A. S. CARPENTER A N D D. F. TW ISS, D unlop Rubber Co., B irm ingh am , England
C ONSIDERABLE difficulty has been experienced in assessing the ability of sheets of rubberlike materials to resist permeation by gases. The slowness of the process in the most im portant case of the “perm anent” gases neces
sitates special experimental conditions to give a compara
tively great permeation rate, and so introduces several factors which tend to decrease the trustworthiness of the results.
Among the difficulties introduced m ay be mentioned the preparation of suitable test pieces (thin, uniform sheets free from pinholes), accurate tem perature control of bulky ap
paratus, the prevention of deformation of the sheets, and the measurement of the quantities of gas passed after reason
able periods of time. Moreover, in investigations which do not approximate the conditions under which the material is to be employed in practice, interfering factors may be introduced which m ay make results worthless even though they m ay be reproducible. For example, when the experi
m ental conditions are such as to give comparatively rapid permeation rates, rate of solution and of evaporation of gas may be influential factors, whereas under the conditions in which the material is to be used in practice, the governing factor m ay be the rate of diffusion of dissolved gas in the material.
The indirect method of assessing permeability, outlined below, overcomes in a large measure the difficulties of the direct method, and also gives independent values for the two factors which are shown to govern permeability.
Diffusion o f Gases
Gases are soluble in rubberlike substances. When such substances are placed in a gas, solution takes place a t the surfaces and dissolved gas diffuses into the interior. If the gas pressure is maintained constant, or allowed to become
constant, an equilibrium is attained a t which dissolved gas is uniformly distributed throughout the material. This satura
tion concentration is dependent upon the chemical nature of the solute and of the solvent, upon the tem perature, and upon the gas pressure. If, after equilibrium has been attained, the gas pressure is changed isothermally and m aintained a t a new value, gas dissolves or evaporates a t the surface, according, respectively, as the new pressure is greater or less than the original pressure, and diffusion takes place in the body of the solvent m aterial until a new equilibrium is reached at which dissolved gas is once more uniformly distributed and a t the saturation concentration corresponding w ith the new gas pressure. We m ay regard the dissolution of free gas, or the evaporation of dissolved gas in an infinitesimally thin layer, as occurring instantaneously with change of gas pres
sure, and assume the gas dissolved in the surface layer to be always in equilibrium w ith the external gas. As, however, in the interior of the solvent m aterial (as distinct from the infinitesimally thin surface layer) change of concentration can take place only by the movement of dissolved gas by diffusion under a concentration gradient, the concentration of gas in the interior is a function of time.
When a solute is n o t uniformly distributed throughout a solvent medium the change of concentration which takes place a t any point due to diffusion is given by the Fick's law relationship
&c _ u R>2c i . d’c l
5i -
k
L a? +Bp
+ a ? J (1>w h ere c = c o n c e n tra tio n
£ *= tim e
k = d iffu sio n c o n s ta n t
x, y, a n d z = c o o rd in a te s o f th e p o in t w ith re s p e c t to rec
ta n g u la r a x es
100 IN D U STR IA L AND E N G IN E E R IN G C H EM ISTR Y VOL. 12, NO. 2 The diffusion constant is defined by the fundamental dif
fusion relationship
dQ = -fc X A x | x d l (2)
where Q = quantity of solute I = distance
A = cross-section area perpendicular to the direction in which I is measured: its dimensions are (length)2
(time)"1 T h e d iffu s io n constant is char
acteristic of any pair of substances, solute and solvent, and is dependent upon temperature.
J u s tif ic a tio n for the assumption of Equations 1 and 2, th a t the diffusion constant is inde
p e n d e n t of th e actual concentra
tion, is given below.
Perm eation through a U niform Sh eet The mechanism of permeation here described, involving solution, diffusion, and evaporation, has been clearly in
dicated by Daynes (1). Consider a sheet of a rubberlike substance, of uniform thickness, L, and cross-sectional area, A, separating two reservoirs of gas maintained a t different constant pressures, pi and p2. Gas dissolves a t the surfaces and diffuses into the interior. No static equilibrium can be reached as the saturation gas concentrations, corresponding with the two gas pressures, are maintained in the respective faces and a constant concentration difference is maintained between them. A dynamic equilibrium will, however, be attained a t which gas passes through the sheet a t a constant rate.
We may assume Henry’s law and write c = hp, where p = gas pressure, c = equilibrium gas concentration in the material—
i. e., the solubility at gas pressure p—and h = proportionality constant characteristic of the solute and solvent. The simpler diffusion Equation 2 may be applied and used to show, first, that
c = Ci + £ (Ct — C i)
where c is the gas concentration at a point distant I from the face in contact with the gas at pressure pi, and ci and Ca are the equilibrium gas concentrations corresponding with gas pressures Pi and pt, respectively—i. e., Ci = hpx and c, = hp2—that is, the concentration varies linearly with distance from one face.
Secondly, it can be deduced that
Q = - k A MP2 - Pi) (3)
In this relationship, the factors characteristic of the materials are k and h, and with standardized experimental conditions we may write
Q = (constant) X kh (4)
Or, in a comparison of the resistance offered by sheets of ma
terials to the passage of gases,
Qa • Qb* Qc ■ • kaha : kbhb ‘ kche —
-where Qo, Qb, Qc are the quantities of the gases passing, and Kah?, kbhb, kchc --- are the corresponding products of the dif
fusion constant and the Henry’s law constant for the correspond
ing combinations of rubberlike substance and gas.
Absorption b y a Block
_ Considering now the change in the total quantity of gas dissolved in a solid block of rubberlike material, when after attaining equilibrium in the gas a t one pressure, the pressure is instantaneously and isothermally changed and maintained a t a new value, the total quantity of gas, q, dissolved a t any time is given by
- / / / • X dx X dy X dz (5) where c, the concentration of gas dissolved in the element of volume dx X dy X dz a t the point x, y, z, is given by the Fick’s law relationship, Equation 1. I t is possible, under the above conditions of gas pressure, and limiting con
sideration to certain geometric shapes for the block of ma
terial, to integrate the Fick’s law equation and evaluate c as a function of the product kt and x, y, and z, and further, to carry out the integrations to determine q as a function of kt and of the dimensions of the block of material.
The particular geometric shape of the block which we con
sider here (Figure 1) is th a t of a right-rectangular prism of dimensions X , Y, and Z, and we discuss the three particular cases in which (1) two pairs of opposite faces are impermeable, the pair of faces of dimensions Y Z only being permeable;
(2) one pair of opposite faces is impermeable, the two pairs of faces of dimensions Y Z and X Z only being permeable; and (3) all six faces are permeable.
The condition of impermeability, or negligible permeability, is easily experimentally realized either by the obvious method of the application of impermeable coatings—e. g., of tinfoil—
by means of a suitable adhesive, or by employing test pieces the dimensions of which are great in certain directions—
for example, case 1 is m et by employing a sheet of relatively thin material of great area and case 2 by a rectangular- sectioned prism of great length.
In the first case, diffusion of dissolved gas m ay be regarded as taking place in one direction only (parallel to the axis of x); in the second case, in two directions a t right angles (parallel to the axes of x and y) ; and in the third case, in three directions a t right angles (parallel to the axes of x, y, and z). In spite of the simplicity of these conditions, in no case is it possible to obtain a simple expression for c, the
1.0 0 .9 0 .8 0.7 0.6
x (DISTANCE FROM SURFACE AS A FRACTION OF THE THICKNESS.)
Fi g u r e 2 . Va r i a t i o n o f Fr a c t i o n a l Co n c e n t r a t i o n o f Ga s i n a Sh e e t w i t h Di s t a n c e f r o m Su r f a c e
FEBRUARY 15, 1940 ANALYTICAL E D IT IO N 101 concentration of gas a t the point x, y, z, from the Ficlc’s
law equation; solutions can, however, be obtained in series form. These m ay be written
sheet of material in which the diffusion constant of the gas is h —i. e., kt = kih—is given by
Cj = _£— £°_ = i cœ Co c t = £ — Sl = i
Ln' A'1 1 X
e s i n e
Co
4 V ~ 'l
i t / j n
1 6 V ^ J _ e -* (« * T ‘ + nl & ) ‘ t 2 / j m n
in m y x X sine
(6)
(7 ) + •
- _ c - Co _ 1 _ « V U - ~ i ( “ f ' + m ’ f ’ + n , ł > ) ‘
= c „ — Co T » / A m n 6
s i n e l ^ r X X s i n e m y y X s i n e n ^ z (8 )
where Z, m, and n are positive odd integers; Co = concentration of gas in the block a t equilibrium under the original gas pressure—i. e., the uniform concentra
tion a t zero time—c» = concentration of gas in the block a t equilibrium under the new gas pressure— i. e., the uniform concentration after infinite time— Ch C2, and Ci are the fractional increases in concentration a t the point (x , y, z) in cases 1, 2, and 3, respectively.
The summations may be carried out by assuming particular values for the product kt and for x, y, and z.
For example, the fractional increase in gas concentra
tion after a time tu a t points one quarter of the way through a sheet of material—i. e., x = X— in which the diffusion constant of the gas is h — i. e., kt = hti—is given by
The series is rapidly convergent and can be summed by evaluating the first few terms.
4 r i -i* r e X« s i n e -r +x a.
T L l 4
1 —3* ^ k i t i . o X , - e A 1 s i n e 3 ^ +
1 -5 * h t i . , x , - e A» sine 5 7 + -
5 4
The series is rapidly convergent and can be summed by evaluating the first few terms. Figure 2, which was derived in this way, shows the gas concentration a t all points through a sheet after several time intervals.
The quantities of gas absorbed a t any time are obtained from relationships 6, 7, and 8 by integration (see Equation 5), the results again being in series form. They m ay be w ritten
Fi g u r e 3 . Va r i a t i o n o f Fr a c t i o n a l Qu a n t i t y o f Ga s Ab s o r b e d b y Sh e e t, Sq u a r e- Se c t i o n Ro d, a n d Cu b e
w i t h Ti m e
I t is to be noted th a t Qi> Q'-i and Qi are independent of go.
q„ , k, and I and are dependent only upon x, y, z, and the product kt. T h at is, two identical blocks of materials in which the diffusion constants of the gases are h and kt reach the same degree of fractional saturation after times ti and tt which bear the relation ii/£j = kt/kt.
Relationships 9, 10, and 11 may be simplified, as m ay also 6, 7, and 8, by considering test pieces the dimensions of which are equal in the directions in which diffusion takes place and by choosing units of length such th a t these dimensions are equal to r units; or, in other words, by considering, for example, (1) an infinite sheet of thickness 7r units of length (X = x); (2) an infinite rod of square cross section tt X t (units of length)2, (X = Y = 7r); and (3) a cube, t X t X 1 (units of length)*, (X = Y = Z = x).
Relationships 9, 10, and 11 reduce to
<h
<2.
<3.
go _
g» ~ go g - go
?» — go g - go g~ — go
1
- S 3
(9 )1 “ ï E s s r * e~* * + (10) . 5 1 2 V ^ 1
tt« Z - j l W n 1 e ‘
4* m * - — n*
Y * “ Z*
(ii)
y
and
w h e r e
Qi = 1 — S Q, = 1 - 5»
Q, = 1
-= 1 " V ' 1
X 2 n 2
(12)
(1 3 ) (1 4 )
— n * k t
where I, m, and n are positive odd integers; go = quantity of gas in the block a t equilibrium under the original gas pressure—
i. e., a t zero time—q„ — quantity of gas in the block at equilibrium under the new gas pressure—i. e., after infinite time—and Qt, Q2, and Qt are the fractional increases in the quantities of gas absorbed in cases 1, 2, and 3, respectively.
The summations may be carried out by assuming particular values for the product kt. For example, the fractional in
crease in the quantity of gas absorbed after a time h by a
and n is a positive odd integer.
Table I gives the values of Qu Qj, and Qs for a range of values of kt; it also includes the values of Qi2for these values of kt. I t is seen from the table that, up to a value of about 0.6 for Qt, the value of Qi2remains substantially constant. This is a purely empirical relationship. Figure 3 gives the values of Table I plotted graphically. This constancy of in the experimental results of investigations covering only an initial fraction of the absorption has led to an erroneous assumption th a t this law applies over the whole absorption period; the
102 infinite time—i. e., a t saturation—is q„
v(Pi - Pi)h
sumption has been explained as being due to swelling (6).
All the foregoing considerations have been based upon the assumption th a t the diffusion constant is independent of concentration. This assumption is justified on two grounds.
In the first place, diffusion arises out of the molecular move
ment, and the diffusion law, stated in Equation 2, is a neces
sary consequence of the kinetic theory, assuming only th at there is no m utual interference between the diffusing mole
cules—i. e,( assuming conditions comparable with those under which the gas laws may be applied to solutions. In the case of the “perm anent” gases with which we are here primarily concerned, the solubilities are so low under all normal condi
tions of gas pressure and tem perature as to be well within the range of noninterference between their molecules. For example, soft rubber a t 30° C. dissolves about 5.5 X 10 2 cc.
of nitrogen measured a t normal tem perature and pressure per cc., per atmosphere pressure. The saturation concentration of nitrogen, therefore, a t 30° C. under a pressure of 3 atmos
pheres (the highest pressure which would normally be m et with in practice) is 0.0074 gram molecule per liter. In the second place, there is ample experimental evidence th a t the solubility of gases in rubberlike substances obeys H enry’s law 02, 5, 7); furthermore, the available experimental evidence shows th a t permeation rates are proportional to pressure dif
ference (1, 4). I t is readily realized th a t these conditions necessitate the independence of the diffusion constant of concentration.
A pplication of Absorption Experim ents to M easu rem ent o f Perm eability
Relationships 12, 13, and 14 show th a t in all cases the fractional increase in the quantity of gas absorbed is de
pendent only upon the product of the diffusion constant and time. Furthermore, the total quantity of gas absorbed a t equilibrium is independent of the diffusion constant and of time. Therefore experimental results giving the relation
ship between the quantity of gas absorbed and time can be used to evaluate k and h independently and hence to evaluate the product, kh, which has been shown to be directly pro values of the fractional increase in the quality of gas absorbed are compared w ith the corresponding values of the product kt (hh, hU, k3t3, etc.), deduced from the appropriate theo
retical relationships 12,13, or 14. This is most conveniently carried out by means of the graph of Figure 3. The cor of the test piece, measured in the direction in which diffusion takes place, are v units. If these dimensions, measured in the ordinary units, are a cm. the value of the diffusion con
stan t deduced as above m ust be multiplied by (a/ir)* to give its value with centimeters as the units of length.
Fi g u r e 4 . Va r i a t i o n o f Fr a c t i o n a l Qu a n t i t y o f Ga s
FEBRUARY 15, 1940 ANALYTICAL E D IT IO N 103
in order th a t this method may be employed to determine k.
In connection with the alteration of the numerical value of k with adjustm ent of the units of length, a point which arises is the theoretical comparison of the absorption of gas by specimens of different essential dimensions. We consider here only the cases of sheets of great area relative to their thicknesses, of square-sectioned prisms of great length relative to their cross-sectional dimensions, and of cubes. Equations 9, 10, and 11 are modified to
Qi = 1
where i = 1, 2, or 3 according, respectively, as diffusion of dissolved gas takes place in one direction only, or in two or three directions a t right angles and where k is the value of the diffusion constant with the same units of length which are used to measure a, the dimensions of the test piece in the direction or directions in which diffusion takes place. Figure 4 shows the variation of the fractional quantity of a gas absorbed by uniform sheets of different thicknesses. In this graph, one particular thickness is chosen for reference and the thicknesses of the other sheets are given as multiples (X) of this thickness.
The curves are all on the same scale, the factor ~ being necessary in the abscissas in order to retain a constant abso
lute value for k with various thicknesses of specimen.
Ap plicability o f M ethod to Nonhom ogeneous Rubberlike Products
One of the most common and most obvious methods for reducing the permeability of sheets of rubberlike materials to gas is compounding with finely divided inert substances such as clay, whiting, and barytes. The reduction of perme
ability by the incorporation of these substances may be considered as arising out of a t least two separable effects.
First, they replace some of the solute material and thus reduce the cross-sectional area through which gas m ay pass;
secondly, they cause the gas to take a devious path and so increase the effective thickness.
Considering the first effect, it can be shown th a t if a fractional volume, 6, of the continuous medium is replaced by uniformly dispersed particles, the fractional area of any cross section oc
cupied by the particles is also 0. This result is independent of particle shape or size, provided only th a t the numbers of particles in the volume and area considered are sufficiently great to allow permeability by absorption experiments, the reduction due to this effect is quantitatively allowed for in the reduced H enry’s law constant.
The second effect of fillers, the increased effective thickness of the sheet, is less readily assessed, as particle shape and
orienta-tion have to be taken into account. If, however, we regard the reduction of permeability by this effect as arising not out of increased effective sheet thickness, but out of a reduced diffusion constant, it is readily seen th a t the diffusion constant obtained from permeation experimente, although having no exact physical interpretation, can be applied quantitatively to the assessment of permeability of sheet material, provided only th a t in the case of anisometric filler particles, their orientation with respect to the direction or directions in which diffusion takes place in the absorption experiments is the same as th a t of permeation under service conditions. On account of this provision it is preferable, in the absorption experiments, to cut sections from the sheet m aterial and to allow diffusion to take place only in the ap propriate direction.
A third possible effect of fillers arises out of experimental results given below (Table IV). W ith diatomite-compounded rubber mixings the solubility of nitrogen in the rubber (as op
posed to the mixing) decreases with increasing proportions of diatomite (see Table II).
Rubber removed by adsorption is calculated from the relative solubility values assuming th a t adsorption of rubber by diatomite provides a complete explanation for the irregularity in solubility
I t is suggested th a t the diatomite m ay remove some of the nibber from the rubber phase by adsorption, so th a t it is not free to dissolve gas. The reduction of permeability by diatomite will, therefore, be greater than th a t anticipated from the volumes of filler incorporated, owing to the effective reduction in the quan
tity of available rubber and the corresponding increase in particle size of the diatomite.
Barytes does not show a similar adsorption effect. On the other hand, the incorporation of carbon black in rubber mixings increases the solubility of gases by adsorption of gas and this method of measuring permeability cannot be applied, without further consideration, to rubberlike substances compounded with carbon black. following the absorption or desorption when the gas in which a test piece of specified shape is immersed is subjected to a pressure increase or decrease. Those found most con
venient of manipulation are described below.
venient of manipulation are described below.