Langmuir 1995, 21, 639-642
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Thermal Expansion of Liquid Foams M. A. Fortes Departamento de Engenharia de Materiais, Instituto Superior Thcnico, Lisboa, Portugal Received August 8, 1994. I n Final Form: October 26, 1994@ Various contributions to the thermal expansion of a liquid foam are discussed, which are related to the transfer of molecules between the liquid and the gas as the temperature changes. Experimental measurementsof the thermal expansion of two types of aqueous foams are presented which can be interpreted in terms of the change with temperature of the equilibrium water vapor pressure. Because of this effect the foams expand considerably more (by a factor of -1.4 under normal conditions) than what would be predicted for a closed ideal gas. 1. Introduction Liquid foams have attracted much attention in recent years not only because of their practical importance but also because they are paradigmatic of cellular systems, a class of systems of widespread i n t e r e ~ t . l - ~ One of the basic properties of a liquid foam that has not been sufficiently discussed in the literature is its thermal expansion. It is usually accepted that the thermal expansion is, except for minor corrections, that of the dispersed gas in the foam. Assuming that the gas has ideal behavior, the coefficient of thermal expansion,
where Vis the volume of the foam, T i s temperature, and P is the external pressure, would be
1 T
a(T)= -
(2)
Strictly this result implies a negligeable volume fraction (1- 4) of liquid, i.e. a dry froth, in which case the total volume coincides with the volume of gas. On the other hand, eq 2 only applies if the number of moles in the gas is a constant, and this may not be the case in a foam because a temperature change may induce variation in the number of moles in the gas phase. The gas pressure varies from cell to cell in a foam, because-of the curvature of the liquid films. Its average value, Pi, can be related to the external pressure, P, through the following equation first derived by Derajuin for a monodispersed foam.*
2 Pi= P i-ysv 3
(3)
where y is the liquid-gas surface tension and SV is the area of films per unit volume. Sv-l is of the same order of magnitude as the average cell diameter. The last term in eq 3 is due to the pressure difference across the peripheric films which are convex when seen from the outside of the foam because the films meet at 120"a t triple junctions. The applicability of eq 3 to polydispersed foams was subsequently discussed by Morrison and Ross.5 Abstract published inAdvanceACSAbstrmts, January 1,1995. (1) Weaire, D.; Rivier, N. Contemporuly Phys. 1982,25, 59. (2) Gibson, L.;kshby, M. F.Cellular Solids: Structureand Properties, Pergamon Press: Oxford, 1988. (3) Glazier, J.;Weaire, D. J.Phys.: Condens. Mutter 1992,4, 1867. (4) Derjaguin, B. V. Kolloid. 2. 1933, 64, 1. (5)Morrison, I. D.; Ross, S. J.J.Colloid Interface Sci. 1983,96,97. @
0743-7463/95/2411-0639$09.00/0
The volume fraction ofgas in a foam, 4, can vary between 1 and -0.75, when the cells change from polyhedral to spherical.6 Neglecting the volume change of the liquid, the equation of state of a foam can then be written as6 (4)
where Vis the total volume of the foam, SVis the area of films per unit volume of the foam, n is the number of moles of (ideal) gas in the foam, and R is the gas constant. Printed also includes a second additive term to the pressure, related to the osmotic pressure ofthe foam. This extra term also depends on 4 but is usually negligeable6 and will not be considered further. Indeed, also the is negligeable in normal foams capillarity term (2/3)y(S~/4) a t atmospheric pressure (P = lo5 N m-2). For example, for a foam with y = 40 m J m-2, 4 = 1, and 30 pm (which is a finelly dispersed foam) the capillarity term is of order of lo3 N m-2. 2. The Coefficient of Thermal Expansion When the temperature is changed a t constant pressure, P, there may be a change in the number of moles, n, of gas, i.e. n = n ( n ,a possibility that has been generally ignored. In fact, there are various processes that cause transfer of molecules between the liquid and the gas as the temperature is changed, even though the system as a whole is closed. These processes will be discussed below. Neglecting the change with temperature of the capillarity term in eq (4) gives the following for the coefficient of thermal expansion, dT):
(5) The coefficient d T ) therefore deviates from 1/T if there is exchange of molecules between the liquid and the gas as T changes. Equation 5 can be put in the form
d(ln n ) Ta(T)= 1 + d(ln T ) Before analyzing various possible contributions to the term d(ln n)ld(ln r), experimental results will be presented which show that in common foams this quantity is on the order of unity and cannot therefore be neglected. 3. Experimental Results Two types of aqueous foams were used in the experiments: a commercial dishwashing detergent solution and (6) Princen, H. M. Lungmuir 1988,4, 164.
0 1995 American Chemical Society
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640 Langmuir, Vol. 11, No. 2, 1995
a commercial shaving cream foam. The first foam was produced by blowing air through a pipete into a dilute solution. The bubbles were forced to enter a cylindrical glass tube which was then closed a t the end through which the bubbles penetrated. The average diameter ofthis foam was on the order of a few milimeters. Some measurements were carried out with the dishwater foam after coarsening to the extent that a bamboo structure developed in the tube, with successive films perpendicular to the tube axis. The average separation of the films in the bamboo was on the order of the tube diameter. The gas phase of the shaving cream foam is a mixture of hydrocarbons. The foam (density around 0.07 g ~ m - initial ~; average cell diameter -35 pm) was introduced in a cylindrical tube through one end, which was then closed. The other end of the tubes containing the froths was tapered to reduce evaporation of the liquid in the course of the experiments. The diameters ofthe tubes varied between 0.7 and 1.5cm in different experiments. The tubes were always kept horizontal and were rotated from time to time to avoid heterogeneities related to liquid drainage. In each experiment, the lengths of the foams in the tube (Lo and L1, respectively) were measured upon equilibration a t two different temperatures (TOand TI, with T1 > TO). The lengths were measured with a ruler. There was some uncertainty about the location of the free surface of the foams. The error in L is around 2 mm except in the bamboo foam, where it is smaller. Experiments were done for the following pairs of TO,T1 (in degrees Celsius): 8,23; 18,35; 18,45; and 23,45. A refrigerator and an oven were used to obtain nonroom temperatures. The temperatures were measured with an accuracy of 0.5 K. A slight change offoam length occurred when the doors of the oven and refrigerator were open; to minimize the error, the equilibrium length was measured with a ruler placed inside the oven or refrigerator. Experiments a t higher temperatures were difficult because of evaporation or decomposition of the foams. Temperature equilibration was relatively rapid, lasting for not more than 15 min. No changes in length a t constant temperature were observed following this equilibration period. The coarsening of the foams during this period was negligeable but was observed in both foams if successive experiments were performed lasting for a considerably longer period. The shaving cream foam was also found to absorb air a t the surface exposed to air, originating a slight expansion of the cells closer to the surface, but the resulting expansion of the foam was negligeable in the time duration of one experiment. For each experiment, i.e., for each pair oftemperatures, TO,T1, the quantity 1 defined as
(7) was calculated. Since the volume of the foam is proportional to L (neglecting the expansion ofthe glass tube) the quantity defined in eq 7 is in fact an average TdT) in the corresponding temperature interval and should be unity for no exchange of molecules between the liquid and the (ideal) gas. For a given pair, TO,T1, the coefficient 1 was found to be, within experimental error, the same for the two foams and independent of tube diameter and of the degree of coarsening developed (including the bamboo structure of the detergent froth). Figure 1 shows the experimental values of 1 for the various experiments. Because of the limited number of temperatures used, the temperature used in the plot is the average temperature in the interval
TO,TI. The quantity 1 increases with increasing temperature, from 1.3 to 2.2. It is considerably larger than unity, reflecting a large increase in the number of moles of gas as the temperature increases. In the following section the possible origin of this effect is discussed. 4. Variation of the Number of Moles in the Gas There are two possible contributions to the variation with temperature of the number of moles of gas, n, in the foams. They are related to gas solubility changes and to evaporatiodcondensation of water. A third possibility is related to the adsorption of the gases in the foam on the film surfaces. It is expected that the quantity absorbed decreases with increasing temperature. No data on this effect seems to be available, and it is therefore impossible to predict its contribution to expansion. But the effect, if detectable, should depend on the surface area of the foam, SV,and this is not observed experimentally. Therefore, only the effects of solubility and evaporation will be considered. In the analysis it will be assumed that there is only one gas in the foam in addition to water vapor and that thermodynamic equilibrium is achieved a t each temperature. The analysis could easily be generalized to more than one gas (as is the case with both foams used in the experiments). 4.1. Solubility Effect. Let p be the average partial pressure of the gas in the cglls. In the following analysis it will be assumed t h a t p =Pi G P,i.e. the partial pressure of water vapor is neglected. The solubility x (in moles of gas per mole of liquid) will be related top through Henry’s law p=Hx
(8)
where H is Henry’s constant. H is independent ofp but in general decreases with increasing temperature. It is convenient to introduce the coefficient p defined by
1dH 9=-?Iz Using data of solubilities of gases in water’ it can be concluded that this coefficient is not much affected by the nature of the gas and is fairly independent of temperature. As the temperature increases the equilibrium concentration, x , decreases. This concentration is given by
(10) where n(L) is the number of moles of gas in solution and N is the number of moles of solvent (water). The change (dn) in the number of moles n in the gas due to a temperature change (dT) is dn = -dn(G) = %v dT
(11)
where use was made of eqs 8 and 9. Note that the gas pressure, p e P , is a constant. The mass of the foam is NM,, where M , is the molar mass of water (neglecting the mass of other solutes and of the gas) and its volume is nVMlq5, where VMis the molar volume of the gas. Thus, the density, @F, of the foam is approximately (7) Wilhelm, E.; Battino, R.; Wilcock, R. J. Chem. Reu. 1977,77,219.
Thermal Expansion of Liquid Foams
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Eliminating N from eqs 11 and 12 and noting that ~ V M = R T yield (13)
The solubilities of gases in aqueous solutions of tensioactives are appreciably larger than the corresponding solubilities in pure water.8 Nevertheless, the values of q are not much different from those for pure water. Typically q is on the order of 0.03 K-l. The values of H vary considerably. For pure water at 298 OK, the values of H range from 4 x lo9 N m-2 for nitrogen to 1.3 x lo8N m-2 for COZ. The estimated value of d(ln n)/d(ln T ) for the shaving cream foam (with 8 = 0.07 g ~ m - is~ on ) order of 0.01. It is still lower for the other foam. The effect of solubility change is therefore very small in the foams used in the experiments, but may be large for foams with very soluble gases. 4.2. Evaporation of Water. Let n, be the number of moles ofwater vapor and nI the number of moles of another gas (or gases) in the gas phase of a foam. The equilibrium partial pressure of water vapor, p,, is
Pw=P
-
‘t
0
20
40
oc
(To*T1)/2 Figure 1. Experimental values of the quantity I = AldlAlnT plotted as a function of average temperature in interval A T 0, detergent foam; x, shaving cream foam. 1
6
(14)
where P is the external pressure (changes in pressure from cell to cell will be neglected) and f is the mole fraction of water vapor or 3
2
As T changes, nI is unchanged, but n, andp, change. The change in n equals the change in n,. Then, a t constant total pressure, P,
(16) Dividing by pwand introducing the mole fraction f lead to
1
0
20
40
60
80
oc
1 Figure 2. Calculatedvalues of Ta(T) as a function of T, taking into account the contribution from water exchange between the liquid and gas phases of a foam, for three values of the pressure. Only the curve for 0.2 atm is shown up to the maximum permissible temperature. The bars are average experimentalvalues o f l (eq 7) obtained with the shavingcream foam at atmospheric pressure for various temperature intervals.
P,= 212.2 bar and T, = 647.3 K are the critical pressure
and temperature of water, respectively, and A, B , C,and D are known constants. For a given external pressure, P, the mole fraction, f , and the coefficient of thermal expansion, a(T), can be calculated as a function of T, using eq 5,17, and 18.It will be assumed that the equilibrium vapor pressure of water is not significantly affected by the solutes in water (i.e. the tensioactives and other additives1.l’ In Figure 2 is plotted the quantity Ta(T) as a function of T for three external pressures, including atmospheric pressure. This quantity is unity for a closed ideal gas. The experimental values of the quantity A defined by eq 7 and shown in Figure 1 are also indicated in Figure
(8)Ownby, D.W.; King, A. D., Jr. J. Colloid Interface Sci. 1964,101, 271. (9)See, for example, Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The properties ofgases and liquids, 4th ed; McGraw-Hill: NY,1988;Chapter 7 and Appendix A.
(10)Wagner, W. A new correlation method for thermodynamic data applied to the vapor-pressure curve of argon, nitrogen and water. Watson, J. T.R., trans. andEd.; IUPAC Thermodynamic Tables;Project Centre: London, 1977. (11)McBain, J. W.; O’Connor, J. J. J.Am. Chem.SOC.1940,62,2855.
Various equations have been proposed to relate the equilibrium vapor pressurep, of water with t e m ~ e r a t u r e . ~ Wagner’s four-parameter equationlowill be used here, in the form
In(p,JP,) = (1 - y)-l[Ay
+ By15 + Cy3+ Dy61 (18a)
where y = 1 - (T/T,)
(18b)
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642 Langmuir, Vol. 11, No. 2, 1995
2 in the form ofhorizontal bars covering the corresponding temperature intervals. Note that A should be compared to a n average TdT)in the relevant temperature interval. The good fit of the experimental results to the curve of Figure 2 for atmospheric pressure indicates that the major contribution to the “anomalous”thermal expansion of the foams used in the experiments is attributable to water evaporatiodcondensation, the gas in the cells being saturated with water vapor a t each temperature.
5. Conclusions The coefficient, d T ) , of thermal expansion of liquid foams deviates considerably from the value 1IT expected for a closed ideal gas. There is a strong contribution to thermal expansion due to the increase in the number of moles in the gas phase as the temperatures increases. The experimental values of TdT)obtained for two aqueous foams increase from 1.3to 2.2 as the temperature increases
from roughly 10 to 35 “C. The observed increase in the number of moles of gas with increasing temperature is due to evaporation of water. Other contributions to expansion that may be important in other foams are those related to gas solubility and eventually also to adsorption of the gas on the film surfaces, both leading to a n increase in thermal expansion compared to ideal gas behavior. As a result ofthis exchange of matter between the liquid and the gas, the form (4)of the equation of state of a foam has to be complemented with a n equation giving the variation ofthe number of gas moles, n, with temperature and pressure.
Acknowledgment. The author acknowledges helpful discussions with Professor E. F. Murillo. This research was supported by the EC Science Programme, Contract NO. SCl*-CT92-0777. LA9406230
