19798
J. Phys. Chem. B 2005, 109, 19798-19805
New Equation of State for Thin Foam Films Cosima Stubenrauch*,† and Reinhard Strey‡ School of Chemical and Bioprocess Engineering, UniVersity College Dublin, Belfield, Dublin 4, Ireland, and Institut fu¨r Physikalische Chemie, UniVersita¨t zu Ko¨ln, Luxemburger Str. 116, D-50939 Ko¨ln, Germany ReceiVed: May 21, 2005; In Final Form: August 12, 2005
We recently constructed phase diagrams for thin foam films stabilized with nonionic surfactants by treating disjoining pressure (Π) versus thickness (h) curves of foam films like p-Vm isotherms of real gases. However, the correlations were qualitative rather than quantitative. In the present paper, an empirically derived equation of state for a quantitative description of foam film properties is proposed. New experimental data are provided which clearly illustrate under which conditions (a) subcritical, critical, and supercritical films are formed, (b) the common black film (CBF) to Newton black film (NBF) transition is reversible, and (c) CBF and NBF coexist.
1. Introduction The stability of well-drained quasi-static foam films (thickness 240 nm2. In other words, at h < 15 nm, the distances between two charges and between the two surfaces are similar so that we can no longer speak of a homogeneous surface charge distribution, which is one of the key issues of the DLVO theory. Thus, the lower the qDLVO values, the larger the film thickness below which the classical DLVO theory is no longer applicable.21 (2) At h < 20 nm, the van der Waals attraction starts to play an important role, which is usually calculated with the Hamaker
New Equation of State for Thin Foam Films
J. Phys. Chem. B, Vol. 109, No. 42, 2005 19801
TABLE 1: Parameters with Which the Curves Shown in Figures 4 and 5 Were Calculated C10E4 K/10-3 N m C-1 a/10-13 N b/10-9 m κ-1/10-9 m c1/10-4 M q1/10-3 C m-2 C1/107 Pa qDLVO/10-3 C m-2 c2/10-4 M q2/10-3 C m-2 C2/107 Pa qDLVO/10-3 C m-2 ccr/10-4 M qcr/10-3 C m-2 Ccr/107 Pa qDLVO/10-3 C m-2 c3/10-4 M q3/10-3 C m-2 C3/107 Pa qDLVO/10-3 C m-2
35 4.0 4.0 30
1.1 1.30 3.40 1.17 5.0 1.10 3.10 0.95
C10E6
C10E8
35 2.5 4.6 30 1.0 1.20 3.3 1.17 6.0 0.77 2.3 0.67 6.0 < ccr < 9.0 0.65 2.0 b 9.0 0.53 1.6 a
35 2.5 5.2 30 4.0 0.80 2.3 a 8.0 0.68 2.1 a 8.0 < ccr < 12.0 0.55 1.8 b 12.0 0.49 1.3 a
C12E6 35 2.0 4.0 30
0.1 3.3c 4.2 1.7 0.5 0.62 1.8 0.67 1.0 0.52 1.4 b
a Nonfittable. b No experimental data available. c The upper limit for q in a nonionic system was estimated to be 1.0-1.7 mC m-2 for celectrolyte ) 10-4 M and pH 5.5.2,8-10 However, this limit (a) is only an estimation, (b) has not yet been determined accurately, and (c) is in the same order of magnitude. In light of these arguments, q ) 3.3 mC m-2 is acceptable and does not question the overall concept.
constant A ) 3.7 × 10-20 J for the air-water-air system. This approximation is definitely not valid in the case of large surfactants such as C10E6 and C10E8, respectively, where the interaction should be calculated using (at least) a five-layer model (e.g., air-surfactant-water-surfactant-air), which is not an easy task. What is clear, however, is that the A values of the real system are lower than 3.7 × 10-20 J, which results in less attractive interactions.4 (3) The main reason, however, is most likely a contribution arising from the large headgroups, the range of which is unknown due to the lack of an appropriate model. On one hand, the NBFs of C10E6 and C10E8 cannot be regarded as incompressible hard walls, as their thicknesses decrease with increasing pressure. On the other hand, the brush theory of de Gennes22 is not applicable either, as hcore < 2hhead is not fulfilled (hcore is the thickness of the film core consisting of water and the headgroups; hhead is the length of the surfactant headgroup). The experimental observation of a continuous transition from a mainly electrostatically stabilized (CBF-like) film to an entropically stabilized (NBF-like) film only tells us that the range of steric repulsion between the headgroups is comparable to, or even larger than, the range of the van der Waals attraction. In conclusion, one can say that combining the classical DLVO theory with a hard-wall approach for the NBF only describes the Π-h curves of a limited number of systems. In the case of nonionic surfactants, we are restricted to surfactants with small headgroups, while, for typical ionic surfactants, the classical DLVO approach can only be used for low surface charge densities and low electrolyte concentrations (see discussion in refs 13 and 23). One way to overcome the mentioned limitations is to use our concept of treating Π-h curves of foam films like p-Vm isotherms of real gases, which will be discussed in the following. 3. New Equation of State The basis for our empirical approach toward an appropriate equation of state was the classical van der Waals equation for
a real gas which we simply transformed into a “van der Waals” equation for a foam film by substituting the gas constant (R) by the fundamental constant (K), the pressure (p) by Π, the molar volume (Vm) by h, and the temperature (T) by q (see Appendix II, eq A3). The general idea was promising, as it allowed us to describe the Π-h curves of C10E8 adequately. However, in contrast to the classical DLVO approach discussed above, it was now the Π-h curves of the shorter homologues C10E6 and C10E4 that could not be described (see Figure A2). The main reason for this mismatch is the second term of eq A3 which represents the long-range electrostatic component of the disjoining pressure (Πelec). In Figure A2, it is clearly seen that Πelec is proportional to h-1 only at small thicknesses, while an h-1 decay is too weak at larger thicknesses. To find an alternative for the second term of eq A3, we drew on some aspects of the DLVO theory. It is known that under certain conditions (small potentials in the middle of the film and very little overlap between the two double layers) the electrostatic repulsion between two charged interfaces in a 1:1 electrolyte can be expressed as
Πelec ) 64RTc tanh2
Fψ exp(-κh) 4RT
(1)
where T is the temperature, c the electrolyte concentration, F the Faraday constant, ψ the surface potential, and κ-1 the Debye length. Note that ψ and q are connected via the Grahame equation, that is, q ) x80RTc sinh(Fψ/2RT).4 On the basis of this knowledge, we replaced the second term of eq A3 by the exponential C exp(-κh), thus obtaining eq 2
Π)
Kqb a + C exp(-κh) - 2 h(h - b) h
(2)
Finally, we used eq 2 to describe the experimental Π-h curves. The results are shown in Figure 4, and the parameters with which the curves were calculated are listed in Table 1. Comparing Figure 4 with Figures 3 and A2, respectively, we believe it is fair to say that eq 2 is a major advance with regard to a generally valid equation of state for foam films. Not only are the experimental data described in surprising detail, but in all cases, supercritical, critical, and subcritical Π-h curves could be calculated. The finding that qcr decreases with increasing size of the headgroup, that is, increasing thickness of the NBF, is in agreement with the results of our previews work (Figure 3 in ref 10). Following the concept explained in ref 9, we were even able to schematically draw the spinodal and the binodal of the phase diagrams between which the two-phase region (shadowed gray) is located. Three general comments have to be made with regard to the quality of our new approach. (1) According to the calculations, the Π-h curve of C10E4 at 1.1 × 10-4 M is near-critical while that of C10E6 at 1.0 × 10-4 M is supercritical. However, in both cases, a film rupture instead of a continuous thinning is observed. The reason for this behavior is the stability of foam films which is not solely determined by interaction forces (see discussions in refs 24 and 25) but decreases with decreasing surfactant concentration. This, however, is not taken into account either in eq 2 or in any other theory developed so far. (2) The Π-h curve of C10E4 at 5.0 × 10-4 M can be described satisfactorily with eq 2 at small and large thicknesses. Around 20 nm, however, discrepancies between the calculated and measured data are seen. We attribute this deviation to the
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Figure 5. Same experimental data as those shown in Figure 1. The solid lines are calculated according to the new equation of state (eq 2) with the parameters listed in Table 1. The bold solid line (c ) 0.5 × 10-4 M) is the critical Π-h curve with qcr ) 0.62 mC m-2. The gray area indicates the two-phase region between the spinodal and the binodal, the common maximum of which is the critical point (schematic drawing). The dashed horizontal line is a tie line obtained via the Maxwell construction.
Figure 4. Same experimental data as those shown in Figure 2. The solid lines are calculated according to the new equation of state (eq 2) with the parameters listed in Table 1. The bold solid lines are the critical Π-h curves with qcr ) 1.30 mC m-2 for C10E4, qcr ) 0.65 mC m-2 for C10E6, and qcr ) 0.55 mC m-2 for C10E8. The gray area indicates the two-phase region between the spinodal and the binodal, the common maximum of which is the critical point (schematic drawings). The dashed horizontal line is a tie line obtained via the Maxwell construction.
fact that no equilibrium was reached at these particular pressures. Note that we measured this Π-h curve at a time when we were not aware of near-critical phenomena in foam films.17 Thus, it is very likely that we simply “overshot” the pressure at which the transition to the NBF occurs. (3) The Π-h curves of C10E6 at 9.0 × 10-4 M and of C10E8 at 12.0 × 10-4 M are subcritical. Having a look at eq 2, one sees that the Debye length (κ-1) which was kept constant for all calculations (Table 1) is part of the second term. However, the composition of the coexisting films and thus κ-1 are unknown so that the calculated subcritical curves are to be seen as guides to the eyes rather than descriptions of the experimental data. To understand the great value of eq 2 and its close similarity to the original van der Waals equation, we have to look at each of the three contributions more closely. First Term of eq 2. The first term describes the short-range repulsion which we considered to be a hard wall in our previous work. According to eq 2, however, the short-range repulsion is
a function of both q and h and it is only for h f b where this term approaches infinitely high values, that is, the hard-wall limit. Having the analogy to the van der Waals fluid in mind (see Appendix II, eq A2), it is obvious that the parameter b in eq 2 is the analogon to the excluded volume of the van der Waals fluid. It seems reasonable to consider the thickness of a completely compressed film (water-free surfactant bilayer) as the “excluded volume of a foam film”. The following values for C10E8 may help to illustrate this claim: the thickness of the NBF is ∼8 nm, the length of two extended C10E8 chains is ∼8.3 nm, and the best fitted value of b was 5.2 nm (Table 1). Thus, theoretically, it should be possible to reduce the NBF thickness down to ∼5 nm if very high pressures are applied. Practically, however, we do not expect to obtain such a thin NBF, as it will definitely rupture before reaching the final “compressed” stage. A further reasonable finding is that decreasing the headgroup size from E8 down to E4 decreases the excluded volume of the foam film (b) from 5.2 to 4.0 nm (Table 1). Second Term of eq 2. We argued that the second term of eq 2 describes the electrostatic repulsion. However, having a closer look at eq 2, one realizes that both the first and second terms are a function of q and thus should both contribute to Πelec. (For the second term, q comes into play via ψ of the prefactor C according to eq 1.) Thus, we have two different, superposing decays of Πelec(h), the combination of which obviously allows us to describe the data adequately (Figure 4). Moreover, a comparison of the q values obtained with eq 2 and the classical DLVO theory, respectively, reveals the high reliability of our new approach. On average, the respective q values deviate less than 10% (Table 1). Third Term of eq 2. The third, and only attractive, term in eq 2 is a much weaker function of h than the commonly used van der Waals term where the attractive contribution to Π decreases with h-3. Thus, the maximum of the respective Π-h curves is shifted toward larger h values compared to the classical DLVO approach. In analogy to the van der Waals fluid, the parameter a represents attractive interactions without giving any specific physical origin to these interactions. (In the case of a foam film, it is the interaction between the two monolayers which is relevant.) The finding that a does not change
New Equation of State for Thin Foam Films significantly from C10E4 to C10E8 (Table 1) is simply due to the fact that the third term is relatively insensitive to changes in a. This observation is in agreement with the small changes in the Π-h curves that one observes if the Hamaker constant (A) is varied (Figure 4 in ref 26). Let us come back to the beginning of the present paper, namely, to the experimental Π-h curves of C12E6 (Figure 1), and describe these data with eq 2. The results are seen in Figure 5 and Table 1. As was the case for the C10Ej surfactants, eq 2 not only describes the data very well but also leads to supercritical, critical, and subcritical curves. According to these calculations, the Π-h curve measured for 0.1 × 10-4 M C12E6 is indeed supercritical. Again, the reason for the discrepancies between experimental and calculated data seen at 0.5 × 10-4 M is most likely due to the fact that no equilibrium was reached (see C10E4 data). Comparing the results obtained for C12E6 on one hand and for C10Ej on the other, one would expect that the parameter b of C12E6 is close to that of C10E6, namely, 5 instead of 4 nm (Table 1). However, the NBF thickness of C12E6 was experimentally determined to be ∼5 nm while that of C10E6 was ∼7.5 nm. It is this difference which is the reason for the different b values. (It is not understood yet why a NBF stabilized by C12E6 is thinner than a NBF stabilized by C10E6.)
4. Conclusions A theoretical description of experimental Π-h curves has always been a challenge. Using the classical DLVO approach, for example, we are faced with three problems. First, there is no term to account for the electrostatic repulsion in the case of a non-homogeneous surface charge distribution, that is, at film thicknesses that are comparable to the distance between two charges. Second, the exact value of the Hamaker constant is not known, which leads to inaccurate calculations of the van der Waals attraction. Third, there is no reliable description of the repulsion arising from large headgroups. A way out of this dilemma is to treat Π-h curves of foam films like p-Vm isotherms of real gases. On the basis of this concept, we developed an appropriate equation of state with which the experimental Π-h curves of four different nonionic surfactants can be described satisfactorily. Moreover, we are now able to calculate the conditions under which subcritical, critical, and supercritical foam films appear, thus being able to construct phase diagrams and predict phase transitions. So far, our investigations have been restricted to foam films stabilized by nonionic surfactants. The next challenge is to apply our new concept to films that are stabilized by ionic surfactants as well as to films that contain polyelectrolytes. Once it has been confirmed that the new equation of state is generally applicable, a scaling description for thin foam films in terms of corresponding states can be derived. Last but not least, the use of the same approach for the description of phase transitions in Langmuir-Blodgett films and Gibbs films seems to be feasible and reasonable. Acknowledgment. C.S. is indebted to the DFG and the Fond der Chemischen Industrie for financial support. The authors wish to thank Dr. Judith Schulze-Schlarmann for measuring the Π-h curves as well as Dipl.-Phys. Natalie Buchavzov for measuring the surface tension isotherms. Illuminating discussions with Prof. Ben Widom are gratefully acknowledged.
J. Phys. Chem. B, Vol. 109, No. 42, 2005 19803 Appendix I Materials and Cleaning Procedure. The nonionic surfactant hexaethylene glycol dodecyl ether (C12E6) was purchased from Fluka (Germany), tetraethylene glycol decyl ether (C10E4) was purchased from Bachem (Germany), and C10E6 and C10E8 were purchased from Sigma (Germany). All surfactants were used as received after checking their purity by measuring the surface tensions as a function of the concentration (c) at 22 °C (see below). Sodium chloride was obtained from Merck (Germany) and roasted at 500 °C before use to remove organic impurities. Water used for the preparation of all solutions was purified with a Millipore Milli-Q Plus 185 water purification system. All glassware (except the film holders of the TFPB) was cleaned with Deconex from Borer Chemie (as a replacement for chromic sulfuric acid) and rinsed thoroughly with Milli-Q water before use. The film holders used in the TFPB (see below) were boiled twice in acetone and six times in water, and at least 0.5 L of hot water was sucked through each disk. Solutions with different surfactant concentrations were prepared in 10-4 M NaCl background electrolyte. It was made sure that the pH of the investigated solutions was equal to the usual pH of 5.5 ( 0.3, which is due to the dissolution of CO2 present in air. Surface Tension Measurements. In Figure A1, the surface tensions (σ) of aqueous C10Ej solutions with j ) 4, 6, and 8 are presented as a function of the surfactant concentration (c) at 22 °C. The σ-c curves were measured by the DuNou¨y ring method using a Kru¨ss K10ST tensiometer. From the experimental data, the following cmc values were determined: 8.6 × 10-4 M for C10E4, 8.9 × 10-4 M for C10E6, and 10.4 × 10-4 M for C10E8. Moreover, the data were fitted to the reorientation model27 (solid lines) to get the minimal surface area (Amin) of each surfactant, namely, 0.50 nm2 for C10E4, 0.54 nm2 for C10E6, and 0.66 nm2 for C10E8. The respective data for C12E6 are published in ref 12. Thin Film Pressure Balance Technique. The disjoining pressure (Π) was measured as a function of the film thickness (h) with the thin film pressure balance (TFPB) technique. Experimental details have been published only recently.19 In brief, with a TFPB, free-standing horizontal liquid films are investigated. These films are formed in a film holder that consists of a glass tube, which is connected to a porous glass frit so that the solution is free to move. A hole with a diameter
Figure A1. Surface tension (σ) as a function of the surfactant concentration (c) at T ) 22 ( 1 °C for the three nonionic polyoxyethylene decyl ethers C10E4, C10E6, and C10E8. The error is about the size of the symbols. The lines represent the best fit according to the reorientation model.
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of 1-2 mm, in which the film is formed, is drilled in the frit. The film holder is sealed in a cell in which the gas pressure can be adjusted in a controlled manner. Π(h) curves are generated by interferometrically measuring the equivalent film thickness (heq) after applying a fixed pressure in the cell. The true film thickness (h) is obtained according to the three-layer model28 where the film is considered as a solution layer of refractive index ns plus two adsorbed surfactant layers of different refractive indices, that is,
(
h ) heq - 2htail
)
ntail2 - ns2 ns2 - 1
- 2hhead
(
)
nhead2 - ns2 ns2 - 1
(A1)
where htail and hhead are the thicknesses of the hydrocarbon tail and the headgroup, respectively, and ntail and nhead are the corresponding refractive indices. For the C10Ej series, it holds that htail ) 1.1 nm29 and hhead ) 1.45 nm (E4), 1.2 nm (E6), and 1.65 (E8) as derived from neutron reflectivity measurements.30 ntail was approximated to that of ndecane (1.4102) and nhead to that of the corresponding ethylene glycols, namely, nE4 ) 1.460, nE6 ) 1.465, and nE8 ) 1.4650. The respective data for C12E6 are published in ref 12. The error margins in the reported Π(h) curves are (30 Pa which is a result of uncertainty in the measurement of hydrostatic pressure in the glass tube of the film holder and (5% for the film thickness. The latter results from the uncertainty in the determination of the equilibrium intensity from which the film thickness is calculated.19 Appendix II van der Waals Approach to Π-h Curves. Rewriting the classical van der Waals equation for a real gas leads to the following expression
p)
a RT RTb - 2 + Vm(Vm - b) Vm Vm
(A2)
with which we can distinguish between one attractive contribution and two repulsive ones, as is explained in detail in ref 9. What is of relevance for the paper at hand is the fact that we can also identify one attractive (van der Waals) contribution and two repulsive (steric and electrostatic) ones for the foam film. We know from our previous work that the pressure (p), the molar volume (Vm), and the temperature (T) of a real gas correspond to Π, h, and q, respectively, of a foam film. Thus, replacing p, Vm, and T in eq A2 by Π, h, and q leads to
Π)
Kq a Kqb - 2 + h h(h - b) h
Figure A2. Same experimental data as those shown in Figure 2. The lines are calculated according to eq A3, which is simply the Van der Waals equation of a real gas where the gas constant (R) is substituted by the fundamental constant (K), the pressure (π) by Π, the molar volume (Vm) by h, and the temperature (T) by q. (top) The Π-h curves at the two lower concentrations were nonfittable; at c ) 7.5 × 10-4 M, no CBF but a NBF is directly formed (vertical dotted line at h ) 5 nm). (middle) The Π-h curve at the highest concentration was nonfittable. (bottom) The Π-h curve at the highest concentration was nonfittable. For further details, see text.
TABLE A1: Parameters with Which the Curves Shown in Figure A2 Were Calculated K/10-3 N m C-1 a/10-13 N b/10-9 m c1/10-4 M q1/10-3 C m-2 qDLVO/10-3 C m-2 c2/10-4 M q2/10-3 C m-2 qDLVO/10-3 C m-2
(A3)
where K can be considered as a fundamental constant for foam films in analogy to the gas constant (R), while a and b are specific to a particular surfactant in analogy to the van der Waals coefficients that are specific to a particular gas. Unfortunately, it is only the Π-h curves of C10E8 at the two lower concentrations that can be described satisfactorily with eq A3. The results are shown in Figure A2, and the parameters with which the curves were calculated are listed in Table A1.
a
C10E4
C10E6
C10E8
30 3.0 4.4 1.1 a 1.17 5.0 1.20 0.95
30 3.0 4.8 1.0 1.10 1.17 6.0 0.80 0.67
30 3.0 5.2 4.0 0.89 a 8.0 0.77 a
Nonfittable.
References and Notes (1) Mysels, K. J.; Jones, M. N. Discuss. Faraday Soc. 1966, 42, 42. (2) Exerowa, D.; Scheludko, A. C. R. Acad. Bulg. Sci. 1971, 24, 47. (3) Bergeron, V.; Radke, C. J. Langmuir 1992, 8, 3020. (4) Israelachvili, J. N. Intermolecular & Surface Forces, 2nd ed.; Academic Press: San Diego, CA, 1991.
New Equation of State for Thin Foam Films (5) Israelachvili, J. N.; Wennerstro¨m, H. J. Phys. Chem. 1992, 96, 520. (6) Bergeron, V. J. Phys.: Condens. Matter 1999, 11, R215. (7) Exerowa, D.; Kruglyakov, P. M. Foam and Foam FilmssTheory, Experiment, Application; Studies in Interface Science, Vol. 5; Elsevier: Amsterdam, The Netherlands, 1998. (8) Stubenrauch, C.; v. Klitzing, R. J. Phys.: Condens. Matter 2003, 15, R1197. (9) Stubenrauch, C.; Kashchiev, D.; Strey, R. J. Colloid Interface Sci. 2004, 280, 244 and references therein. (10) Stubenrauch, C.; Strey, R. Langmuir 2004, 20, 5185. (11) Stubenrauch, C. ChemPhysChem 2005, 6, 35. (12) Stubenrauch, C.; Rojas, O. J.; Schlarmann, J.; Claesson, P. M. Langmuir 2004, 20, 4977. (13) Exerowa, D.; Kolarov, T.; Khristov, K. Colloids Surf. 1987, 22, 171. (14) Casteletto, V.; Cantat, I.; Sarker, D.; Bausch, R.; Bonn, D.; Meunier, J. Phys. ReV. Lett. 2003, 90, 048302. (15) Kolarov, T.; Cohen, R.; Exerowa, D. Colloids Surf. 1989, 42, 49. (16) Karraker, K. Ph.D. Thesis, University of California, Berkeley, 1999. (17) Schlarmann, J.; Stubenrauch, C.; Strey, R. Phys. Chem. Chem. Phys. 2003, 5, 184. (18) Stubenrauch, C.; Schlarmann, J.; Strey, R. Phys. Chem. Chem. Phys. 2003, 5, 2736.
J. Phys. Chem. B, Vol. 109, No. 42, 2005 19805 (19) Stubenrauch, C.; Schlarmann, J.; Strey, R. Phys. Chem. Chem. Phys. 2002, 4, 4504. (20) Waltermo, Å.; Manev, E.; Pugh, R.; Claesson, P. J. Dispersion Sci. Technol. 1994, 15, 273. (21) Foret, L.; Ku¨hn, R.; Wu¨rger, A. Phys. ReV. Lett. 2002, 89, 156102. (22) de Gennes, P. G. AdV. Colloid Interface Sci. 1987, 27, 189. (23) Mishra, N. C.; Muruganathan, R. M.; Mu¨ller, H.-J.; Krustev, R. Colloids Surf., A 2005, 256, 77. (24) Bergeron, V. Langmuir 1997, 13, 3474. (25) Stubenrauch, C.; Miller, R. J. Phys. Chem. B 2004, 108, 6412. (26) Bergeron, V.; Waltermo, Å.; Claesson, P. M. Langmuir 1996, 12, 1336. (27) Fainerman, V. B.; Lucassen-Reynders, E. H.; Miller, R. Colloids Surf., A 1998, 143, 141. (28) Duyvis, E. M. Ph.D. Thesis, Utrecht, 1962. (29) Persson, C. M.; Claesson, P. M.; Lunkenheimer, K. J. Colloid Interface Sci. 2002, 251, 182. (30) Lu, J. R.; Li, Z. X.; Thomas, R. K.; Binks, B. P.; Crichton, D.; Fletcher, P. D. I.; McNab, J. R.; Penfold, J. J. Phys. Chem. B 1998, 102, 5785.
