Molecular diameters

The next point of interest has to do with the question of how deep the surface region or region of appreciably unbalanced forces is. This depends primarily on the range of intermolecular forces and, except where ions are involved, the principal force between molecules is of the so-called van der Waals type (see Section VI-1). This type of force decreases with about the seventh power of the intermolecular distance and, consequently, it is only the first shell or two of nearest neighbors whose interaction with a given molecule is of importance. In other words, a molecule experiences essentially symmetrical forces once it is a few molecular diameters away from the surface, and the thickness of the surface region is of this order of magnitude (see Ref. 23, for example). (Certain aspects of this conclusion need modification and are discussed in Sections X-6C and XVII-5.)... 

Fig. Ill-13. (a) Plots of molecular density versus distance normal to the interface a is molecular diameter. Upper plot a dielectric liquid. Lower plot as calculated for liquid mercury. (From Ref. 122.) (b) Equilibrium density profiles for atoms A and B in a rare-gas-like mixmre for which o,bb/ o,aa = 0.4 and q,ab is given by Eq. III-56. Atoms A and B have the same a (of Eq. m-46) and the same molecular weight of SO g/mol the solution mole fraction is jcb = 0.047. Note the strong adsorption of B at the interface. [Reprinted with permission from D. J. Lee, M. M. Telo de Gama, and K. E. Gubbins, J. Phys. Chem., 89, 1514 (1985) (Ref. 88). Copyright 1985, American Chemical Society.]...

The parameter should be unity if molecular diameters also obey a geometric mean law [193] and is often omitted. Equation X-44, if applied to the Young equation with omission of leads to the relationship [192]... 

In the second picture, an interfacial layer or region persists over several molecular diameters due to a more slowly decaying interaction potential with the solid (note Section X-7C). This situation would then be more like the physical adsorption of vapors (see Chapter XVII), which become multilayer near the saturation vapor pressure (e.g.. Fig. X-15). Adsorption from solution, from this point of view, corresponds to a partition between bulk and interfacial phases here the Polanyi potential concept may be used (see Sections X-7C, XI-1 A, and XVII-7). 

Figure A2.4.1. Radial distribution fiinction g(R ) for water (dashed curve) at 4 °C and 1 atm and for liquid argon (fiill curve) at 84.25 K and 0.71 atm as functions of the reduced distance R = R/a, where a is the molecular diameter from [1].

At distances of a few molecular diameters, the interaction will be dominated by electric multipole interactions for dipolar molecules, such as water, the dominant tenn will be the dipole-dipole interaction ... 

The well defined contact geometry and the ionic structure of the mica surface favours observation of structural and solvation forces. Besides a monotonic entropic repulsion one may observe superimposed periodic force modulations. It is commonly believed that these modulations are due to a metastable layering at surface separations below some 3-10 molecular diameters. These diflftise layers are very difficult to observe with other teclmiques [92]. The periodicity of these oscillatory forces is regularly found to correspond to the characteristic molecular diameter. Figure Bl.20.7 shows a typical measurement of solvation forces in the case of ethanol between mica. 

The correct treatment of boundaries and boundary effects is crucial to simulation methods because it enables macroscopic properties to be calculated from simulations using relatively small numbers of particles. The importance of boundary effects can be illustrated by considering the following simple example. Suppose we have a cube of volume 1 litre which is filled with water at room temperature. The cube contains approximately 3.3 X 10 molecules. Interactions with the walls can extend up to 10 molecular diameters into the fluid. The diameter of the water molecule is approximately 2.8 A and so the number of water molecules that are interacting with the boundary is about 2 x 10. So only about one in 1.5 million water molecules is influenced by interactions with the walls of the container. The number of particles in a Monte Carlo or molecular dynamics simulation is far fewer than 10 -10 and is frequently less than 1000. In a system of 1000 water molecules most, if not all of them, would be within the influence of the walls of the boundary. Clecirly, a simulation of 1000 water molecules in a vessel would not be an appropriate way to derive bulk properties. The alternative is to dispense with the container altogether. Now, approximately three-quarters of the molecules would be at the surface of the sample rather than being in the bulk. Such a situation would be relevcUit to studies of liquid drops, but not to studies of bulk phenomena. 

In general there are two factors capable of bringing about the reduction in chemical potential of the adsorbate, which is responsible for capillary condensation the proximity of the solid surface on the one hand (adsorption effect) and the curvature of the liquid meniscus on the other (Kelvin effect). From considerations advanced in Chapter 1 the adsorption effect should be limited to a distance of a few molecular diameters from the surface of the solid. Only at distances in excess of this would the film acquire the completely liquid-like properties which would enable its angle of contact with the bulk liquid to become zero thinner films would differ in structure from the bulk liquid and should therefore display a finite angle of contact with it. 

If a solid contains micropores—pores which are no more than a few molecular diameters in width—the potential fields from neighbouring walls will overlap and the interaction energy of the solid with a gas molecule will be correspondingly enhanced. This will result in a distortion of the isotherm, especially at low relative pressures, in the direction of increased adsorption there is indeed considerable evidence that the interaction may be strong enough to bring about a complete filling of the pores at a quite low relative pressure. 

Further evidence pointing in the same direction was provided by Pierce, Wiley and Smith, who found that on steam activation of a particular char at 900°C the saturation uptake increased three-fold, yet the isotherm was still of Type I. They argued that even if the width of the pores was only two molecular diameters before activation, it would increase, by removal of oxides, during the activation so that the second Type I isotherm would correspond to pores more than two molecular diameters wide. (The alternative explanation, that activation produced new pores of the same width as the old, seems unlikely.)... 

The ratio 0/0 is thus a measure of the enhancement of the energy of adsorption in a micropore as compared with that on an open surface. In curve (i) of Fig. 4.9 this ratio is plotted as a function of d/r and, as is seen, the enhancement is still appreciable when d = l-Sr, but has almost disappeared when d = 2r , i.e. when the slit is only two molecular diameters wide. Even when d/r = 1, which corresponds to a single molecule tightly packed into the width of the slit, the enhancement is only 1-6-fold. The effect... 

Micropore Diffusion. In very small pores in which the pore diameter is not much greater than the molecular diameter the diffusing molecule never escapes from the force field of the pore wall. Under these conditions steric effects and the effects of nonuniformity in the potential field become dominant and the Knudsen mechanism no longer appHes. Diffusion occurs by an activated process involving jumps from site to site, just as in surface diffusion, and the diffusivity becomes strongly dependent on both temperature and concentration. 

Fig. 8. Variation of activation energy with kinetic molecular diameter for diffusion in 4A 2eohte (A), 5A 2eohte (0)> carbon molecular sieve (MSC-5A) (A). Kinetic diameters are estimated from the van der Waals co-volumes. From ref. 7. To convert kj to kcal divide by 4.184.

The channels in zeoHtes are only a few molecular diameters in size, and overlapping potential fields from opposite walls result in a flat adsorption isotherm, which is characterized by a long horizontal section as the relative pressure approaches unity (Fig. 6). The adsorption isotherms do not exhibit hysteresis as do those in many other microporous adsorbents. Adsorption and desorption are reversible, and the contour of the desorption isotherm foUows that of adsorption. 

Physical and ionic adsorption may be either monolayer or multilayer (12). Capillary stmctures in which the diameters of the capillaries are small, ie, one to two molecular diameters, exhibit a marked hysteresis effect on desorption. Sorbed surfactant solutes do not necessarily cover ah. of a sohd iaterface and their presence does not preclude adsorption of solvent molecules. The strength of surfactant sorption generally foUows the order cationic > anionic > nonionic. Surfaces to which this rule apphes include metals, glass, plastics, textiles (13), paper, and many minerals. The pH is an important modifying factor in the adsorption of all ionic surfactants but especially for amphoteric surfactants which are least soluble at their isoelectric point. The speed and degree of adsorption are increased by the presence of dissolved inorganic salts in surfactant solutions (14). 

Fig. 16. Diffusivities of penetrants in rigid (A) and plasticized ( ) poly(vinyl chloride) versus molecular diameter at 30°C (31).

Fig. 8. Internal volume (FQ that is accessible to sugars as functions of the cotton molecular diameters (33). (a) Batting A, greige , scoured—bleached , caustic mercerized H, Hquid ammonia treated, (b) Fabric 0> scoured—bleached V> cross-linked.

The solidification speed of salol is about 2.3 mm mim at 10°C. Using eqn. (6.15) estimate the energy barrier q that must be crossed by molecules moving from liquid sites to solid sites. The melting point of salol is 43°C and its latent heat of fusion is 3.2 x 10 ° J molecule F Assume that the molecular diameter is about 1 nm. 

In numerous applications of polymeric materials multilayers of films are used. This practice is found in microelectronic, aeronautical, and biomedical applications to name a few. Developing good adhesion between these layers requires interdiffusion of the molecules at the interfaces between the layers over size scales comparable to the molecular diameter (tens of nm). In addition, these interfaces are buried within the specimen. Aside from this practical aspect, interdififlision over short distances holds the key for critically evaluating current theories of polymer difllision. Theories of polymer interdiffusion predict specific shapes for the concentration profile of segments across the interface as a function of time. Interdiffiision studies on bilayered specimen comprised of a layer of polystyrene (PS) on a layer of perdeuterated (PS) d-PS, can be used as a model system that will capture the fundamental physics of the problem. Initially, the bilayer will have a sharp interface, which upon annealing will broaden with time. 

Adsorption of supercritical gases takes place predominantly in pores which are less than four or five molecular diameters in width. As the pore width increases, the forces responsible for the adsorption process decrease rapidly such that the equilibrium adsorption diminishes to that of a plane surface. Thus, any pores with widths greater than 2 nm (meso- and macropores) are not useful for enhancement of methane storage, but may be necessary for transport into and out of the adsorbent micropores. To maximize adsorption storage of methane, it is necessary to maximize the fractional volume of the micropores (<2 nm pore wall separation) per unit volume of adsorbent. Macropore volume and void volume in a storage system (adsorbent packed storage vessel) should be minimized [18, 19]. 

Among all the low energy interactions, London dispersion forces are considered as the main contributors to the physical adsorption mechanism. They are ubiquitous and their range of interaction is in the order 2 molecular diameters. For this reason, this mechanism is always operative and effective only in the topmost surface layers of a material. It is this low level of adhesion energy combined with the viscoelastic properties of the silicone matrix that has been exploited in silicone release coatings and in silicone molds used to release 3-dimensional objects. However, most adhesive applications require much higher energies of adhesion and other mechanisms need to be involved. 

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