Liquid sorption by rubber sheets and evaporation; models and experiments

Polymer Testing 6 (1986) 253-265

Liquid Sorption by Rubber Sheets and Evaporation; Models and Experiments

Y. K h a t i r , J. B o u z o n a n d J. M. V e r g n a u d Laboratory of Materials and Chemical Engineering, UER of Sciences, University of St-Etienne, 23 Dr P. Michelon, Saint-Etienne 42100, France

SUMMARY The sorption of liquid by rubber sheets has been considered as a diffusion process controlled by a constant diffusivity. In spite of the swelling, the thickness of the sheet may be assumed to remain constant for integrating Fick's law as diffusion proceeds. The evaporation of liquid from the rubber is diffusion controlled. No analytical solution can be obtained in general cases. A model is described, based on an explicit method with finite differences, taking into account the diffusivity and rate of evaporation. This model gives results for desorption in good agreement with experiments in the case of motionless air and with stirring. Results with high stirring can also be obtained by analytical solution of Fick's law integrated by taking zero for the concentration of liquid on the rubber faces at the start of the process.

INTRODUCTION The action of a liquid on rubber may result in absorption of liquid by the solid, followed sometimes by extraction of rubber constituents. Usually, absorption is greater than extraction, the resulting increase in volume being expressed as swelling of the rubber. The volume change is a good measure of the resistance of the rubber to the liquid; a high degree of swelling clearly indicates that a rubber is not 253 Polymer Testing 0142-9418/86/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1986. Printed in Northern Ireland

254

Y. Khatir, J. Bouzon, J. M. Vergnaud

suitable for use in the environment of the given liquid. Also, the degree of swelling can be related to the state of cure of the rubber. 1-3 Several standardized test procedures 4-6 are concerned with the measurement of change in mechanical properties (tensile strength, hardness) and dimensions. Because swelling is relatively simple to measure, changes in volume or mass are readily followed while the test piece is immersed in the test liquid at the chosen temperature. The increase in weight of the rubber and hence the rate of uptake of liquid are considered to be the result of a diffusion process controlled by either a diffusion coefficient constant or a concentration-dependent diffusivity. 7 The first case is sometimes easy to solve: for instance, when the sheet is initially free of liquid and is placed in the liquid, the liquid concentration on each surface immediately attains a value of concentration corresponding to the equilibrium uptake concentration and remains constant afterwards. The extraction of plasticizer or additives from plasticized PVC has been explained by a complicated process with a simultaneous diffusion of the solvent into and plasticizer out of the plasticized PVC. 8-1° In this case, no analytical solution can be given and modelling using numerical analysis and a computer is the only possibility. 11 Desorption is the complementary problem in which all the solute is at first uniformly distributed through the rubber sheet and diffuses out of the rubber. If free and immobilized solute are considered to be initially in equilibrium everywhere in the sheet, the mathematical solution presented for sorption also describes desorption, provided the following conditions are chosen: the diffusivity does not depend on the local liquid concentration; the concentration of the solute is zero on the rubber faces as soon as desorption starts. An interesting case of desorption is diffusion-controlled evaporation of a liquid initially located in the rubber. The evaporation process is diffusioncontrolled in the sense that the rate of evaporation depends largely on the rate at which solvent is supplied to the evaporating surface by internal diffusion. It is possible to express the rate of loss of liquid in terms of a simple expression only in these two cases: (1) the surface of the sheet is assumed to reach equilibrium with the outside when evaporation commences, and the concentration at the surface falls immediately to zero if the atmosphere is free of vapour;

Liquid sorption by rubber sheets and evaporation

255

(2) for the condition that the rate of evaporation from the surface equals the rate of transfer by diffusion to the surface, the concentration of the vapour at the equilibrium outside the rubber must be known.7 The purpose of this work has been to study processes of sorption of liquid by rubber sheets, and evaporation of liquid out of the same sheets. In both cases, matter transfers are found to be controlled by diffusion, with a constant diffusivity in spite of the importance of the swelling. So the sorption process can be expressed by a simple well-known equation. The process of evaporation of the liquid cannot be expressed mathematically, because only the rate of evaporation can be measured. A model has been described for this difficult case, taking into account the matter transfer obtained by diffusion through the rubber and the evaporation from the rubber surface. No drastic assumption has been imposed for the concentration of liquid on the rubber faces. This model is available for different rubbers and shapes.

THEORETICAL

Assumptions (a) (b) (c)

(d)

A thin plane sheet of rubber is considered with a onedimensional diffusion. The diffusion takes place under transient conditions with a constant diffusivity. During sorption, the concentration of liquid on the rubber face reaches the equilibrium value as soon as the rubber is immersed in the liquid. At the beginning of desorption, the concentration of liquid throughout the rubber is constant.

Equations for sorption The equation of one-dimensional diffusion 8C 32C at = D $x 2

(1)

Y. Khatir, J. Bouzon,

256

with initial and boundary

conditions

J. M. Vergnaud

as follows:

t = 0,

O
c=o

(2)

t b- 0,

o
c = c*.x c = c,

(3)

x = 0, x = 1, has the solution for concentration

of liquid:

c, - c* n

(

c,

c (- 1)” sin (2m + l)Jrx exp _ (2m + 1)%r2D t ( > 4= m&l(2m + 1) 1 l2

>

The total amount of diffusing liquid which has entered the sheet at time t is a function of the corresponding quantity after infinite time:’

Numerical analysis for desorption Take the sheet to occupy the space 0
AIR

RUl313ER SPACE v2 1 n-l

1 0

n+l

n

-r-l-l------T-rr--

(i+ltl)ti Ci;ZO I

I :

Fig. 1.

( ___ : , Time-space

Ci+l+n

1

(

diagram.

,---.

;

I

Liquid sorption by rubber sheets and evaporation

257

Inside the rubber As previously shown, 1~ the concentration at the time (i + 1) At is as follows: 1 C

(6)

Ci+l'n = ~1 [ i,n--I "~- (M - 2)C n "~- Ci, n+l]

with the dimensionless number: (Ax) 2 1 -At D

M-

(7)

Rubber faces By considering the matter balance of liquid in the rubber slice of thickness Ax/2 next to the face, we have:

D(ac)

ac

Ax

(8)

which becomes with finite differences: I

At

(9)

Ci+l.i/4 - Ci.1/4 = --M ( fi, l - C,o) - 2 f ~ x

By supposing that: Ci+,,,l 4 -- Ci.i,4 = ~4[Ci+i.O-- Ci, o]

I

(lO)

"~~ [C/+I,/- C,I]

eqn. (9) can be rewritten: At

Ci+I ' 0 = Ci.o "~- 3f [Ci.I -- Ci+,,'] "Ji-- ~8 Ice', -- Ci'O] -- ~3F Ax

(11)

A m o u n t of liquid evaporated The amount of diffusing substance which remained in the sheet at time i At after evaporation is obtained by integrating eqn. (11). Mi = 4aC;,o+ 9Ci,, + 2 ~'~ Ci~,+ C~,. Ax z

(12)

At the beginning of the evaporation, the concentration of liquid on the rubber faces is obtained from eqn. (11), and this concentration is

258

Y. Khatir, J. Bouzon, J. M. Vergnaud

positive. It decreases regularly from the initial concentration to zero. After this time, the concentration on the rubber faces is kept equal to zero. EXPERIMENTAL

Rubber samples Rubber sheets of different thicknesses (1.5-4.8 mm) were prepared and cured by pressing rubber in a steel mould operated by a power press under a pressure of 50 bars. Conditions for rubber cure have been previously determined from earlier studies 9'1° in order to make sure that vulcanizates have a high state of cure. Several discs (38 mm in diameter) were cut from these sheets (Table 1).

TABLE 1 Rubber Sample and Cure Conditions Samp~thickness (ram)

1.5 4-8

Temperature of mould (°C) Time (rain)

170 170

10 15

Test condition for sorption All experiments have been carried out with two rubber discs immersed in about 300 ml of toluene (Rh6ne-Poulenc) in a flask of 500 ml capacity, using either motionless liquid or a controlled rate of stirring. Test specimens were removed at intervals of time and excess liquid on the surfaces removed. After weighing, the specimens were returned to the flask.

Test condition for desorption-evaporation Samples having the largest quantity of liquid resulting from a preceding sorption were exposed either to motionless air or to air with a forced convection. The rubber sheets were vertical and the

Liquid sorption by rubber sheets and evaporation

259

temperature constant at 20°C. The weight of the samples was determined at various times.

RESULTS Determination of diffusion coefficients and evaporation rate

In eqn. (5), the thickness l of the sheet is assumed to remain constant as diffusion proceeds. In fact the sheet swells and the thickness increases as the liquid enters. Equation (5) can still be used, provided we take a frame of reference fixed with respect to the rubber itself. Thus we take the basic volume of the sheet to be its volume in the absence of liquid and use the unit of length such that it contains, per unit area, unit basic volume of the rubber. Then the thickness of the sheet, measured in these units, is constant and equal to the original unswollen thickness?

Use of half-time o f process The value of t/l 2 for which M,/M= = 1/2 is approximately given by eqn. (13), the error being about 0.001%. 5 D = 0.049(~)

(13)

The half-time of the sorption process is observed experimentally and the constant diffusion coefficient is determined from eqn. (13).

Use of final rates of sorption and desorption In the later stages of diffusion in the sheet, only the first term in the series of eqn. (5) needs be considered, and we have: M~ - 214,_ 8 M~ 3r2 exp (

ar2Dt] ~5 ]

(14)

Results for the diffusion coefficient Results obtained by using these two different methods are about the same for the diffusion coefficient, as shown in Table 2, either for the sorption or desorption process.

260

Y. Khatir, J. Bouzon, J. M. Vergnaud

TABLE 2 Diffusion Coefficient D Process

Sorption Sorption Sorption Desorption

Thickness (cm)

D x 107 Equation (cruZ~s) (M®/MI)IO0

0.15 0-15 0-48 0.15

(13) (14) (13) (14)

5 5 5.3 5.1

338 338 320 338

The rate of evaporation has been determined by using the initial rate of desorption, when the process may be assumed to be controlled by evaporation. It is possible to deduce this value from the initial gradient of the desorption curve when plotted against the time, as shown in eqn. (15). F=

dM dt

for

t~0

(15)

The values of the rate of evaporation d e p e n d largely on the air velocity and stirring (Table 3).

TABLE 3 Rate of Evaporation F in Films 0.15 cm Thick Conditions

Motionless air Stirred air

F(g/cm 2s)

10.4 × 10-s 38.5 × 10-s

Sorption by a swelling sheet

Figure 2 illustrates the validity of the assumption concerned with constant diffusion coefficient. The three curves correspond to the experimental uptake of toluene (expressed in weight % of the initial weight of rubber), the uptake calculated by using the series of eqn. (5), and the uptake obtained from numerical analysis. As can be

Liquid sorption by rubber sheets and evaporation

261

i 1001

r~)loo

"/ •

60

f

•

"

20

0

Fig. 2.

I[~-

'

200

-

-

300

T~r~nY

T o l u e n e absorption by 0.15 cm thick rubber sheet. O, Analytical solution; , numerical analysis; ~r, experimental curve. D = 5 x 10 7 cm2/s.

seen, the values obtained from these three methods can be accommodated on a single curve. Theoretical results obtained by using eqn. (5) are also in good agreement with experiment for a different value of the thickness of the sheet, as shown in Fig. 3. The physical process of sorption is clearly demonstrated and described mathematically in terms of a constant diffusion coefficient and constant thickness. The diffusion coefficients are about the same for different thicknesses of sheets made of the same rubber. The known t/l 2 law is approximately verified.

Desorption and evaporation process Three different curves are drawn in Fig. 4 obtained with rubber sheet 0-15 cm thick exposed to motionless air: the experimental curve, the theoretical curve determined with the help of eqn. (5), and the theoretical curve obtained from our model.

262

100

~00

,.~_'_~

80

60

40

20

0

10

lime
2b

30

Fig. 3. T o l u e n e absorption by 0.48 cm thick rubber sheet. O , Analytical solution; , numerical analysis; * , experimental curve. D = 5.3 x 10 -7 cm2/s.

iO0

.(~)10o

80 60 40. 20 .

.

.

.

.

.

.

.

.

.

.

.

.

.

""n)

Fig. 4, T o l u e n e desorption from 0.15 cm thick rubber sheet in motionless air. 0 , Analytical solution; , numerical analysis; * , experimental curve. D = 5.1 × 10 -7 cm2/s; F = 10.4 × 10 -5 g/cm 2 s.

Liquid sorption by rubber sheets and evaporation

263

As shown in this Figure, experimental results and theoretical results calculated using our model are in very good agreement throughout the whole process. There is a large difference between the values of the liquid desorbed when they are measured and calculated with the help of eqn. (5). The reason for this difference has to be found in the value of the liquid concentration of zero at the rubber faces, as soon as desorption takes place. In fact, at the beginning of the process the matter transfer by diffusion is higher than that obtained by evaporation from the rubber faces and the total transfer is controlled by a surface effect. As the liquid concentration decreases at the rubber faces, the relative importance of the matter transfer by diffusion also decreases. Finally, when the concentration of liquid at the rubber faces tends to zero, the three curves are well superimposed. In the case of stirred air, the experimental curve agrees well with results calculated with the help of our model (Fig. 5). The effect of agitation on desorption rate is of importance at the beginning of the operation. With higher values of the rate of evaporation, the liquid

100

80

60

40

20

11meCmn) . . . . .

~do

. . . .

260

. . . .

360'

'---'-

Fig. 5. T o l u e n e d e s o r p t i o n f r o m 0 . 1 5 c m thick r u b b e r s h e e t in stirred air. O, Analytical solution; , n u m e r i c a l analysis; ~', e x p e r i m e n t a l curve. D = 5 . 1 x 10 _7 cm2/s; F = 38.5 × 10 5 g/cm 2 s.

264

Y. Khatir, J. Bouzon, J. M. Vergnaud

concentration at the rubber faces falls more quickly to zero. The limitation of the effect of agitation on desorption rate is shown by the curve drawn using the analytical equation (4) (obtained by assuming liquid concentration is zero as soon as the evaporation starts). This equation must be slightly transformed for desorption: the left-hand side of the equation becomes Ct/C~.

CONCLUSIONS This paper has been concerned with some methods of determination of kinetics of sorption of liquid by rubber and desorption by an evaporation process. The sorption process of liquid entering the rubber is well described by diffusion with a constant diffusivity; sorption experiments are in good agreement with the theoretical results calculated by using the analytical solution of Fick's laws obtained with the appropriate boundary conditions. The evaporation process of liquid is diffusion-controlled, and the rate of evaporation depends largely on the rate at which the liquid is supplied to the evaporating surface by internal diffusion. No analytical solution can be obtained for this problem, but a model using numerical analysis has been found to be of interest for determining the loss in weight as a function of time. This model takes into account a constant diffusivity of the liquid through the rubber mass, and a constant rate of evaporation of the liquid. In the case of desorption, the effect of stirring on the rate of the evaporation process is shown to be of importance. This is due to the fact that the rate of evaporation depends largely on the motion of air. For very fast stirring, the rate of evaporation is limited by the value calculated by using the analytical solution of Fick's law obtained by taking zero for the concentration of liquid at the rubber faces at the start of the process.

REFERENCES 1. Brown, R. P. (1979). Physical Testing of Rubbers, Applied Science Publishers, London, pp. 268-76.

Liquid sorption by rubber sheets and evaporation

265

2. Hands, D. and Horsfall, F. (1980). 'A new method for simulating industrial cure processes', RAPRA Research Report 44. 3. Brown, R. P. (1970). RAPRA Research Report 191. 4. ISO 1817, 1975. Resistance to Liquids. Methods of Test. 5. BS 903, Part A16, 1971. The resistance of vulcanised rubber to liquids. 6. ASTM D471--75. Rubber Property. Effect of Liquids. 7. Crank, J. (1975). The Mathematics of Diffusion, 2nd edn. Clarendon Press, Oxford. 8. Frisch, H. L. (1978). Simultaneous nonlinear diffusion of a solvent and organic penetrant in a polymer, J. Polym. Sci., 16, 1651-64. 9. Messadi, D. and Vergnaud, J. M. (1981). Simultaneous diffusion of benzyl alcohol into PVC and of plasticizer from polymer into liquid, J. Appl. Polym. Sci., 26, 2315-24. 10. Vergnaud, J. M. (1983). Scientific aspects of plasticizer migration from plasticized PVC into liquids, Polym. Plast. Technol. Eng., 20, 1, 1-22. 11. Taverdet, J. L. and Vergnaud, J. M. (1984). Study of transfer process of liquid into and plasticizer out of plasticized PVC by using short tests, J. Appl. Polym. Sci., 29, 3391-400. 12. Sadr-Bazaz, A., Granger, R. and Vergnaud, J. M. (1984). Calculation of profiles of temperature and state of cure within the rubber mass in injection molding process, J. Appl. Polym. Sci., 29, 955-63. 13. Abdul, M. and Vergnaud, J. M. (1984). Vulcanization progress in rubber sheets during cooling in motionless air after extraction from the mold, Thermochim. Acta, 76, 161-70.