Molecular forces, Laplace

In his thesis van der Waals expressed his desire to determine a quantity that plays a peculiar role in Laplace s theory of capillarity. He was referring to a molecular pressure, a measure for the cohesion of matter. He was—in the Newtonian tradition—looking for a way of grasping inter-molecular forces, the forces that would appear in his own equation of state. 

This effect assumes importance only at very small radii, but it has some applications in the treatment of nucleation theory where the excess surface energy of small clusters is involved (see Section IX-2). An intrinsic difficulty with equations such as 111-20 is that the treatment, if not modelistic and hence partly empirical, assumes a continuous medium, yet the effect does not become important until curvature comparable to molecular dimensions is reached. Fisher and Israelachvili [24] measured the force due to the Laplace pressure for a pendular ring of liquid between crossed mica cylinders and concluded that for several organic liquids the effective surface tension remained unchanged... 

The molecular theory of surface tension was dealt with by Laplace (1749-1827). But, as a result of the clarification of the nature, of intermolecular forces by quantum mechanics and of the more recent developments in the study of molecular distribution in liquids, the nature and value of surface tension have been better understood from a molecular viewpoint. Surface tension is closely associated with a sudden, but continuous change in the density from the value for bulk liquid to the value for die gaseous state in traversing the surface. See Fig. 2. As a result of this inhomogeneity, the stress across a strip parallel to the boundary—pu per unit area—is different from that across a strip perpendicular to die boundary—pr per unit area. This is in contrast with die case of homogeneous fluid in which the stress across any elementary plane has the same value regardless of the direction of die plane,... 

We see that a calculation of Ar involves a Laplace transform of the time-dependent friction kernel. This may typically be determined in a molecular dynamics (MD) simulation, where the autocorrelation function of the random force (R(O)R(t)) may be determined, which then allows us to determine (f) using the fluctuation-dissipation theorem in Eq. (11.58). Note that Eq. (11.85) is an implicit equation for Ar that in general must be solved by iteration. In the absence of friction we see from Eq. (11.85)... 

In a polymeric system, it would be reasonable to examine the possibility that the free volume concept described earlier, which explains so well phenomena like molecular diffusion and viscosity, might perhaps also explain nucleation phenomena. The critical radius re of a stable bubble can be obtained from a simple mechanical-force balance, yielding the Laplace equation ... 

It is possible to investigate other properties of liquid surfaces by Laplace s method, and much of the treatment of surface tension in physic works is concerned with such mathematical calculations, but the matter will not be carried further here, since the fundamental assumptions in the theory are questionable. Einstein showed that the radius of molecular action is of the order of the molecular diameter, so that only actually adjacent molecules will exert forces on one another, and the surface layer is a particular phase which is one molecule in thickness. This idea has received much support from experiments on surface films by Langmuir, mentioned in 19.VIII G, and it is now part of the stock-in-trade of physical chemists. Raman and Ramdas, from the behaviour of polarised light reflected from a very clean liquid surface, concluded that the surface layer was about 10 cm. thick, i.e. unimolecular. 

It is well known that the attraction between two portions of a fluid decreases very rapidly with the distance and may be taken as zero when this distance exceeds a limiting value, the so-called range of molecular action. According to Laplace, surface tension, y, is a force acting tangentially to the interfacial area, which equals the integral of the difference between the external pressure, and the tangential pressure, p. ... 

In Eq. [7], the frequency-dependent friction is the Laplace transform of the time-dependent friction The presence of the Laplace transform means that the time-dependence of the friction must be known in order to determine the Laplace transform. This friction can be readily determined from molecular dynamics simulations in the approximation where the motion along the reaction coordinate is fixed at x = 0. (A discussion of some subtle, but important, aspects of this approximation is given by Carter et al. ) In that case, the random force R(t) can be calculated from equilibrium dynamics in the presence of this one constraint. From R(t), the time-dependent friction (t) can be calculated and the implicit Eq. [7] solved. The result gives the Grote—Hynes value of the transmission coefficient for that system. 

If the viscous forces are significantly smaller than the surface tension (as expressed by a capillary number Ca much less than 1 Ca = U rfljL, with U being the velocity of the contact line and t] the viscosity of the liquid), the shape of a liquid drop can be reasonably well approximated by a spherical cap. Laplace pressure Fl (where Fl = 2 yi K and the curvature K = l/R for a spherical cap of radius R and thus Fl = 2 yiJR) will be able to assure the same curvature everywhere (with the possible exception of the region very close to the contact line, where long-range molecular interactions, e.g., van der Waals forces, may become important). 

This idea goes back to Francis Bacon (see Vol. II, p. 413), Willis, and Stahl. Wohler in a letter to Liebig said he could not understand the cause of the molecular motion, to which Liebig replied that it is a general mechanical consequence of the transmission of motion Laplace and Berthollet, he says, had shown that an atom (molecule) set in motion by any kind of force transmits its motion to another atom with which it is in contact. This is a law of dynamics of general applicability when the resistance (force, vital force, affinity, electric force, force of cohesion) opposing the motion is insufficient to prevent it. ... 

It results from the work of Laplace on capillary action, that the liquids exert on themselves, under the terms of mumal atfraction their molecules, a normal pressure on the snrface at each point that this pressure can be viewed as emanating from a surface layer having a thickness equal to the sensitive range of activity of the molecular attraction and is, consequently, extremely small finally that this same pressure depends on the curvature of surface at the point considered, and that if one indicates by P the pressure, the force per unit area exerted on a plane surface, by A a constant, and by R and R the principal radii of curvature, i.e. those of maximum and minimum curvature of surface at the same point, the pressure corresponding to this point has as a value, always with respect to the unit of area. 

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