. Landel, and J. D. Ferry, J. Am. Chem. SOC., 77, 3701 (1955). 19. R. Shimha and R. F. Boyer, J. Chem. Phys., 37, 1003 (1962). 20. S. M. Aharoni, J. Macromol. Sci., Phys., B10, 663 (1974). 21. R. C. Reid, J. M. Prausnitz, and T. K. Shenvood, The Properties of Gases and Liquids, 4th

22. S. S. Kulkarni and S. A. Stern, J. Polym. Sci., Polym. Phys. Ed., 21, 441 (1983). 23. A. S. Michaels and H. J. Bixler, J. Polym. Sci., 60, 413 (1961). 24. R. A. Svehla, NASA Tech. Rep. R-132, Lewis Research Center, Cleveland, OH, 1962. 25. C.-L. Lee, S. A. Stern, J. E. Mark, and E. Hoffman, Final Report No. GRI-87/003, Gas

26. R. M. Barrer, J. A. Barrie, and N. K. Raman, Polymer, 3, 595 (1962). 27. W. L. Robb, Ann. N.Y. Acad. Sci., 146, 119 (1968). 28. V. M. Shah, B. J. Hardy, and S. A. Stern, J. Polym. Sci., Polym. Phys. Ed., 24, 2033

(1983).

(1986).

ed., McGraw-Hill, New York, 1986.

Research Institute, Chicago, IL, September 1986, Part I, Table 6.

(1986).

Received December 12, 1988 Accepted June 2, 1989

Permeation, diffusion, and solution of cyclopropane in silicone rubber

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