J. Phys. Chem. 1992, 96, 5086-5093
5086
density profiles computed from the simulation by applying the trapezoid rule. Using the density profiles for the long range correction predicts a surface tension that is 1-2 dyn/cm lower than assuming a step function profile for which all integrals can be evaluated analytically to obtain the expression for a one component atomic Lennard-Jones system in ref 21: Registry No. Decane, 124-18-5; eicosane, 112-95-8.
References and Notes (1) Rowlinson, J. S.; Widom, B. Molecular Theory of Cupillariry; Clarendon Press: Oxford, 1982; p 327. (2) Poser, C. I.; Sanchez, I. C. J. Colloid Interface Sci. 1979,69, 539-548. (3) Evans, R. Adv. Phys. 1979, 28, 143-200. (4) Evans, R. J. Phys.: Condens. Matter 1990, 2, 8989-9007. ( 5 ) Velasco, E.; Tarazona, P. Phys. Rev. A 1990, 42, 7340-7346. (6) Velasco, E.; Tarazona, P. J . Chem. Phys. 1989, 91, 7916-7924. (7) Vanderlick, T. K.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1989, 90, 2422-2436. ( 8 ) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 178. (9) Scheutjens, J. M. H. M.; Fleer, G. J. J . Phys. Chem. 1979,83, 1619. (10) Theodorou, D. N. Macromolecules 1987, 21, 1391-1400. (1 1) Theodorou, D. N. Macromolecules 1989, 22, 4578-4589. (12) Theodorou, D. N. Macromolecules 1988, 21, 1400-1410. (13) Szleifer, I. J . Chem. Phys. 1990, 92, 6940-6952. (14) Henderson, D.; Abraham, F. F.; Barker, J. A. Mol. Phys. 1976, 31, 1291-1 295. (15) Sarman, S. J . Chem. Phys. 1990, 92,4447-4455. (16) Yethiraj, A.; Hall, C. K. J . Chem. Phys. 1991, 95, 3749-55. (17) Chen, S. H., personal communication. (18) Lee, J. K.; Barker, J. A. J . Chem. Phys. 1974,60, 1976-1980. (19) Kalos, M. H.; Percus, J. K.; Rao, M. J. Stat. Phys. 1977, 17, 111. (20) Lee, D. J.; Telo da Gama, M. M.; Gubbins, K. E. Mol. Phys. 1984, 53, 1113-1130. (21) Salomons, E.; Mareschal, M. J . Phys.: Condens. Matter 1991, 3, 3645-3661. (22) Holcomb, C. D.; Clancy, P.; Thompson, S. M.; Zollweg, J. A. Fluid Phase Equil., in press. (23) Barker, J. A.; Fisher, R. A,; Watts, R. 0. Mol. Phys. 1971, 21, 657-673. (24) D'Evelyn, M. P.; Rice, S. A. J . Chem. Phys. 1983, 78, 5225.
(25) Harris, J. G.; Gryko, J.; Rice, S. A. J . Chem. Phys. 1987, 87, 3069-308 1. (26) Chapela, G. A.; Saville, G.; Thompson, S. M.; Rowlinson, J. S. J . Chem. Soc., Faraday Trans. 2 1977, 73, 1133. (27) Townsend, R. M.; Gryko, J.; Rice, S. A. J. Chem. Phys. 1985,82, 4391. (28) Matsumoto, M.; Kataoka, Y. J. Chem. Phys. 1989,90.2398-2407. (29) Weber, T. A.; Helfand, E. J . Chem. Phys. 1980, 72, 4014-4018. (30) Linse, P. J. Chem. Phys. 1987,86, 4177-4187. (31) Carpenter, I. L.; Hehre, W. J. J . Chem. Phys. 1990, 94, 531-536. (32) Xia, T. K.; Ouyang, J.; Ribarsky, W. M.; Landman, U. Interfacial Alkane Films, preprint. (33) Vacatello, M.; Yoon, D. Y.; Laskowski, B., preprint. (34) Mansfield, K. F.; Theodorou, D. N. Macromolecules 1990, 23, 4430-4445. (35) Mansfield, K. F.; Theodorou, D. N. Macromolecules 1991, 24, 4295-4305. (36) Harris, J. G.; Wang, Y. Polym. Prepr. 1992, 30, 539-540. (37) Jorgensen, W. L.; Madura, J. D.; Swenson, C. J. J. Am. Chem. SOC. 1984, 106,6638-6646. (38) Weiner, S. J.; Kollman, P. A.; Case, D. A.; Chandra Singh, U.; Ghio, C.; Alagona, G.; Profeta, S.;Weiner, P. J. Am. Chem. Soc. 1984, 106, 765-784. (39) Compton, D. A. C.; Montero, S.; Murphy, W. F. J . Phys. Chem. 1980,84, 3587-3591. (40) Murphy, W. F.; Femandez-Sanchez,J. M.; Raghavachari, K. J. Phys. Chem. 1991, 95, 1124-1 139. (41) Andersen, H. C. J. Comput. Phys. 1983, 52, 24-34. (42) Andrea, T. A.; Swope, W. C.; Andersen, H. C. J . Chem. Phys. 1983, 79, 4576-4584. (43) Berendsen, H. J. C.; Pastma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J. R. J. Chem. Phys. 1984,81, 3684-3690. (44) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1990. (45) Ciccotti, G.; Ryckaert, J. P. Comput. Phys. Rep. 1986,4, 345-392. (46) Madden, W. G. J. Chem. Phys. 1987.87, 1405-1422. (47) Plischke, M.; Henderson, D. J . Chem. Phys. 1986, 84, 2846-2853. (48) Harris, J. G., unpublished computations. (49) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic Press: Boston, 1986; p 556. (50) de Pablo, J. J.; Prauznitz, J. Fluid Phase Equilibria, in press. (51) Toxvaerd, S. J. Chem. Phys. 1990.93, 4290-4295. (52) Jasper, J. J. J . Phys. Chem. Ref.Dara 1972, 1 , 841-980. (53) Vargaftik, N. B. Tables on the Thermophysical Properties of Liquids and Gases, 2nd ed.; Halsted Press: New York, 1975.
Osclllatory Solvation Forces: A Comparison of Theory and Experiment Phil Attard* and John L. Parker Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National University, Canberra, A.C.T., Australia 2601 (Received: January 16, 1992)
Measurements are reported for the osciUatoryforce between mica surfaces in octamethyltetracyclosiloxane(OMCTS). Improved experimental resolution is achieved by solid-state distance control and force measurement and by an automated data acquisition system. The period of the oscillations is measured to be 0.82 f 0.02 nm. Theoretical calculations are shown to quantitatively describe the magnitude of the force in the case of hard-sphere and Lennard-Jones models of the OMCTS molecule, with the period being 0.74 and 0.78 nm, respectively. A rigorous proof of the Derjaguin approximation is also given.
I. Introduction The measurement of the force between mica surfaces in liquid octamethyltetracyclosiloxane (OMCTS) by Horn and Israelachvili,I and the demonstration that this force was oscillatory, was of great significance. For the fmt time it had become possible to measure directly surface forces with molecular resolution. The direct force measurement technique now plays an essential role in the characterization of many systems of practical and technological importance. Furthermore, it continues to stimulate theoretical analysis, not only because of the molecular detail it provides about the behavior of surfaces and fluids but also because the measurements are particularly clear and unambiguous, and
hence they are ideal tests of theoretical models and approximations. The oscillatory force in OMCTS, and those since measured in other fluids?-' are attributed to the free energy change associated with the packing of molecules between smooth surfaces a t small separations. Early theoretical predictions, both simulationss-10 and integral equations,I1J2stimulated the experimental measurements. Since then there have been a number of theoretical discussions of oscillatory structural forces using models and a p proximations of varying In particular, Vanderlick et al. have described the broad behavior of OMCTScyclohexane mixtures in terms of a one-dimensional hard-rod The theories indicate that oscillatory forces arise from
0022-365419212096-5086$03.00/00 1992 American Chemical Society
Oscillatory Solvation Forces molecular size effects in dense fluids between smooth surfaces, and in this sense there is qualitative agreement between theory and experiment. Both disciplines have advanced to such a degree in recent years that it now appears possible to seek a quantitative comparison of the two. The original surface forces apparatus relied on mechanical springs and optical interferometry.28 Improvements on this basic design have been made,29and more recently, the force measurement method has been replaced by solid-state techniques that utilize piezoelectric material^.^^.^' The distance resolution of the new device is about 0.05 nm, and the force resolution is of the order of 0.01 mN m-1.30.31Further, an automated data acquisition system reduces the subjectivity of the measurement, and the large volume of data collected allows for meaningful fits and error estimates. The concomitant improvements in theory are largely of a practical nature. It has been demonstrated that the hypernetted chain (HNC) approximation, and its variants, provide accurate and reliable results of the oscillatory structural forces between macrospheres and walls in hard-sphere f l ~ i d s . ~Second, ~,~~a statistical mechanical derivation of the Derjaguin approximation32y33has been given,24and it has been explicitly shown that it accurately relates the force between curved surfaces in fluids with short-rangerepulsions to the interaction free energy between This last property is crucial to the interpretation planar of the experimental data since the measurements are carried out between crossed cylinders. It is important that these two techniques, which considerably simplify the computation of the properties of model fluids, have been shown to be reliable. It means that any parameters adjusted in the theory to secure agreement with experiment have an unambiguous interpretation in the context of the particular model of the experimental system and that they are not merely artifacts of the statistical mechanical approximation used to solve the model. This paper is concerned with a quantitative comparison of theory and experiment for oscillatory solvation forces. New accurate measurements are reported for the mica-OMCTS system using a radically new force measurement t e ~ h n i q u e . ~ ~Theoretical -~' results are obtained for macrocavities in hard-sphere fluids and for Lennard-Jones macrospheres in Lennard-Jones fluids. The Ornstein-Zernike equation is solved with the HNC and with the reference HNC approximations. Quantitativeagreement between the measurements and the calculations is obtained. In the model fluids there is one adjustable parameter, essentially the diameter of the molecule. The measured period of the oscillations in OMCTS is found to be 0.82 f 0.02 nm. The hard-sphere diameter required to fit the magnitude of the force data is 0.74 nm, and the effective Lennard-Jones diameter is 0.78 nm. A rigorous proof of the Derjaguin approximation is appended.
The Journal of Physical Chemistry, Vol. 96, No. 12, 1992 5087 Piezo Voltage P, (V) -28
-26
-24
-22
-20 t -1.4
I
7 . w r (
0
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'
D. For the case of large macrospheres where the distance between their surfaces is very much less than their diameter, the Derjaguin a p p r o x i m a t i ~ n is ~ ~known , ~ ~ to be a c c ~ r a t e , and ~ ~ ,the ~ ~force between the cavities can be related to the interaction free energy between planar wells (see Appendix). In this work, macrospheres of diameter 30 times that of the solvent are used, and results for separations up to 10 solvent diameters are given. The properties of the hard-sphere fluid have a trivial dependence on temperature and on solvent diameter and a nontrivial dependence on the packing fraction (essentially the dimensionless product of the number density and the diameter cubed). Thus, considering the measured quantities for OMCTS given above, there is one free parameter for the hard-sphere fluid, namely, the diameter of the molecule. 2. Lennard-Jones. In this subsection, a Lennard-Jones model of the mica-OMCTS system is discussed. The OMCTS molecule is modeled as a Lennard-Jones atom, and the mica is modeled as a structureless macrosphere composed of Lennard-Jones atoms. The Lennard-Jones pair potential between atoms of species i and j has the form
Here po is the density of the mica continuum comprising the macrosphere, and pi is the density of OMCTS. The macrosphere potentials are the sum of the interactions due to the mica elements in the continuum, less the sum of the interactions due to OMCTS molecules over the same spherical region. This procedure is necessary because one requires the change in the total free energy with macrosphere position, and it gives rise to the factors of density in the above coefficients. It may be seen that there are five parameters in the above: p l , all01,poao, and pd0. The density of OMCTS is p1 = 1.94 X lC2' m-3?9 The strength of the short-range repulsion in OMCTS, a1will be regarded as a free parameter, which essentially determines the size of the OMCTS molecule. For simplicity, the quantity poao will be equated to pIaI.As noted above, the results presented below are not sensitive to the short-range mica-OMCTS repulsion. The two dispersion coefficients will be determined from the known Hamaker constants of mica and OMCTS. The van der Waals force per unit area between semiinfinite half-spaces is, in the Hamaker approximation
Foo(h)/ A = - H / 12rhz
(3.7)
This may be compared to the asymptotic pressure between Lennard-Jones half-spaces across Lennard-Jones fluidsz4
(3.2) The r4 term is the usual dispersion attraction, and the pi are related to the atomic polarizabilitiesand ionization strengths. The r-12term accounts for the short-range repulsion between atoms; the particular product form assumed for this coefficient between unlike atoms is not important for the results reported here. The macrosphere-OMCTS pair potential is derived by integrating the Lennard-Jones pair potential over a sphere of radius R. One has (with t = (9+ sz - 2rs cos V@)(r;R)= 2 + x R d s s 2 S0* d 6 sin 6 t" r3 = -47r r>R 3 (? - R2)3'
(3.3)
+ 63rC'R2+ 4 5 ? P + 5R6), r > R (9- R2)'
(3.4)
and
27r 4R3(15P =90
Only the dispersion interaction between the macrospheres need be considered, and one has
d 6 ) ( r ; R ) 2 r x R d s s z 10* d 6 sin 6 V @ ) ( f )
(3.8) From the mica-air-mica Hamaker constant,40H = 1 X one deduces that (since for air, p1 = 0) poj30
= 1.0 X lo-'' J
p1/31
= 6.54 X lo-" J
J,
(3.9) From this and the mica-OMCTS-mica Hamaker constant,40H = 1.23 X J, one obtains (3.10)
with the value of p1 given above. Note that this last result follows from the positive square root in eq 3.8. This choice, which corresponds to pl& < &,,, is the correct root, since mica is solvophilic for OMCTS. As mentioned above, there is now only one adjustable parameter in the Lennard-Jones model, namely, the size of the OMCTS molecule. B. "heoretical Approach and Computational Algoritbm. The above models prescribe the pair potentials of a binary fluid mixture. This consists of solvent (OMCTS) and macrospherical solutes at infinite dilution. The mean force between the macrospheres will be compared to the measured force between mica surfaces via the Derjaguin approximation. The former can be determined from the Ornstein-Zernike equations for the pair correlation functions of the binary mixture.23 These relate the total (h) and the direct (c) correlation functions. In Fourier space they are for the solvent-solvent (3.11a) for the solutesolvent
In view of the above results, the pair potentials actually used in the Lennard-Jones model were, for the solventsolvent a12
@I2
u11(r) = - - r12 P
(3.6a)
(3.11b) and for the solutesolute
for the macrospheresolvent u01(d
=
Another equation that relates the pair correlation functions is required, and this is the closure hjj(r) = -1 exp[-8wij(r)] (3.12)
+
and for the macrosphere-macrosphere um(r) =
f m
.r t 2 R ~
-(popo- p1P1)2W'6'(r;R)
r > 2R
(3.6~)
where 8 = I/kBT,kB being Boltzmann's constant and T being the absolute temperature. The potential of mean force is given by Wij(r) ujj(r) - kBT[hij(r) - C j j ( r ) + d,j(r)] (3.13)
5090 The Journal of Physical Chemistry, Vol. 96, No. 12, 1992
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Attard and Parker
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Radial Separation (nm) Figure 2. Model potentials as a function of the radial separation between a pair of OMCTS molecules, as used in the theoretical calculations. The solid line is the hard-sphere potential, and the dashed line is the LennardJones potential. The arrow indicates the period of the oscillatory force obtained for the Lennard-Jones fluid. The insert shows a space filling model of the OMCTS molecule drawn in its flattest configuration.
where the pair potentials for each model were defined above and where d&) is the bridge function which must be approximated. For the hard-sphere model, this is taken to be the Padi5 approximant formed from the first two terms in its density expansion23 du(r) =
P,
24;’(r)
1 - &dff(r)/df)(r)
(HNCP)
(3.14)
For the Lennard-Jones model the bridge function is simply neglected, which is the hypernetted-chain approximation, and djj(r) = 0 (HNC) (3.15) Note that the total correlation function is related to the radial distribution function, h&) = g&) - 1, and since the latter vanishes in the region near contact, the closure equation (3.12) is replaced by the exact condition inside the core, ho(r) = -1. The computational algorithm is to solve by iteration eqs 3.1 1, 3.12,3.13, and either 3.14 or 3.15, using the pair potentials either eq 3.1 or eq 3.6. The fast Fourier transform method is used, and only continuous functions are ever tran~formed.~~ The number of mesh points used was 213, and the mesh spacing was Ar = O.O1dl. Note that the solutesolute functions are not required until the other pair functions have converged. The macrosphere was 30 times the diameter of the solvent. The experimental data are routinely plotted as the measured force between crossed cylinders divided by their radius. According to the Derjaguin a p p r ~ x i m a t i o n , ~this ~ . ~is~the interaction free energy per unit area between planes, divided by 27r. For the interacting macrospheres of the theoretical calculations, the Derjaguin approximation yields (see Appendix) (3.16) where Em(h) is the interaction free energy per unit area between planes.
IV. Results Figure 1 presents the raw voltage data from an inward run and the return outward run and the derived force curve for the inward run. The piezotube voltage, which drives the surface separation, is the abscissa, and the coil voltage, which balances the surface force, is the ordinate. The drift discussed in section IIB is evident in the upper voltage plot. The large separation data is not zero and has a small nonzero slope. The fact that the outward run does not return to the large separation starting point (hysteresis) indicates that drift has occurred during the time between the runs. Note that the inward run mainly records the maxima and that the first minimum dominates the outward run. The straight lines from the maxima of the inward run to part of the way up the next oscillation are jumps due to the instabilities that occur when the
-0.1
-0.15 4
5
6
7
8
9
1
0
h/o Figure 3. Planar interaction free energy per unit area between mica surfaces in OMCTS. Measurements from five separate force runs are shown by the various symbols. The calculated force in the model hardsphere fluid (solid curve) and in the Lennard-Jones fluid (dashed line) are also shown. In each case the separation has been scaled to give unit period for the oscillatory force. A five-point boxcar average has been used to smooth the experimental data.
derivative of the surface force exceeds the glue elastic constant. An analogous jump is evident from the first minimum of the outward run. The marked slope up the face of the maxima of the inward run,and the parallel slope down the face of the outward run before the jump, are due to the compliance of the glue. The lower part of Figure 1 show the force derived from the voltage calibration. This data has been corrected for thermal drift and for glue elasticity as discussed in section IIB. The distance between successive maxima was measured to be a = 0.82 nm, as indicated. Figure 2 displays the hard-sphere and the Lennard-Jones pair potentials used for the theoretical calculations, in each case with the actual parameter fitted to the measured data. The diameter of the hard sphere is 0.74 nm, and the well depth of the Lennard-Jones potential is -0.2kBT,occurring at 0.9 nm. The arrow at 0.78 nm indicates the period of the oscillations calculated for the force in the Lennard-Jones fluid. A realistic model of the OMCTS molecule is also shown in Figure 2. The atoms are drawn to scale with their van der Waals radii, and the correct bond angles and lengths have been used. Even in this configuration, which displays the maximum asymmetry of the molecule, the departure from sphericity is relatively small. Figure 3 compares the smoothed experimental data from several runs, corrected for drift and for glue elasticity as described in section IIB, with the results of theoretical calculations for the hard-sphere and for the Lennard-Jones fluids. In all cases the separations have been scaled so that the oscillations have unit period. Hence, the most important point of comparison is the amplitude of the force and its rate of decay. It may be seen that there is good agreement between theory and experiment for the depth of the three minima between h = 6a and h = 8a (contact is at h = 0). This is also the case for the maxima in each case. The first measured minimum is not as deep as the theoretical curves and is somewhat displaced. At this relatively large adhesion, this could be an indication of surface flattening or of nonlinear contributions to the glue elasticity. Note that it was in order to avoid large surface deformations that the maximum force applied was limited. Consequently, neither the full height of the fifth maximum nor the inner oscillations were measured in these experiments. The scatter beyond h = 8a gives an indication of the experimental resolution; nonequilibrium data due to jumps have not been eliminated in this regime.
Oscillatory Solvation Forces The theoretical calculations for the hard-sphere and for the Lennard-Jones models are shown by the solid and by the dashed curves in Figure 3. The hard spheres had a diameter of dl = 0.74 nm, giving a reduced number density of pld13= 0.80. For the J mI2. Lennard-Jones fluid, the fitted value was l y I = 4 X This, together with PI = 1 X J m6, eq 3.10, corresponds to a potential well of -0).2kBT at 0.9 nm. The scaling factor required for unit period of oscillations was u = 0.78 nm, at which separation the OMCTS pair potential has the value ull(u) = 0.7kBT. It is interesting to note that the oscillations in the Lennard-Jones fluid have a period significantly larger than a nominal core diameter, =0.7 nm, where the potential is 5kBT. Because the definition of the core size of the Lennard-Jones fluid is arbitrary, it is difficult to give a reduced number density. Using the distance between oscillations, u = 0.78 nm, one would have p1u13= 0.85, which is somewhat higher than one would calculate using the 5kBTcutoff above, namely 0.62. Another consequence of the soft repulsion is the small increase in distance between oscillations at increasing separations, which has been seen in earlier measurements.' The amplitude of the measured force is fitted equally well by both theoretical models when plotted with unit period. However, based on the calibration of the driving piezotube, the oscillations in OMCTS have a period measured to be 0.82 f 0.02 nm. The theoretical curves correspond to periods of 0.74 nm for the hard-sphere fluid and 0.78 nm for the Lennard-Jones fluid. If one were to increase the hard-sphere diameter at fixed number density, the predicted force would increase in magnitude and in decay rate. On this basis the Lennard-Jones potential is a better model for OMCTS than is the hard-sphere potential. Indeed, even though it obviously grossly simplifies the real intermolecular potential, the Lennard-Jones pair potential evidently retains those essential elements of reality that give rise to the measured oscillatory forces. The broad agreement between the two models indicates that the origin of the measured forces lies in the size (i.e., core repulsion) of the OMCTS molecule, a fact supported by the lack of an evident van der Waals attraction, which is automatically included in the Lennard-Jones calculations. (The van der Waals attraction can be seen in calculations using a slightly smaller diameter for the Lennard-Jones model; the forces in this case no longer oscillate about zero.) Figure 4 displays the magnitude of the measured and the calculated forces (as determined from the extrema) on a logarithmic plot. It is noticeable that the data are roughly linear, which indicates that the force law corresponds to an exponentially damped sinusoid. For the Lennard-Jones atom, an even-cdd effect is apparent. (Minima are located approximately at integral multiples of u, and the maxima are at half-integral values.) This is also observable in other theoretical calculations at smaller separations. The calculated decay length of the theoretical curves that correspond to Figure 3 is 0.78 nm. The present experimental data are fitted by the same decay length, which is somewhat shorter than earlier possibly because there is less surface deformation at the smaller applied loads used in the present study. Figure 4 includes the present data measured with the new a p p a r a t u ~ , ~and ~ f ' earlier data obtained by Christenson and B10mP4using the original apparatus developed by Israelachvili.28 It can be seen that there is quantitative agreement between the two sets of force data. This is an important observation because the new apparatus has a quite different measurement method and data acquisition system. That the independent methods give the same force law substantiatesthe results. The data of Christenson and B l ~ m are~the ~ most accurate obtained with the original apparatus. It is evident that the present data display significantly less scatter and are in fact enveloped by the earlier results. Note that the earlier measurement^^^ were not corrected for glue elasticity or for thermal drift (which would have been manifest as a drift in force rather than in the directly measured surface separation). V. Conclusion This paper has been concerned with the quantitative theoretical
The Journal of Physical Chemistry, Vol. 96, No. 12, 1992 5091
h (W 1
0.1 h
9
E
3w 0.01
0.00
Figure 4. Magnitude of the extrema of the free energy plotted on a
logarithmicscale against the surface separation. The solid symbols are data from the present experiments (squares are maxima, diamonds are minima), and the crossed symbols are from Christenson and Blom3' (crosses are maxima, asterisks are minima). The theoretical lines join consecutive extrema. The solid curve is for the hard-sphere fluid, and the dashed curve is for the Lennard-Jones fluid. description of new accurate experimental measurements of the oscillatory solvation force between mica surfaces in liquid octamethyltetracyclosiloxane(OMCTS). It has been shown that the magnitude of the measured surface force was well-described by a hard-sphere and a Lennard-Jones model of the OMCTS molecule. In the models the single parameter was the size of the molecule, and this was varied to obtain the right amplitude of the force. For the case of the hard-sphere fluid, the consequent period of the oscillations was 0.74 nm, and for the Lennard-Jones fluid it was 0.78 nm. This indicates that the Lennard-Jones model is a better model of the experimental system where the period was measured to be 0.82 f 0.02 nm. OMCTS is a large, slightly oblate, molecule. It is obviously a great simplifcation to model it with a pairwise additive, spherically symmetric Lennard-Jones potential. It is impossible that such a potential simultaneously account for all the physical properties of OMCTS. For example, the bulk pressure is greatly overestimated by this model. However, the potential may describe a particular set of measurements provided it contains the essential physical mechanisms that give rise to the data. This appears to be the case for the net pressure, which is the oscillatory solvation force discussed here. The fact that the hard-sphere and the Lennard-Jones models both predict similar forces to the measured ones indicates that the force arises from excluded-volume effects. Further, the softer nature of the core repulsion in the LennardJones fluid is necessary to obtain simultaneously the magnitude and period of the measured force. On the other hand, the longranged van der Waals force does not appear to be an important contribution to the oscillatory solvation forces. It is also unlikely that the specific surfacefluid potential, which was here taken to be that due to a Lennard-Jones continuum, plays a major role for this dense liquid. (Sarman2*has discussed surface effects in comparatively dilute Lennard-Jones fluids.) This paper unites results from a number of sources. There is quite good agreement between the earlier measurements of Christenson and B l ~ m and~ those ~ obtained here with the new surface forces a~paratus.3~3~~ Although the improved experimental resolution is evident in the present data, there remains quantitative agreement between the different techniques. Further, there can be no doubt that the measurements are fundamentally sound, because it would not have been possible to fit the results with the present theories if there were some systematic error in the ex-
5092
The Journal of Physical Chemistry, Vol. 96, No. 12, 1992
perimental technique. The quantitative agreement between theory and experiment for the oscillatory solvation forces described here is quite satisfying. Acknowledgment. We thank Hugo Christenson for providing his original data. Appendix. Rigorous Derivation of the Derjaguin Approximation This appendix is concerned with the statistical mechanical basis of the Derjaguin appr0ximation.~2~~ Closely following the analysis of Attard et it is shown that the approximation is exact in the large radius limit. Here the bridge function is analyzed explicitly, whereas earlier the hypernetted chain approximation was utilized, and the exact result that included the bridge function was merely stated.24 The potential of mean force between macrospherical solutes, species 0, radius R, is woo(r12;R) = uoo(r12;R) - k~Tboo(r12;R)- k~Tdoo(rl2;R) (AI) where u, b, and d a r e the pair potential, the series function, and the bridge function, respectively. One can write the latter two as a convolution integral of the form
Attard and Parker One is interested in separations small compared to the macrosphere radius, x % rI2- 2R
